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1 Introduction and preliminaries

1.1 Measure of noncompactness

We start with listing of some notations and preliminaries that we shall need to express our results. Throughout the paper, we denote the set of real numbers, the set of natural numbers, and . Let be a real Banach space with zero element . By we denote the closed ball centered at with radius . The symbol stands for the ball . If is a nonempty subset of , then and denote the closure and closed convex hull of , respectively, and diam as diameter of the set . Moreover, let us denote by the family of all nonempty and bounded subsets of and by its subfamily consisting of all relatively compact sets. We also denote as a family of all nonempty, bounded, closed and convex subsets of .

We now recall the concept of measure of noncompactness.

Definition 1. (See [5].) A mapping is said to be a measure of noncompactness ( for brief) in if it satisfies the following conditions:

• The family is nonempty, and ;

• Monotonicity:;

• Invariance under closure:

• Invariance under passage to the convex hull:

• Convexity: for

• , where

• Cantor’s intersection property: If is a sequence of nonempty, closed sets in such that and, then the set is nonempty and compact.

The family defined in axiom (i) is called the kernel of the .

One of the properties of the is . Indeed, from the inequality for we infer that .

The well-known measure of noncompactness is due to Kuratowski [15], which is the map given as

In 1930, Schauder [20] generalized Brouwer’s fixed point theorem to Banach spaces as follows.

Theorem 1. Let be a unbounded subset of a Banach space . Then every compact, continuous map has at least one fixed point.

We recall that the mapping is said to be a compact operator if is continuous and maps bounded sets into relatively compact sets, where and are normed linear spaces, and is a subset of .

In 1955, Darbo [8] used the notion of measure of noncompactness to establish an extension of Schauder’s fixed point problem as below.

Theorem 2. Let be a subset of a Banach space , and let be a continuous and -set contraction operator, that is, there exists a constant with

for any , where is an on . Then has a fixed point.

The following well-known theorem was proved in 1967 by Sadovskii [19], it is a generalization of Darbo’s fixed point theorem.

Theorem 3. Let be a subset of a Banach space , and let be a continuous and -condensing operator, that is,

for any , where is an on . Then has a fixed point.

1.2 Best proximity theory

It is well understood that a mapping on a nonempty subset of possesses a fixed point if is nonempty. If is fixed point free, then in this case, we intend to find the element in so that and have smallest distance. In this case, the point is a best approximant for . The credit of pioneering best approximation theory goes to Ky Fan (1969) (refer [6] and references therein for more details of best approximation theory). But the problem arises when is mapped into another subset of by . In this case the problem is to find a point, which estimates the distance between these two sets and . Such points are known as best proximity points.

Let us take two nonempty subsets and of . It is to be assume that a pair satisfies a property if and individually satisfy that property. For example, we say a pair is compact if and only if and are compact. For the pair , we will define

It is worth noticing that the pair may be empty, but in particular, if is a nonempty, convex and weakly compact pair in , then is also nonempty, convex and weakly compact. If and , then the pair is called proximinal.

A mapping is called cyclicif and , and if and , then is noncyclic. is called relatively nonexpansive if it satisfies whenever and . In special case, if , then is called nonexpansive self-mapping. We consider a best proximity point for a cyclic mapping , which is defined as a point satisfying

In case of a noncyclic mapping , we consider existence of a pair for which and . Such pairs are called best proximity

Eldred et al. in [9] coined the idea of cyclic (noncyclic) relatively nonexpansive mappings and obtained best proximity point (pair) results. In doing so, they have used the concept, which is called as proximal normal structure (in short, PNS). In 2017, Gabeleh [11] proved that every convex and compact (nonempty) pair in a Banach space has PNS by using a concept of proximal diametral sequences. Considering this fact, Gabeleh obtains following result. Recall that the compactness of means that is compact.

Theorem 4. (See [12].) Let be a Banach space, and let . Assume that is a relatively nonexpansive cyclic mapping, then has a best proximity point, provided is compact and .

Before stating the result for noncyclic mappings, let us recall a mathematical concept of strict convexity of Banach spaces. A Banach space is strictly convex if for and ,

holds. The space and Hilbert spaces are examples of strictly convex Banach spaces.

Theorem 5. (See [12].) Let be a strictly convex Banach space, and let . Assume that is a relatively nonexpansive noncyclic mapping. If is compact and , thenhas a best proximity pair

Recently, several works appeared (see [12–14, 16, 18, 21]) in which best proximity point (pair) results are obtained using measure of noncompactness.

