Open Access

Nonlocal boundary value problems for second-order nonlinear Hahn integro-difference equations with integral boundary conditions

Advances in Difference Equations20172017:170

https://doi.org/10.1186/s13662-017-1228-9

Received: 27 March 2017

Accepted: 2 June 2017

Published: 17 June 2017

Abstract

In this paper, we study a boundary value problem for second-order nonlinear Hahn integro-difference equations with nonlocal integral boundary conditions. Our problem contains two Hahn difference operators and a Hahn integral. The existence and uniqueness of solutions is obtained by using the Banach fixed point theorem, and the existence of at least one solution is established by using the Leray-Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem. Illustrative examples are also presented to show the applicability of our results.

Keywords

Hahn difference equations boundary value problems existence uniqueness fixed point theorems

MSC

39A10 39A13 39A70

1 Introduction

A quantum calculus substitute the classical derivative by a difference operator, which allows one to deal with sets of non-differentiable functions. There are many different types of quantum difference operators such as h-calculus, q-calculus, Hahn’s calculus, forward quantum calculus and backward quantum calculus. These operators are also found in many applications of mathematical areas such as orthogonal polynomials, basic hypergeometric functions, combinatorics, the calculus of variations and the theory of relativity. Some recent results in quantum calculus can be found in [18] and the references cited therein.

Hahn [9] introduced his difference operator \(D_{q,\omega }\) (see Definition 2.1) where \(q\in (0,1)\) and \(\omega >0\) are fixed, which unifies (in the limit) the two best-known and most-used quantum difference operators: the Jackson q-difference derivative \(D_{q}\), where \(q\in (0,1)\) (cf. [1012]); and the forward difference \(D_{\omega }\) where \(\omega >0\) (cf. [2, 13, 14]). The Hahn difference operator is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems (cf. [1517]).

The right inverse of the Hahn difference operator was introduced by Aldwoah [18, 19] who defined the right inverse of \(D_{q,\omega }\) in the terms of both the Jackson q-integral containing the right inverse of \(D_{q}\) and the Nörlund sum involving the right inverse of \(\Delta_{\omega }\).

Malinowska and Torres [20, 21] introduced the Hahn quantum variational calculus, while Malinowska and Martins [22] studied the generalized transversality conditions for the Hahn quantum variational calculus. Hamza et al. [23, 24] studied the theory of linear Hahn difference equations, and investigated the existence and uniqueness results of the initial value problems with Hahn difference equations using the method of successive approximations.

Recently, Sitthiwirattham [25] initiated the study of boundary value problems for Hahn difference equations by considering the boundary value problem consisting of the nonlinear Hahn difference equation supplemented with nonlocal three-point boundary conditions of the form:
$$\begin{aligned}& \begin{aligned}&D^{2}_{q,\omega } x(t)+f\bigl(t,x(t),D_{p,\theta }x(pt+ \theta)\bigr)=0,\quad t\in [\omega_{0},T]_{q,\omega }, \\ &x(\omega_{0})=\varphi (x), \\ &x(T)=\lambda x(\eta),\qquad \eta \in (\omega_{0},T)_{q,\omega }, \end{aligned} \end{aligned}$$
(1.1)
where \(0< q<1\), \(0<\omega <T\), \(\omega_{0}:=\frac{\omega }{1-q}\), \(1\leq \lambda <\frac{T-\omega_{0}}{\eta -\omega_{0}}\), \(p=q^{m}\), \(m\in {\mathbb{N}}\), \(\theta =\omega ( \frac{1-p}{1-q} ) \), \(f: [\omega_{0},T]_{q,\omega }\times {\mathbb{R}}\times {\mathbb{R}} \rightarrow {\mathbb{R}}\) is a given function, and \(\varphi: C ( [ \omega_{0},T]_{q,\omega },{\mathbb{R}} ) \rightarrow {\mathbb{R}}\) is a given functional. He proved existence results for (1.1) by using the Banach and Krasnoselskii fixed point theorems and also gave some numerical examples.
In this paper, motivated by the above papers, we continue the study of boundary value problems for Hahn difference equations by considering the nonlinear boundary value problem for Hahn integro-difference equations with nonlocal integral boundary conditions of the form
$$\begin{aligned}& \begin{aligned} &D^{2}_{q,\omega } x(t)=f\bigl(t,x(t),D_{p,\theta }x(pt+ \theta), \varPsi_{p,\theta }x(pt+\theta)\bigr),\quad t\in [\omega_{0},T]_{q,\omega }, \\ &x(\omega_{0})=x(T), \\ &x(\eta)=\mu \int_{\omega_{0}}^{T}g(s)x(s)\,d_{q,\omega }s,\quad \eta \in ( \omega_{0},T)_{q,\omega }, \end{aligned} \end{aligned}$$
(1.2)
where \(0< q<1\), \(0<\omega <T\), \(\omega_{0}:=\frac{\omega }{1-q}\), \(\mu \int_{\omega_{0}}^{T} g(r) \,d_{q,\omega }r\neq 1\), \(\mu \in \mathbb{R}\), \(p=q^{m}\), \(m\in {\mathbb{N}}\), \(\theta =\omega ( \frac{1-p}{1-q} ) \), \(f\in C([\omega_{0},T]_{q,\omega }\times {\mathbb{R}}\times {\mathbb{R}} \times {\mathbb{R}},{\mathbb{R}})\), and \(g\in C([\omega_{0},T]_{q, \omega },{\mathbb{R}}^{+})\) are given functions, and for \(\varphi \in C([\omega_{0},T]_{q,\omega }\times [\omega_{0},T]_{q,\omega },[0, \infty))\)
$$ \varPsi_{p,\theta } x(t):= \int_{\omega_{0}}^{t}\varphi (t,ps+\theta)x(ps+ \theta)\,d_{p,\theta } s. $$
(1.3)
Existence and uniqueness results are proved by using fixed point theorems. Also many special cases and examples are presented.

The paper is organized as follows: In Section 2, we briefly recall some definitions and lemmas that are used in this research. In Section 3, we prove an existence and uniqueness result by using the Banach fixed point theorem and two existence result via the Leray-Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem, respectively.

2 Preliminaries

In the following, there are notations, definitions, and lemmas which are used in proving the main results.

Definition 2.1

[9]

For \(0< q<1\), \(\omega >0\) and f defined on an interval \(I\subseteq {\mathbb{R}}\) containing \(\omega_{0}:=\frac{\omega }{1-q}\), the Hahn difference of f is defined by
$$D_{q,\omega }f(t)=\frac{f(qt+\omega)-f(t)}{t(q-1)+\omega }\quad \mbox{for } t\neq \omega_{0}, $$
and \(D_{q,\omega }f(\omega_{0})=f'(\omega_{0})\), provided that f is differentiable at \(\omega_{0}\). We call \(D_{q,\omega }f\) the q, ω-derivative of f, and say that f is q, ω-differentiable on I.
This operator unifies and generalizes two well-known difference operators. The first is Jackson q-difference operator defined by
$$ D_{q,0}f(t)=\textstyle\begin{cases} \frac{f(t)-f(qt)}{t(1-q)}, &t\ne 0, \\ f'(0), &t=0, \end{cases} $$
(2.1)
provident that \(f'(0)\) exists. Here f is supposed to be defined on a q-geometric set \(A\subset {\mathbb{R}}\), for which \(qt\in A\) whenever \(t\in A\).
The second operator is the forward difference operator
$$ D_{1,\omega }f(t)=\frac{f(t+\omega)-f(t)}{\omega }, $$
(2.2)
where \(\omega >0\) is fixed.
Letting \(a,b\in I \subseteq \mathbb{R} \) with \(a<\omega_{0}<b\) and \([k]_{q}=\frac{1-q^{k}}{1-q}\), \(k\in {\mathbb{N}}_{0}:={\mathbb{N}} \cup \{0\}\), we define the q, ω-interval by
$$\begin{aligned} {[a,b]}_{q,\omega } :=& \bigl\{ q^{k} a+\omega [k]_{q} : k\in {\mathbb{N}} _{0} \bigr\} \cup \bigl\{ q^{k} b+\omega [k]_{q} : k\in {\mathbb{N}} _{0} \bigr\} \cup \{ \omega_{0} \} \\ =&{[a,\omega_{0}]}_{q,\omega } \cup {[\omega_{0},b]}_{q,\omega } \\ =&{(a,b)}_{q,\omega } \cup \{a,b \} = {[a,b)}_{q,\omega } \cup \{b \} = {(a,b]}_{q,\omega } \cup \{a \}. \end{aligned}$$

Example 2.1

The interval \({[\frac{1}{2},20]}_{\frac{1}{4},3}\) can be expressed by
$${ \biggl[ \frac{1}{2},20 \biggr] }_{\frac{1}{4},3}= \biggl\{ \frac{1}{2}, \frac{25}{8}, \frac{121}{32}, \frac{505}{128}, \frac{2{,}041}{512}, \frac{8{,}185}{2{,}048},\ldots \biggr\} \cup \biggl\{ 20, 8, 5, \frac{17}{4}, \frac{65}{16}, \frac{257}{64},\ldots \biggr\} \cup \{4 \}. $$

An essential function which plays an important role in Hahn’s calculus is \(h(t)=qt+\omega \). This function is normally taken to be defined on an interval I, which contains the number \(\omega_{0}=\frac{\omega }{1-q}\). Note that h is a contraction, \(h(I)\subseteq I\), \(h(t)< t\) for \(t>\omega_{0}\), \(h(t)>t\) for \(t<\omega_{0}\), and \(h(\omega_{0})=\omega_{0}\). One can see that the kth-order iteration of \(h(t)\) is given by \(h^{k}(t)=q^{k} t+ \omega [k]_{q}\), \(t\in I\). Observe that, for each \(s\in [a,b]_{q,\omega }\), the sequence \(\{q^{k} s+\omega [k]_{q} \}_{k=0}^{\infty }\) is uniformly convergent to \(\omega_{0}\).

