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On a fractional Caputo–Hadamard problem with boundary value conditions via different orders of the Hadamard fractional operators

Abstract

We investigate the existence of solutions for a Caputo–Hadamard fractional integro-differential equation with boundary value conditions involving the Hadamard fractional operators via different orders. By using the Krasnoselskii’s fixed point theorem, the Leray–Schauder nonlinear alternative, and the Banach contraction principle, we prove our main results. Also, we provide three examples to illustrate our main results.

Introduction

In recent decades, it has become clear to researchers that studying different types of fractional differential equations is of particular importance. This is a tool to complete our modeling information.

In fact, some practical instances done in the framework of the concepts and notions of the fractional calculus show us the power of this branch of mathematics in the modeling of different natural phenomena. In the meantime, fractional differential equations and inclusions of different types play an important role to reach desired practical goals. More precisely, in recent years, some researches invoked these fractional equations to model some processes and patterns via newly defined fractional operators (see, for example, [14]). The techniques used in these initial value problems are based on the analytical and the existence methods. In the following, some researchers designed new fractional models and investigated them via numerical techniques (see, for example, [514]). Therefore, the fractional calculus has been created a powerful tool for researchers to achieve more exact findings in other applied sciences. Also for further study, notice that a lot of works about different types of fractional integro-differential equations have been published (see, for example, [1545]), q-difference equations (see, for example, [4648]), integro-differential equations involving the Caputo–Fabrizio or the Caputo–Hadamard derivatives (see, for example, [4951]), hybrid equations (see, for example, [52]), approximate solutions of different fractional equations (see, for example, [53, 54]), and modern models (see, for example, [55]).

In 2014, Ahmad et al. investigated the existence of solutions for the nonlinear fractional q-difference equation equipped with four-point nonlocal integral boundary conditions

$$ \textstyle\begin{cases} {}^{c}\mathcal{D}_{q}^{\beta }( {}^{c}\mathcal{D}_{q}^{\gamma }+ \lambda )u(t) = f(t,u(t)), \\ u(0)=a\mathcal{I}_{q}^{\alpha -1}u(\eta ), \qquad u(1)=b\mathcal{I}_{q}^{ \alpha -1}u(\sigma ), \end{cases} $$

where \(t\in [0,1]\), \(q\in (0,1)\), \(\lambda \in \mathbb{R}\), \(0< \eta , \sigma < 1\) and \(\alpha > 2\); \({}^{c}\mathcal{D}_{q}^{\vartheta }\) denotes the Caputo q-fractional derivative of order \(\vartheta \in \{ \beta , \gamma : \beta , \gamma \in (0,1] \}\), \(\mathcal{I}_{q}^{\alpha }\) denotes the Riemann–Liouville q-fractional integral of order α, and \(f: [0,1] \times \mathbb{R} \to \mathbb{R}\) is a continuous function [46]. In 2016, Niyom et al. reviewed the problem

$$ \textstyle\begin{cases} (\lambda \mathcal{D}^{\alpha }+ (1-\lambda )\mathcal{D}^{\beta })u(t)=f(t, u(t)), \\ u(0)=0, \qquad \mu \mathcal{D}^{\gamma _{1}}u(T)+(1-\mu )\mathcal{D}^{ \gamma _{2}}u(T)=\gamma _{3}, \end{cases} $$

where \(T>0\), \(t\in [0,T]\), \(1< \alpha , \beta < 2\), \(0< \gamma _{1} \), \(\gamma _{2} < \alpha -\beta \), \(\mathcal{D}^{\phi }\) is the Riemann–Liouville fractional derivative of order \(\phi \in \{ \alpha , \beta , \gamma _{1} , \gamma _{2} \} \), \(0< \lambda \leq 1\), \(0 \leq \mu \leq 1\), \(\gamma _{3} \in \mathbb{R,}\) and \(f\in C([0,T] \times \mathbb{R}, \mathbb{R})\) [56]. In the same year, Ahmad et al. extended the boundary value problem presented by Niyom to the sequential fractional integro-differential equation of the form

$$ \textstyle\begin{cases} ( {}^{c}\mathcal{D}^{q} + k {}^{c}\mathcal{D}^{q-1} )x(t)=f(t, x(t),{}^{c} \mathcal{D}^{\beta }x(t) , \mathcal{I}^{\gamma }x(t)), \\ x(0)=0, \qquad x'(0)=0, \qquad \sum_{i=1}^{m} a_{i} x( \xi _{i} ) = \lambda \mathcal{I}^{\delta }x( \eta ), \end{cases} $$

where \(t\in [0,1]\), \(\xi _{i}, \eta \in (0,1)\), \(q \in (2 , 3]\), \(\beta , \gamma \in (0,1)\), \(k, \delta >0\), λ, \(a_{i}\) (\(i=1, \dots , m\)) are real constants, \({}^{c}\mathcal{D}^{(\cdot )}\) denotes the Caputo derivative of the fractional order \((\cdot )\), and \(f: [0,1] \times \mathbb{R}^{3} \to \mathbb{R}\) is a continuous function [57].

By using main ideas of the aforementioned articles, we investigate the Caputo–Hadamard fractional integro-differential equation of different orders:

$$ \bigl[ \kappa {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\varrho }+ (1-\kappa ) {}^{\mathrm{CH}} \mathcal{D}_{1^{+}}^{\varpi } \bigr] w(t)= \alpha \psi \bigl(t,w(t) \bigr)+ \beta {}^{H} \mathcal{I}_{1^{+}}^{\mu }\varphi \bigl(t,w(t) \bigr), $$
(1)

with mixed Hadamard and Caputo–Hadamard boundary value conditions

$$ \textstyle\begin{cases} w(1) = 0, \qquad {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\delta } w(e) = 0, \\ {}^{\mathrm{CH}}\mathcal{D}_{1^{+}} w(1) = 0 , \qquad \frac{1}{\varGamma ( \vartheta )} \int _{1}^{e} ( \ln \frac{e}{s})^{ \vartheta - 1} w(s) \frac{ \mathrm{d}s}{s} = 0, \end{cases} $$
(2)

where \(t\in [1,e]\), \(\varrho , \varpi \in (3, 4]\), \(\delta \in (1,2]\), \(\kappa \in (0,1]\), \(\mu , \vartheta > 0\) with \(\delta + \vartheta \neq 0\) and also \(\alpha , \beta \in \mathbb{R}^{+}\). The notation \({}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\nu }\) denotes the Caputo–Hadamard fractional derivative of order \(\nu \in \{ \varrho , \varpi \}\) and \({}^{H}\mathcal{I}^{\mu }\) is the Hadamard fractional integral of order μ. Moreover, functions \(\psi , \varphi : [1,e]\times \mathbb{R} \to \mathbb{R}\) are continuous. Note that the integro-differential equation (1) contains the Caputo–Hadamard derivatives of fractional orders ϱ and ϖ and a Hadamard integral of fractional order μ, while the Caputo–Hadamard derivative of order δ and Hadamard integral of order ϑ are involved in the boundary value conditions (2).

It should also be noted that boundary value conditions given in this paper are general and cover many different special cases. This new type of the modeling is an abstract idea and can include various existing natural processes in the future studies. Therefore, the main purpose of this manuscript is to focus on the existence results and provide some necessary conditions for the analytical investigation and so the practical aspects of boundary value problem (1)–(2) is not our main desire here. To reach our main aim, we apply three different fixed point theorems to establish the existence and uniqueness results. These analytical results guarantee the convergence of the numerical methods to desired solution with the least error, and so this can be a reliable criterion for modeling real processes.

The rest of the paper is arranged by follows. In the next section, we recall some basic notions and definitions which are necessary in the sequel. In Sect. 3, our main existence results are presented by three different analytical techniques such as the Krasnoselskii’s fixed point theorem, the Leray–Schauder nonlinear alternative, and the Banach contraction principle. In Sect. 4, we examine the validity of our theoretical findings by providing three illustrative examples. In Sect. 5, the conclusion is stated.

Preliminaries

In this section, we recall some important and basic definitions on the fractional operators.

Definition 1

([58, 59])

Let \(\varrho \geq 0\). The Hadamard fractional integral of a continuous function \(w:(a,b)\to \mathbb{R}\) of order ϱ is defined by \(({}^{H}\mathcal{I}_{a^{+}}^{0} w)(t)=w(t) \) and

$$ \bigl({}^{H}\mathcal{I}_{a^{+}}^{\varrho }w \bigr) (t)= \frac{1}{\varGamma (\varrho )} \int _{a}^{t} \biggl( \ln \frac{t}{s} \biggr)^{( \varrho -1)}w(s) \frac{\mathrm{d}s}{s} $$

provided that the right-hand side integral exists.