1.3Concepts from fractional calculus

We present some concepts and outcomes from fractional calculus, which will be used in application part of this article. Let . Let denotes the space of all continuous functions on . We denote by , the spaces of Lebesgue-integrable functions on . See [10] for more details on fractional calculus.

The left-sided Riemann–Liouville fractional integrals and derivatives are defined as follows.

Definition 2. Let . The integral

is called left-sided Riemann–Liouville fractional integral of order of the function .

Definition 3. The left-sided Riemann–Liouville fractional derivative of order of is defined as the following expression:

provided the right-hand side exists.

We have following results for above power functions.

Lemma 1. For , we have

Lemma 2. For and , we have

Definition 4. (See [10].) The left-sided Hilfer fractional derivative operator of order and type is defined by

Remark 1. The Hilfer derivative is considered as an interpolator between the Riemann– Liouville and Caputo derivative since

The differential equations with fractional derivatives gain a lot of importance in recent years. For proving existence of solutions for such equations, the fixed point theory and the concept of measure of noncompactness is of immense importance. For more applications of fixed point theorems and , we refer the readers to following works [1–3, 22] and references therein.

In this article, we first present the results proving existence of best proximity points (pairs) for some new variants of -contractive mappings. These conclusions extend some of recent results in the literature. As an application, we prove existence of optimum solutions for the differential equations of arbitrary fractional order involving the left-sided Hilfer fractional differential operator.

2 Main results

We start with defining the following notion introduced in [17, 24].

Definition 5. Let be a family of all functions such that:

(F1) is strictly increasing;

(F2) for each sequences .

Moreover, denotes the set of all mappings such that

We refer the interested readers to the chapter [23] for review of class of -contractive conditions. The authors give fixed point existence result established by using such contraction condition together with measure of noncompactness. Moreover, the applicability of these results in the theory of functional equations is discussed.

We define a new notion of cyclic (noncyclic) contractive operator using these two classes of functions. Throughout this section, is an on , and .

Definition 6. An operator is said to be cyclic (noncyclic) contractive if there exist and a lower semi-continuous function : such that implies

for every proximinal and invariant pair with dist.

If , then the operator is called a cyclic (noncyclic) -contractive operator. We now state the first main existence result.

Theorem 6.Let be a Banach space, and let be a relatively non- expansive cyclic contractive operator. If , then has a best proximity

Proof. Note that is proximinal. Also if , there exists an element such that . Since is relatively nonexpansive,

which gives , that is, . Similarl, , and so is cyclic on .

We start with assumption and and define a sequence pair as and for all . We claim that

We have . Therefore,. Continuing this pattern, we get by using induction. Similarly, we can see that for all . Thus for all . Hence, we get a decreasing sequence of nonempty, closed and convex pairs in . Moreover,and . Therefore for all , the pair is -invariant. By a similar manner we can see that is also -invariant for all .

Besides, if is such that , then and

Next, we show that the pair is proximinal using mathematical induction. Obviously, for , the pair is proximinal. Suppose that is proximinal. We show that is also proximinal. Let be an arbitrary member in . Then it is represented as with and . Due toproximinality of the pair , there exists for such that. Take . Then and

This means that the pair is proximinal, and induction does the rest to prove that is proximinal for all .

It is worth noticing that if for some , then the relatively nonexpansive mapping is compact, and the result follows from Theorem 4.

So we assume for all . Since , there exists and such that for every . As is contractive operator, we have

For all , we deduce that

that is,

Therefore, as , and by (F2) we must have

That is, . Now, let Using property (vii) of Definition 1, the pair is nonempty, convex, compact and -invariant with dist . Therefore, admits a best proximity point in , and this completes the proof.

If we put in Theorem 6, then we have following result for contractive mapping.

Corollary 1. Let be a Banach space, and let be a relatively nonexpansive cyclic contractive operator. If , then has a best proximity point.

Corollary 2. Let be a Banach space, and let be a relatively non-expansive cyclic operator, which satisfies

If , then has a best proximity point.

Proof. If we set and , then the proof follows from Theorem 6.

It is noteworthy here that if we consider in above corollary, then we get a particular case of Darbo-type best proximity point theorem.

The second existence result is for relatively nonexpansive noncyclic contractive operator.

Theorem 7. Let be a strictly convex Banach space, and let be a relatively nonexpansive noncyclic contractive operator. If is nonempty, then has a best proximity pair.