If f is q, ω-differentiable n times on q, ω-interval \(I_{q,\omega }\), we define the higher-order derivatives by
$$D^{n}_{q,\omega }f ( s):=D_{q,\omega }D^{n-1}_{q,\omega } f ( s), $$
where \(D^{0}_{q,\omega }f(s):=f(s)\), \(s\in I_{q,\omega }\subset {\mathbb{R}}\).

Next, we introduce the right inverse of the operator \(D_{q,\omega }\), the so-called q, ω-integral operator.

Definition 2.2

[18]

Let I be any closed interval of \(\mathbb{R}\) containing a, b and \(\omega_{0}\). Assuming that \(f:I\rightarrow \mathbb{R}\) is a given function, we define the q, ω-integral of f from a to b by
$$\int_{a}^{b} f(t)\,d_{q,\omega }t:= \int_{\omega_{0}}^{b} f(t)\,d_{q, \omega }t- \int_{\omega_{0}}^{a} f(t)\,d_{q,\omega }t, $$
where
$$\int_{\omega_{0}}^{t} f(s)\,d_{q,\omega }s= \mathcal{I}_{q,\omega }f(t)= \bigl[t(1-q)-\omega \bigr]\sum _{k=0}^{\infty } q^{k} f \bigl( tq^{k}+ \omega [k]_{q} \bigr),\quad x\in I, $$
is the convergent series at \(x=a\) and \(x=b\). f is called q, ω-integrable on \([a,b]\) and the sum to the right hand side of the above equation will be called the Jackson-Nörlund sum.

We note that the actual domain of the function f is \([a,b]_{q, \omega }\subset I\).

The following lemma is the fundamental theorem of Hahn calculus.

Lemma 2.1

[18]

Let \(f:I\rightarrow \mathbb{R}\) be continuous at \(\omega_{0}\). Define
$$F(x):= \int_{\omega_{0}}^{x} f(t)\,d_{q,\omega }t,\quad x\in I, $$
then F is continuous at \(\omega_{0}\). Furthermore, \(D_{q,\omega_{0}}F(x)\) exists for every \(x\in I\) and
$$D_{q,\omega }F(x)=f(x). $$
Conversely,
$$\int_{a}^{b} D_{q,\omega }F(t)\,d_{q,\omega }t=F(b)-F(a) \quad \textit{for all }a,b\in I. $$

Next, we give some auxiliary lemmas for simplifying calculations.

Lemma 2.2

[25]

Let \(0< q<1\), \(\omega >0\) and \(x:I\rightarrow \mathbb{R}\) be continuous at \(\omega_{0}\). Then
$$\begin{aligned} \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{r} x(s) \,d_{q,\omega }s \,d_{q,\omega }r =& \int_{\omega_{0}}^{t} \int_{qs+\omega }^{t} x(s) \,d_{q,\omega }r \,d_{q,\omega }s. \end{aligned}$$

Remark 2.1

Observe that
$$\begin{aligned} \int_{\omega_{0}}^{t} \int_{qs+\omega }^{t} x(s) \,d_{q,\omega }r \,d_{q,\omega }s =& \int_{\omega_{0}}^{t}\bigl(t-(qs+\omega)\bigr)x(s) \,d_{q, \omega }s. \end{aligned}$$

Lemma 2.3

[25]

Let \(0< q<1\) and \(\omega >0\), then
$$\begin{aligned} \int_{\omega_{0}}^{t} d_{q,\omega }s=t- \omega_{0}\quad \textit{and}\quad \int_{\omega_{0}}^{t} \bigl[t-(qs+\omega)\bigr] \,d_{q,\omega }s=\frac{ ( t- \omega_{0} ) ^{2}}{1+q}. \end{aligned}$$

Lemma 2.4

Let \(0< q<1\) and \(\omega >0\), then the following equation holds:
$$\begin{aligned} \underbrace{ \int_{\omega_{0}}^{t}\cdot \cdot \cdot \int_{\omega_{0}} ^{t}}_{n\ \mathrm{times}} d_{q,\omega }s\cdot \cdot \cdot \,d_{q,\omega }r=\frac{(t- \omega_{0})^{n}}{[n]_{q}!}, \end{aligned}$$
(2.3)
where \([n]_{q}!=\prod_{k=1}^{n}\frac{1-q^{k}}{1-q}\).

Proof

Mathematical induction will be used in our proof as follows. For \(n=1\), we have
$$\begin{aligned} \int_{\omega_{0}}^{t} d_{q,\omega }s= \bigl[t(1-q)-\omega \bigr]\sum_{k=0}^{\infty } q^{k}=(t- \omega_{0}), \end{aligned}$$
which means that equation (2.3) is true for \(n=1\).
Suppose that equation (2.3) holds for \(n=k\). Hence, for \(n=k+1\), we have
$$\begin{aligned} \underbrace{ \int_{\omega_{0}}^{t}\cdot \cdot \cdot \int_{\omega_{0}} ^{t}}_{k+1\ \mathrm{times}} d_{q,\omega }s\cdot \cdot \cdot \,d_{q,\omega }r =& \int_{\omega_{0}}^{t}\frac{(r-\omega_{0})^{k}}{[k]_{q}!} \,d_{q,\omega }r \\ =&\frac{ [t(1-q)-\omega ]}{[k]_{q}!}\sum_{k=0}^{\infty } q ^{k} \bigl( tq^{k}+\omega [k]_{q}- \omega_{0} \bigr) ^{k} \\ =&\frac{(1-q)(t-\omega_{0})}{[k]_{q}!}\sum_{k=0}^{\infty } q^{k} \bigl( tq^{k}-\omega_{0}q^{k} \bigr) ^{k} \\ =&\frac{(1-q)(t-\omega_{0})}{[k]_{q}!}\sum_{k=0}^{\infty } q^{k}q {^{k^{2}}} ( t-\omega_{0} ) ^{k} \\ =&\frac{(1-q)(t-\omega_{0})^{k+1}}{[k]_{q}!(1-q^{k+1})} \\ =&\frac{(t-\omega_{0})^{k+1}}{[k+1]_{q}!}. \end{aligned}$$
Thus, equation (2.3) holds for \(n=k+1\). By the principle of induction, equation (2.3) is true for all \(n\in {\mathbb{N}}\). □

The following lemma deals with the linear variant of problem (1.2) and gives a presentation of the solution.

Lemma 2.5

Let \(\mu \int_{\omega_{0}}^{T} g(r)\,d_{q,\omega } r \neq 1\) and \(h\in C ( [\omega_{0},T]_{q,\omega },{\mathbb{R}} ) \) be a given function. Then the function x is a solution of the problem
$$\begin{aligned}& D^{2}_{q,\omega } x(t)=h(t),\quad t\in [ \omega_{0},T]_{q,\omega }, \end{aligned}$$
(2.4)
$$\begin{aligned}& x(\omega_{0})=x(T),\qquad x(\eta)=\mu \mathcal{I}_{q,\omega } (gx) (T),\quad \eta \in (\omega_{0},T)_{q,\omega }, \end{aligned}$$
(2.5)
if and only if
$$\begin{aligned} x(t) =& \int_{\omega_{0}}^{t}\bigl[t-(qs+\omega) \bigr]h(s)\,d_{q,\omega }s-\frac{(t- \omega_{0})}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr]h(s)\,d_{q,\omega }s \\ &{}+\frac{1}{\varOmega } \biggl[\mu \int_{\omega_{0}}^{T} \int_{\omega _{0}}^{r}g(r)\bigl[r-(qs+\omega) \bigr]h(s)\,d_{q,\omega }s\,d_{q,\omega }r \\ &{}- \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega) \bigr]h(s)\,d_{q,\omega }s+\frac{1}{T- \omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] h(s)\,d_{q,\omega }s \\ & {}\times\biggl(\eta -\omega_{0}-\mu \int_{\omega_{0}}^{T}g(r) (r-\omega_{0})\,d_{q,\omega }r \biggr) \biggr], \end{aligned}$$
(2.6)
where
$$ \varOmega =1-\mu \int_{\omega_{0}}^{T}g(r)\,d_{q,\omega }r. $$
(2.7)