Note that the semigroup property is satisfied by the Hadamard fractional integral as follows: \({}^{H}\mathcal{I}_{a^{+}}^{\varpi } {}^{H}\mathcal{I}_{a^{+}}^{\varrho }w(t)={}^{H} \mathcal{I}_{a^{+}}^{\varpi + \varrho } w(t)\) for \(\varrho , \varpi \in \mathbb{R}^{+}\). Also, we have

$$ {}^{H}\mathcal{I}_{a^{+}}^{\varrho } \biggl(\ln \frac{t}{a} \biggr)^{\varpi }= \frac{\varGamma (\varpi +1)}{\varGamma (\varrho +\varpi +1)} \biggl( \ln \frac{t}{a} \biggr)^{\varrho +\varpi } $$

for \(\varrho , \varpi \geq 0\) and \(t>a\) [58, 59]. It is clear that \({}^{H}\mathcal{I}_{a^{+}}^{\varrho }1=\frac{1}{\varGamma (\varrho +1)}( \ln \frac{t}{a})^{\varrho }\) for all \(t>a\) by putting \(\varpi = 0\) [59].

Definition 2

([58, 59])

Let \(n = [\varrho ] +1 \) and \(n-1 < \varrho \leq n\). The Hadamard fractional derivative of order ϱ for a continuous function \(w : (a, b) \to \mathbb{R}\) is defined by

$$ \bigl({}^{H}\mathcal{D}_{a^{+}}^{\varrho }w \bigr) (t)= \frac{1}{\varGamma ( n - \varrho )} \biggl( t \frac{ \mathrm{d}t}{t} \biggr)^{n} \int _{a}^{t} \biggl( \ln \frac{t}{s} \biggr)^{(n - \varrho -1)}w(s) \frac{\mathrm{d}s}{s} $$

provided that the right-hand side integral exists.

Definition 3

([51, 58])

Let \(\mathrm{AC}_{\theta }^{n} [a,b] = \{ w: [a,b] \to \mathbb{R} : \theta ^{n-1}w(t) \in \mathrm{AC}[a,b], \theta = t \frac{ \mathrm{d} }{ \mathrm{d}t} \}\). The Caputo–Hadamard fractional derivative of order ϱ for an absolutely continuous function \(w \in \mathrm{AC}_{\theta }^{n}([a, b], \mathbb{R})\) is defined by

$$ \bigl({}^{\mathrm{CH}}\mathcal{D}_{a^{+}}^{\varrho }w \bigr) (t)= \frac{1}{\varGamma ( n - \varrho )} \int _{a}^{t} \biggl( \ln \frac{t}{s} \biggr)^{(n - \varrho -1)} \biggl(t \frac{ \mathrm{d}t}{t} \biggr)^{n} w(s) \frac{\mathrm{d}s}{s} $$

whenever the right-hand side integral exists.

Assume that \(w \in \mathrm{AC}_{\theta }^{n}([a, b], \mathbb{R})\) and \(n-1 < \varrho \leq n\). It has been proved that the solution of the Caputo–Hadamard fractional differential equation \(({}^{\mathrm{CH}}\mathcal{D}_{a^{+}}^{\varrho }w)(t)= 0\) is in the form \(w(t) = \sum_{k=0}^{n-1} c_{k} ( \ln \frac{t}{a})^{k} \), and we have

$$ {}^{H}\mathcal{I}_{a^{+}}^{\varrho } {}^{\mathrm{CH}} \mathcal{D}_{a^{+}}^{\varrho }w(t) = w(t) + c_{0} + c_{1} \biggl( \ln \frac{t}{a} \biggr) + c_{2} \biggl( \ln \frac{t}{a} \biggr)^{2} + \cdots + c_{n-1} \biggl( \ln \frac{t}{a} \biggr)^{n-1} $$

for all \(t>a\) [58, 59]. We need the following results.

Lemma 4

(Krasnoselskii’s, [60])

LetMbe a closed, bounded, convex, and nonempty subset of a Banach space\(\mathcal{E}\). Consider two operators\(\varUpsilon _{1}\)and\(\varUpsilon _{2}\)fromMinto\(\mathcal{E}\)such that

  1. (i)

    \(\varUpsilon _{1} w_{1} +\varUpsilon _{2} w_{2} \in M\)for all\(w_{1},w_{2}\in M\),

  2. (ii)

    \(\varUpsilon _{1}\)is compact and continuous,

  3. (iii)

    \(\varUpsilon _{2}\)is a contraction map.

Then there exists\(z\in M\)such that\(z=\varUpsilon _{1}z+\varUpsilon _{2}z\).

Lemma 5

([61])

Let\(\mathcal{E}\)be a Banach space, Ca closed, convex subset of\(\mathcal{E}\), \(\mathcal{U}\)an open subset ofC, and\(0\in \mathcal{U}\). Suppose that\(\varUpsilon :\overline{ \mathcal{U} }\to C\)is a continuous and compact map (that is, \(\varUpsilon (\overline{ \mathcal{U} })\)is a relatively compact subset ofC). Thenϒhas a fixed point in\(\overline{ \mathcal{U} }\)or there is a\(w\in \partial \mathcal{U}\) (the boundary of\(\mathcal{U}\)inC) and\(\lambda \in (0,1)\)with\(w=\lambda \varUpsilon (w)\).

Lemma 6

([62])

Let\(\mathcal{E}\)be a Banach space andMa closed subset of\(\mathcal{E}\). Suppose that\(\varUpsilon : M\to M\)is a contraction. Thenϒhas a unique fixed point inM.

Main results

Here, we are ready to prove our main results. We first characterize the structure of the solutions of the problem (1)–(2). Consider the Banach space \(\mathcal{E} = \{ w: w(t) \in C([1,e], \mathbb{R}) \} \) with the norm \(\Vert w\Vert _{\mathcal{E}} = \sup_{t\in [1,e]} \vert w(t) \vert \). We first provide our key lemma.

Lemma 7

Let\(\phi (t)\in \mathcal{E}\). Then\(w_{0}\)is a solution for the Caputo–Hadamard problem

$$ \textstyle\begin{cases} [ \kappa {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\varrho }+ (1-\kappa ) {}^{\mathrm{CH}} \mathcal{D}_{1^{+}}^{\varpi } ] w(t)= \phi (t) \quad (t\in [1,e]), \\ w(1) = 0, \qquad {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\delta } w(e) = 0, \\ {}^{\mathrm{CH}}\mathcal{D}_{1^{+}} w(1) = 0 , \qquad \frac{1}{\varGamma ( \vartheta )} \int _{1}^{e} ( \ln \frac{e}{s})^{ \vartheta - 1} w(s) \frac{ \mathrm{d}s}{s} = 0 \end{cases} $$
(3)

if and only if\(w_{0}\)is a solution for the fractional integral equation

$$\begin{aligned} w(t) =& \frac{(\kappa -1)}{\kappa \varGamma (\varrho -\varpi )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{\varrho -\varpi -1}w(s) \frac{\mathrm{d}s}{s} + \frac{1}{\kappa \varGamma (\varrho )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{ \varrho -1}\phi (s) \frac{ \mathrm{d}s }{s} \\ &{}+ \frac{( 1- \kappa ) [ 3 \varGamma (4 + \vartheta ) (\ln t)^{2} + (\delta -3)\varGamma (4 - \vartheta ) ( \ln t)^{3} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \\ &{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} w(s) \frac{\mathrm{d}s}{s} \\ &{}+ \frac{( 1- \kappa ) \varGamma (4- \delta ) [ \varGamma (4- \vartheta ) ( \ln t)^{3} - 3 \varGamma (3+\vartheta )(\ln t)^{2}] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta )} \\ &{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} w(s) \frac{\mathrm{d}s}{s} \\ &{}+ \frac{ (3-\delta )\varGamma (4- \vartheta ) ( \ln t)^{3} - 3 \varGamma (4+ \vartheta ) (\ln t)^{2} }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \phi (s) \frac{\mathrm{d}s}{s} \\ &{}+ \frac{ \varGamma (4- \delta ) [ 3 \varGamma (3+\vartheta ) ( \ln t)^{2} - \varGamma (4-\vartheta ) (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta )} \\ &{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \phi (s) \frac{\mathrm{d}s}{s} . \end{aligned}$$
(4)

Proof

Let \(w_{0}\) be a solution for the Caputo–Hadamard problem (3). Then, we have

$$ \kappa {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\varrho }w_{0}(t) + (1-\kappa ) {}^{\mathrm{CH}} \mathcal{D}_{1^{+}}^{\varpi }w_{0}(t) = \phi (t) $$

and so \({}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\varrho }w_{0}(t) = \frac{\kappa -1}{ \kappa } {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\varpi }w_{0}(t) + \frac{1}{\kappa } \phi (t)\). By using the Hadamard fractional integral of order ϱ, we obtain

$$\begin{aligned} w_{0}(t) &= \frac{\kappa -1}{ \kappa } {}^{H}\mathcal{I}_{1^{+}}^{\varrho } {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\varpi }w_{0}(t) + \frac{1}{\kappa } {}^{H}\mathcal{I}_{1^{+}}^{\varrho }\phi (t) \\ &\quad{}+ b_{0} + b_{1} (\ln t) + b_{2} ( \ln t)^{2} + b_{3} ( \ln t)^{3} , \end{aligned}$$

where \(b_{0} \), \(b_{1} \), \(b_{2} \), and \(b_{3}\) are some real constants. Hence,

$$\begin{aligned} w_{0}(t) &= \frac{\kappa -1}{ \kappa \varGamma ( \varrho - \varpi ) } \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{ \varrho - \varpi - 1} w_{0}(s) \frac{ \mathrm{d}s}{s} \\ &\quad{}+ \frac{1}{\kappa \varGamma ( \varrho )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{ \varrho - 1} \phi (s) \frac{ \mathrm{d}s}{s} \\ &\quad{}+ b_{0} + b_{1} (\ln t) + b_{2} ( \ln t)^{2} + b_{3} ( \ln t)^{3} . \end{aligned}$$
(5)

