Skip to main content

On solution of generalized proportional fractional integral via a new fixed point theorem

Abstract

The aim of this paper is the solvability of generalized proportional fractional(GPF) integral equation at Banach space \(\mathbb{E}\). Herein, we have established a new fixed point theorem which is then applied to the GPF integral equation in order to establish the existence of solution on the Banach space. At last, we have illustrated a genuine example that verified our theorem and gave a strong support to prove it.

Introduction

In 1930 Kuratowski [1] introduced the notion of a measure of noncompactness. In functional analysis, this idea is particularly important in metric fixed point theory and operator equation theory in Banach spaces. The theory of infinite systems of fractional integral equations (FIEs) plays a pivotal role in different fields, which includes various implications in the scaling system theory, the theory of algorithms, etc. There are many real life problems which can be formulated by infinite systems of integral equations with fractional order in a very effective manner.

In recent times, the fixed point theory (FPT) has applications in various scientific fields. Also, FPT can be applied seeking solutions for FIE.

Different real life situations which are formulated via FIEs can be studied using FPT and measure of noncompactness (MNC) (see [224]).

Let a real Banach space \(( \mathbb{E}, \Vert . \Vert ) \) and \(B(x,r)= \lbrace y \in \mathbb{E}: \Vert y-x \Vert \leq r \rbrace \). If \(\Omega (\neq \phi ) \subseteq \mathbb{E}\). Also, Ω̄ and ConvΩ represent the closure and convex closure of Ω. Moreover, let

  1. a.

    \(\mathfrak{M}_{\mathbb{E}}=\) collection of all nonempty and bounded subsets of \(\mathbb{E}\),

  2. b.

    \(\mathfrak{N}_{\mathbb{E}}=\) collection of all relatively compact sets,

  3. c.

    \(\mathbb{R}=\) collection of all real numbers,

    and

  4. d.

    \(\mathbb{R}_{+}=\) collection of all nonnegative real numbers.

The following definition of an MNC is given in [25].

Definition 1.1

A function \(\Pi:\mathfrak{M}_{\mathbb{E}} \rightarrow [0,\infty )\) is called an MNC in \(\mathbb{E}\) if it satisfies the following conditions:

  1. (i)

    The family ker \(\Pi = \lbrace \Omega \in \mathfrak{M}_{\mathbb{E}}: \Pi ( \Omega )=0 \rbrace \) is nonempty and ker \(\Pi \subset \mathfrak{N}_{\mathbb{E}}\).

  2. (ii)

    \(\Omega \subseteq \Omega _{1} \implies \Pi ( \Omega ) \leq \Pi ( \Omega _{1} )\).

  3. (iii)

    \(\Pi ( \bar{\Omega } )=\Pi ( \Omega )\).

  4. (iv)

    \(\Pi ( \operatorname{Conv} \Omega )=\Pi ( \Omega )\).

  5. (v)

    \(\Pi ( \rho \Omega + (1- \rho )P ) \leq \rho \Pi ( \Omega )+ (1- \rho )\Pi ( P )\) for \(\rho \in [ 0, 1 ]\).

  6. (vi)

    If \(\Omega _{n} \in \mathfrak{M}_{\mathbb{E}}, \Omega _{n}= \bar{\Omega }_{n}, \Omega _{n+1} \subset \Omega _{n}\) for \(n=1,2,3,\ldots \) and \(\lim_{n \rightarrow \infty }\Pi ( \Omega _{n} )=0\) then \(\Omega _{\infty } =\bigcap_{n=1}^{\infty }\Omega _{n} \neq \phi \).

The kerΠ family is kernel of measure Π. Note that the intersection set \(\Omega _{\infty }\) from (vi) is a member of the family kerΠ. In fact, since \(\Pi (\Omega _{\infty }) \leq \Pi (\Omega _{n})\) for any n, we conclude that \(\Pi (\Omega _{\infty })=0\). This gives \(\Omega _{\infty } \in \ker \Pi \).

The fixed point principle and theorem play a key role in the theory of fixed point.

Theorem 1.2

(Shauder [26])

Let \(\mathbb{V}\) be a nonempty, closed, and convex subset of a Banach space \(\mathbb{E}\). Then every compact, continuous map \(\varUpsilon: \mathbb{V} \rightarrow \mathbb{V}\) has at least one fixed point(\(\mathbb{F}\mathbb{P}\)) in \(\mathbb{V}\).

Theorem 1.3

(Darbo [27])

Let V be a nonempty, bounded, closed, and convex(NBCC) subset of a Banach space \(\mathbb{E}\). Let \(\varUpsilon: V \rightarrow V\) be a continuous mapping. Assume that there is a constant \(p\in [ 0,1 ) \) such that

$$\begin{aligned} \eta (\varUpsilon \Omega )\leq p\eta (\Omega ), \quad\Omega \subseteq V, \end{aligned}$$

where η is an arbitrary MNC. Then ϒ has an \(\mathbb{F}\mathbb{P}\) in V.

We introduced the following generalization of the Banach contraction principle, in which we get a variety of contractive inequalities by substituting different functions g.

Theorem 1.4

Let \(( \gamma,d ) \) be a complete metric space. Also, let \(J: \gamma \mapsto \gamma \) be a continuous self-mapping. Suppose that there exists a function \(g:\mathbb{R_{+}} \rightarrow \mathbb{R_{+}}\) such that \(\lim_{t \rightarrow o^{+}} g(t)=0\), \(g(0)=0\), and

$$\begin{aligned} d(Jx,Jy)\leq g\bigl(d(x,y)\bigr) - g\bigl(d(Jx,Jy)\bigr); \quad\forall x,y \in \gamma. \end{aligned}$$

Then J has a unique \(\mathbb{F}\mathbb{P}\).

Definition 1.5

([28])

Let \(\mathbb{F}\) be the class of all functions \(F: \mathbb{R}_{+}\times \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} \) satisfying:

  1. (1)

    \(\max \lbrace m_{1},m_{2} \rbrace \leq F(m_{1},m_{2})\) for \(m_{1},m_{2} \geq 0\);

  2. (2)

    F is continuous;

  3. (3)

    \(F(m_{1}+m_{2},n_{1}+n_{2}) \leq F(m_{1},n_{1})+F(m_{2},n_{2})\);

e.g. \(F(m_{1},m_{2}) = m_{1}+m_{2}\).

Main result

Theorem 2.1

Let \(\mathbb{V}\) be an NBCC subset of a Banach space \(\mathbb{E}\), and let \(\varUpsilon: \mathbb{V} \rightarrow \mathbb{V}\) be a continuous operator such that

$$\begin{aligned} F \bigl[\Pi (\varUpsilon X),\phi \bigl(\Pi (\varUpsilon X)\bigr) \bigr] \leq \Delta \bigl[ F \bigl\lbrace \Pi (X),\phi \bigl(\Pi (X)\bigr) \bigr\rbrace \bigr] - \Delta \bigl[ F \bigl\lbrace \Pi (\varUpsilon X),\phi \bigl( \Pi (\varUpsilon X)\bigr) \bigr\rbrace \bigr] \end{aligned}$$
(2.1)

for all \(X\subseteq \mathbb{V}\), where \(\Delta,\phi: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} \) are nondecreasing continuous functions and Π is an arbitrary MNC. Then ϒ has at least one \(\mathbb{F}\mathbb{P}\) in \(\mathbb{V}\).

