Some Variants of Krasnoselskii-Type Fixed Point Results for Equiexpansive Mappings with Applications

Abstract

In this paper, we investigate the Krasnoselskii-type fixed point results for the operator of two variables by assuming that the family is equiexpansive. The results may be considered as variants of the Krasnoselskii fixed point theorem in a general setting. We use our main results to obtain the existence of solutions of a fractional evolution differential equation. An example of a controlled system is given to illustrate the application.

1. Introduction

Fractional evolution equations give a unique way to evaluate the well-posedness of many complicated systems. Differential models with fractional derivatives provide a great tool for the description of memory and hereditary properties. The fractional-order models of real physical systems are always more appropriate than the classical integer order systems. Many fractional order controlled problems and fractional evolution differential equations are recently studied. Their existence results can be seen in [1–5]. Most of the results involve contractive operators with more restrictive conditions. This is the reason that many existence results cover a restrictive class of physical problems.

Krasnoselskii fixed point theorem is a generalized form of Schauder and Banach fixed point theorems. While studying the solutions of delay and neutral differential/integral equations, it has been noticed that the solution can be expressed as a sum of contractive and compact operators. This theorem plays an important role in the existence of solutions of delay integral equations and neutral functional equations. Many generalizations and modifications of the Krasnoselskii fixed point theorem have appeared; for example, see [6–16] and the references therein. The Krasnoselskii fixed point theorem [17] may be stated as follows:

Theorem 1. Let be a Banach space and be a closed convex nonempty subset of . Suppose and map into such that (1) for all in (2) is compact and continuous(3) is contraction mappingThen, there is a in such that

In Krasnoselskii fixed point theorem, there are two operators in which is compact and continuous and is a contraction mapping. It is important to note that there is a very restrict and small class of operators that are contractions; due to this reason, many theorems involving contractive operators are less applicable. Therefore, it is required to cover more applications with different types of operators; in this regard, one of the suitable choices is the class of expansive operators. Also, there are a number of equations in which we cannot decompose the operator as a sum of two or more operators. To overcome this situation, the authors in [9–11] established Krasnoselskii-type results in a more general setting consisting of a single map depending upon two variables. The following result may be seen in [11]:

Theorem 2. Let be a bounded closed convex subset of a given Banach space , and be a mapping of into such that (1) for some , and for all (2)where is completely continuous and maps into . Then, there is an element such that

If we define where is compact and continuous and is contraction then, Theorem 1 is the special case of Theorem 2.

In most fixed point results, there is a condition such that where , is a metric space and is an operator on into No much attention has been given to the case Xiang and Yuan [15] investigated the case and obtained the following important results in this direction.

Theorem 3 (see [15]). Let be a nonempty closed subset of a complete metric space and is an expansive mapping of into such that Then, there is a unique such that

Theorem 4 (see [15]). Let be a Banach space and be a nonempty closed convex subset of . Suppose and are mappings of into such that (1) is continuous such that lies in a compact subset of (2) is an expansive mapping(3) for Then, there is with .

Definition 5 (see [9]). Let be a Banach space and The family is called equicontractive if there is a such that for all in the domain of
In a similar way, we define the following.

Definition 6. Let be a Banach space and The family is called equiexpansive if there is such that for all in the domain of

2. Main Results

Inspired by the above results (Theorems 1, 2, 3, and 4), we study the Krasnoselskii-type fixed point results for the operator with conditions: (i)The family is equiexpansive(ii) for or for

where is a subset of a Banach space and is an operator of into or .

Remark 7. In Theorem 2, the mapping is equicontractive and for , in while in (i) and (ii), the family is equiexpansive and for .

Theorem 8. Suppose be a nonempty closed convex subset of a Banach space and be a continuous mapping of into such that resides in a compact subset of Let be a mapping of into such that the family is equiexpansive and for with Then, there is such that

Proof. For define a mapping of into by The mapping is expansive because and By Theorem 3, there exists a unique point in such that since for each , there is a unique so we can define to obtain for some ); therefore,
we have . Now, for , This means that which shows that is a continuous function of into . Also, is a continuous function of into Since where is any compact set; therefore, and is compact by [18] (page 37). By [19] (page 412), is compact because is relatively compact, and resides in a compact set . Since , therefore, Hence, and is a continuous function of into a compact set By Schauder second theorem [17], there is such that or Since for there is unique such that and also , therefore,

Corollary 9 (see [15]). Let be a Banach space and be a nonempty closed convex subset of . Suppose and are mappings of into such that (1) is continuous such that lies in a compact subset of (2) is an expansive mapping(3) for Then, there is with .

Proof. Define with (Theorem 8). Since and from (3) therefore, Now, since is expansive mapping, therefore, Also, Hence, there is such that

Remark 10. The above corollary is Theorem 2.2 in [15] or above Theorem 4. If we take , zero operator, then, we obtain Theorem 3 or Theorem 2.1 in [15].

Corollary 11. Let be a Banach space and be a nonempty closed convex subset of . Suppose and are mappings of into such that (1) nonexpansive and lies in a compact subset of (2) is expansive mapping(3) for Then, there is with .