Now by using the boundary value conditions and properties of the Hadamard and Caputo–Hadamard fractional operators, we get

$$\begin{aligned}& \begin{aligned} {}^{\mathrm{CH}}\mathcal{D}_{1^{+}} w_{0}(t) &= \frac{\kappa -1}{ \kappa \varGamma ( \varrho - \varpi -1) } \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{ \varrho - \varpi - 2} w_{0}(s) \frac{ \mathrm{d}s}{s} \\ &\quad{}+ \frac{1}{\kappa \varGamma ( \varrho -1 )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{ \varrho - 2} \phi (s) \frac{ \mathrm{d}s}{s} + b_{1} + 2b_{2} ( \ln t) + 3b_{3} ( \ln t)^{2}, \end{aligned} \\& \begin{aligned} {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{\delta }w_{0}(t) &= \frac{\kappa -1}{ \kappa \varGamma ( \varrho - \varpi - \delta ) } \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{ \varrho - \varpi - \delta - 1} w_{0}(s) \frac{ \mathrm{d}s}{s} \\ &\quad{}+ \frac{1}{\kappa \varGamma ( \varrho -\delta )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{ \varrho - \delta - 1} \phi (s) \frac{ \mathrm{d}s}{s} + b_{2} \frac{2}{\varGamma (3-\delta )} ( \ln t)^{2- \delta } \\ &\quad{}+ b_{3} \frac{6}{\varGamma (4-\delta )} ( \ln t)^{3-\delta }, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} {}^{H}\mathcal{I}_{1^{+}}^{\vartheta }w_{0}(t) &= \frac{\kappa -1}{ \kappa \varGamma ( \varrho - \varpi + \vartheta ) } \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{ \varrho - \varpi + \vartheta - 1} w_{0}(s) \frac{ \mathrm{d}s}{s} \\ &\quad{}+ \frac{1}{\kappa \varGamma ( \varrho + \vartheta )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{ \varrho + \vartheta - 1} \phi (s) \frac{ \mathrm{d}s}{s} + b_{0} \frac{1}{\varGamma (1+\vartheta )} ( \ln t)^{\vartheta } \\ &\quad{}+ b_{1} \frac{1}{\varGamma (2+\vartheta )} (\ln t)^{1+ \vartheta } \\ &\quad{}+ b_{2} \frac{2}{\varGamma (3+\vartheta )} ( \ln t)^{ 2+\vartheta } + b_{3} \frac{6}{\varGamma (4+\vartheta )} ( \ln t)^{ 3+ \vartheta } . \end{aligned}$$

By using two first boundary conditions, we obtain \(b_{0} = b_{1} = 0\). By using two other boundary conditions, we obtain

$$\begin{aligned} b_{2} &= \frac{( 1- \kappa ) \varGamma (4 + \vartheta ) }{2 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} w_{0}(s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{( \kappa -1) \varGamma (4- \delta ) }{2 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi - \delta )} \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} w_{0}(s) \frac{\mathrm{d}s}{s} \\ &\quad{}- \frac{ \varGamma (4 + \vartheta ) }{2 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \phi (s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \varGamma (4- \delta ) }{2 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \delta )} \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \phi (s) \frac{\mathrm{d}s}{s} \end{aligned}$$

and

$$\begin{aligned} b_{3} &= \frac{( 1- \kappa )(\delta -3) \varGamma (4- \vartheta ) }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} w_{0}(s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{( 1 - \kappa )\varGamma (4- \delta )\varGamma (4-\vartheta ) }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \varpi - \delta )} \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} w_{0}(s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ (3-\delta )\varGamma (4- \vartheta ) }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \phi (s) \frac{\mathrm{d}s}{s} \\ &\quad{}- \frac{ \varGamma (4- \delta ) \varGamma (4- \vartheta ) }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta )} \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \phi (s) \frac{\mathrm{d}s}{s}. \end{aligned}$$

Now by substituting the values for \(b_{0}\), \(b_{1}\), \(b_{2}\), \(b_{3}\) in equation (5), we see that \(w_{0}\) is a solution for the integral equation. For the converse part, by using some direct calculations, one can see that \(w_{0}\) is a solution for the Caputo–Hadamard problem (3) whenever \(w_{0}\) is a solution for the integral equation (4). This completes the proof. □

Now, consider the operator \(\varUpsilon :\mathcal{E}\to \mathcal{E}\) defined by

$$\begin{aligned} ( \varUpsilon w) (t) &= \frac{(\kappa -1)}{\kappa \varGamma (\varrho -\varpi )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{\varrho -\varpi -1}w(s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{\alpha }{\kappa \varGamma (\varrho )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho -1} \psi \bigl(s , w(s) \bigr) \frac{ \mathrm{d}s }{s} + \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \\ &\quad{}\times \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho + \mu -1} \varphi \bigl(s , w(s) \bigr) \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{( 1- \kappa ) [ 3 \varGamma (4 + \vartheta ) (\ln t)^{2} + (\delta -3)\varGamma (4 - \vartheta ) ( \ln t)^{3} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} w(s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{( 1- \kappa ) \varGamma (4- \delta ) [ \varGamma (4- \vartheta ) ( \ln t)^{3} - 3 \varGamma (3+\vartheta )(\ln t)^{2}] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} w(s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha [ (3-\delta )\varGamma (4- \vartheta ) ( \ln t)^{3} - 3 \varGamma (4+ \vartheta ) (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \psi \bigl(s , w(s) \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta [(3-\delta )\varGamma (4- \vartheta ) ( \ln t)^{3} - 3 \varGamma (4+ \vartheta ) (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta + \mu - 1} \varphi \bigl(s , w(s) \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha \varGamma (4- \delta ) [ 3 \varGamma (3+\vartheta ) ( \ln t)^{2} - \varGamma (4-\vartheta ) (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \psi \bigl(s , w(s) \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta \varGamma (4- \delta ) [ 3 \varGamma (3+\vartheta ) ( \ln t)^{2} - \varGamma (4-\vartheta ) (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \mu - \delta - 1} \varphi \bigl(s, w(s) \bigr) \frac{\mathrm{d}s}{s}, \end{aligned}$$
(6)

where \(w\in \mathcal{E}\) and \(t\in [1,e]\). Put

$$\begin{aligned} \begin{gathered} \mathcal{K}_{0}^{*} := \frac{ \vert \kappa -1 \vert }{\kappa \varGamma (\varrho -\varpi + 1 )} + \frac{( 1- \kappa ) [ \vert 3 \varGamma (4 + \vartheta ) \vert + \vert (\delta -3)\varGamma (4 - \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta + 1 )} \\ \hphantom{\mathcal{K}_{0}^{*} :=}{}+ \frac{( 1- \kappa ) \vert \varGamma (4- \delta ) \vert [ \vert \varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (3+\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta +1 )} , \\ \mathcal{K}_{1}^{*} := \frac{\alpha }{\kappa \varGamma (\varrho +1 )} + \frac{ \alpha [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (4+ \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta +1 )} \\ \hphantom{\mathcal{K}_{1}^{*} :=}{}+ \frac{ \alpha \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert + \vert \varGamma (4-\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta +1 )}, \\ \mathcal{K}_{2}^{*} := \frac{\beta }{\kappa \varGamma (\varrho + \mu +1 )} + \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (4+ \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu +1 )} \\ \hphantom{\mathcal{K}_{2}^{*} :=}{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert + \vert \varGamma (4-\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta +1 )}. \end{gathered} \end{aligned}$$
(7)

Theorem 8

Suppose that\(\psi ,\varphi :[1,e]\times \mathbb{R} \to \mathbb{R} \)are continuous functions such that

\((\mathcal{N}_{1})\):

there is\(L>0\)so that\(\vert \psi (t,w_{1} )- \psi (t,w_{2} )\vert \leq L \vert w_{1}-w_{2} \vert \)for all\(w_{1} , w_{2} \in \mathbb{R}\)and\(t\in [1,e]\),

\((\mathcal{N}_{2})\):

there exists a real-valued continuous functionσon\([1,e]\)such that\(\vert \varphi (t,w)\vert \leq \sigma (t)\)for all\(w\in {\mathbb{R}}\)and\(t\in [1,e]\).

If\(\mathcal{K}_{0}^{*}+L\mathcal{K}_{1}^{*} < 1\), then the Caputo–Hadamard boundary value problem (1)(2) has at least one solution, where\(\mathcal{K}_{0}^{*}\)and\(\mathcal{K}_{1}^{*}\)are given by (7).