Proof

By Lemma 2.2 and Remark 2.1, a general solution for (2.4) can be written as
$$\begin{aligned} x(t) =& \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{r} h(s)\,d_{q,\omega }s\,d _{q,\omega }r+C_{1}(t-\omega_{0})+C_{2} \\ =& \int_{\omega_{0}}^{t} \int_{qs+\omega }^{t} h(s)\,d_{q,\omega }r\,d _{q,\omega }s+C_{1}(t-\omega_{0})+C_{2} \\ =& \int_{\omega_{0}}^{t} \bigl[ t-(qs+\omega) \bigr] h(s)\,d_{q,\omega }s+C_{1}(t-\omega_{0})+C_{2}, \end{aligned}$$
(2.8)
for \(t\in [\omega_{0},T]_{q,\omega }\). Taking the q, ω-integral for (2.8), we obtain, for \(t\in [\omega_{0},T]_{q,\omega }\),
$$\begin{aligned} \mathcal{I}_{q,\omega }x(t) =& \int_{\omega_{0}}^{t} \int_{\omega_{0}} ^{r} \bigl[ r-(qs+\omega) \bigr] h(s)\,d_{q,\omega }s\,d_{q,\omega }r+C _{1} \int_{\omega_{0}}^{t}(r-\omega_{0})\,d_{q,\omega }r \\ &{}+C_{2} \int_{\omega_{0}}^{t}d_{q,\omega }r. \end{aligned}$$
(2.9)
From the boundary conditions of (2.5), we obtain
$$\begin{aligned} C_{1} =&- \frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr]h(s)\,d_{q,\omega }s, \end{aligned}$$
(2.10)
$$\begin{aligned} C_{2} =&\frac{1}{\varOmega } \biggl[\mu \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{r} g(r) \bigl[r-(qs+\omega) \bigr]h(s)\,d_{q,\omega }s\,d_{q, \omega }r- \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega) \bigr]h(s)\,d_{q, \omega }s \\ &{}-\frac{\mu }{T-\omega_{0}} \int_{\omega_{0}}^{T} \bigl[ T-(qs+\omega) \bigr] h(s) \,d_{q,\omega }s \int_{\omega_{0}}^{T}g(r) (r-\omega_{0}) \,d_{q,\omega }s \\ &{}+\frac{\eta -\omega_{0}}{T-\omega_{0}} \int_{\omega_{0}}^{T} \bigl[ T-(qs+ \omega) \bigr] h(s) \,d_{q,\omega }s \biggr], \end{aligned}$$
(2.11)
where Ω is defined as (2.7). Substituting the constants \(C_{1}\), \(C_{2}\) into (2.8), we obtain (2.6).

On the other hand, by taking the second-order q, ω-derivative to (2.6), we have (2.4). It is easy to check that equation (2.6) satisfies (2.5). This completes the proof. □

3 Existence and uniqueness results

In this section, we present the existence and uniqueness of solutions for problem (1.2). Let
$$X=\bigl\{ x\vert x\in C \bigl( [\omega_{0},T]_{q,\omega }, { \mathbb{R}} \bigr) \mbox{ and } D_{p,\theta }x\in C \bigl( [\omega_{0},T]_{q,\omega }, {\mathbb{R}} \bigr) \bigr\} $$
be the Banach space of all continuous functions \(x: [\omega_{0},T]_{q, \omega }\to {\mathbb{R}}\) with the norm defined by
$$\Vert x\Vert _{X}=\Vert x\Vert +\Vert D_{p,\theta }x \Vert , $$
where \(\Vert x\Vert = \max \{ \vert x(t)\vert : t\in [\omega_{0},T]_{q,\omega }\}\) and \(\Vert D_{p,\theta}x \Vert = \max \{\vert D_{p,\theta }x(pt+\theta)\vert : t\in [\omega_{0},T]_{q,\omega } \}\). We define an operator \({\mathcal{F}}:X\rightarrow X\) by
$$\begin{aligned}& ({\mathcal{F}}x) (t) \\& \quad = \int_{\omega_{0}}^{t}\bigl[t-(qs+\omega)\bigr]f \bigl(s,x(s),D_{p,\theta }x(ps+ \theta),\varPsi_{p,\theta }x(ps+\theta) \bigr)\,d_{q,\omega }s \\& \qquad {}-\frac{(t-\omega_{0})}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+ \omega)\bigr]f \bigl(s,x(s),D_{p,\theta }x(ps+\theta),\varPsi_{p,\theta }x(ps+ \theta) \bigr)\,d_{q,\omega }s \\& \qquad {}+\frac{1}{\varOmega } \biggl[\mu \int_{\omega_{0}}^{T} \int_{\omega _{0}}^{r}g(r)\bigl[r-(qs+\omega)\bigr]f \bigl(s,x(s),D_{p,\theta }x(ps+\theta), \varPsi_{p,\theta }x(ps+\theta) \bigr)\,d_{q,\omega }s\,d_{q,\omega }r \\& \qquad {}- \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega)\bigr]f \bigl(s,x(s),D_{p,\theta }x(ps+\theta),\varPsi_{p,\theta }x(ps+\theta) \bigr)\,d_{q,\omega }s \\& \qquad {}+\frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] f \bigl(s,x(s),D _{p,\theta }x(ps+\theta),\varPsi_{p,\theta }x(ps+\theta) \bigr)\,d_{q, \omega }s \\& \qquad {}\times\biggl(\eta -\omega_{0}-\mu \int_{\omega_{0}}^{T}g(r) (r-\omega_{0})\,d_{q,\omega }r \biggr) \biggr], \end{aligned}$$
(3.1)
where \(\varOmega \neq 0\) is defined by (2.3), \(p=q^{m}\), \(m\in {\mathbb{N}} \) and \(\theta =\omega ( \frac{1-p}{1-q} ) \).

Obviously, problem (1.2) has solutions if and only if the operator \(\mathcal{F}\) has fixed points.

3.1 Existence and uniqueness result via Banach’s fixed point theorem

Our first result concerns existence and uniqueness of solutions of problem (1.2) and is based on Banach’s fixed point theorem.

Theorem 3.1

Let
$$ \varphi_{0}= \max \bigl\{ \varphi (t,ps+\theta): (t,ps+\theta)\in [\omega_{0},T]_{q, \omega }\times [ \omega_{0},T]_{q,\omega }\bigr\} . $$
(3.2)
Assume that:
(\(H_{1}\)): 
there exist functions \(h_{i}\in C ( [\omega_{0},T]_{q, \omega },{\mathbb{R}^{+}} )\), \(i=1,2,3\) such that
$$\bigl\vert f(t,x_{1},x_{2},x_{3})-f(t,y_{1},y_{2},y_{3}) \bigr\vert \le h_{1}(t) \vert x_{1}-y_{1} \vert + h_{2}(t)\vert x_{2}-y_{2}\vert + h_{3}(t)\vert x_{3}-y_{3}\vert $$
for each \(t\in [\omega_{0},T]_{q,\omega }\) and \(x_{i}, y_{i}, z_{i} \in {\mathbb{R}}\), \(i=1,2,3\);
(\(H_{2}\)): 

for each \(t\in [\omega_{0},T]_{q,\omega }\), \(0< g(t)< N\);

(\(H_{3}\)): 

\(\mathfrak{S}:=\varPhi_{0}(\varPhi_{1}+\varPhi_{2}) < 1\),

where
$$\begin{aligned}& \begin{aligned}&\varPhi_{0} =\Vert h_{1}\Vert + \Vert h_{2}\Vert +\Vert h_{3}\Vert \varphi_{0}(T-\omega_{0}), \\ &\begin{aligned}\varPhi_{1} ={}&\frac{2(T-\omega_{0})^{2}}{1+q}+\frac{1}{(1+q)\vert \varOmega \vert } \biggl[ \frac{\vert \mu \vert N(T-\omega_{0})^{3}(2+2q+q^{2})}{(1+q)(1+q+q^{2})} \\ &{}+(\eta -\omega_{0}) (T+\eta -2\omega_{0}) \biggr],\end{aligned} \\ &\varPhi_{2} =\frac{(T-\omega_{0})(1+p+p^{2})}{1+q}. \end{aligned} \end{aligned}$$
(3.3)
Then problem (1.2) has a unique solution on \([\omega_{0},T]_{q, \omega }\).