Proof

Let \(\mathbb{V}_{0}=\mathbb{V}\) and construct a sequence \(\lbrace \mathbb{V}_{n}\rbrace \) such that \(\mathbb{V}_{n+1}=\operatorname{Conv}(\varUpsilon \mathbb{V}_{n})\) for all \(n \in \mathbb{N}\). If there exists a positive integer \(N_{0}\in \mathbb{N}\) such that \(\Pi (\mathbb{V}_{N_{0}}) = 0\), so \(\mathbb{V}_{N_{0}}\) is relatively compact. And by Theorem 2.1, we give that ϒ has an \(\mathbb{F}\mathbb{P}\).

If possible, assume that \(\Pi (\mathbb{V}_{n}) > 0\) for all n. Also, we have

$$\begin{aligned} \mathbb{V}_{1}\supseteq \mathbb{V}_{2}\supseteq \cdots \supseteq \mathbb{V}_{n}\supseteq \mathbb{V}_{n+1}\supseteq \ldots \end{aligned}$$

Since the sequence \(\lbrace \Pi (\mathbb{V}_{n}) \rbrace \) is decreasing. So, \(\phi ( \Pi (\mathbb{V}_{n})) \) is decreasing.

Hence, the sequence \(F [\Pi (\mathbb{V}_{n}),\phi (\Pi (\mathbb{V}_{n})) ]\) is decreasing.

Since \(\lim_{n \rightarrow \infty }F [\Pi (\mathbb{V}_{n}), \phi (\Pi (\mathbb{V}_{n})) ]=L\).

By using equation (2.1), we have

$$\begin{aligned} 0 &\leqslant F \bigl[\Pi (\mathbb{V}_{n+1}),\phi \bigl(\Pi ( \mathbb{V}_{n+1})\bigr) \bigr] \\ &=F \bigl[\Pi (\varUpsilon \mathbb{V}_{n}),\phi \bigl(\Pi (\varUpsilon \mathbb{V}_{n})\bigr) \bigr] \\ &\leq \Delta \bigl[ F \bigl\lbrace \Pi (\mathbb{V}_{n}),\phi \bigl(\Pi ( \mathbb{V}_{n})\bigr) \bigr\rbrace \bigr] - \Delta \bigl[ F \bigl\lbrace \Pi (\varUpsilon \mathbb{V}_{n}),\phi \bigl(\Pi (\varUpsilon \mathbb{V}_{n})\bigr) \bigr\rbrace \bigr] \\ &= \Delta \bigl[ F \bigl\lbrace \Pi (\mathbb{V}_{n}),\phi \bigl(\Pi ( \mathbb{V}_{n})\bigr) \bigr\rbrace \bigr] - \Delta \bigl[ F \bigl\lbrace \Pi (\mathbb{V}_{n+1}),\phi \bigl(\Pi ( \mathbb{V}_{n+1})\bigr) \bigr\rbrace \bigr]. \end{aligned}$$

As \(n \rightarrow \infty \), we get

$$\begin{aligned} 0\leqslant L\leqslant \Delta (L) -\Delta (L) = 0, \end{aligned}$$

that is, \(L=0\).

Therefore, \(\lim_{n \rightarrow \infty }\Pi (\mathbb{V}_{n})=0\). According to axiom (vi) of Definition 1.1, we conclude that \(\mathbb{V}_{\infty }=\bigcap_{n=1}^{\infty }\mathbb{V}_{n}\) is an NBCC set, invariant under the mapping ϒ and belongs to kerΠ. By Theorem 1.2, we have ϒ has an \(\mathbb{F}\mathbb{P}\). □

Theorem 2.2

Let V be an NBCC subset of a Banach space \(\mathbb{E}\), and let \(\varUpsilon: \mathbb{V} \rightarrow \mathbb{V}\) be a continuous operator such that

$$\begin{aligned} 2F \bigl[\Pi (\varUpsilon X),\phi \bigl(\Pi (\varUpsilon X)\bigr) \bigr] \leq F \bigl\lbrace \Pi (X),\phi \bigl(\Pi (X)\bigr) \bigr\rbrace \end{aligned}$$
(2.2)

for all \(X\subseteq \mathbb{V}\), where \(\phi: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} \) is a nondecreasing continuous function and Π is an arbitrary MNC. Then ϒ has at least one \(\mathbb{F}\mathbb{P}\) in \(\mathbb{V}\).

Proof

Taking \(\Delta (t) = t\);\(t\geq 0\) in Theorem 2.1. □

The statement in the next corollary is a result of Theorem 2.1.

Corollary 2.3

Let V be an NBCC subset of a Banach space \(\mathbb{E}\), and let \(\varUpsilon: \mathbb{V} \rightarrow \mathbb{V}\) be a continuous operator such that

$$\begin{aligned} 2\Pi (\varUpsilon X) + 2\phi \bigl(\Pi (\varUpsilon X)\bigr) \leq \Pi (X) + \phi \bigl(\Pi (X)\bigr) \end{aligned}$$
(2.3)

for all \(X\subseteq \mathbb{V}\), where \(\phi: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} \) is a nondecreasing continuous function and Π is an arbitrary MNC. Then ϒ has at least one \(\mathbb{F}\mathbb{P}\) in \(\mathbb{V}\).

Proof

Taking \(F(m_{1},m_{2}) = m_{1}+m_{2}\) in Theorem 2.2. So, we get the required result. □

Corollary 2.4

Let V be an NBCC subset of a Banach space \(\mathbb{E}\), and let \(\varUpsilon: \mathbb{V} \rightarrow \mathbb{V}\) be a continuous operator such that

$$\begin{aligned} \Pi (\varUpsilon X) \leq p \Pi (X) \end{aligned}$$
(2.4)

for all \(X\subseteq \mathbb{V}\), where \(p= \frac{1}{2}\in (0,1] \) and Π is an arbitrary MNC. Then ϒ has at least one \(\mathbb{F}\mathbb{P}\) in \(\mathbb{V}\).

Proof

Taking \(\phi (t) = 0\) in Corollary 2.3, we get the required result. □

Theorem 2.5

Let V be an NBCC subset of a Banach space \(\mathbb{E}\), and let \(\varUpsilon: \mathbb{V} \rightarrow \mathbb{V}\) be a continuous operator such that

$$\begin{aligned} F \bigl[\Pi (\varUpsilon X),\phi \bigl(\Pi (\varUpsilon X)\bigr) \bigr] \leq \lambda F \bigl\lbrace \Pi (X),\phi \bigl(\Pi (X)\bigr) \bigr\rbrace \end{aligned}$$
(2.5)

for all \(X\subseteq \mathbb{V}\), where \(\phi: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} \) is a nondecreasing continuous function and Π is an arbitrary MNC, where \(\lambda = \frac{k}{k+1}\in [0,1) \). Then ϒ has at least one \(\mathbb{F}\mathbb{P}\) in \(\mathbb{V}\).

Proof

Taking \(\Delta (t) = k t\) where \(t\geq 0\), \(k\geq 0\) in Theorem 2.1. □

Corollary 2.6

Let V be an NBCC subset of a Banach space \(\mathbb{E}\), and let \(\varUpsilon: \mathbb{V} \rightarrow \mathbb{V}\) be a continuous operator such that

$$\begin{aligned} \Pi (\varUpsilon X) \leq \lambda \Pi (X) \end{aligned}$$
(2.6)

for all \(X\subseteq \mathbb{V}\), where \(\lambda \in (0,1] \) and Π is an arbitrary MNC. Then ϒ has at least one \(\mathbb{F}\mathbb{P}\) in \(\mathbb{V}\).