Proof

Let \(\Vert \sigma \Vert := \sup_{t\in [1,e]} \vert \sigma (t)\vert \) and \(O := \sup_{t\in [1,e]}\vert \psi (t,0)\vert \). Consider the operator \(\varUpsilon :\mathcal{E} \to \mathcal{E}\) and the set \(\mathcal{V}_{r} := \{ w\in \mathcal{E}: \Vert w\Vert _{\mathcal{E}} \leq r \} \) which is a closed, convex, and bounded nonempty subset of Banach space \(\mathcal{E}\), where \(r \geq \frac{\Vert \sigma \Vert \mathcal{K}_{2}^{*} + O \mathcal{K}_{1}^{*}}{1-( \mathcal{K}_{0}^{*} +L\mathcal{K}_{1}^{*})}\) and \(\mathcal{K}_{1}^{*}\) and \(\mathcal{K}_{2}^{*}\) are given by (7). Note that each fixed point of ϒ is a solution for the Caputo–Hadamard problem (1)–(2). Let \(t\in [1,e]\) be given. Then, we have

$$\begin{aligned} (\varUpsilon _{1}w) (t) &= \frac{(\kappa -1)}{\kappa \varGamma (\varrho -\varpi )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{\varrho -\varpi -1}w(s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{\alpha }{\kappa \varGamma (\varrho )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho -1} \psi \bigl(s , w(s) \bigr) \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{( 1- \kappa ) [ 3 \varGamma (4 + \vartheta ) (\ln t)^{2} + (\delta -3)\varGamma (4 - \vartheta ) ( \ln t)^{3} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} w(s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{( 1- \kappa ) \varGamma (4- \delta ) [ \varGamma (4- \vartheta ) ( \ln t)^{3} - 3 \varGamma (3+\vartheta )(\ln t)^{2}] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} w(s) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha [ (3-\delta )\varGamma (4- \vartheta ) ( \ln t)^{3} - 3 \varGamma (4+ \vartheta ) (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \psi \bigl(s , w(s) \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha \varGamma (4- \delta ) [ 3 \varGamma (3+\vartheta ) ( \ln t)^{2} - \varGamma (4-\vartheta ) (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \psi \bigl(s , w(s) \bigr) \frac{\mathrm{d}s}{s} \end{aligned}$$

and

$$\begin{aligned} (\varUpsilon _{2}w) (t) &= \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho + \mu -1} \varphi \bigl(s , w(s) \bigr) \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{ \beta [(3-\delta )\varGamma (4- \vartheta ) ( \ln t)^{3} - 3 \varGamma (4+ \vartheta ) (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta + \mu - 1} \varphi \bigl(s , w(s) \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta \varGamma (4- \delta ) [ 3 \varGamma (3+\vartheta ) ( \ln t)^{2} - \varGamma (4-\vartheta ) (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \mu - \delta - 1} \varphi \bigl(s, w(s) \bigr) \frac{\mathrm{d}s}{s}. \end{aligned}$$

Thus, we have

$$\begin{aligned} & \bigl\vert ( \varUpsilon _{1} w_{1}) (t) + ( \varUpsilon _{2} w_{2}) (t) \bigr\vert \\ &\quad \leq \frac{ \vert \kappa -1 \vert }{\kappa \varGamma (\varrho -\varpi )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho -\varpi -1} \bigl\vert w_{1}(s) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{\alpha }{\kappa \varGamma (\varrho )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho -1} \bigl( \bigl\vert \psi \bigl(s, w_{1}(s) \bigr) - \psi (s , 0) \bigr\vert + \bigl\vert \psi (s , 0 ) \bigr\vert \bigr) \frac{ \mathrm{d}s }{s} \\ &\qquad{}+ \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho + \mu -1} \bigl\vert \varphi \bigl(s , w_{2}(s) \bigr) \bigr\vert \frac{ \mathrm{d}s }{s} \\ &\qquad{}+ \frac{( 1- \kappa ) [ \vert 3 \varGamma (4 + \vartheta ) \vert (\ln t)^{2} + \vert (\delta -3)\varGamma (4 - \vartheta ) \vert ( \ln t)^{3} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} \bigl\vert w_{1}(s) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{( 1- \kappa ) \vert \varGamma (4- \delta ) \vert [ \vert \varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (3+\vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} \bigl\vert w_{1}(s) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{ \alpha [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \bigl( \bigl\vert \psi \bigl(s, w_{1}(s) \bigr) - \psi (s , 0) \bigr\vert + \bigl\vert \psi (s , 0 ) \bigr\vert \bigr) \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta + \mu - 1} \bigl\vert \varphi \bigl(s , w_{2}(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{ \alpha \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \bigl( \bigl\vert \psi \bigl(s, w_{1}(s) \bigr) - \psi (s , 0) \bigr\vert + \bigl\vert \psi (s , 0 ) \bigr\vert \bigr) \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \mu - \delta - 1} \bigl\vert \varphi \bigl(s, w_{2}(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ & \quad \leq \frac{ \vert \kappa -1 \vert }{\kappa \varGamma (\varrho -\varpi + 1 )} \Vert w_{1} \Vert + \frac{\alpha }{\kappa \varGamma (\varrho +1 )} \bigl( L \Vert w_{1} \Vert + O \bigr) + \frac{\beta }{\kappa \varGamma (\varrho + \mu +1 )} \Vert \sigma \Vert \\ &\qquad{}+ \frac{( 1- \kappa ) [ \vert 3 \varGamma (4 + \vartheta ) \vert + \vert (\delta -3)\varGamma (4 - \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta + 1 )} \Vert w_{1} \Vert \\ &\qquad{}+ \frac{( 1- \kappa ) \vert \varGamma (4- \delta ) \vert [ \vert \varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (3+\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta +1 )} \Vert w_{1} \Vert \\ &\qquad{}+ \frac{ \alpha [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (4+ \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta +1 )} \bigl( L \Vert w_{1} \Vert + O \bigr) \\ &\qquad{}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (4+ \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu +1 )} \Vert \sigma \Vert \\ &\qquad{}+ \frac{ \alpha \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert + \vert \varGamma (4-\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta +1 )} \bigl( L \Vert w_{1} \Vert + O \bigr) \\ &\qquad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert + \vert \varGamma (4-\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta +1 )} \Vert \sigma \Vert \\ &\quad = \bigl(\mathcal{K}_{0}^{*} +L \mathcal{K}_{1}^{*} \bigr) \Vert w_{1} \Vert + \mathcal{K}_{2}^{*} \Vert \sigma \Vert + \mathcal{K}_{1}^{*} O \\ & \quad \leq \bigl(\mathcal{K}_{0}^{*} +L \mathcal{K}_{1}^{*} \bigr) r + \mathcal{K}_{2}^{*} \Vert \sigma \Vert + \mathcal{K}_{1}^{*} O \leq r \end{aligned}$$

for all \(w_{1},w_{2}\in \mathcal{V}_{r}\). Hence, \(\Vert \varUpsilon _{1} w_{1} + \varUpsilon _{2} w_{2} \Vert \leq r\) and so \(\varUpsilon _{1} w_{1} + \varUpsilon _{2} w_{2} \in \mathcal{V}_{r}\) for all \(w_{1},w_{2} \in \mathcal{V}_{r}\). Now let \(\{ w_{n} \}_{n \geq 1} \) be a sequence in \(\mathcal{V}_{r}\) with \(w_{n} \to w\) and \(t\in [1,e]\). Then, we have

$$\begin{aligned} & \bigl\vert (\varUpsilon _{2} w_{n}) (t) - (\varUpsilon _{2} w) (t) \bigr\vert \\ &\quad \leq \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho + \mu -1} \bigl\vert \varphi \bigl(s, w_{n}(s) \bigr) - \varphi \bigl(s,w(s) \bigr) \bigr\vert \frac{ \mathrm{d}s }{s} \\ &\qquad{}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta + \mu - 1} \bigl\vert \varphi \bigl(s, w_{n}(s) \bigr) - \varphi \bigl(s,w(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \mu - \delta - 1} \bigl\vert \varphi \bigl(s, w_{n}(s) \bigr) - \varphi \bigl(s,w(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad \leq \frac{\beta }{\kappa \varGamma (\varrho + \mu +1 )} \bigl\vert \varphi \bigl(s, w_{n}(s) \bigr) - \varphi \bigl(s,w(s) \bigr) \bigr\vert \\ &\qquad{}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (4+ \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu +1 )} \bigl\vert \varphi \bigl(s, w_{n}(s) \bigr) - \varphi \bigl(s,w(s) \bigr) \bigr\vert \\ &\qquad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert + \vert \varGamma (4-\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta +1 )} \bigl\vert \varphi \bigl(s, w_{n}(s) \bigr) - \varphi \bigl(s,w(s) \bigr) \bigr\vert . \end{aligned}$$