Proof. Let be such that . Since is relatively nonexpansive noncyclic mapping,

which gives , that is,. Similarly, and so is noncyclic on .

Let us define a pair as and with and . We have that . Therefore, . Thus . Continuing this pattern, we get by using induction. Similarly, we can see that for all. Hence we get a decreasing sequence of nonempty, closed and convex pairs in . Also, . and Therefore, for all , the pair is -invariant. From the proof of Theorem 6 we have is a proximinal pair such that for all .

Following the proof of Theorem 6, if for some , then the relatively nonexpansive mapping is compact, and the result follows from Theorem 5.

So we assume that for all . In view of the fact that , there exist and such that for every . Since is contractive operator,

Thus, for all , we obtain

that is,

This implies that as , and by (F2) we have

Thereby, . Now, let Using property (vii) of Definition 1, the pair is nonempty, convex, compact and -invariant with dist Therefore, has a best proximity pair.

If we set in Theorem 7, then we have the following result for contractive mapping.

Corollary 3. Let be a strictly convex Banach space, and let be a relatively nonexpansive noncyclic contractive operator. If is nonempty, then has a best proximity pair.

Corollary 4. Let be a strictly convex Banach space, and let be a relatively nonexpansive noncyclic operator, which satisfies

If , thenhas a best proximity pair.

Proof. If we set and , then the proof follows from Theorem 7.

It is noteworthy here that if we consider in above corollary, then we get a particular case of Darbo-type best proximity pair theorem.

3 Application

In this section, we establish the existence of an optimal solution of the following problem involving systems of Hilfer fractional differential equations with initial conditions.

Let and be positive real numbers, , and let be a Banach space.

Let and be closed balls in , where .

We consider the following system of Hilfer fractional differential equation of arbitrary order with initial conditions:

where is the left-sided Hilfer fractional differential operator, ; the state takes the values from Banach space and are given mappings satisfying some assumptions. The following result establishes the equivalence of (1) with the integral equation.

Lemma 3. (See [10].) The initial value problem (1) is equivalent to the following integral equation:

Let and let be a Banach space of continuous mappings from into endowed with supremum norm. Let

So is a nonempty, bounded, closed and convex pair in . Now, for every and , we have . Therefore dist, which ensures that is nonempty. Now, let us define the operator as follows:

Lemma 4. The operator defined by (3)is cyclic if and are bounded and continuous such that .

Proof. Let and set . We have

Applying on both sides and applying Lemma 1, we get

Here by Lemma 2. Therefore , which means . Similarly, one can show that . Thus is cyclic operator.

We say that is an optimal solution for system (1) and (2), provided that , that is, is a best proximity point of the operator defined in (3).

Assumptions. We consider the following hypotheses to prove the existence of optimal solutions to the differential equations.

(A1) Let be any . For any bounded pair , there exist , a nondecreasing function and such that , implies

And

The following result is the mean-value theorem for fractional differential, which we have rewritten according to our notations.

Theorem 8. (See [7].) Let and be given as above. Let be integrable on , and let and be the infimum and supremum of , respectively, on . Then there exists a point in such that

Then we give the following result.

Theorem 9. Under notations defined above, the hypotheses of Lemma 4 and assumptions (A1) and (A2), the system of Hilfer fractional differential equation (1)-(2)has an optimal solution.

Proof. It is clear that system (1)–(2) has an optimal solution if the operator defined in (3) has a best proximity point.

From Lemma 4, is a cyclic operator. It follows trivially that is a bounded subset of . We prove that is also an equicontinuous subset of . For with and , we observe that

As , right-hand side tends to . Thus as . Thus is equicontinuous. With the similar argument, we can prove that is bounded and equicontinuous subset of . Thus the application of Arzela–Ascoli theorem concludes that is relatively compact.

Next, we show that is relatively nonexpansive. For any , we have

and thereby, . Therefore is relatively nonexpansive.

At last, let be nonempty, closed, convex and proximinal pair, bn which is -invariant and such that dist. By using a generalized version of Arzela–Ascoli theorem(see Ambrosetti [4]) and assumption (A1) we get

So, in view of Theorem 8, it follows that

Therefore, we conclude that satisfies all the hypotheses of Theorem 6, and so the operator has a best proximity point , which is an optimal solution for system (1) and (2).

Acknowledgments

The authors would like to thank the reviewers for their valuable comments and suggestions.

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