Proof

Taking \(F(m_{1},m_{2}) = m_{1}+m_{2}\) and \(\phi (t)\equiv 0\) in Theorem 2.5. So, we get the result which is Darbo’s fixed point theorem. □

Definition 2.7

([29])

An element \((\mathcal{A},\mathcal{B} ) \in \mathcal{X} \times \mathcal{X}\) is called a coupled fixed point of a mapping \(\mathcal{T}:\mathcal{X} \times \mathcal{X} \rightarrow \mathcal{X} \) if \(\mathcal{T} (\mathcal{A},\mathcal{B} ) = \mathcal{A}\) and \(\mathcal{T} (\mathcal{B},\mathcal{A} ) = \mathcal{B}\).

Theorem 2.8

([25])

Suppose that \(\Pi _{1}, \Pi _{2},\dots, \Pi _{n}\) is the MNC in \(\mathbb{E}_{1}, \mathbb{E}_{2},\dots, \mathbb{E}_{n}\) respectively. Moreover, suppose that the function \(\mathcal{X}: \mathbb{R}^{n}_{+} \rightarrow \mathbb{R}_{+}\) is convex and \(\mathcal{F} (y_{1},y_{2},\dots, y_{n}) = 0\) \(y_{t} = 0\) for \(t = 1, 2,\dots,n\), then \(\Pi (\mathcal{X}) = \mathcal{F}(\Pi _{1}(\mathcal{X}_{1}), \Pi _{2}( \mathcal{X}_{2}), \dots, \Pi _{n}(\mathcal{X}_{n}))\) defines an MNC in \(\mathbb{E}_{1}, \mathbb{E}_{2},\dots, \mathbb{E}_{n}\), where \(\mathcal{X}_{t}\) denotes the natural projection of \(\mathcal{X}\) into \(\mathbb{E}_{t}\) for \(t = 1, 2,\dots, n\).

Example 2.9

([25])

Let Π be an MNC on \(\mathbb{E}\). Define \(\mathcal{F}(\mathcal{A},\mathcal{B} ) = \mathcal{A}+\mathcal{B}; \mathcal{A},\mathcal{B} \in \mathbb{R}_{+}\). Then \(\mathcal{F}\) has all the properties mentioned in Theorem 2.8. Hence, \(\Pi ^{cf}(\mathcal{X}) = \Pi _{1}(\mathcal{X}_{1}) + \Pi _{2}( \mathcal{X}_{2})\) is an MNC in the space \(\mathbb{E}\times \mathbb{E}\), where \(\mathcal{X}_{t}\), \(t = 1, 2\), denotes the natural projections of \(\mathcal{X}\).

Definition 2.10

([30])

Suppose that G is the set of all functions \(\mu:\mathbb{R}_{+} \rightarrow \mathbb{R}\) satisfying the following conditions:

  1. (1)

    μ is a continuous strictly increasing function.

  2. (2)

    \(\lim_{n \rightarrow \infty }\mu (s_{n}) = -\infty \) \(\lim_{n \rightarrow \infty } s_{n} = 0\) for all \({s_{n}} \subseteq \mathbb{R}_{+}\).

For example,

  1. i.

    \(\mu _{1}(s) = \ln (s)\),

  2. ii.

    \(\mu _{2}(s) = 1 - \frac{1}{s^{t}}\), \(t > 0\).

Theorem 2.11

Let \(\mathbb{V}\) be an NBCC subset of a Banach space \(\mathbb{E}\), and let \(\varUpsilon: \mathbb{V}\times \mathbb{V} \rightarrow \mathbb{V}\) be a continuous operator such that

$$\begin{aligned} \mu \bigl[F \bigl\lbrace \Pi \bigl(\varUpsilon (s_{1} \times s_{2})\bigr),\phi \bigl(\Pi \bigl( \varUpsilon (s_{1} \times s_{2})\bigr)\bigr) \bigr\rbrace \bigr] \leq \frac{\Delta }{2} \bigl[\mu \bigl\lbrace \Pi (s_{1}\times s_{2}) +\phi \bigl( \Pi (s_{1}\times s_{2})\bigr) \bigr\rbrace \bigr] \end{aligned}$$
(2.7)

for all \(s_{1}, s_{2}\subseteq \mathbb{V}\), where Δ, F, and ϕ are as in Theorem 2.1and Π is an arbitrary MNC. In addition, we assume \(\mu (\mathcal{A}+\mathcal{B}) \leq \mu (\mathcal{A})+\mu ( \mathcal{B}) \); \(\mathcal{A},\mathcal{B} \geq 0\) and \(\phi (\mathcal{A}+\mathcal{B}) \leq \phi (\mathcal{A})+\phi ( \mathcal{B})\); \(\mathcal{A},\mathcal{B} \geq 0\). Then ϒ has at least a couple of \(\mathbb{F}\mathbb{P}\) in \(\mathbb{V}\).

Proof

Consider a mapping \(\varUpsilon ^{cf}: \mathbb{V}\times \mathbb{V} \rightarrow \mathbb{V}\times \mathbb{V}\) by \(\varUpsilon ^{cf}(\mathcal{A},\mathcal{B}) = (\varUpsilon ( \mathcal{A},\mathcal{B}), \varUpsilon (\mathcal{B},\mathcal{A}))\); \(\mathcal{A},\mathcal{B}\in \mathbb{V}\). It is trivial that \(\varUpsilon ^{cf}\) is continuous.

Let \(s\subseteq \mathbb{V}\times \mathbb{V}\) be nonempty. We have \(\Pi ^{cf}(s) = \Pi (s_{1}) + \Pi (s_{2})\) is an MNC, where \(s_{1}, s_{2}\) are the natural projections of s into \(\mathbb{E}\).