Proof

Denote \(\hat{f}_{x}(t):= f(t,x(t),D_{p,\theta }x(pt+ \theta),\varPsi_{p,\theta }x(pt+\theta))\). Then we have
$$\begin{aligned} \bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \le &h_{1}(t)\bigl\vert x(t)-y(t)\bigr\vert +h_{2}(t) \bigl\vert D_{p,\theta }x(pt+\theta)-D_{p,\theta }y(pt+\theta)\bigr\vert \\ &{}+h_{3}(t)\bigl\vert \varPsi_{p,\theta }x(pt+\theta)- \varPsi_{p,\theta }y(pt+\theta)\bigr\vert \\ \le &\Vert h_{1}\Vert \Vert x-y\Vert +\Vert h_{2}\Vert \Vert D_{p,\theta }x-D_{p,\theta }y\Vert + \Vert h_{3}\Vert \varphi_{0}(T-\omega_{0}) \Vert x-y\Vert \\ \le &\Vert h_{1}\Vert \Vert x-y\Vert _{X}+\Vert h_{2}\Vert \Vert x-y\Vert _{X}+\Vert h_{3}\Vert \varphi_{0}(T- \omega_{0})\Vert x-y\Vert _{X} \\ =& \bigl(\Vert h_{1}\Vert +\Vert h_{2}\Vert + \Vert h_{3}\Vert \varphi_{0}(T-\omega_{0}) \bigr) \Vert x-y\Vert _{X}. \end{aligned}$$
Using Lemma 2.3, for each \(t\in [\omega_{0},T]_{q, \omega }\) and \(x,y\in X\) we have
$$\begin{aligned}& \bigl\vert ({\mathcal{F}}x) (t)-({\mathcal{F}}y) (t)\bigr\vert \\& \quad \leq \int_{\omega_{0}}^{t}\bigl[t-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s+ \frac{(t-\omega_{0})}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \\& \qquad {}+\frac{1}{\vert \varOmega \vert } \biggl[ \biggl\vert \mu \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{r}g(r)\bigl[r-(qs+\omega)\bigr] \bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s\,d_{q,\omega }r \\& \qquad {}- \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)- \hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \biggr\vert \\& \qquad {}+\frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] \bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \biggl\vert \eta -\omega_{0}-\mu \int_{\omega_{0}}^{T}g(r) (r-\omega_{0})\,d_{q,\omega }r \biggr\vert \biggr] \\& \quad \leq \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s+ \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] \bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \\& \qquad {}+\frac{1}{\vert \varOmega \vert } \biggl[ \vert \mu \vert N \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{r}\bigl[r-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s\,d_{q,\omega }r \\& \qquad {}+ \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \\& \qquad {}+\frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] \bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \biggl((\eta -\omega_{0}) \\& \qquad {}+\vert \mu \vert N \int_{\omega_{0}}^{T}(r-\omega_{0})\,d_{q,\omega }r \biggr) \biggr] \\& \quad \leq \biggl\{ \frac{2(T-\omega_{0})^{2}}{1+q}+\frac{1}{\vert \varOmega \vert } \biggl[\frac{\vert \mu \vert N(T-\omega_{0})^{3}}{(1+q)(1+q+q^{2})}+ \frac{( \eta -\omega_{0})^{2}}{1+q} \\& \qquad {}+\frac{T-\omega_{0}}{1+q} \biggl((\eta -\omega_{0})+\frac{\vert \mu \vert N(T- \omega_{0})^{2}}{1+q} \biggr) \biggr] \biggr\} \\& \qquad {} \times\bigl(\Vert h_{1}\Vert +\Vert h_{2}\Vert + \Vert h_{3}\Vert \varphi_{0}(T-\omega_{0}) \bigr) \Vert x-y\Vert _{X} \\& \quad \leq \biggl\{ \frac{2(T-\omega_{0})^{2}}{1+q}+\frac{1}{(1+q)\vert \varOmega \vert } \biggl[ \frac{\vert \mu \vert N(T-\omega_{0})^{3}(2+2q+q^{2})}{(1+q)(1+q+q^{2})} \\& \qquad {}+(\eta -\omega_{0}) (T+\eta -2\omega_{0}) \biggr] \biggr\} \bigl(\Vert h_{1}\Vert + \Vert h_{2}\Vert +\Vert h_{3}\Vert \varphi_{0}(T-\omega_{0}) \bigr) \Vert x-y\Vert _{X} \\& \quad =\varPhi_{0}\varPhi_{1} \Vert x-y\Vert _{X}. \end{aligned}$$
Taking the p, θ-derivative for (3.1) where \(p=q^{m}\), \(m \in {\mathbb{N}}\) and \(\theta =\omega ( \frac{1-p}{1-q} ) \), we obtain
$$\begin{aligned}& \bigl\vert (D_{p,\theta }{\mathcal{F}}x) (pt+\theta)-(D_{p,\theta }{ \mathcal{F}}y) (pt+\theta) \bigr\vert \\& \quad \leq \biggl\vert \frac{1}{[-(pt+\theta)(1-p)+\theta ]} \biggl\{ \biggl( \int_{\omega_{0}}^{p(pt+\theta)+\theta } \bigl[p(pt+\theta)+\theta -(qs+ \omega)\bigr]\bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t) \bigr\vert \,d_{q,\omega }s \\& \qquad {}- \int_{\omega_{0}}^{pt+\theta } \bigl[(pt+\theta)-(qs+\omega)\bigr] \bigl\vert \hat{f} _{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \biggr)+\frac{(pt+\theta)(1-p)+ \theta }{T-\omega_{0}} \\& \qquad {}\times \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)-\hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \biggr\} \biggr\vert \\& \quad \leq \biggl\{ \frac{1}{p(1-p)(t-\omega_{0})} \biggl\vert \int_{\omega _{0}}^{pt+\theta } \bigl[(pt+\theta)-(qs+\omega) \bigr]\,d_{q,\omega }s \\& \qquad {}- \int_{\omega_{0}}^{p^{2}t+(p+1)\theta } \bigl[p^{2}t+(p+1) \theta -(qs+\omega)\bigr]\,d_{q,\omega }s \biggr\vert \\& \qquad {} +\frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega) \bigr]\,d_{q,\omega }s \biggr\} \\& \qquad {}\times \bigl(\Vert h_{1}\Vert +\Vert h_{2}\Vert + \Vert h_{3}\Vert \varphi_{0}(T-\omega_{0}) \bigr) \Vert x-y\Vert _{X} \\& \quad \leq \biggl\{ \frac{(pt+\theta -\omega_{0})^{2}-(p^{2}t+(p+1) \theta -\omega_{0})^{2}}{p(1-p)(t-\omega_{0})(1+q)}+\frac{T-\omega _{0}}{1+q} \biggr\} \\& \qquad {}\times \bigl(\Vert h_{1}\Vert +\Vert h_{2}\Vert + \Vert h_{3}\Vert \varphi_{0}(T-\omega_{0}) \bigr) \Vert x-y\Vert _{X} \\& \quad \leq \frac{(T-\omega_{0})(1+p+p^{2})}{1+q} \bigl(\Vert h_{1}\Vert +\Vert h_{2}\Vert + \Vert h_{3}\Vert \varphi_{0}(T- \omega_{0}) \bigr)\Vert x-y\Vert _{X} \\& \quad = \varPhi_{0}\varPhi_{2} \Vert x-y\Vert _{X}. \end{aligned}$$
Therefore
$$\Vert {\mathcal{F}}x-{\mathcal{F}}y\Vert _{X}\le \varPhi_{0}(\varPhi_{1}+ \varPhi_{2}) \Vert x-y \Vert _{X}. $$
This implies, by (\(H_{3}\)), that \({\mathcal{F}}\) is a contraction. Therefore, by Banach’s fixed point theorem, \({\mathcal{F}}\) has a unique fixed point, which is the unique solution of problem (1.2) on \([\omega_{0},T]_{q,\omega }\). The proof is completed. □

Corollary 3.1

If \(h_{i}=L\), \(i=1,2,3\) with \(L<\frac{1}{[2+\varphi_{0}(T-\omega_{0})](\varPhi_{1}+\varPhi_{2})}\) in Theorem  3.1, then problem (1.2) has a unique solution on \([\omega_{0},T]_{q,\omega }\).