Since φ is continuous, \(\Vert \varUpsilon _{2} w_{n}-\varUpsilon _{2} w \Vert \to 0\), and so the operator \(\varUpsilon _{2}\) is continuous on the open ball \(\mathcal{V}_{r}\). Now, we show that \(\varUpsilon _{2}\) is uniformly bounded. Let \(w\in \mathcal{V}_{r}\) and \(t\in [1,e]\). Then, we get

$$\begin{aligned} \bigl\vert (\varUpsilon _{2} w) (t) \bigr\vert &\leq \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho + \mu -1} \bigl\vert \varphi \bigl(s,w(s) \bigr) \bigr\vert \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta + \mu - 1} \bigl\vert \varphi \bigl(s,w(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \mu - \delta - 1} \bigl\vert \varphi \bigl(s,w(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\leq \frac{\beta }{\kappa \varGamma (\varrho + \mu +1 )} \Vert \sigma \Vert + \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (4+ \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu +1 )} \Vert \sigma \Vert \\ &\quad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert + \vert \varGamma (4-\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta +1 )} \Vert \sigma \Vert \\ &\leq \Vert \sigma \Vert \biggl[ \frac{\beta }{\kappa \varGamma (\varrho + \mu +1 )} + \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (4+ \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu +1 )} \\ &\quad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert + \vert \varGamma (4-\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta +1 )} \biggr] \\ &= \mathcal{K}_{2}^{*} \Vert \sigma \Vert \end{aligned}$$

which implies that \(\Vert \varUpsilon _{2} w\Vert \leq \mathcal{K}_{2}^{*} \Vert \sigma \Vert \). This shows that \(\varUpsilon _{2}\) is uniformly bounded. Here, we prove that \(\varUpsilon _{2}\) is equicontinuous. Let \(t_{1}, t_{2} \in [1,e]\) with \(t_{1} < t_{2}\). We show that \(\varUpsilon _{2}\) maps bounded sets into equicontinuous sets. For each \(w\in \mathcal{V}_{r}\), we have

$$\begin{aligned}& \bigl\vert (\varUpsilon _{2} w) (t_{2}) - (\varUpsilon _{2} w) (t_{1}) \bigr\vert \\& \quad \leq \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \int _{1}^{t_{1}} \biggl[ \biggl(\ln \frac{t_{2}}{s} \biggr)^{\varrho + \mu -1} - \biggl(\ln \frac{t_{1}}{s} \biggr)^{ \varrho + \mu -1} \biggr] \bigl\vert \varphi \bigl(s, w(s) \bigr) \bigr\vert \frac{ \mathrm{d}s }{s} \\& \qquad {}+ \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \int _{t_{1}}^{t_{2}} \biggl( \ln \frac{t_{2}}{s} \biggr)^{\varrho + \mu -1} \bigl\vert \varphi \bigl(s, w(s) \bigr) \bigr\vert \frac{ \mathrm{d}s }{s} \\& \qquad {}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert [ ( \ln t_{2})^{3} - ( \ln t_{1})^{3} ] + 3 \vert \varGamma (4+ \vartheta ) \vert [ (\ln t_{2})^{2} - (\ln t_{1})^{2} ] ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu )} \\& \qquad {}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta + \mu - 1} \bigl\vert \varphi \bigl(s,w(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\& \qquad {}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert [ ( \ln t_{2})^{2} - ( \ln t_{1})^{2} ] + \vert \varGamma (4-\vartheta ) \vert [ (\ln t_{2})^{3} - (\ln t_{1})^{3} ] ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta )} \\& \qquad {}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \mu - \delta - 1} \bigl\vert \varphi \bigl(s,w(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\& \quad \leq \Vert \sigma \Vert \biggl( \frac{2\beta }{\kappa \varGamma (\varrho + \mu +1 )} \biggl( \ln \frac{t_{2}}{t_{1}} \biggr)^{\varrho + \mu } + \frac{\beta }{\kappa \varGamma (\varrho + \mu +1 )} \bigl\vert ( \ln t_{2} )^{ \varrho + \mu } - ( \ln t_{1} )^{\varrho + \mu } \bigr\vert \\& \qquad {}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert [ ( \ln t_{2})^{3} - ( \ln t_{1})^{3} ] + 3 \vert \varGamma (4+ \vartheta ) \vert [ (\ln t_{2})^{2} - (\ln t_{1})^{2} ] ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu +1 )} \\& \qquad {}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert [ ( \ln t_{2})^{2} - ( \ln t_{1})^{2} ] + \vert \varGamma (4-\vartheta ) \vert [ (\ln t_{2})^{3} - (\ln t_{1})^{3} ] ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta +1)} \biggr). \end{aligned}$$

Note that the right-hand side is independent of \(w\in \mathcal{V}_{r}\) and converges to zero as \(t_{1}\to t_{2}\). This means that \(\varUpsilon _{2}\) is equicontinuous. Consequently, the operator \(\varUpsilon _{2}\) is relatively compact on \(\mathcal{V}_{r}\) and, by using the Arzela–Ascoli theorem, we conclude that \(\varUpsilon _{2}\) is completely continuous. Hence, \(\varUpsilon _{2}\) is compact on the open ball \(\mathcal{V}_{r}\). Now, we show that \(\varUpsilon _{1}\) is a contraction. Let \(w_{1},w_{2}\in \mathcal{V}_{r}\) and \(t\in [1,e]\). Then, we have

$$\begin{aligned} & \bigl\vert (\varUpsilon _{1} w_{1}) (t) - (\varUpsilon _{1} w_{2}) (t) \bigr\vert \\ &\quad \leq \frac{ \vert \kappa -1 \vert }{\kappa \varGamma (\varrho -\varpi )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho -\varpi -1} \bigl\vert w_{1}(s) - w_{2}(s) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{\alpha }{\kappa \varGamma (\varrho )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho -1} \bigl\vert \psi \bigl(s, w_{1}(s) \bigr) - \psi \bigl(s, w_{2}(s) \bigr) \bigr\vert \frac{ \mathrm{d}s }{s} \\ &\qquad{}+ \frac{( 1- \kappa ) [ \vert 3 \varGamma (4 + \vartheta ) \vert (\ln t)^{2} + \vert (\delta -3)\varGamma (4 - \vartheta ) \vert ( \ln t)^{3} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} \bigl\vert w_{1}(s) - w_{2}(s) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{( 1- \kappa ) \vert \varGamma (4- \delta ) \vert [ \vert \varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (3+\vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} \bigl\vert w_{1}(s) - w_{2}(s) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{ \alpha [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \bigl\vert \psi \bigl(s, w_{1}(s) \bigr) - \psi \bigl(s, w_{2}(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\qquad{}+ \frac{ \alpha \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta )} \\ &\qquad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \bigl\vert \psi \bigl(s, w_{1}(s) \bigr) - \psi \bigl(s, w_{2}(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad \leq \biggl[ \frac{ \vert \kappa -1 \vert }{\kappa \varGamma (\varrho -\varpi + 1 )} + \frac{( 1- \kappa ) [ \vert 3 \varGamma (4 + \vartheta ) \vert + \vert (\delta -3)\varGamma (4 - \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta + 1 )} \\ &\qquad{}+ \frac{( 1- \kappa ) \vert \varGamma (4- \delta ) \vert [ \vert \varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (3+\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta +1 )} + \frac{L\alpha }{\kappa \varGamma (\varrho +1 )} \\ &\qquad{}+ \frac{ L\alpha [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert + 3 \vert \varGamma (4+ \vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta +1 )} \\ &\qquad{}+ \frac{ L\alpha \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert + \vert \varGamma (4-\vartheta ) \vert ] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta +1 )} \biggr] \Vert w_{1}-w_{2} \Vert \\ &\quad = \bigl(\mathcal{K}_{0}^{*} +L \mathcal{K}_{1}^{*} \bigr) \Vert w_{1}-w_{2} \Vert . \end{aligned}$$

Since \(\mathcal{K}_{0}^{*} +L \mathcal{K}_{1}^{*} < 1\), \(\varUpsilon _{1}\) is a contraction. Note that \(\varUpsilon =\varUpsilon _{1}+\varUpsilon _{2}\). Now, by using Lemma 4, the operator ϒ has a fixed point which is a solution for the Caputo–Hadamard boundary value problem (1)–(2). □

Here, we are going to investigate the existence of solutions for the Caputo–Hadamard problem (1)–(2) by considering different conditions.

Theorem 9

Suppose that\(\psi , \varphi : [1,e]\times \mathbb{R} \to \mathbb{R} \)are continuous functions such that

\((\mathcal{N}_{3})\):

there are continuous nondecreasing functions\(\xi _{1}, \xi _{2}:[0,\infty )\rightarrow (0,\infty )\)and two maps\(\theta _{1}, \theta _{2} \in C([0, 1],\mathbb{R}^{+})\)such that\(\vert \psi (t,w) \vert \leq \theta _{1}(t)\xi _{1}( \vert w \vert )\)and\(\vert \varphi (t,w) \vert \leq \theta _{2}(t)\xi _{2}( \vert w \vert )\)for all\((t,w)\in [1,e]\times \mathbb{R}\),

\((\mathcal{N}_{4})\):

\(\mathcal{K}_{0}^{*} <1\)and there is a constant\(\varXi >0\)such that\(\frac{(1-\mathcal{K}_{0}^{*})\varXi }{\mathcal{K}_{1}^{*} \Vert \theta _{1} \Vert \xi _{1} (\varXi )+\mathcal{K}_{2}^{*} \Vert \theta _{2} \Vert \xi _{2} (\varXi )} > 1\), where\(\mathcal{K}_{0}^{*}\), \(\mathcal{K}_{1}^{*}\), \(\mathcal{K}_{2}^{*}\)are defined by (7).

Then the Caputo–Hadamard problem (1)(2) has at least one solution.