We get

$$\begin{aligned} &\mu \bigl[F \bigl\lbrace \Pi ^{cf}\bigl(\varUpsilon ^{cf} (s) \bigr),\phi \bigl(\Pi ^{cf}\bigl( \varUpsilon ^{cf} (s)\bigr) \bigr) \bigr\rbrace \bigr] \\ &\quad\leqslant \mu \bigl[F \bigl\lbrace \Pi ^{cf}\bigl(\varUpsilon (s_{1}\times s_{2}) \times \varUpsilon (s_{2} \times s_{1})\bigr),\phi \bigl(\Pi ^{cf}\bigl(\varUpsilon (s_{1} \times s_{2})\times \varUpsilon (s_{2} \times s_{1})\bigr)\bigr) \bigr\rbrace \bigr] \\ &\quad=\mu \bigl[F \bigl\lbrace \Pi \bigl(\varUpsilon (s_{1}\times s_{2})\bigr)+\Pi \bigl( \varUpsilon (s_{2}\times s_{1})\bigr),\phi \bigl(\Pi \bigl(\varUpsilon (s_{1} \times s_{2})\bigr)+\Pi \bigl(\varUpsilon (s_{2}\times s_{1})\bigr)\bigr) \bigr\rbrace \bigr] \\ &\quad\leq \mu \bigl[F \bigl\lbrace \Pi \bigl(\varUpsilon (s_{1}\times s_{2})\bigr)+\Pi \bigl( \varUpsilon (s_{2}\times s_{1})\bigr),\phi \bigl(\Pi \bigl(\varUpsilon (s_{1} \times s_{2})\bigr)\bigr)+\phi \bigl(\Pi \bigl(\varUpsilon (s_{2} \times s_{1})\bigr)\bigr) \bigr\rbrace \bigr] \\ &\quad\leq \mu \bigl[F \bigl\lbrace \Pi \bigl(\varUpsilon (s_{1}\times s_{2})\bigr),\phi \bigl( \Pi \bigl(\varUpsilon (s_{1}\times s_{2})\bigr)\bigr) \bigr\rbrace \bigr]\\ &\qquad{}+\mu \bigl[F \bigl\lbrace \Pi \bigl(\varUpsilon (s_{2}\times s_{1})\bigr),\phi \bigl(\Pi \bigl( \varUpsilon (s_{2}\times s_{1})\bigr)\bigr) \bigr\rbrace \bigr] \\ &\quad\leq \Delta \bigl[\mu \bigl\lbrace \Pi (s_{1})+\Pi (s_{2})+\phi \bigl(\Pi (s_{1})+ \Pi (s_{2}) \bigr) \bigr\rbrace \bigr] \\ &\quad= \Delta \bigl[\mu \bigl\lbrace \Pi ^{cf}(s)+\phi \bigl(\Pi ^{cf}(s)\bigr) \bigr\rbrace \bigr] \\ &\quad= \Delta \bigl[\mu \bigl\lbrace F\bigl(\Pi ^{cf}(s),\phi \bigl(\Pi ^{cf}(s)\bigr)\bigr) \bigr\rbrace \bigr]. \end{aligned}$$

By Theorem 2.1, we conclude that \(\varUpsilon ^{cf}\) has minimum of one fixed point in \(\mathbb{V}\times \mathbb{V}\). That is, ϒ has minimum of one coupled fixed point. □

Measure of noncompactness on \(C([0,T])\)

Consider the space \(\mathbf{E}=C(I)\) which is the set of real continuous functions on I, where \(I=[0,T]\). Then E is a Banach space with the norm

$$\begin{aligned} \Vert \varrho \Vert =\sup \bigl\lbrace \bigl\vert \varrho ( \varsigma ) \bigr\vert :\varsigma \in I \bigr\rbrace , \quad\varrho \in \mathbf{E}. \end{aligned}$$

Let \(\varUpsilon (\neq \phi ) \subseteq \mathbf{E}\) be bounded. For \(\varrho \in \varUpsilon \) and \(\epsilon >0\), denote by \(\omega (\varrho,\epsilon )\) the modulus of the continuity of ϱ, i.e.,

$$\begin{aligned} \omega (\varrho,\epsilon )=\sup \bigl\lbrace \bigl\vert \varrho ( \varsigma _{1})-\varrho (\varsigma _{2}) \bigr\vert : \varsigma _{1}, \varsigma _{2} \in I, \vert \varsigma _{1}-\varsigma _{1} \vert \leq \epsilon \bigr\rbrace . \end{aligned}$$

Further, we define

$$\begin{aligned} \omega (\varUpsilon,\epsilon )=\sup \bigl\lbrace \omega (\varrho, \epsilon ): \varrho \in \varUpsilon \bigr\rbrace ;\qquad\omega _{0}( \varUpsilon )=\lim _{\epsilon \rightarrow 0}\omega ( \varUpsilon,\epsilon ). \end{aligned}$$

It is well known that the function \(\omega _{0}\) is an MNC in E such that the Hausdorff measure of noncompactness χ is given by \(\chi (\varUpsilon )=\frac{1}{2}\omega _{0}(\varUpsilon )\) (see [25]).

Solvability of fractional integral equation

For \(\rho \in (0,1]\) and \(\alpha \in \mathbb{C}, \operatorname{Re}(\alpha )>0\), we define the left GPF integral of f defined by [31]

$$\begin{aligned} \bigl({}_{a}I^{\alpha, \rho } f \bigr) (t)= \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \int _{a}^{t}e^{ \frac{ ( \rho -1 )(t-\tau ) }{\rho }}(t-\tau )^{\alpha -1}f( \tau ) \,d \tau. \end{aligned}$$

In this part, we study the following fractional integral equation:

$$\begin{aligned} \mathcal{Z}(\varsigma )=\Delta \bigl(\varsigma,\mathcal{L} \bigl( \varsigma, \mathcal{Z}(\varsigma )\bigr), \bigl({}_{0}I^{\alpha, \rho } \mathcal{Z} \bigr) (\varsigma ) \bigr), \end{aligned}$$
(4.1)

where \(\alpha >1, \rho \in (0,1], \varsigma \in I=[0,T]\).

Let

$$\begin{aligned} B_{d_{0}}= \bigl\lbrace \mathcal{Z}\in \mathbf{E}: \Vert \mathcal{Z} \Vert \leq d_{0} \bigr\rbrace . \end{aligned}$$

Assume that

  1. (A)

    \(\Delta:I \times \mathbb{R}^{2} \rightarrow \mathbb{R}, \mathcal{L}: I \times \mathbb{R}\rightarrow \mathbb{R}\) is continuous, and there exist constants \(\delta _{1}, \delta _{2}, \delta _{3} \geq 0\) satisfying

    $$\begin{aligned} \bigl\vert \Delta (\varsigma,\mathcal{L},I_{1})-\Delta (\varsigma, \bar{\mathcal{L}},\bar{I}_{1}) \bigr\vert \leq \delta _{1} \vert \mathcal{L}-\bar{\mathcal{L}} \vert +\delta _{2} \vert I_{1}- \bar{I}_{1} \vert , \quad \varsigma \in I; \mathcal{L},I_{1}, \bar{\mathcal{L}},\bar{I}_{1} \in \mathbb{R} \end{aligned}$$

    and

    $$\begin{aligned} \bigl\vert \mathcal{L}(\varsigma, J_{1})-\mathcal{L}(\varsigma, J_{2}) \bigr\vert \leq \delta _{3} \vert J_{1}-J_{2} \vert ,\quad J_{1},J_{2}\in \mathbb{R}. \end{aligned}$$
  2. (B)

    There exists \(d_{0}>0\) satisfying

    $$\begin{aligned} \bar{\Delta }=\sup \bigl\lbrace \bigl\vert \Delta (\varsigma, \mathcal{L},I_{1}) \bigr\vert :\varsigma \in I,\mathcal{L}\in [- \hat{\mathcal{L}},\hat{\mathcal{L}}],I_{1}\in [-\hat{\mathcal{I}}, \hat{\mathcal{I}}] \bigr\rbrace \leq d_{0} \end{aligned}$$

    and

    $$\begin{aligned} \delta _{1}\delta _{3}< 1, \end{aligned}$$

    where

    $$\begin{aligned} \hat{\mathcal{L}}=\sup \bigl\lbrace \bigl\vert \mathcal{L} \bigl( \varsigma, \mathcal{Z}(\varsigma ) \bigr) \bigr\vert : \varsigma \in I, \mathcal{Z}( \varsigma )\in [-d_{0},d_{0}] \bigr\rbrace \end{aligned}$$

    and

    $$\begin{aligned} \hat{\mathcal{I}}=\sup \bigl\lbrace \bigl\vert \bigl({}_{0}I^{\alpha, \rho } \mathcal{Z} \bigr) (\varsigma ) \bigr\vert : \varsigma \in I, \mathcal{Z}( \varsigma )\in [-d_{0},d_{0}] \bigr\rbrace . \end{aligned}$$
  3. (C)

    \(\vert \Delta (\varsigma,0,0 ) \vert =0, \mathcal{L}(\varsigma, 0) =0\).