Example 3.1

Consider the following boundary value problem for second-order Hahn integro-difference equation:
$$ \textstyle\begin{cases} D_{1/4,3/2}^{2} x(t) = f(t,x(t), D_{p,\theta } x(pt+\theta), \varPsi_{p,\theta } x(pt+\theta)), \\ x(2)=x(4),\qquad x(1{,}025/512)=(4/3) \int_{2}^{4} e^{\cos (2\pi s)} x(s)\,d_{1/4,3/2}s, \end{cases} $$
(3.4)
where
$$\begin{aligned}& f\bigl(t,x(t),D_{p,\theta } x(pt+\theta), \varPsi_{p,\theta } x(pt+ \theta)\bigr) \\& \quad =\frac{1}{(30+t^{3})(1+\vert x(t)\vert )} \bigl[e^{-\sin^{2}(2\pi t)}\bigl(x^{2}+2\vert x \vert \bigr)+e ^{-\cos^{2}(2\pi t)}\vert D_{1/256,255/128} x\vert \\& \qquad {}+e^{-t^{2}}\vert \varPsi_{1/256,255/128} x\vert \bigr] \end{aligned}$$
(3.5)
and
$$\varPsi_{1/256,255/128} x(t)= \int_{2}^{t} \frac{e^{-((1/256)s+ 255/128-t)}}{15} x\bigl((1/256)s+ 255/128\bigr)\,d_{1/256,255/128}s. $$
Here \(q=1/4\), \(\omega =3/2\), \(\omega_{0}=2\), \(p=1/256\), \(m=4\), \(\theta =255/128\), \(T=4\), \(\eta =1{,}025/512\), \(\mu =4/3\), \(g(t)=e^{ \cos (2\pi t)}\), \(\varphi (t,ps+\theta)= \frac{e^{-((1/256)s+ 255/128-t)}}{15}\). By direct computation, we find that
$$\begin{aligned}& \bigl\vert f(t,x,D_{p,\theta } x,\varPsi_{p,\theta } x)-f(t,y,D_{p,\theta } y,\varPsi_{p,\theta } y)\bigr\vert \\& \quad \leq \frac{2e^{-\sin^{2}(2\pi t)}}{30+t^{3}}\Vert x-y\Vert +\frac{e^{-\cos ^{2}(2\pi t)}}{30+t^{3}}\Vert D_{p,\theta }x-D_{p,\theta }y\Vert + \frac{e ^{-t^{2}}}{30+t^{3}}\Vert \varPsi_{p,\theta }x-\varPsi_{p,\theta }y\Vert , \end{aligned}$$
\(\varphi_{0}=1/15\), and (\(H_{1}\)) is satisfied with \(h_{1}(t)=\frac{2e ^{-\sin^{2}(2\pi t)}}{30+t^{3}}\), \(h_{2}(t)=\frac{e^{-\cos^{2}(2 \pi t)}}{30+t^{3}}\), \(h_{3}(t)=\frac{e^{-t^{2}}}{30+t^{3}}\). So, \(\varPhi_{0} = 0.06238\). From \(0< g(t)< e\), (\(H_{2}\)) is satisfied with \(N=e\). By the given data we find \(\varPhi_{1}= 13.66619\), \(\varPhi _{2}=1.60627\). Therefore, we can compute that
$$\mathfrak{S}=\varPhi_{0}(\varPhi_{1}+\varPhi_{2}) = 0.9527. $$
Hence, by Theorem 3.1, problem (3.4) with (3.5) has a unique solution on \([2,4]_{1/4,3/2}\).

3.2 Existence result via Leray-Schauder’s nonlinear alternative

Lemma 3.1

Nonlinear alternative for single valued maps [26]

Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and \(0\in U\). Suppose that \(F:\overline{U}\to C\) is a continuous, compact (that is, \(F( \overline{U})\) is a relatively compact subset of C) map. Then either
  1. (i)

    F has a fixed point in , or

     
  2. (ii)

    there is a \(u\in \partial U\) (the boundary of U in C) and \(\lambda \in (0,1)\) with \(u=\lambda F(u)\).

     

Theorem 3.2

Let \(f:[\omega_{0},T]_{q,\omega }\times \mathbb{R}\times \mathbb{R} \times \mathbb{R}\rightarrow \mathbb{R}\) be a continuous function. In addition, we assume that:
\({(H_{4})}\)
there exist a function \(m\in C([\omega_{0},T]_{q, \omega }, {\mathbb{R}})\), and continuous nondecreasing functions \(\psi_{i}: {\mathbb{R}}^{+}\to { \mathbb{R}}^{+}\), \(i=1,2,3\) such that
$$\bigl\vert f(t,x,y,z)\bigr\vert \le m(t) \bigl(\psi_{1}\bigl( \vert x\vert \bigr)+\psi_{2}\bigl(\vert y\vert \bigr)+ \psi_{3}\bigl(\vert z\vert \bigr)\bigr), $$
\(\forall (t,x,y,z) \in [\omega_{0},T]_{q,\omega } \times {\mathbb{R}} \times {\mathbb{R}}\times {\mathbb{R}}\);
\({(H_{5})}\)
there exists a constant \(M>0\) such that
$$\frac{M}{ (\psi_{1}(M)+\psi_{2}(M)+\psi_{3}(\varphi_{0}(T-\omega _{0})M)) \Vert m\Vert (\varPhi_{1}+\varPhi_{2})} >1, $$
where \(\varphi_{0}\) is defined by (3.2) and \(\varPhi_{1}\), \(\varPhi_{2}\) are defined by (3.3).

Then the boundary value problem (1.2) has at least one solution on \([\omega_{0},T]_{q,\omega }\).

Proof

Consider the operator \({\mathcal{F}}: X \to X\) defined by (3.1).

