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1 Introduction

Integral equations have multiple practical applications in modelling specific real world problems and different types of real-life situations, e.g., in laws of physics, in the theory of radioactive transmission, in the theory of statistical mechanics, and in the cytotoxic ac- tivity. 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 modelled 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.

Recently, number of articles have been published in connection: with scalar linear im- pulsive Riemann–Liouville fractional differential equations with constant delay-explicit solutions, coupled systems of integral equations of Urysohn Volterra–Chandrasekhar mixed type, noninstantaneous impulsive fractional integro–differential equations, frac- tional differential equations, and proximity theory (the readers can consult the papers [14, 16–18, 22] and references therein).

In [14], Gabeleh and Künzi have established that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. They have also discussed the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions.

In [16], Harjani et al. established sufficient condition about the length of the interval for the existence and uniqueness of mild solutions to a fractional boundary value problem with Sturm–Liouville boundary conditions when the data function is of Lipschitzian type. Moreover, they have presented an application of our result to the eigenvalues problem and its connection with a Lyapunov-type inequality.

In [17], Hristova and Tersian have studied Riemann–Liouville fractional differential equations with a constant delay and impulses. Also, they have studied the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point of im- pulse. The initial conditions as well as impulsive conditions are defined in an appropriate way for both cases. The explicit solutions are obtained and applied to the study of finite- time stability.

In [18], Kataria et al. have established the existence of mild solution for noninstan- taneous impulsive fractional-order integro–differential equations with local and nonlocal conditions in Banach space. Existence results with local and nonlocal conditions are ob- tained through operator semigroup theory using generalized Banach contraction theorem and Krasnoselskii’s fixed point theorem, respectively. Finally, illustrations are added to validate derived results.

In [22], Nabil has studied the solvability of a coupled system of integral equations of Urysohn Volterra–Chandrasekhar mixed type. To realize the existence of a solution of those mixed systems, he has use the Perov’s fixed point combined with the Leray– Schauder fixed point approach in generalized Banach algebra spaces.

Different real-life situations, which are modeled via FIEs, can be studied using FPT and measure of noncompactness (MNC) (see [2, 3, 5, 7, 10, 11, 13, 19, 21, 23–26]).

The following notations will be used denotes a Banach space the closure of $\textcolor{red}{}\mathrm{\Omega }$; Conv $\textcolor{red}{}\mathrm{\Omega \; -}$ the convex closure of $\textcolor{red}{}\mathrm{\Omega }$; $m_E$ - the family of all nonempty and bounded subsets of the subfamily consisting of all relatively compact sets; the set of real numbers;.

Definition 1. A mapping is said to be an MNC in if the following hold:

1. (i) If and , then $\textcolor{red}{}\mathrm{\Omega }$ is relatively compact;

2. (ii) and

3. (iii)

4. (iv)

5. (v)

6. vi)

7. (vii)

Here denotes the kernel of for

Theorem 1 [Schauder theorem]. (See [1].) Let be a BS, and let be closed and convex. If is continuous and compact, the it admits at least one fixed point.

Theorem 2 [Darbo theorem]. (See [9].) Let be a Banach space and be nonempty, bounded, closed, and convex (NBCC). Let be continuous, and let there exist a constant with

Then ∆ has a fixed point.

With the help of following concepts, we establish our fixed point theorem.

Definition 2. (See [24].) Let functions . Then the pair is called a pair of shifting distance functions (SDF) if:

1. 1. For , then

   2. For with for all

   Examples of SDF are:

   (i)

   (ii)

Definition 3. (See [12].) will denote the family of all maps with:

1. 1.

2. 2. k is continuous and nondecreasing;

3. 3.

   For example:

2 New results

Theorem 3. Let be a BS and be NBCC. Also, let be a continuous mapping with

foris a continuous mapping, and is an arbitrary MNC. Moreover, is nondecreasing withiff . Then admits a fixed point in .

Proof. Define a sequence. Then . Similary is compact. Applying Theorem 1, we observe that admits a fixed point.

If we have

which gives

Since , we have

Clearly, the sequence is nonnegative and nonincreasing; thus, we can find an such that

We claim that .

Therefore,

which gives

Letting , we get . Hence, , , wich gives

which gives

By using the properties of and $\textcolor{red}{}k$ we get . Since , by Definition 1 we get is nonempty, closed, and convex. Also, $C_{\infty }$ is invariant under. Thus, Theorem 1 implies that has a fixed point in .