  4. (D)

    There exists a positive solution \(d_{0}\) of the inequality

    $$\begin{aligned} \delta _{1}\delta _{3}r+ \frac{\delta _{2}rT^{\alpha }}{\rho ^{\alpha }\Gamma (\alpha +1)}.e^{ \frac{(\rho -1)T}{\rho }} \leq r. \end{aligned}$$

Theorem 4.1

If conditions (A)(D) hold, then Eq. (4.1) has a solution in \(\mathbf{E}=C(I)\).

Proof

Define the operator \(\mathcal{T}: \mathbf{E} \rightarrow \mathbf{E}\) as follows:

$$\begin{aligned} (\mathcal{T} \mathcal{Z}) (\varsigma )= \Delta \bigl(\varsigma,\mathcal{L} \bigl(\varsigma, \mathcal{Z}(\varsigma )\bigr), \bigl({}_{0}I^{\alpha, \rho } \mathcal{Z} \bigr) (\varsigma ) \bigr). \end{aligned}$$

Step 1: We prove that the function \(\mathcal{Q}\) maps \(B_{d_{0}}\) into \(B_{d_{0}}\). Let \(\varUpsilon \in B_{d_{0}}\). We have

$$\begin{aligned} & \bigl\vert (\mathcal{T} \mathcal{Z}) (\varsigma ) \bigr\vert \\ &\quad \leq \bigl\vert \Delta \bigl(\varsigma,\mathcal{L} \bigl(\varsigma, \mathcal{Z}( \varsigma )\bigr), \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) ( \varsigma ) \bigr)-\Delta (\varsigma,0,0 ) \bigr\vert + \bigl\vert \Delta ( \varsigma,0,0 ) \bigr\vert \\ & \quad\leq \delta _{1} \bigl\vert \mathcal{L}\bigl(\varsigma, \mathcal{Z}( \varsigma )\bigr)-0 \bigr\vert + \delta _{2} \bigl\vert \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) (\varsigma )-0 \bigr\vert \\ &\quad \leq \delta _{1}\delta _{3} \bigl\vert \mathcal{Z}( \varsigma ) \bigr\vert + \delta _{2} \bigl\vert \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) ( \varsigma ) \bigr\vert . \end{aligned}$$

Also,

$$\begin{aligned} & \bigl\vert \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) ( \varsigma ) \bigr\vert \\ &\quad= \biggl\vert \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \int _{0}^{\varsigma }e^{ \frac{ ( \rho -1 )(\varsigma -\tau ) }{\rho }}(\varsigma - \tau )^{\alpha -1}\mathcal{Z}(\tau ) \,d \tau \biggr\vert \\ &\quad\leq \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \int _{0}^{\varsigma }e^{ \frac{ ( \rho -1 )(\varsigma -\tau ) }{\rho }}(\varsigma - \tau )^{\alpha -1} \bigl\vert \mathcal{Z}(\tau ) \bigr\vert \,d \tau \\ &\quad \leq \frac{d_{0}e^{\frac{ ( \rho -1 )T }{\rho }}}{\rho ^{\alpha }\Gamma (\alpha )} \int _{0}^{\varsigma }(\varsigma -\tau )^{\alpha -1} \,d \tau \\ &\quad\leq \frac{d_{0}T^{\alpha }e^{\frac{ ( \rho -1 )T }{\rho }}}{\rho ^{\alpha }\Gamma (\alpha +1)}. \end{aligned}$$

Hence, \(\Vert \mathcal{T} \Vert < d_{0}\) gives

$$\begin{aligned} \Vert \mathcal{T} \Vert \leq \delta _{1}\delta _{3}d_{0}+ \frac{\delta _{2}d_{0}T^{\alpha }}{\rho ^{\alpha }\Gamma (\alpha +1)}.e^{ \frac{(\rho -1)T}{\rho }}\leq d_{0}. \end{aligned}$$

Due to assumption (D), \(\mathcal{T}\) maps \(B_{d_{0}}\) into \(B_{d_{0}}\).

Step 2: We prove that \(\mathcal{T}\) is continuous on \(B_{d_{0}}\). Let \(\epsilon >0\) and \(\mathcal{Z}, \bar{\mathcal{Z}} \in B_{r_{0}}\) such that \(\Vert \mathcal{Z} - \bar{\mathcal{Z}} \Vert < \epsilon \). We have

$$\begin{aligned} & \bigl\vert ( \mathcal{T} \mathcal{Z} ) (\varsigma )- ( \mathcal{T} \bar{ \mathcal{Z}} ) (\varsigma ) \bigr\vert \\ & \quad\leq \bigl\vert \Delta \bigl(\varsigma,\mathcal{L}\bigl(\varsigma, \mathcal{Z}(\varsigma )\bigr), \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) (\varsigma ) \bigr)-\Delta \bigl(\varsigma,\mathcal{L}\bigl( \varsigma, \bar{\mathcal{Z}}(\varsigma )\bigr), \bigl({}_{0}I^{\alpha, \rho }\bar{ \mathcal{Z}} \bigr) (\varsigma ) \bigr) \bigr\vert \\ &\quad \leq \delta _{1} \bigl\vert \mathcal{L}\bigl(\varsigma, \mathcal{Z}( \varsigma )\bigr)-\mathcal{L}\bigl(\varsigma, \bar{\mathcal{Z}}( \varsigma )\bigr) \bigr\vert + \delta _{2} \bigl\vert \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) (\varsigma )- \bigl({}_{0}I^{\alpha, \rho }\bar{\mathcal{Z}} \bigr) (\varsigma ) \bigr\vert . \end{aligned}$$

Also,

$$\begin{aligned} & \bigl\vert \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) ( \varsigma )- \bigl({}_{0}I^{\alpha, \rho }\bar{\mathcal{Z}} \bigr) ( \varsigma ) \bigr\vert \\ &\quad= \biggl\vert \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \int _{0}^{\varsigma }e^{ \frac{ ( \rho -1 )(\varsigma -\tau ) }{\rho }}(\varsigma - \tau )^{\alpha -1} \bigl\lbrace \mathcal{Z}(\tau )-\bar{\mathcal{Z}}( \tau ) \bigr\rbrace d \tau \biggr\vert \\ &\quad\leq \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \int _{0}^{\varsigma }e^{ \frac{ ( \rho -1 )(\varsigma -\tau ) }{\rho }}(\varsigma - \tau )^{\alpha -1} \bigl\vert \mathcal{Z}(\tau )-\bar{\mathcal{Z}}(\tau ) \bigr\vert \,d \tau \\ & \quad< \frac{\epsilon T^{\alpha }e^{\frac{ ( \rho -1 )T }{\rho }}}{\rho ^{\alpha }\Gamma (\alpha +1)} . \end{aligned}$$

Hence, \(\Vert \mathcal{Z} - \bar{\mathcal{Z} } \Vert <\epsilon \) gives

$$\begin{aligned} \bigl\vert ( \mathcal{T}\mathcal{Z} ) (\varsigma )- ( \mathcal{T} \bar{ \mathcal{Z}} ) (\varsigma ) \bigr\vert < \delta _{1} \delta _{3}\epsilon + \frac{\epsilon T^{\alpha }e^{\frac{ ( \rho -1 )T }{\rho }}}{\rho ^{\alpha }\Gamma (\alpha +1)} . \end{aligned}$$

As \(\epsilon \rightarrow 0\) we get \(\vert ( \mathcal{T}\mathcal{Z} )(\varsigma )- ( \mathcal{T} \bar{\mathcal{Z}} )(\varsigma ) \vert \rightarrow 0\). This shows that \(\mathcal{T}\) is continuous on \(B_{d_{0}}\).