Firstly, we show that the operator \({\mathcal{F}}\) maps bounded sets into bounded sets in the space X. Let \(B_{r}=\{x\in X : \Vert x\Vert _{X} \le r\}\), \(r>0\). For any \(x\in B_{r}\), putting
$$\hat{f}_{x}(t):=f\bigl(t,x(t),D_{p,\theta }x(pt+\theta), \varPsi_{p,\theta }x(pt+\theta)\bigr), $$
and using the inequalities
$$\begin{aligned} \bigl\vert \hat{f}_{x}(t)\bigr\vert \le & m(t) \bigl( \psi_{1}\bigl(\bigl\vert x(t)\bigr\vert \bigr)+\psi_{2} \bigl(\bigl\vert D_{p,\theta }x(pt+\theta)\bigr\vert \bigr) + \psi_{3}\bigl(\bigl\vert \varPsi_{p,\theta }x(pt+\theta)\bigr\vert \bigr) \bigr) \\ \le & m(t) \bigl(\psi_{1}\bigl(\Vert x\Vert \bigr)+ \psi_{2}\bigl(\Vert D_{p,\theta }x\Vert \bigr) +\psi _{3}\bigl(\varphi_{0}(T-\omega_{0})\Vert x \Vert \bigr) \bigr) \\ \le & \Vert m\Vert \bigl(\psi_{1}\bigl(\Vert x\Vert _{X}\bigr)+\psi_{2}\bigl(\Vert x\Vert _{X} \bigr) +\psi_{3}\bigl( \varphi_{0}(T-\omega_{0}) \Vert x\Vert _{X}\bigr) \bigr) \\ \le & \Vert m\Vert \bigl(\psi_{1}(r)+\psi_{2}(r) + \psi_{3}\bigl(\varphi_{0}(T- \omega_{0})r\bigr) \bigr), \end{aligned}$$
we have
$$\begin{aligned}& \bigl\vert ({\mathcal{F}}x) (t)\bigr\vert \\& \quad \leq \int_{\omega_{0}}^{t}\bigl[t-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q, \omega }s+\frac{(t-\omega_{0})}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+ \omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s \\& \qquad {}+\frac{1}{\vert \varOmega \vert } \biggl[ \biggl\vert \mu \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{r}g(r)\bigl[r-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s\,d_{q,\omega }r \\& \qquad {}- \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q, \omega }s \biggr\vert \\& \qquad {}+\frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] \bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s \biggl\vert \eta - \omega_{0}-\mu \int_{\omega_{0}}^{T}g(r) (r-\omega_{0})\,d_{q,\omega }r \biggr\vert \biggr] \\& \quad \leq \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q, \omega }s+ \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q, \omega }s \\& \qquad {}+\frac{1}{\vert \varOmega \vert } \biggl[ \vert \mu \vert N \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{r}\bigl[r-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s\,d _{q,\omega }r + \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega)\bigr]\bigl\vert \hat{f} _{x}(t)\bigr\vert \,d_{q,\omega }s \\& \qquad {}+\frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] \bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s \biggl((\eta - \omega_{0})+\vert \mu \vert N \int_{\omega_{0}}^{T}(r-\omega_{0})\,d_{q,\omega }r \biggr) \biggr] \\& \quad \leq \biggl\{ \frac{2(T-\omega_{0})^{2}}{1+q}+\frac{1}{\vert \varOmega \vert } \biggl[\frac{\vert \mu \vert N(T-\omega_{0})^{3}}{(1+q)(1+q+q^{2})}+ \frac{( \eta -\omega_{0})^{2}}{1+q}+\frac{T-\omega_{0}}{1+q} \\& \qquad {}\times\biggl((\eta -\omega_{0})+\frac{\vert \mu \vert N(T-\omega_{0})^{2}}{1+q} \biggr) \biggr] \biggr\} \Vert m\Vert \bigl(\psi_{1}(r)+\psi_{2}(r) + \psi_{3}\bigl( \varphi_{0}(T-\omega_{0})r\bigr) \bigr) \\& \quad \leq \biggl\{ \frac{2(T-\omega_{0})^{2}}{1+q}+\frac{1}{(1+q)\vert \varOmega \vert } \biggl[ \frac{\vert \mu \vert N(T-\omega_{0})^{3}(2+2q+q^{2})}{(1+q)(1+q+q^{2})} \\& \qquad {}+(\eta -\omega_{0}) (T+\eta -2\omega_{0}) \biggr] \biggr\} \Vert m\Vert \bigl(\psi _{1}(r)+\psi_{2}(r) + \psi_{3}\bigl(\varphi_{0}(T-\omega_{0})r\bigr) \bigr). \end{aligned}$$
Taking the p, θ-derivative for (3.1) where \(p=q^{m}\), \(m\in {\mathbb{N}}\) and \(\theta =\omega ( \frac{1-p}{1-q} ) \), we obtain
$$\begin{aligned}& \bigl\vert (D_{p,\theta }{\mathcal{F}}x) (pt+\theta)\bigr\vert \\& \quad \leq \biggl\vert \frac{1}{[-(pt+\theta)(1-p)+\theta ]} \biggl\{ \biggl( \int_{\omega_{0}}^{p(pt+\theta)+\theta } \bigl[p(pt+\theta)+\theta -(qs+ \omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s \\& \qquad {}- \int_{\omega_{0}}^{pt+\theta } \bigl[(pt+\theta)-(qs+\omega)\bigr] \bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s \biggr) \\& \qquad {}+\frac{(pt+\theta)(1-p)+\theta }{T-\omega_{0}} \int_{\omega_{0}} ^{T}\bigl[T-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s \biggr\} \biggr\vert \\& \quad \leq \biggl\{ \frac{(pt+\theta -\omega_{0})^{2}-(p^{2}t+(p+1) \theta -\omega_{0})^{2}}{p(1-p)(t-\omega_{0})(1+q)}+\frac{T-\omega _{0}}{1+q} \biggr\} \\& \qquad {}\times \Vert m\Vert \bigl(\psi_{1}(r)+\psi_{2}(r) + \psi_{3}\bigl(\varphi_{0}(T-\omega _{0})r\bigr) \bigr) \\& \quad \leq \biggl\{ \frac{(1+p+p^{2})(T-\omega_{0})}{1+q} \biggr\} \Vert m\Vert \bigl( \psi_{1}(r)+\psi_{2}(r) +\psi_{3}\bigl( \varphi_{0}(T-\omega_{0})r\bigr) \bigr). \end{aligned}$$
Consequently
$$\Vert Fx\Vert _{X} \le \Vert m\Vert \bigl( \psi_{1}(r)+\psi_{2}(r) +\psi_{3}\bigl(\varphi _{0}(T-\omega_{0})r\bigr) \bigr) (\varPhi_{1}+ \varPhi_{2}). $$
Next, we shall show that \({\mathcal{F}}: B_{r}\to B_{r}\) is equicontinuous. For any \(t_{1}, t_{2} \in [\omega_{0},T]_{q,\omega }\), \(t_{1}< t_{2}\). Then we have
$$\begin{aligned}& \bigl\vert ({\mathcal{F}}x) (t_{2})-({\mathcal{F}}x) (t_{1})\bigr\vert \\& \quad \leq \int_{\omega_{0}}^{t_{2}}\bigl[t_{2}-(qs+\omega) \bigr]\bigl\vert \hat{f}_{x}(s)\bigr\vert \,d_{q,\omega }s- \int_{\omega_{0}}^{t_{1}}\bigl[t_{1}-(qs+\omega) \bigr]\bigl\vert \hat{f}_{x}(s)\bigr\vert \,d_{q,\omega }s \\& \qquad {}+\frac{\vert t_{2}-t_{1}\vert }{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+ \omega)\bigr]\bigl\vert \hat{f}_{x}(s)\bigr\vert \,d_{q,\omega }s \\& \quad \leq \int_{\omega_{0}}^{t_{1}}[t_{2}-t_{1}]\bigl\vert \hat{f}_{x}(s)\bigr\vert \,d_{q, \omega }s+ \int_{t_{1}}^{t_{2}}\bigl[t_{2}-(qs+\omega) \bigr]\bigl\vert \hat{f}_{x}(s)\bigr\vert \,d_{q,\omega }s \\& \qquad {}+\frac{\vert t_{2}-t_{1}\vert }{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+ \omega)\bigr]\bigl\vert \hat{f}_{x}(s)\bigr\vert \,d_{q,\omega }s \\& \quad \leq \biggl( \int_{\omega_{0}}^{t_{1}}[t_{2}-t_{1}]\,d_{q,\omega }s+ \int_{t_{1}}^{t_{2}}\bigl[t_{2}-(qs+\omega) \bigr]\,d_{q,\omega }s+\frac{\vert t_{2}-t_{1}\vert (T-\omega_{0})}{1+q} \biggr) \\& \qquad {}\times \Vert m\Vert \bigl(\psi_{1}(r)+\psi_{2}(r) + \psi_{3}\bigl(\varphi_{0}(T-\omega _{0})r\bigr) \bigr) \end{aligned}$$
and
$$\begin{aligned}& \bigl\vert (D_{p,\theta }{\mathcal{F}}x) (pt_{2}+\theta)-(D_{p,\theta }{\mathcal{F}}x) (pt_{1}+\theta)\bigr\vert \\& \quad \le \biggl\vert \frac{1}{[-(pt_{2}+\theta)(1-p)+\theta ]} \biggl\{ \biggl( \int_{\omega_{0}}^{p(pt_{2}+\theta)+\theta } \bigl[p(pt_{2}+\theta)+ \theta -(qs+\omega)\bigr] \,d_{q,\omega }s \\& \qquad {}- \int_{\omega_{0}}^{pt_{2}+\theta } \bigl[(pt_{2}+\theta)-(qs+ \omega)\bigr] \,d_{q,\omega }s \biggr) \\& \qquad {}+\frac{(pt_{2}+\theta)(1-p)+\theta }{T-\omega_{0}} \int_{\omega_{0}} ^{T}\bigl[T-(qs+\omega)\bigr] \,d_{q,\omega }s \biggr\} \\& \qquad {}-\frac{1}{[-(pt_{1}+\theta)(1-p)+\theta ]} \biggl\{ \biggl( \int_{\omega_{0}}^{p(pt_{1}+\theta)+\theta } \bigl[p(pt_{1}+\theta)+ \theta -(qs+\omega)\bigr] \,d_{q,\omega }s \\& \qquad {}- \int_{\omega_{0}}^{pt_{1}+\theta } \bigl[(pt_{1}+\theta)-(qs+ \omega)\bigr] \,d_{q,\omega }s \biggr) \\& \qquad {}+\frac{(pt_{1}+\theta)(1-p)+\theta }{T-\omega_{0}} \int_{\omega_{0}} ^{T}\bigl[T-(qs+\omega)\bigr] \,d_{q,\omega }s \biggr\} \biggr\vert \\& \qquad {}\times \Vert m\Vert \bigl(\psi_{1}(r)+\psi_{2}(r) + \psi_{3}\bigl(\varphi_{0}(T-\omega _{0})r\bigr) \bigr) \\& \quad \leq \biggl\{ \frac{(pt_{2}+\theta -\omega_{0})^{2}-(p^{2}t_{2}+(p+1) \theta -\omega_{0})^{2}}{p(1-p)(t_{2}-\omega_{0})(1+q)} \\& \qquad {}-\frac{(pt_{1}+\theta -\omega_{0})^{2}-(p^{2}t_{1}+(p+1)\theta - \omega_{0})^{2}}{p(1-p)(t_{1}-\omega_{0})(1+q)} \biggr\} \\& \qquad {}\times \Vert m\Vert \bigl(\psi_{1}(r)+\psi_{2}(r) + \psi_{3}\bigl(\varphi_{0}(T-\omega _{0})r\bigr) \bigr) \\& \quad \leq \biggl\{ \frac{p(1+p)(t_{2}-t_{1})}{1+q} \biggr\} \Vert m\Vert \bigl(\psi _{1}(r)+\psi_{2}(r) +\psi_{3}\bigl( \varphi_{0}(T-\omega_{0})r\bigr) \bigr). \end{aligned}$$
Hence
$$\max_{x\in \bar{B}_{r}}\bigl\vert ({\mathcal{F}}x) (t_{2})-({ \mathcal{F}}x) (t_{1})\bigr\vert +\max_{x\in \bar{B}_{r}} \bigl\vert D_{p,\theta }({\mathcal{F}}x) (pt_{2}+\theta)-D_{p,\theta }({\mathcal{F}}x) (pt_{1}+\theta)\bigr\vert \to 0, $$
as \(t_{2}\to t_{1}\) and the limit is independent of \(x\in \bar{B}_{r}\). Therefore the operator \({\mathcal{F}}: B_{r}\to B_{r}\) is equicontinuous and uniformly bounded. The Arzelá-Ascoli theorem implies that \({\mathcal{F}}\) is completely continuous.

The result will follow from the Leray-Schauder nonlinear alternative (Lemma 3.1) once we have proved the boundedness of the set of all solutions to equations \(x=\lambda {\mathcal{F}} x\) for \(\lambda \in (0,1)\).