Proof

We first show that the operator ϒ maps bounded sets of \(\mathcal{E}\) into bounded sets. Let \(\epsilon >0 \), \(\mathcal{B}_{\epsilon }=\{w\in \mathcal{E}: \Vert w\Vert \leq \epsilon \}\) and \(t\in [1, e]\). Then, we have

$$\begin{aligned} \bigl\vert (\varUpsilon w) (t) \bigr\vert &\leq \frac{ \vert \kappa -1 \vert }{\kappa \varGamma (\varrho -\varpi )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{\varrho -\varpi -1} \Vert w \Vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{\alpha }{\kappa \varGamma (\varrho )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho -1} \Vert \theta _{1} \Vert \xi _{1} \bigl( \Vert w \Vert \bigr) \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho + \mu -1} \Vert \theta _{2} \Vert \xi _{2} \bigl( \Vert w \Vert \bigr) \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{( 1- \kappa ) [ \vert 3 \varGamma (4 + \vartheta ) \vert (\ln t)^{2} + \vert (\delta -3)\varGamma (4 - \vartheta ) \vert ( \ln t)^{3} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} \Vert w \Vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{( 1- \kappa ) \vert \varGamma (4- \delta ) \vert [ \vert \varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (3+\vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} \Vert w \Vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \Vert \theta _{1} \Vert \xi _{1} \bigl( \Vert w \Vert \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta + \mu - 1} \Vert \theta _{2} \Vert \xi _{2} \bigl( \Vert w \Vert \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \Vert \theta _{1} \Vert \xi _{1} \bigl( \Vert w \Vert \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \mu - \delta - 1} \Vert \theta _{2} \Vert \xi _{2} \bigl( \Vert w \Vert \bigr) \frac{\mathrm{d}s}{s} \\ &\leq \mathcal{K}_{0}^{*} \Vert w \Vert + \mathcal{K}_{1}^{*} \Vert \theta _{1} \Vert \xi _{1} \bigl( \Vert w \Vert \bigr)+\mathcal{K}_{2}^{*} \Vert \theta _{2} \Vert \xi _{2} \bigl( \Vert w \Vert \bigr). \end{aligned}$$

Hence, \(\Vert \varUpsilon w \Vert \leq \mathcal{K}_{0}^{*} \epsilon + \mathcal{K}_{1}^{*} \Vert \theta _{1} \Vert \xi _{1}(\epsilon )+ \mathcal{K}_{2}^{*} \Vert \theta _{2} \Vert \xi _{1}( \epsilon )\). Now, we prove that ϒ maps bounded sets into equicontinuous sets of \(\mathcal{E}\). Let \(t_{1}, t_{2}\in [1, e]\) with \(t_{1} < t_{2}\) and \(w\in \mathcal{B}_{\epsilon }\). Then, we get

$$\begin{aligned}& \bigl\vert (\varUpsilon w) (t_{2}) - (\varUpsilon w) (t_{1}) \bigr\vert \\& \quad \leq \frac{ \vert \kappa -1 \vert \epsilon }{\kappa \varGamma (\varrho -\varpi )} \biggl[ \int _{1}^{t_{1}} \biggl[ \biggl(\ln \frac{t_{2}}{s} \biggr)^{\varrho + \varpi -1} - \biggl(\ln \frac{t_{1}}{s} \biggr)^{\varrho + \varpi -1} \biggr] \frac{\mathrm{d}s}{s} + \int _{t_{1}}^{t_{2}} \biggl(\ln \frac{t_{2}}{s} \biggr)^{ \varrho + \varpi -1} \frac{\mathrm{d}s}{s} \biggr] \\& \qquad {}+ \frac{\alpha \Vert \theta _{1} \Vert \xi _{1} ( \epsilon ) }{\kappa \varGamma (\varrho )} \biggl[ \int _{1}^{t_{1}} \biggl[ \biggl(\ln \frac{t_{2}}{s} \biggr)^{\varrho -1} - \biggl( \ln \frac{t_{1}}{s} \biggr)^{\varrho -1} \biggr] \frac{\mathrm{d}s}{s} + \int _{t_{1}}^{t_{2}} \biggl(\ln \frac{t_{2}}{s} \biggr)^{\varrho -1} \frac{\mathrm{d}s}{s} \biggr] \\& \qquad {}+ \frac{\beta \Vert \theta _{2} \Vert \xi _{2} ( \epsilon ) }{\kappa \varGamma (\varrho + \mu )} \biggl[ \int _{1}^{t_{1}} \biggl[ \biggl(\ln \frac{t_{2}}{s} \biggr)^{\varrho + \mu -1} - \biggl(\ln \frac{t_{1}}{s} \biggr)^{\varrho + \mu -1} \biggr] \frac{\mathrm{d}s}{s} + \int _{t_{1}}^{t_{2}} \biggl(\ln \frac{t_{2}}{s} \biggr)^{ \varrho + \mu -1} \frac{\mathrm{d}s}{s} \biggr] \\& \qquad {}+ \frac{( 1- \kappa ) [ \vert 3 \varGamma (4 + \vartheta ) \vert [(\ln t_{2})^{2} - (\ln t_{1})^{2} ] + \vert (\delta -3)\varGamma (4 - \vartheta ) \vert [ ( \ln t_{2})^{3} - ( \ln t_{1})^{3} ] ] \epsilon }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta +1 )} \\& \qquad {}+ \bigl(( 1- \kappa ) \bigl\vert \varGamma (4- \delta ) \bigr\vert \bigl[ \bigl\vert \varGamma (4- \vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{3} - ( \ln t_{1})^{3} \bigr] \\& \qquad {}+ 3 \bigl\vert \varGamma (3+\vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{2} - ( \ln t_{1})^{2} \bigr] \bigr]\epsilon \bigr) \\& \qquad {}/\bigl(6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta +1 )\bigr) \\& \qquad {}+ \bigl( \alpha \bigl[ \bigl\vert (3-\delta )\varGamma (4- \vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{3} - ( \ln t_{1})^{3} \bigr] \\& \qquad {}+ 3 \bigl\vert \varGamma (4+ \vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{2} - ( \ln t_{1})^{2} \bigr] \bigr] \Vert \theta _{1} \Vert \xi _{1} ( \epsilon ) \bigr) \\& \qquad {}/\bigl(6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta +1 )\bigr) \\& \qquad {}+ \bigl( \beta \bigl[ \bigl\vert (3-\delta )\varGamma (4- \vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{3} - ( \ln t_{1})^{3} \bigr] \\& \qquad {}+ 3 \bigl\vert \varGamma (4+ \vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{2} - ( \ln t_{1})^{2} \bigr] \bigr] \Vert \theta _{2} \Vert \xi _{2} ( \epsilon ) \bigr) \\& \qquad {}/\bigl(6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu +1 )\bigr) \\& \qquad {}+ \bigl( \alpha \bigl\vert \varGamma (4- \delta ) \bigr\vert \bigl[ 3 \bigl\vert \varGamma (3+\vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{2} - ( \ln t_{1})^{2} \bigr] \\& \qquad {}+ \bigl\vert \varGamma (4-\vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{3} - ( \ln t_{1})^{3} \bigr] \bigr] \Vert \theta _{1} \Vert \xi _{1} ( \epsilon ) \bigr) \\& \qquad {}/\bigl(6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta +1 )\bigr) \\& \qquad {}+ \bigl( \beta \bigl\vert \varGamma (4- \delta ) \bigr\vert \bigl[ 3 \bigl\vert \varGamma (3+\vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{2} - ( \ln t_{1})^{2} \bigr] \\& \qquad {}+ \bigl\vert \varGamma (4-\vartheta ) \bigr\vert \bigl[ ( \ln t_{2})^{3} - ( \ln t_{1})^{3} \bigr] \bigr] \Vert \theta _{2} \Vert \xi _{2} ( \epsilon ) \bigr) \\& \qquad {}/\bigl(6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta +1 )\bigr). \end{aligned}$$

Note that the right-hand side tends to zero independently of \(w\in \mathcal{B}_{\epsilon }\) as \(t_{2}\rightarrow t_{1}\). By using the Arzela–Ascoli theorem, we deduce that \(\varUpsilon : \mathcal{E} \rightarrow \mathcal{E} \) is completely continuous. Here, we prove that the set of all solutions of the equation \(w=\lambda (\varUpsilon w)\) is bounded for each \(\lambda \in [ 0,1]\). Let \(\lambda \in [0,1]\), w be such that \(w=\lambda ( \varUpsilon w)\) and \(t\in [1, e]\). Then by using computations used in the first step, we obtain \(\Vert w \Vert \leq \mathcal{K}_{0}^{*} \Vert w\Vert + \mathcal{K}_{1}^{*} \Vert \theta _{1} \Vert \xi _{1}( \Vert w\Vert )+ \mathcal{K}_{2}^{*} \Vert \theta _{2} \Vert \xi _{2}( \Vert w \Vert )\). Thus, we conclude that \(\frac{(1- \mathcal{K}_{0}^{*} ) \Vert w\Vert }{ \mathcal{K}_{1}^{*} \Vert \theta _{1} \Vert \xi _{1}( \Vert w\Vert )+ \mathcal{K}_{2}^{*} \Vert \theta _{2} \Vert \xi _{2}( \Vert w \Vert ) } \leq 1\). By using the assumption \((\mathcal{N}_{4})\), we can choose a number \(\varXi > 0 \) such that \(\Vert w\Vert \neq \varXi \) and \(\frac{(1-\mathcal{K}_{0}^{*})\varXi }{\mathcal{K}_{1}^{*} \Vert \theta _{1} \Vert \xi _{1} (\varXi )+\mathcal{K}_{2}^{*} \Vert \theta _{2} \Vert \xi _{2} (\varXi )} > 1\). Consider the set \(\mathcal{U} = \{ w\in \mathcal{E} : \Vert w\Vert < \varXi \}\). Note that the operator \(\varUpsilon :\overline{\mathcal{U}}\rightarrow \mathcal{E}\) is continuous and completely continuous and also we can not find \(w\in \partial \mathcal{U}\) such that \(w=\lambda (\varUpsilon w)\) holds for some \(\lambda \in (0,1)\). Now, by using Lemma 5, the operator ϒ has a fixed point in \(\overline{ \mathcal{U} }\) which is a solution for the Caputo–Hadamard fractional integro-differential boundary value problem (1)–(2). □

Now by using the Banach contraction principle, we review the Caputo–Hadamard problem (1)–(2) under some different conditions.