Step 3: An estimate of \(\mathcal{T}\) with respect to \(\omega _{0}\): Assume that \(\varOmega (\neq \phi ) \subseteq B_{d_{0}}\). Let \(\epsilon >0\) be arbitrary and choose \(\mathcal{Z} \in \varOmega \) and \(\varsigma _{1}, \varsigma _{2} \in I\) such that \(\vert \varsigma _{2}-\varsigma _{1} \vert \leq \epsilon \) and \(\varsigma _{2} \geq \varsigma _{1}\).

Now,

$$\begin{aligned} & \bigl\vert ( \mathcal{T}\mathcal{Z} ) (\varsigma _{2})- ( \mathcal{T}\mathcal{Z} ) (\varsigma _{1}) \bigr\vert \\ &\quad = \bigl\vert \Delta \bigl(\varsigma _{2},\mathcal{L} \bigl(\varsigma _{2}, \mathcal{Z}(\varsigma _{2})\bigr), \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{2}) \bigr)-\Delta \bigl(\varsigma _{1},\mathcal{L} \bigl(\varsigma _{1}, \mathcal{Z}(\varsigma _{1})\bigr), \bigl({}_{0}I^{ \alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{1}) \bigr)\bigr\vert \\ &\quad\leq \bigl\vert \Delta \bigl(\varsigma _{2},\mathcal{L} \bigl(\varsigma _{2}, \mathcal{Z}(\varsigma _{2})\bigr), \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{2}) \bigr)-\Delta \bigl(\varsigma _{2},\mathcal{L} \bigl(\varsigma _{2}, \mathcal{Z}(\varsigma _{2})\bigr), \bigl({}_{0}I^{ \alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{1}) \bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert \Delta \bigl(\varsigma _{2},\mathcal{L} \bigl( \varsigma _{2}, \mathcal{Z}(\varsigma _{2})\bigr), \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{1}) \bigr)- \Delta \bigl(\varsigma _{2},\mathcal{L} \bigl(\varsigma _{1}, \mathcal{Z}(\varsigma _{1})\bigr), \bigl({}_{0}I^{ \alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{1}) \bigr) \bigr\vert \\ &\quad{}+\bigl\vert \Delta \bigl(\varsigma _{2},\mathcal{L} \bigl( \varsigma _{1}, \mathcal{Z}(\varsigma _{1})\bigr), \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{1}) \bigr)-\Delta \bigl(\varsigma _{1},\mathcal{L} \bigl(\varsigma _{1}, \mathcal{Z}(\varsigma _{1})\bigr), \bigl({}_{0}I^{ \alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{1}) \bigr) \bigr\vert \\ & \quad\leq \delta _{2} \bigl\vert \bigl({}_{0}I^{\alpha, \rho } \mathcal{Z} \bigr) (\varsigma _{2})- \bigl({}_{0}I^{\alpha, \rho } \mathcal{Z} \bigr) (\varsigma _{1}) \bigr\vert +\delta _{1} \bigl\vert \mathcal{L} \bigl( \varsigma _{2}, \mathcal{Z}(\varsigma _{2}) \bigr)-\mathcal{L} \bigl( \varsigma _{1}, \mathcal{Z}(\varsigma _{1}) \bigr) \bigr\vert + \omega _{ \Delta }(I, \epsilon ) \\ &\quad\leq \delta _{2} \bigl\vert \bigl({}_{0}I^{\alpha, \rho } \mathcal{Z} \bigr) (\varsigma _{2})- \bigl({}_{0}I^{\alpha, \rho } \mathcal{Z} \bigr) (\varsigma _{1}) \bigr\vert +\delta _{1}\delta _{3} \bigl\vert \mathcal{Z}(\varsigma _{2})- \mathcal{Z}(\varsigma _{1}) \bigr\vert + \omega _{\Delta }(I, \epsilon ), \end{aligned}$$

where

ω Δ (I,ϵ)=sup { | Δ ( ς 2 , L , I 1 ) Δ ( ς 1 , L , I 1 ) | : | ς 2 ς 1 | ϵ ; ς 1 , ς 2 I ; L [ L ˆ , L ˆ ] ; I 1 [ I ˆ , I ˆ ] } .

Also,

$$\begin{aligned} & \bigl\vert \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) ( \varsigma _{2})- \bigl({}_{0}I^{\alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{1}) \bigr\vert \\ &\quad = \biggl\vert \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \int _{0}^{ \varsigma _{2}}e^{ \frac{ ( \rho -1 )(\varsigma _{2}-\tau ) }{\rho }}( \varsigma _{2}-\tau )^{\alpha -1}\mathcal{Z}(\tau ) \,d \tau\\ &\qquad{} - \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \int _{0}^{\varsigma _{1}}e^{ \frac{ ( \rho -1 )(\varsigma _{1}-\tau ) }{\rho }}( \varsigma _{1}-\tau )^{\alpha -1}\mathcal{Z}(\tau ) \,d \tau \biggr\vert \\ &\quad \leq \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \biggl\vert \int _{0}^{ \varsigma _{2}}e^{ \frac{ ( \rho -1 )(\varsigma _{2}-\tau ) }{\rho }}( \varsigma _{2}-\tau )^{\alpha -1}\mathcal{Z}(\tau ) \,d \tau - \int _{0}^{ \varsigma _{1}}e^{ \frac{ ( \rho -1 )(\varsigma _{1}-\tau ) }{\rho }}( \varsigma _{1}-\tau )^{\alpha -1}\mathcal{Z}(\tau ) \,d \tau \biggr\vert \\ &\quad \leq \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \biggl\vert \int _{0}^{ \varsigma _{2}}e^{ \frac{ ( \rho -1 )(\varsigma _{2}-\tau ) }{\rho }}( \varsigma _{2}-\tau )^{\alpha -1}\mathcal{Z}(\tau ) \,d \tau - \int _{0}^{ \varsigma _{1}}e^{ \frac{ ( \rho -1 )(\varsigma _{2}-\tau ) }{\rho }}( \varsigma _{2}-\tau )^{\alpha -1}\mathcal{Z}(\tau ) \,d \tau \biggr\vert \\ &\qquad{}+ \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \biggl\vert \int _{0}^{ \varsigma _{1}}e^{ \frac{ ( \rho -1 )(\varsigma _{2}-\tau ) }{\rho }}( \varsigma _{2}-\tau )^{\alpha -1}\mathcal{Z}(\tau ) \,d \tau - \int _{0}^{ \varsigma _{1}}e^{ \frac{ ( \rho -1 )(\varsigma _{1}-\tau ) }{\rho }}( \varsigma _{1}-\tau )^{\alpha -1}\mathcal{Z}(\tau ) \,d \tau \biggr\vert \\ & \quad\leq \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \int _{\varsigma _{1}}^{ \varsigma _{2}}e^{ \frac{ ( \rho -1 )(\varsigma _{2}-\tau ) }{\rho }}( \varsigma _{2}-\tau )^{\alpha -1} \bigl\vert \mathcal{Z}(\tau ) \bigr\vert \,d \tau \\ &\qquad{}+ \frac{1}{\rho ^{\alpha }\Gamma (\alpha )} \int _{0}^{\varsigma _{1}} \bigl\vert \bigl( e^{ \frac{ ( \rho -1 )(\varsigma _{2}-\tau ) }{\rho }}( \varsigma _{2}-\tau )^{\alpha -1} - e^{ \frac{ ( \rho -1 )(\varsigma _{1}-\tau ) }{\rho }}( \varsigma _{1}-\tau )^{\alpha -1} \bigr)\mathcal{Z}(\tau ) \bigr\vert \,d \tau \\ &\quad \leq \frac{-e^{\frac{(\rho -1)T}{\rho }}}{\rho ^{\alpha }\Gamma (\alpha +1)} \Vert \mathcal{Z} \Vert (\varsigma _{2}-\varsigma _{1})^{ \alpha } \\ &\qquad{}+ \frac{ \Vert \mathcal{Z} \Vert }{\rho ^{\alpha }\Gamma (\alpha )} \int _{0}^{\varsigma _{1}} \bigl\vert e^{ \frac{ ( \rho -1 )(\varsigma _{2}-\tau ) }{\rho }}( \varsigma _{2}-\tau )^{\alpha -1} - e^{ \frac{ ( \rho -1 )(\varsigma _{1}-\tau ) }{\rho }}( \varsigma _{1}-\tau )^{\alpha -1} \bigr\vert \,d \tau. \end{aligned}$$