Let x be a solution. Then, for \(t\in [\omega_{0},T]_{q,\omega }\), and using the computations in proving that \({\mathcal{F}}\) is bounded, for \(\lambda \in (0,1)\), let \(x=\lambda {\mathcal{F}}x\). Then we have
$$\Vert x\Vert _{X}\le \bigl(\psi_{1}\bigl(\Vert x \Vert _{X}\bigr)+\psi_{2}\bigl(\Vert x\Vert _{X}\bigr)+\psi_{3}\bigl( \varphi_{0}(T- \omega_{0})\Vert x\Vert _{X}\bigr) \bigr)\Vert m \Vert (\varPhi_{1}+\varPhi _{2}), $$
or
$$\frac{\Vert x\Vert _{X}}{ (\psi_{1}(\Vert x\Vert _{X})+\psi_{2}(\Vert x\Vert _{X})+\psi _{3}(\varphi_{0}(T-\omega_{0})\Vert x\Vert _{X}))\Vert m\Vert (\varPhi_{1}+ \varPhi_{2})}\le 1. $$
In view of (\(H_{5}\)), there exists M such that \(\Vert x\Vert _{X} \ne M\). Let us set
$$U = \bigl\{ x \in C\bigl([\omega_{0},T]_{q,\omega }, {\mathbb{R}} \bigr): \Vert x\Vert _{X}< M \bigr\} . $$
Note that the operator \({\mathcal{F}}:\overline{U} \to C([\omega_{0},T]_{q, \omega }, \mathbb{R})\) is continuous and completely continuous. From the choice of U, there is no \(x \in \partial U\) such that \(x =\lambda {\mathcal{F}}x\) for some \(\lambda \in (0,1)\). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that \({\mathcal{F}}\) has a fixed point \(x \in \overline{U}\) which is a solution of problem (1.2). This completes the proof. □

Corollary 3.2

Suppose that a continuous function f satisfies \(\vert f(t,x,y,z)\vert \leq K(\vert x\vert +\vert y\vert +\vert z\vert )\), \(K\geq 0\). If \(K[2+\varphi_{0}(T- \omega_{0})](\varPhi_{1}+\varPhi_{2})<1\), then problem (1.2) has at least one solution on \([\omega_{0},T]_{q,\omega }\).

Example 3.2

Consider the following boundary value problem for second-order Hahn integro-difference equation:
$$ \textstyle\begin{cases} D_{1/3,4/3}^{2} x(t) =f(t,x(t),D_{p,\theta } x(pt+\theta), \varPsi_{p,\theta } x(pt+\theta)), \\ x(2)=x(4),\qquad x(1{,}460/729)=(5/3) \int_{2}^{4} e^{\cos (2\pi s)} x(s)\,d_{1/3,4/3}s, \end{cases} $$
(3.6)
where
$$\begin{aligned}& f\bigl(t,x(t),D_{p,\theta } x(pt+\theta), \varPsi_{p,\theta } x(pt+ \theta)\bigr) \\& \quad =\frac{1}{(t+1)^{2}} \biggl[\frac{1}{10}e^{-\vert x\vert }\sin \vert x \vert + \frac{\vert D_{1/81,160/81} x\vert ^{2}}{12 (1+\vert D_{1/81,160/81} x\vert )}+ \frac{\vert \varPsi_{1/81,160/81} x\vert ^{5}}{15 (1+\vert \varPsi_{1/81,160/81} x\vert ^{4} )}+3 \biggr] \end{aligned}$$
(3.7)
and
$$\varPsi_{1/81,160/81} x(t)= \int_{2}^{t} \frac{e^{-((1/81)s+ 160/81-t)}}{150} x\bigl((1/81)s+ 160/81\bigr)\,d_{1/81,160/81}s. $$
Here \(q=1/3\), \(\omega =4/3\), \(\omega_{0}=2\), \(p=1/81\), \(m=4\), \(\theta =160/81\), \(T=4\), \(\eta =1{,}460/729\), \(\mu =5/3\), \(g(t)=e^{ \cos (2\pi t)}\), \(\varphi (t,ps+\theta)= \frac{e^{-((1/81)s+ 160/81-t)}}{150}\). Since
$$\begin{aligned}& \bigl\vert f(t,x,D_{p,\theta } x,\varPsi_{p,\theta } x)\bigr\vert \\& \quad \leq \frac{1}{(t+1)^{2}} \biggl[\frac{1}{10}\vert x\vert + \frac{1}{12} \vert D_{1/81,160/81} x\vert +\frac{1}{15}\vert \varPsi_{1/81,160/81} x\vert +3 \biggr], \end{aligned}$$
\(\varphi_{0}=1/150\), (\(H_{4}\)) is satisfied with \(m(t)=\frac{1}{(t+1)^{2}}\), \(\psi_{1}(\vert x\vert )=\frac{1}{10}\vert x\vert \), \(\psi_{2}(\vert D_{1/81,160/81} x\vert )= \frac{1}{12}\vert D_{1/81,160/81} x\vert +1\), \(\psi_{3}(\vert \varPsi_{1/81,160/81} x\vert )=\frac{1}{15}\vert \varPsi_{1/81,160/81} x\vert +2\). In addition, we see that \(0< g(t)< e\), then (\(H_{2}\)) is satisfied with \(N=e\). From the above information, we find that \(\varPhi_{1}= 13.86461\), \(\varPhi_{2}= 1.51875\). Therefore, there exists a constant \(M> 7.48455\) satisfying (\(H_{5}\)). Hence, by Theorem 3.2, problem (3.6) with (3.7) has at least one solution on \([2,4]_{1/3,4/3}\).

3.3 Existence result via Krasnoselskii’s fixed point theorem

The final existence result is based on Krasnoselskii’s fixed point theorem.

Lemma 3.2

Krasnoselskii’s fixed point theorem [27]

Let S be a closed, convex and nonempty subset of a Banach space X. Let A, B be the operators such that (a) \(Ax+By\in S\) whenever \(x,y\in S\); (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists \(z \in S\) such that \(z = Az+Bz\).

Theorem 3.3

Assume that (\(H_{1}\)) and (\(H_{2}\)) hold. In addition we assume that:
(\(H_{6}\)): 

\(\vert f(t,x,y,z)\vert \leq \mu (t)\), for each \(t\in [\omega_{0},T]_{q, \omega }\), \(x, y, z\in {\mathbb{R}}\) and \(\mu \in C([\omega_{0},T], \mathbb{R^{+}})\).

Then problem (1.2) has at least one solution on \([\omega_{0},T]_{q, \omega }\), provided
$$ \frac{T-\omega_{0}}{1+q}(1+T-\omega_{0}) \varPhi_{0}< 1, $$
(3.8)
where \(\varPhi_{0}\) is defined by (3.3).