Theorem 4. Let be a BS and be NBCC. Also, let : be a continuous mapping with

foris a continuous mapping, and ϑ is an arbitrary MNC. Moreover, is nondecreasing with and . Then admits a fixed point in

Proof. Taking in Theorem 3, the result follows.

Corollary 1. Let be a BS and be NBCC. Also, let be a continuous mapping with

for , where , and is an arbitrary MNC. Moreover, is nondecreasing with . Thenhas a fixed point in

Proof. The result follows by taking in Theorem 4.

Remark 1. If we take , then , and Theorem 2 follows as a spaecial case.

Definition 4. (See [8].) An element is called a coupled fixed point of a mapping .

Theorem 5. (See [4].) Suppose , respectively, and the function is convex and if and only if . Thendefines an MNC in denotes the natural projection of .

Example 1. (See [4].) Let be an MNC on and . Then is an MNC in , where, denotes the natural projections of $\textcolor{red}{}\gamma$ .

Theorem 6. Let be a BS and be NBCC. Also, let be a continuous with

for all , where is a continuous function satisfying, and is an arbitrary MNC. Moreover,is nondecreasing such that , and . Then admits a coupled fixed point in .

Proof. We observe that is an MNC on for any bounded subset , where are natural projections of $\textcolor{red}{}V$. Consider defined by . It is trivial to see that is continuous. Let . We obtain

By Theorem 3 we conclude that has a fixed point in , i.e., has a coupled fixed

Corollary 2. Let be a BS and be NBCC. Also, let be a continuous function satisfying

for all , where is a continuous function satisfying , and is an arbitrary MNC. Also, is nondecreasing such that if and only if and satisfy and . Then has a coupled fixed point in .

Proof. We obtain the desired result by choosing in Theorem 6.

2.1 Measure of noncompactness

Banas´ and Krajewska [6] introduced the notions of tempering sequence and space of tempered sequences. Namely, a fixed positive nonincreasing real sequence is called a tempering sequence.

Recently, Rabbani et al. [26] denoted by $\textcolor{red}{}W$ a collection of all real or complex sequence . Clearly, $\textcolor{red}{}W$ space over $\textcolor{red}{}R$ or $\textcolor{red}{}C$, and this space is denoted by . It is easy to observe that is a Banach space with the norm

If for all .

A Hausdorff MNC for a nonempty bounded set can be given by (see [26])

Let us denote by the space of all continuous functions on , with the values in, which is also a Banach space with the norm

where

Let be a bounded subset of

Thus, an MNC for can be defined by

3 Infinite systems of mixed type fractional integral equations

Let . The Hadamard fractional integral of order , applied to the function, is defined by [15]

Therefore, we have

Let be a real number. The Riemann– Liouville fractional integral of order is defined by [20]

Consider the following infinite system of mixed-type fractional integral equations:

where , where $\textcolor{red}{}E$ is a Banach sequence space.

Assume:

(i) The fuctions are continuous and satisfy

And are continuous with

for , are continuous fuction

Also,

converger to zero for all

with be converger for all

(ii) The fuctions are continuos, and

Also,

(iii) Define an operator were

Finally, let

Theorem 7. If conditions (1).(3)hold, then equation (5)admits a solution in

Proof. For arbitrary fixed

Therefore,

implies

Hence,

Consider the operator given by

where

By assumption (iii), hence, is a self-mapping on

Let be such that

Then, for arbitrary fixed

Since are continuous for all for all

and

Therefore

Thus,; hence, is continuous on

Finally,

i.e.,

Therefore,

Thus, by assumption (i) and Remark 1 one gets that admits a fixed point in

Example 2. Consider

where . Here

Let converges to zero.

Let for some fixed . Then

is convergent as is convergent for . Therefore for fixed

i.e.,

It is obvious that ${\mathrm{\Lambda }}_{\rho }$ is continuous for all and

also,. Moreover,

i.e.,, and also,$l_{\rho }$ is continuous for all It can be observed that are all continuous, and

which gives

The functions are continuous, and

Also,

are convergent for .

Thus, all conditions (1)–(3) of Theorem 7 are satisfied, hence, equation (6) admits a solution in .

Acknowledgments

We thank the editor for his kind support. We are also grateful to the learned referees for useful suggestions, which helped us to improve the text in several places.

References

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