As \(\epsilon \rightarrow 0\), then \(\varsigma _{2} \rightarrow \varsigma _{1}\), and so \(\vert ({}_{0}I^{\alpha, \rho }\mathcal{Z} )(\varsigma _{2})- ({}_{0}I^{\alpha, \rho }\mathcal{Z} )(\varsigma _{1}) \vert \rightarrow 0\).

Hence,

$$\begin{aligned} & \bigl\vert ( \mathcal{TZ} ) (\varsigma _{2})- ( \mathcal{TZ} ) ( \varsigma _{1}) \bigr\vert \\ &\quad \leq \delta _{2} \bigl\vert \bigl({}_{0}I^{\alpha, \rho } \mathcal{Z} \bigr) (\varsigma _{2})- \bigl({}_{0}I^{\alpha, \rho } \mathcal{Z} \bigr) (\varsigma _{1}) \bigr\vert +\delta _{1}\delta _{3}\omega ( \mathcal{Z},\epsilon )+\omega _{\Delta }(I, \epsilon ) \end{aligned}$$

gives

$$\begin{aligned} \omega (\mathcal{TZ},\epsilon )\leq \delta _{2} \bigl\vert \bigl({}_{0}I^{ \alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{2})- \bigl({}_{0}I^{ \alpha, \rho }\mathcal{Z} \bigr) (\varsigma _{1}) \bigr\vert +\delta _{1} \delta _{3} \omega (\mathcal{Z},\epsilon )+\omega _{\Delta }(I, \epsilon ). \end{aligned}$$

By the uniform continuity of Δ on \(I \times [-\hat{\mathcal{L}},\hat{\mathcal{L}}]\times [- \hat{\mathcal{I}},\hat{\mathcal{I}}] \), we have \(\omega _{\Delta }(I,\epsilon )\rightarrow 0\) as \(\epsilon \rightarrow 0\).

Taking \(\sup_{\mathcal{Z} \in \varOmega }\) and \(\epsilon \rightarrow 0\), we get

$$\begin{aligned} \omega _{0}(\mathcal{T} \varOmega )\leq \delta _{1}\delta _{3} \omega _{0}(\varOmega ). \end{aligned}$$

Thus, by Corollary 2.6, \(\mathcal{Q}\) has a fixed point in \(\varOmega \subseteq B_{d_{0}}\), i.e., equation (4.1) has a solution in E. □

Example 4.2

Consider the following equation:

$$\begin{aligned} \mathcal{Z}(\varsigma )= \frac{\mathcal{Z}(\varsigma )}{7+\varsigma ^{2}}+ \frac{ ({}_{0}I^{2,\frac{1}{2}} \mathcal{Z} )(\varsigma )}{10} \end{aligned}$$
(4.2)

for \(\varsigma \in [0,2]=I\).

We have

$$\begin{aligned} \bigl({}_{0}I^{2,\frac{1}{2}} \mathcal{Z} \bigr) (\varsigma )= \frac{4}{\Gamma (2)} \int _{0}^{\varsigma }e^{-(\varsigma -\tau )} (\varsigma -\tau ) \mathcal{Z}(\tau )\,d\tau. \end{aligned}$$

Also, \(\Delta (\varsigma,\mathcal{L},\mathcal{I}_{1})=\mathcal{L}+ \frac{\mathcal{I}_{1}}{10} \) and \(\mathcal{L}(\varsigma, \mathcal{Z})= \frac{\mathcal{Z}}{7+\varsigma ^{2}}\). It is trivial that both \(\Delta, \mathcal{L}\) are continuous satisfying

$$\begin{aligned} \bigl\vert \mathcal{L}(\varsigma, J_{1})-\mathcal{L}(\varsigma, J_{2}) \bigr\vert \leq \frac{ \vert J_{1}-J_{2} \vert }{8} \end{aligned}$$

and

$$\begin{aligned} \bigl\vert \Delta (\varsigma,\mathcal{L},\mathcal{I}_{1})-\Delta ( \varsigma,\bar{\mathcal{L}},\bar{\mathcal{I}}_{1}) \bigr\vert \leq \vert \mathcal{U}-\bar{\mathcal{U}} \vert +\frac{1}{10} \vert \mathcal{I}_{1}- \bar{\mathcal{I}}_{1} \vert . \end{aligned}$$

Therefore, \(\delta _{1}=1, \delta _{2}=\frac{1}{10}, \delta _{3}=\frac{1}{8}\), and \(\delta _{1}\delta _{3}=\frac{1}{8}<1\).

If \(\Vert \mathcal{Z} \Vert \leq d_{0}\), then

$$\begin{aligned} \hat{\mathcal{L}}=\frac{d_{0}}{8} \end{aligned}$$

and

$$\begin{aligned} \hat{\mathcal{I}}=\frac{8d_{0}}{e^{2}}. \end{aligned}$$

Further,

$$\begin{aligned} \bigl\vert \Delta (\varsigma,\mathcal{L},\mathcal{I}_{1}) \bigr\vert \leq \frac{d_{0}}{8}+\frac{8d_{0}}{10e^{2}}\leq d_{0}. \end{aligned}$$

If we choose \(d_{0}=2\), then

$$\begin{aligned} \hat{\mathcal{L}}=\frac{1}{4},\qquad \hat{\mathcal{I}}=\frac{16}{e^{2}}, \end{aligned}$$

which gives

$$\begin{aligned} \bar{\Delta }\leq 2. \end{aligned}$$

On the other hand, assumption (D) is also satisfied for \(d_{0}=2\).