Proof

Consider the operator \({\mathcal{F}}:X\rightarrow X\) defined by (3.1) as
$$\begin{aligned} ({\mathcal{F}}x) (t)=({\mathcal{F}_{1}}x) (t)+({ \mathcal{F}_{2}}x) (t),\quad t\in [\omega_{0},T]_{q,\omega }, \end{aligned}$$
(3.9)
where
$$\begin{aligned} ({\mathcal{F}_{1}}x) (t) =& -\frac{(t-\omega_{0})}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+ \omega)\bigr] \hat{f}_{x}(s)\,d_{q,\omega }s \end{aligned}$$
and
$$\begin{aligned}& ({\mathcal{F}_{2}}x) (t) \\& \quad = \int_{\omega_{0}}^{t}\bigl[t-(qs+\omega)\bigr] \hat{f}_{x}(s)\,d_{q,\omega }s \\& \qquad {}+\frac{1}{\varOmega } \biggl[\mu \int_{\omega_{0}}^{T} \int_{\omega _{0}}^{r}g(r)\bigl[r-(qs+\omega)\bigr] \hat{f}_{x}(s)\,d_{q,\omega }s\,d_{q,\omega }r- \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega)\bigr] \hat{f}_{x}(s)\,d_{q, \omega }s \\& \qquad {}+\frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] \hat{f}_{x}(s)\,d_{q,\omega }s \biggl(\eta -\omega_{0}-\mu \int_{\omega _{0}}^{T}g(r) (r-\omega_{0})\,d_{q,\omega }r \biggr) \biggr]. \end{aligned}$$
Setting \(\max \{\mu (t): t\in [\omega_{0},T]_{q,\omega }\}=\Vert \mu \Vert \) and choosing \(\rho \ge \Vert \mu \Vert (\varPhi_{1}+\varPhi_{2})\) we consider \(B_{\rho }=\{x\in C([\omega_{0},T]_{q,\omega }, {\mathbb{R}}): \Vert x\Vert _{X}\le \rho \}\). For any \(x,y\in B_{\rho }\) we have
$$\begin{aligned}& \bigl\vert ({\mathcal{F}_{1}}x) (t)+({\mathcal{F}_{2}}y) (t)\bigr\vert \\& \quad \leq \int_{\omega_{0}}^{t}\bigl[t-(qs+\omega)\bigr]\bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q, \omega }s+\frac{(t-\omega_{0})}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+ \omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s \\& \qquad {}+\frac{1}{\vert \varOmega \vert } \biggl[ \biggl\vert \mu \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{r}g(r)\bigl[r-(qs+\omega)\bigr]\bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s\,d_{q,\omega }r \\& \qquad {}- \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega)\bigr]\bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q, \omega }s \biggr\vert \\& \qquad {}+\frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] \bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \biggl\vert \eta - \omega_{0}-\mu \int_{\omega_{0}}^{T}g(r) (r-\omega_{0})\,d_{q,\omega }r \biggr\vert \biggr] \\& \quad \leq \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr]\bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q, \omega }s+ \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q, \omega }s \\& \qquad {}+\frac{1}{\vert \varOmega \vert } \biggl[ \vert \mu \vert N \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{r}\bigl[r-(qs+\omega)\bigr]\bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s\,d _{q,\omega }r + \int_{\omega_{0}}^{\eta }\bigl[\eta -(qs+\omega)\bigr]\bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \\& \qquad {}+\frac{1}{T-\omega_{0}} \int_{\omega_{0}}^{T}\bigl[T-(qs+\omega)\bigr] \bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \biggl((\eta - \omega_{0})+\vert \mu \vert N \int_{\omega_{0}}^{T}(r-\omega_{0})\,d_{q,\omega }r \biggr) \biggr] \\& \quad \leq \Vert \mu \Vert \biggl\{ \frac{2(T-\omega_{0})^{2}}{1+q}+\frac{1}{\vert \varOmega \vert } \biggl[ \frac{\vert \mu \vert N(T-\omega_{0})^{3}}{(1+q)(1+q+q^{2})}+\frac{( \eta -\omega_{0})^{2}}{1+q} \\& \qquad {}+\frac{T-\omega_{0}}{1+q} \biggl((\eta -\omega_{0})+\frac{\vert \mu \vert N(T- \omega_{0})^{2}}{1+q} \biggr) \biggr] \biggr\} . \end{aligned}$$
Taking the p, θ-derivative for (3.9) where \(p=q^{m}\), \(m\in {\mathbb{N}}\) and \(\theta =\omega ( \frac{1-p}{1-q} ) \), we obtain
$$\begin{aligned}& \bigl\vert (D_{p,\theta }{\mathcal{F}_{1}}x) (pt+\theta)+(D_{p,\theta }{\mathcal{F}_{2}}y) (pt+\theta)\bigr\vert \\& \quad \leq \biggl\vert \frac{1}{[-(pt+\theta)(1-p)+\theta ]} \biggl\{ \biggl( \int_{\omega_{0}}^{p(pt+\theta)+\theta } \bigl[p(pt+\theta)+\theta -(qs+ \omega)\bigr]\bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \\& \qquad {}- \int_{\omega_{0}}^{pt+\theta } \bigl[(pt+\theta)-(qs+\omega)\bigr] \bigl\vert \hat{f}_{y}(t)\bigr\vert \,d_{q,\omega }s \biggr) \\& \qquad {}+\frac{(pt+\theta)(1-p)+\theta }{T-\omega_{0}} \int_{\omega_{0}} ^{T}\bigl[T-(qs+\omega)\bigr]\bigl\vert \hat{f}_{x}(t)\bigr\vert \,d_{q,\omega }s \biggr\} \biggr\vert \\& \quad \leq \Vert \mu \Vert \biggl\{ \frac{(1+p+p^{2})(T-\omega_{0})}{1+q} \biggr\} . \end{aligned}$$
Consequently
$$\Vert {\mathcal{F}_{1}}x+{\mathcal{F}_{2}}y\Vert _{X} \le \Vert \mu \Vert (\varPhi _{1}+ \varPhi_{2})\le \rho, $$
which shows that \({\mathcal{F}_{1}}x+{\mathcal{F}_{2}}y\in B_{\rho }\).
It is easy to prove that
$$\Vert {\mathcal{F}_{1}}x-{\mathcal{F}_{1}}y\Vert \le \frac{(T-\omega_{0})^{2}}{1+q}\varPhi_{0}\Vert x-y\Vert _{X},\qquad \Vert D_{p,\theta }{\mathcal{F}_{1}}x-D_{p,\theta }{ \mathcal{F}_{1}}y\Vert \le \frac{T- \omega_{0}}{1+q}\varPhi_{0} \Vert x-y\Vert _{X}, $$
and consequently
$$\Vert {\mathcal{F}_{1}}x-{\mathcal{F}_{1}}y\Vert _{X}\le \frac{T-\omega_{0}}{1+q}(1+T-\omega_{0}) \varPhi_{0}\Vert x-y\Vert _{X}, $$
which implies, by (3.8), that \({\mathcal{F}_{1}}\) is a contraction mapping.

Continuity of f implies that the operator \({\mathcal{F}_{2}}\) is continuous. Also, \({\mathcal{F}_{2}}\) is uniformly bounded on \(B_{\rho }\) and equicontinuous, as proved in Theorem 3.2. So \({\mathcal{F}_{2}}\) is relatively compact on \(B_{\rho }\). Hence, by the Arzelá-Ascoli theorem, \({\mathcal{F}_{2}}\) is compact on \(B_{\rho }\). Thus all the assumptions of Lemma 3.2 are satisfied. So the conclusion of Lemma 3.2 implies that problem (1.2) has at least one solution on \([\omega_{0},T]_{q, \omega }\). □

Corollary 3.3

Suppose that a continuous function f satisfies \((H_{1})\) with \(h_{i}=L\), \(i=1,2,3\), and \(\vert f(t,x,y,z)\vert \leq K\), \(K> 0\). If \(\frac{T- \omega_{0}}{1+q}(1+T-\omega_{0})(2+\varphi_{0}(T-\omega_{0}))L<1\) then problem (1.2) has at least one solution on \([\omega_{0},T]_{q, \omega }\).

Example 3.3

Consider the boundary value problem for second-order Hahn integro-difference equation in Example 3.1 with
$$\begin{aligned}& f\bigl(t,x(t),D_{p,\theta } x(pt+\theta), \varPsi_{p,\theta } x(pt+ \theta)\bigr) \\& \quad =\frac{1}{(25+t^{3})(1+\vert x(t)\vert )} \bigl[e^{-\sin^{2}(2\pi t)}\bigl(x^{2}+2\vert x \vert \bigr)+e ^{-\cos^{2}(2\pi t)}\vert D_{1/256,255/128} x\vert \\& \qquad {}+e^{-t^{2}}\vert \varPsi_{1/256,255/128} x\vert \bigr]. \end{aligned}$$
(3.10)
Now, we see that
$$\begin{aligned}& \bigl\vert f(t,x,D_{p,\theta } x,\varPsi_{p,\theta } x)-f(t,y,D_{p,\theta } y,\varPsi_{p,\theta } y)\bigr\vert \\& \quad \leq \frac{2e^{-\sin^{2}(2\pi t)}}{25+t^{3}}\Vert x-y\Vert +\frac{e^{-\cos ^{2}(2\pi t)}}{25+t^{3}}\Vert D_{p,\theta }x-D_{p,\theta }y\Vert + \frac{e ^{-t^{2}}}{25+t^{3}}\Vert \varPsi_{p,\theta }x-\varPsi_{p,\theta }y\Vert . \end{aligned}$$
Then (\(H_{1}\)) is satisfied with \(h_{1}(t)=\frac{2e^{-\sin^{2}(2 \pi t)}}{25+t^{3}}\), \(h_{2}(t)=\frac{e^{-\cos^{2}(2\pi t)}}{25+t^{3}}\), \(h_{3}(t)=\frac{e^{-t^{2}}}{25+t^{3}}\). Therefore, we can find that \(\varPhi_{0} = 0.07183\), and
$$\mathfrak{S}=\varPhi_{0}(\varPhi_{1}+\varPhi_{2}) = 1.09702 > 1. $$
Thus, Theorem 3.1 cannot be applied in this case. However, (\(H_{6}\)) is satisfied with \(\mu (t)=\frac{1}{25+t^{3}}\), by \(\vert f(t,x,D_{p,\theta } x,\varPsi_{p,\theta } x)\vert \leq \frac{1}{25+t^{3}}\). Indeed, we find that
$$\frac{T-\omega_{0}}{(1+q)}(1+T-\omega_{0})\varPhi_{0} = 0.34478< 1. $$
Then, by using Theorem 3.3, problem (3.4) with (3.10) has at least one solution on \([9/2,8]_{2/3,3/2}\).

Declarations

Acknowledgements

This research is partially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok
(2)
Department of Mathematics, University of Ioannina
(3)
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematices, Faculty of Science, King Abdulaziz University
(4)
Centre of Excellence in Mathematics, CHE
(5)
Mathematics Department, Faculty of Science and Technology, Suan Dusit University

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