Theorem 10

Suppose that\(\psi :[1,e]\times {\mathbb{R}}\to {\mathbb{R}}\)is a continuous function satisfying assumption\((\mathcal{N}_{1})\). Assume that the function\(\varphi :[1,e]\times {\mathbb{R}}\to {\mathbb{R}}\)satisfies the following condition:

\((\mathcal{N}_{5})\):

there is a positive constantsuch that for each\(\vert \varphi (t,w_{1})- \varphi (t,w_{2})\vert \leq \tilde{L} \vert w_{1}-w_{2} \vert \)for all\(w_{1},w_{2}\in {\mathbb{R}}\)and\(t\in [1,e]\).

If\(\mathcal{K}_{0}^{*} + L \mathcal{K}_{1}^{*} + \tilde{L} \mathcal{K}_{2}^{*} <1\), then the Caputo–Hadamard problem (1)(2) has a unique solution, where\(\mathcal{K}_{0}^{*}\), \(\mathcal{K}_{1}^{*}\), and\(\mathcal{K}_{2}^{*}\)are given by (7).

Proof

Put \(K^{*}=\sup_{t\in [1,e]} \vert \psi (t,0) \vert < \infty \) and \(N^{*}=\sup_{t\in [1,e]} \vert \varphi (t,0) \vert <\infty \). Choose \(r > 0\) such that \(r \geq \frac{ N^{*} \mathcal{K}_{2}^{*} + K^{*} \mathcal{K}_{1}^{*} }{1-(\mathcal{K}_{0}^{*} + L \mathcal{K}_{1}^{*} + \tilde{L} \mathcal{K}_{2}^{*} )}\). Let \(\mathcal{B}_{r}=\{ w\in \mathcal{E}: \Vert w\Vert \leq r \}\). We show that \(\varUpsilon \mathcal{B}_{r} \subset \mathcal{B}_{r} \). Let \(w\in \mathcal{B}_{r}\). By using assumptions \((\mathcal{N}_{1})\) and \((\mathcal{N}_{5})\), we have

$$\begin{aligned} \Vert \varUpsilon w \Vert &\leq \frac{ \vert \kappa -1 \vert }{\kappa \varGamma (\varrho -\varpi )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{\varrho -\varpi -1} \Vert w \Vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{\alpha }{\kappa \varGamma (\varrho )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho -1} \bigl(L \Vert w \Vert + K^{*} \bigr) \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho + \mu -1} \bigl(\tilde{L} \Vert w \Vert + N^{*} \bigr) \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{( 1- \kappa ) [ \vert 3 \varGamma (4 + \vartheta ) \vert (\ln t)^{2} + \vert (\delta -3)\varGamma (4 - \vartheta ) \vert ( \ln t)^{3} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} \Vert w \Vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{( 1- \kappa ) \vert \varGamma (4- \delta ) \vert [ \vert \varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (3+\vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} \Vert w \Vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \bigl(L \Vert w \Vert + K^{*} \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta + \mu - 1} \bigl(\tilde{L} \Vert w \Vert + N^{*} \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \bigl(L \Vert w \Vert + K^{*} \bigr) \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \mu - \delta - 1} \bigl( \tilde{L} \Vert w \Vert + N^{*} \bigr) \frac{\mathrm{d}s}{s} \\ &\leq \bigl(\mathcal{K}_{0}^{*} + L\mathcal{K}_{1}^{*} + \tilde{L} \mathcal{K}_{2}^{*} \bigr)r+ \mathcal{K}_{2}^{*} N^{*} +\mathcal{K}_{1}^{*} K^{*} < r. \end{aligned}$$

Hence, \(\varUpsilon \mathcal{B}_{r}\subset \mathcal{B}_{r}\). Let \(t\in [1,e]\) and \(w_{1},w_{2}\in \mathbb{R}\). Then, we have

$$\begin{aligned} \bigl\Vert (\varUpsilon w_{1}) (t) - (\varUpsilon w_{2}) (t) \bigr\Vert &\leq \frac{ \vert \kappa -1 \vert }{\kappa \varGamma (\varrho -\varpi )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{\varrho -\varpi -1} \bigl\vert w_{1}(s) - w_{2}(s) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{\alpha }{\kappa \varGamma (\varrho )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho -1} \bigl\vert \psi \bigl(s, w_{1}(s) \bigr) - \psi \bigl(s, w_{2}(s) \bigr) \bigr\vert \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{\beta }{\kappa \varGamma (\varrho + \mu )} \int _{1}^{t} \biggl(\ln \frac{t}{s} \biggr)^{\varrho + \mu -1} \bigl\vert \varphi \bigl(s, w_{1}(s) \bigr) - \varphi \bigl(s, w_{2}(s) \bigr) \bigr\vert \frac{ \mathrm{d}s }{s} \\ &\quad{}+ \frac{( 1- \kappa ) [ \vert 3 \varGamma (4 + \vartheta ) \vert (\ln t)^{2} + \vert (\delta -3)\varGamma (4 - \vartheta ) \vert ( \ln t)^{3} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho - \varpi + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi + \vartheta - 1} \bigl\vert w_{1}(s) - w_{2}(s) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{( 1- \kappa ) \vert \varGamma (4- \delta ) \vert [ \vert \varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (3+\vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma (3 + \vartheta ) \varGamma ( \varrho - \varpi - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \varpi - \delta - 1} \bigl\vert w_{1}(s) - w_{2}(s) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta - 1} \bigl\vert \psi \bigl(s , w_{1}(s) \bigr) - \psi \bigl(s , w_{2}(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta [ \vert (3-\delta )\varGamma (4- \vartheta ) \vert ( \ln t)^{3} + 3 \vert \varGamma (4+ \vartheta ) \vert (\ln t)^{2} ] }{6 \kappa (\delta + \vartheta ) \varGamma ( \varrho + \vartheta + \mu )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \vartheta + \mu - 1} \bigl\vert \varphi \bigl(s , w_{1}(s) \bigr) - \varphi \bigl(s , w_{2}(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \alpha \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho - \delta - 1} \bigl\vert \psi \bigl(s , w_{1}(s) \bigr) - \psi \bigl(s , w_{2}(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad{}+ \frac{ \beta \vert \varGamma (4- \delta ) \vert [ 3 \vert \varGamma (3+\vartheta ) \vert ( \ln t)^{2} + \vert \varGamma (4-\vartheta ) \vert (\ln t)^{3}] }{6 \kappa (\delta + \vartheta ) \varGamma (3+\vartheta ) \varGamma ( \varrho + \mu - \delta )} \\ &\quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{s} \biggr)^{\varrho + \mu - \delta - 1} \bigl\vert \varphi \bigl(s , w_{1}(s) \bigr) - \varphi \bigl(s , w_{2}(s) \bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\leq \bigl(\mathcal{K}_{0}^{*} +L \mathcal{K}_{1}^{*} + \tilde{L} \mathcal{K}_{2}^{*} \bigr) \Vert w_{1} - w_{2} \Vert . \end{aligned}$$

Since we have \(\mathcal{K}_{0}^{*} +L \mathcal{K}_{1}^{*} + \tilde{L} \mathcal{K}_{2}^{*} <1 \), ϒ is a contraction. By using the Banach contraction principle, ϒ has a unique fixed point which is the unique solution of the Caputo–Hadamard problem (1)–(2). This completes the proof. □

Examples

In this section, we provide three numerical examples to examine the validity of our theoretical findings. To do this, we consider constants \(\kappa = 0.78\), \(\alpha = 0.69\), \(\beta = 0.73\), \(\varrho = 3.95\), \(\varpi = 3.87\), \(\mu = 1.3\), \(\delta = 1.92\), and \(\vartheta = 0.001\) with \(\delta + \vartheta = 1.921 \neq 0 \) for our examples. The next example illustrates Theorem 8.