Proof. Define with (Theorem 8). Since and from (3) for therefore, Now, since is expansive mapping, therefore, Also, from (1), is continuous and Hence, there is such that

Theorem 12. Suppose be a nonempty closed convex set such that lies in a compact subset of a Banach space . be a mapping of into such that the family is equiexpansive and for . Let be a continuous mapping of into a metric space with Then, there is such that

Proof. For define a mapping of into by The mapping is expansive and By (Theorem 3), there exists a unique point in say such that . Now, This means that In the above expression, the continuity of implies the continuity of . Since lies in a compact subset of and is continuous, therefore, lies in a compact subset of . So by Schauder second theorem [17], there is such that Since for there is unique such that and also , therefore, .

Corollary 13. Let be a nonempty closed convex set such that lies in a compact subset of a Banach space . be a mapping of into such that (1) for (2), (3)where be a continuous mapping of into ; then, there is such that

Proof. Take (where is a Banach space) in the above theorem.

Remark 14. The above corollary is a variant of Theorem 2 or Theorem 8 in [11].

Corollary 15. Let be a Banach space and be a nonempty closed convex set such that lies in a compact subset of . Suppose and are mappings of into such that (1) is a continuous mapping(2) is an expansive mapping(3) for Then, there is such that .

Proof. Define (in the above Corollary 13). Since and from (3) , therefore, Now, since is expansive mapping, therefore, Also, Hence, there is such that

Theorem 16. Suppose be a nonempty closed convex subset of a Banach space and be a continuous mapping of into such that is contained in a compact subset of . Let be a mapping of into such that the family equiexpansive and for with Then, there exists such that

Proof. For define a mapping of into by The mapping is expansive because Also, By Theorem 3, there exists a unique point say such that . Now, This means that Since is continuous, so is a continuous function of into . Also, is a continuous function of into and resides in a compact subset of . So by Schauder second theorem [17], there is such that or Since for there is unique such that and also , therefore,

Remark 17. In the above theorem, is a mapping of into while in Theorem 8, is a mapping of into .

Corollary 18. Let be a Banach space and be a nonempty closed convex subset of . Suppose is a mapping of into and is mapping of into such that (1) is continuous such that lies in a compact subset of (2) is an expansive mapping(3) implies Then, there exists with .

Proof. Taking with (in the above Theorem 16), we can obtain the required result.

Theorem 19. Suppose be a nonempty closed convex subset of a Banach space and be a continuous mapping of into such that is a precompact subset of a metric space Let be a continuous mapping of into such that the family equiexpansive and for . Then, there is such that

Proof. For define a mapping of into by The mapping is expansive because Also, By Theorem 3, there is a unique point in say such that We show that the mapping of into is continuous. Let be a sequence in such that converges to in . Then, and therefore, Since is continuous, therefore, is continuous. The operator maps into and is continuous. Since is relatively compact, therefore, by [19] (page 412), is relatively compact. Let be the convex hull of . By [20] (page 195), is relatively compact. Since is closed and convex, so , and by [21] (page 415), . Let ; then, is a subset of and Since also, therefore, is compact and convex. So by first Schauder theorem [17], there is a point such that Thus, by similar arguments given in the proof of Theorem 5.

Remark 20. The above theorem is a variant of Theorem 2.2 in [9].

3. Application

Let be a Banach space with norm for denote a closed interval in Let be the Banach space of all continuous functions from into with norm where Let be a closed and convex subset of . Consider the following controlled system:

The mild solution corresponding to of is given by [22]: where

is the probability density function defined on ; for more details about the notations, we refer the readers to [22]. Also, and and are strongly continuous for and are compact for each

We can write in the form where

Assume that the following conditions hold:

is continuous and expansive, i.e., for all and some

is continuous

for each

where is a closed ball with radius and center at in

We show that is equiexpansive. For a fixed and any consider which shows that is equiexpansive.

We prove compactness of Continuity of and implies the continuity of ; we show that is uniformly bounded on For consider which implies that is uniformly bounded. To prove the equicontinuity of , we consider for which is independent of , and as . Therefore, is equicontinuous. By Arzela-Ascoli theorem, is a compact operator. From for each . All conditions of Theorem 19 are satisfied to obtain the such that which is the solution of controlled problem

Finally, we summarize the above discussion in the following theorem.

Theorem 21. Let be a Banach space and be a closed ball with center at and radius Assume that and satisfy –. Then, there exists a mild solution of controlled problem

Example 22. Consider the following controlled system: where , let and consider the operator defined by with domain

Clearly, is densely defined in and is the infinitesimal generator of a resolvent family on [4] and let Define the operators by

Clearly, which shows that is equiexpansive. Note that and are continuous and compact. All conditions of Theorem 21 are satisfied to obtain the solution of the given controlled problem.

4. Conclusion

Some variants of the Krasnoselskii-type fixed point theorem are presented using the notion of newly defined equiexpansive mappings. As an application, we present an existence result for controlled problem with less conditions. This problem is also a generalized kind of Cauchy problem for evolution differential equations.

Data Availability

No data were used to support the results; the results are proved completely using mathematical proofs.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The third author gratefully acknowledges with thanks the Department of Research Affairs at UAEU. This article is supported by the grant: UPAR-2019, Fund No. 31S397.