We observe that all the assumption from (A)–(D) of Theorem 4.1 are satisfied. By Theorem 4.1, it can be said that equation (4.2) has a solution in \(\mathbf{E}=C(I)\).

Availability of data and materials

Not applicable.

References

  1. 1.

    Kuratowski, K.: Sur les espaces complets. Fundam. Math. 15, 301–309 (1930)

    Article  Google Scholar 

  2. 2.

    Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solutions of fractional differential equations via Geraghty type hybrid contractions. Appl. Comput. Math. 20, 2 (2021)

    Google Scholar 

  3. 3.

    Afshari, H., Alsulami, H.H., Karapinar, E.: On the extended multivalued Geraghty type contractions. J. Nonlinear Sci. Appl. 9, 4695–4706 (2016). https://doi.org/10.22436/jnsa.009.06.108

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Afshari, H., Kalantari, S., Baleanu, D.: Solution of fractional differential equations via \(\alpha -\psi \)-Geraghty type mappings. Adv. Differ. Equ. 2018, 347 (2018). https://doi.org/10.1186/s13662-018-1807-4

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional differential equations via coupled fixed point. Electron. J. Differ. Equ. 2015, 286 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Altun, I., Turkoglu, D.: A fixed point theorem for mapping satisfying a general contractive condition of operator type. J. Comput. Anal. Appl. 9(1), 9–14 (2007)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Arab, R., Nashine, H.K., Can, N.H., Binh, T.T.: Solvability of functional-integral equations (fractional order) using measure of noncompactness. Adv. Differ. Equ. 2020, Article ID 12 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Banaś, J., Jleli, M., Mursaleen, M., Samet, B., Vetro, C.: Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness. Springer, Berlin (2017)

    Book  Google Scholar 

  9. 9.

    Banaś, J., Mursaleen, M.: Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, India (2014)

    Book  Google Scholar 

  10. 10.

    Darwish, M.A., Sadarangani, K.: On a quadratic integral equation with supremum involving Erdélyi-Kober fractional order. Math. Nachr. 288(5–6), 566–576 (2015)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Das, A., Hazarika, B., Arab, R., Agarwal, R.P., Nashine, H.K.: Solvability of infinite systems of fractional differential equations in the space of tempered sequences. Filomat 33(17), 5519–5530 (2019)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Das, A., Hazarkia, B., Mursaleen, M.: Application of measure of noncompactness for solvability of the infinite system of integral equations in two variables in \(\ell _{p} ( 1< p< \infty )\). Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(1), 31–40 (2019)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Das, A., Hazarika, B., Panda, S.K., Vijayakumar, V.: An existence result for an infinite system of implicit fractional integral equations via generalized Darbo’s fixed point theorem. Comput. Appl. Math. 40, 143 (2021). https://doi.org/10.1007/s40314-021-01537-z

    MathSciNet  Article  Google Scholar 

  14. 14.

    Mursaleen, M., Mohiuddine, S.A.: Applications of measures of noncompactness to the infinite system of differential equations in lp spaces. Nonlinear Anal., Theory Methods Appl. 75, 2111–2115 (2012)

    Article  Google Scholar 

  15. 15.

    Nashine, H.K., Arab, R., Agarwal, R.P., Haghigh, A.S.: Type fixed and coupled fixed point results and its application to integral equation. Period. Math. Hung. 77, 94–107 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Rabbani, M., Das, A., Hazarika, B., Arab, R.: Measure of noncompactness of a new space of tempered sequences and its application on fractional differential equations. Chaos Solitons Fractals 140, 110221 (2020)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Rabbani, M., Das, A., Hazarika, B., Arab, R.: Existence of solution for two dimensional nonlinear fractional integral equation by measure of noncompactness and iterative algorithm to solve it. J. Comput. Appl. Math. 370, 112654 (2020)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Nguyen, P.D.: Note on a Allen–Cahn equation with Caputo–Fabrizio derivative. Res. Nonlinear Anal. 4(3), 179–185 (2021)

    Google Scholar 

  19. 19.

    Ardjouni, A.: Asymptotic stability in Caputo–Hadamard fractional dynamic equations. Res. Nonlinear Anal. 4(2), 77–86 (2021)

    Article  Google Scholar 

  20. 20.

    Jangid, K., Purohit, S.D., Nisar, K.S., Abdeljawad, T.: Certain generalized fractional integral inequalities. Adv. Theory Nonlinear Anal. Appl. 4(4), 252–259 (2020)

    Google Scholar 

  21. 21.

    Abu Jarad, E.S.A., Abu Jarad, M.H.A., Abdeljawad, T., Jarad, F.: Some properties for certain subclasses of analytic functions associated with k-integral operators. Adv. Theory Nonlinear Anal. Appl. 4(4), 459–482 (2020)

    MATH  Google Scholar 

  22. 22.

    Lazreg, J.E., Abbas, S., Benchohra, M., Karapınar, E.: Impulsive Caputo–Fabrizio fractional differential equations in b-metric spaces. https://doi.org/10.1515/math-2021-0040

  23. 23.

    Sevinik-Adıgüzel, R., Aksoy, Ü., Karapınar, E., Erhan, I.M.: Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions. RACSAM 115, 155 (2021)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Maharaj, S.D., Chaisi, M.: New anisotropic models from isotropic solutions. Math. Methods Appl. Sci. 29, 67–83 (2006)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Banaś, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Dekker, New York (1980)

    MATH  Google Scholar 

  26. 26.

    Agarwal, R.P., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  27. 27.

    Darbo, G.: Punti uniti in trasformazioni a codominio non compatto (Italian). Rend. Semin. Mat. Univ. Padova 24, 84–92 (1955)

    MATH  Google Scholar 

  28. 28.

    Das, A., Hazarika, B., Kumam, P.: Some new generalization of Darbo’s fixed point theorem and its application on integral equations. Mathematics 7, 214 (2019). https://doi.org/10.3390/math7030214

    Article  Google Scholar 

  29. 29.

    Chang, S.S., Huang, Y.J.: Coupled fixed point theorems with applications. J. Korean Math. Soc. 33(3), 575–585 (1996)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Mohammadi, B., Haghighi, A.S., Khorshidi, M., De la Sen, M., Parvaneh, V.: Existence of solutions for a system of integral equations using a generalization of Darbo’s fixed point theorem. Mathematics 8, 492 (2020). https://doi.org/10.3390/math8040492

    Article  Google Scholar 

  31. 31.

    Jarad, F., Abdeljawad, T., Alzabut, J.: Generalized fractional derivatives generated by a class of local proportional derivatives. Eur. Phys. J. Spec. Top. 226, 3457–3471 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

Not applicable.

Funding

Not applicable.

Author information

Affiliations

Authors

Contributions

All authors contributed equally to this article. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Thabet Abdeljawad.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Das, A., Suwan, I., Deuri, B.C. et al. On solution of generalized proportional fractional integral via a new fixed point theorem. Adv Differ Equ 2021, 427 (2021). https://doi.org/10.1186/s13662-021-03589-1

Download citation

MSC

  • 45G05
  • 47H08
  • 47H09
  • 47H10

Keywords

  • Measure of noncompactness (MNC)
  • Fixed point theorem
  • Generalized proportional fractional integral