Example 1

Consider the Caputo–Hadamard fractional integro-differential equation

$$ \bigl[ 0.78 {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{3.95} + 0.22 {}^{\mathrm{CH}} \mathcal{D}_{1^{+}}^{3.87} \bigr] w(t)= 0.69 \frac{0.01 t \vert w(t) \vert }{7 + \vert w(t) \vert } + 0.73 {}^{H} \mathcal{I}_{1^{+}}^{1.3} \ln t \bigl(\sin w(t) \bigr) $$
(8)

with boundary value conditions

$$ \textstyle\begin{cases} w(1) = 0, \qquad {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{1.92 } w(e) = 0, \\ {}^{\mathrm{CH}}\mathcal{D}_{1^{+}} w(1) = 0 , \qquad \frac{1}{\varGamma ( 0.001 )} \int _{1}^{e} ( \ln \frac{e}{s})^{0.001 - 1} w(s) \frac{ \mathrm{d}s}{s} = 0, \end{cases} $$
(9)

where \(t\in [1,e]\). Define the continuous functions \(\psi ,\varphi : [1,e] \times \mathbb{R} \to \mathbb{R}\) by \(\psi (t,w) = \frac{0.99 t \vert w\vert }{7 + \vert w \vert }\) and \(\varphi (t,w)= \ln t (\sin w)\). Note that \(\vert \psi (t,w_{1})- \psi (t,w_{2}) \vert \leq L\vert w_{1}-w_{2} \vert \) holds for all \(w_{1},w_{2}\in \mathbb{R}\), where \(L= 0.01e >0\). Also, the continuous function \(\sigma (t) = \ln t \) on \([1,e]\) is such that \(\vert \varphi (t,w)\vert \leq \sigma (t) = \ln t\) for all \(w\in \mathbb{R}\). In this case, we have \(\Vert \sigma \Vert = \sup_{t\in [1,e]} \sigma (t) = 1\). Some calculations show that \(\mathcal{K}_{0}^{*} = 0.8968\) and \(\mathcal{K}_{1}^{*} = 0.3559\). Hence, \(\mathcal{K}_{0}^{*} + L \mathcal{K}_{1}^{*} = 0.9064 <1\). By using Theorem 8, the Caputo–Hadamard problem (8)–(9) has a solution.

Next example illustrates Theorem 9.

Example 2

Consider the Caputo–Hadamard fractional integro-differential equation

$$\begin{aligned} \bigl[ 0.78 {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{3.95} + 0.22 {}^{\mathrm{CH}} \mathcal{D}_{1^{+}}^{3.87} \bigr] w(t) &= 0.69 \frac{1 }{16+t } \biggl( \frac{3}{4} + \frac{ \vert w(t) \vert }{ 2 + \vert w(t) \vert } \biggr) \\ &\quad{}+ 0.73 {}^{H}\mathcal{I}_{1^{+}}^{1.3} \frac{1}{3+ \sin \frac{\pi t}{2}} \biggl( \frac{4}{5} + \frac{ \vert w(t) \vert }{3+ \vert w (t) \vert } \biggr) \end{aligned}$$
(10)

with boundary value conditions

$$ \textstyle\begin{cases} w(1) = 0, \qquad {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{1.92 } w(e) = 0, \\ {}^{\mathrm{CH}}\mathcal{D}_{1^{+}} w(1) = 0 , \qquad \frac{1}{\varGamma ( 0.001 )} \int _{1}^{e} ( \ln \frac{e}{s})^{0.001 - 1} w(s) \frac{ \mathrm{d}s}{s} = 0, \end{cases} $$
(11)

where \(t\in [1,e]\). Define continuous maps \(\psi , \varphi : [1,e] \times \mathbb{R} \to \mathbb{R}\) by \(\psi (t,w)= \frac{1 }{16+t } ( \frac{3}{4} + \frac{ \vert w\vert }{ 2 + \vert w\vert } )\) and \(\varphi (t,w)= \frac{1}{3+ \sin \frac{\pi t}{2}} ( \frac{4}{5} + \frac{\vert w\vert }{3+ \vert w \vert } )\). Note that \(\vert \psi (t,w(t)) \vert \leq \frac{1}{16+t}(1+ \Vert w\Vert )\) and \(\vert \varphi (t,w(t)) \vert \leq \frac{1}{3+ \sin \frac{\pi t}{2}} (1+ \Vert w\Vert )\) for all \(w\in \mathbb{R}\) and \(t\in [1,e]\). Put \(\theta _{1}(t) = \frac{1}{16+t}\), \(\theta _{2}(t) = \frac{1}{3+ \sin \frac{\pi t}{2}} \), and \(\xi _{1}(\Vert w\Vert ) = \xi _{2} (\Vert w\Vert ) = 1 + \Vert w \Vert \). Then, we have \(\psi (t , w) \leq \theta _{1} (t) \xi _{1} ( \vert w\vert )\) and \(\varphi (t , w) \leq \theta _{2} (t) \xi _{2} ( \vert w\vert )\). Note that \(\Vert \theta _{1} \Vert = \frac{1}{17} = 0.0588 \), \(\Vert \theta _{2} \Vert = \frac{1}{4} = 0.25 \), and \(\xi _{1}( \varXi ) = \xi _{2} (\varXi ) = 1 + \varXi \). Also, \(\mathcal{K}_{0}^{*} = 0.8968 < 1\), \(\mathcal{K}_{1}^{*} = 0.3559\), and \(\mathcal{K}_{2}^{*} = 0.2995\). By considering assumption (\(\mathcal{N}_{4}\)), choose \(\varXi > 12.76 \). Now by using Theorem 9, the Caputo–Hadamard problem (10)–(11) has a solution.

Next example illustrates Theorem 10.

Example 3

Consider the Caputo–Hadamard fractional integro-differential equation

$$ \begin{aligned}[b] & \bigl[ 0.78 {}^{\mathrm{CH}} \mathcal{D}_{1^{+}}^{3.95} + 0.22 {}^{\mathrm{CH}} \mathcal{D}_{1^{+}}^{3.87} \bigr] w(t) \\ &\quad = 0.69 \frac{ \cos t \vert w (t) \vert }{ 1 + \vert w(t) \vert } + 0.73 {}^{H} \mathcal{I}_{1^{+}}^{1.3} \frac{2}{5+t} \biggl( \frac{ \vert \arctan w(t) \vert }{ \vert \arctan w(t) \vert + 1 } \biggr) \end{aligned} $$
(12)

with boundary value conditions

$$ \textstyle\begin{cases} w(1) = 0, \qquad {}^{\mathrm{CH}}\mathcal{D}_{1^{+}}^{1.92 } w(e) = 0, \\ {}^{\mathrm{CH}}\mathcal{D}_{1^{+}} w(1) = 0 , \qquad \frac{1}{\varGamma ( 0.001 )} \int _{1}^{e} ( \ln \frac{e}{s})^{0.001 - 1} w(s) \frac{ \mathrm{d}s}{s} = 0, \end{cases} $$
(13)

where \(t\in [1,e]\). Define continuous maps \(\psi , \varphi : [1,e] \times \mathbb{R} \to \mathbb{R}\) by \(\psi (t,w)= \frac{ \cos t \vert w\vert }{ 1 + \vert w \vert }\) and \(\varphi (t,w)= \frac{2}{7+t} ( \frac{\vert \arctan w(t) \vert }{\vert \arctan w(t) \vert + 1 } )\). Note that \(\vert \psi (t,w_{1}(t)) - \psi (t,w_{2}(t)) \vert \leq \cos t ( \vert w_{1}(t)-w_{2}(t)\vert )\) and \(\vert \varphi (t,w_{1}(t)) - \varphi (t,w_{2}(t)) \vert \leq \frac{2}{7+t} ( \vert w_{1}(t)-w_{2}(t)\vert )\). Put \(L = \vert \cos (e)\vert = 0.9117\) and \(\tilde{L} = 0.25\). Some calculations show that \(\mathcal{K}_{0}^{*}+L \mathcal{K}_{1}^{*} + \tilde{L} \mathcal{K}_{2}^{*} = 0.98127 <1\). Now by using Theorem 10, the Caputo–Hadamard problem (12)–(13) has a unique solution.

Conclusions

It is known that we should increase our ability for studying of different types of fractional integro-differential equations. In this case, we could create modern software in the future by using advanced modelings of distinct phenomena. In this way, we should try to review different types of fractional integro-differential equations. In this work, we study the existence of solutions for a Caputo–Hadamard fractional integro-differential equation with boundary value conditions involving the Hadamard fractional operators via different orders. Also, we provide three examples to illustrate our main results.

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Acknowledgements

The first and second authors were supported by Azarbaijan Shahid Madani University. Also, the third author was supported by Batman University. The authors express their deep gratitude to unknown referees for their helpful suggestions which improved the final version of this paper.

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Correspondence to Shahram Rezapour.

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Etemad, S., Rezapour, S. & Sakar, F.M. On a fractional Caputo–Hadamard problem with boundary value conditions via different orders of the Hadamard fractional operators. Adv Differ Equ 2020, 272 (2020). https://doi.org/10.1186/s13662-020-02741-7

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MSC

  • 34A08
  • 39A12
  • 39A13

Keywords

  • Boundary value problem
  • Leray–Schauder
  • The Caputo–Hadamard fractional derivative
  • The Hadamard fractional integral
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