Fixed Points of Subadditive Maps with an Application to a System of Volterra–Fredholm Type Integrodifferential Equations

Işık, Hüseyin; Moeini, Bahman; Aydi, Hassen; Mlaiki, Nabil

doi:https://doi.org/10.1155/2019/6925891

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Appendix Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 6925891 | https://doi.org/10.1155/2019/6925891

Fixed Points of Subadditive Maps with an Application to a System of Volterra–Fredholm Type Integrodifferential Equations

Hüseyin Işık,^1,2Bahman Moeini,³Hassen Aydi,^4,5and Nabil Mlaiki⁶

Academic Editor: Leonid Shaikhet

Received11 Jun 2019

Revised14 Sept 2019

Accepted20 Sept 2019

Published04 Nov 2019

Abstract

In this paper, some fixed-point theorems are established for strongly subadditive maps on (where denotes the space of -valued continuous functions on a compact Hausdorff space and is a unital Banach algebra). Finally, the result is applied to prove the existence and uniqueness of a solution for a system of nonlinear integrodifferential equations.

1. Introduction

Volterra–Fredholm integro differential equations [1–10] appear in a number of physical models, and an important question is whether these equations can support periodic solutions. This question has been studied extensively by a number of authors; in fact, they investigated the existence of solutions for a class of these kinds of equations using Schauder’s fixed-point theorem and Green’s function. The distance between two zeros of nontrivial solutions to integrodifferential equations was estimated by Domoshnitsky [11]. Ezzinbi and Ndambomve [12] considered the control system governed by some partial functional integrodifferential equations with finite delay in Banach spaces. On the other hand, the fixed-point theory is one of the most rapidly growing topics of nonlinear functional analysis. It is a vast and interdisciplinary subject whose study belongs to several mathematical domains such as classical analysis, functional analysis, operator theory, topology, and algebraic topology. This topic has grown very rapidly perhaps due to its interesting applications in various fields within and outside mathematics such as integral equations, initial and boundary value problems for ordinary and partial differential equations, and others. In our work, using a tool of fixed-point theory, we will ensure the existence and uniqueness of a solution for a system of nonlinear integrodifferential equations.

In [13], some properties of subadditive separating maps have been investigated. A fixed-point theorem was also given, and it was used to study the existence and uniqueness of a solution of nonlinear integral type equations. Consider and as compact Hausdorff spaces. Take as a unital Banach algebra. Denote by (resp. ) the set of -valued continuous functions on (resp. ). The operator is said to be separating if implies that for all and . Recently, an attractive work on separating maps between different spaces of functions (as well as, operator algebras) is considered (see [14–17] and references therein). The existence, uniqueness, continuation, and other properties of solutions for nonlinear integral and integrodifferential equations are studied in [18–25].

The purpose of this work is to prove some new fixed-point theorems for strongly subadditive maps and to ensure the existence and uniqueness of solutions for the following system of Volterra–Fredholm type integrodifferential equations:where , are continuous and ( is a Banach space with norm ). Note that a solution of problem (1) is as follows:

2. Preliminaries

In the sequel, denote by the Banach space of all continuous functions from J into endowed with the norm

Let and be two compact Hausdorff spaces. Let be a unital Banach algebra with a unit element . For , denote by the cozero set of ζ, i.e., . (resp. ) denotes the topological closure (resp. the compliment) of the set U. For a separating map and , and , for all .

Definition 1 (see [26]). (a)The operator is said to be subadditive, iffor all .(b)Given and a subadditive map , if for any there is so that , for all verifying , then is said to be strongly subadditive.

Example 1. Given a continuous map , take . The operator , defined by , is separating and strongly subadditive. Note that it is not additive.

Definition 2 (see [26]). The operator is said to be pointwise subadditive iffor all and .

Definition 3 (see [26]). Let be a separating map and U be an open subset of . Such U is said to be a vanishing set for if for each , implies that . The support of is given as follows:For , define by for each . Here, . Suppose that for every . Due to the fact that and is separating, we have . The use of is to extend the scalar-valued case to vector-valued case.

Remark 1 (decomposition of the identity). If or , then for each finite cover of open subsets of , there exists a continuous decomposition of identity, , subordinate to the . That is, and , for any [27].
Note that is not equal to zero everywhere, so separates . Also, each compact Hausdorff space is a Tychonoff space. The following remark is based on Theorem 1.1 of [28].

Remark 2. Let be a pointwise subadditive separating map. Then, for each , supp is a singleton.

Theorem 1 (see [13]). Let be a strongly subadditive map. Suppose that the following assumptions hold:(A1)For any , there is such that (A2)There is such that and , for any , where M follows using the strong subadditivity of Then, possesses a unique fixed point and , for any .
Examples A.1. and A.2. given in Appendix support Theorem 1.

3. Main Results

Consider the following statement: Let . Define in a sequence approximately to , i.e., satisfieswhere is a sequence of positive numbers, for which .

Theorem 2. Let be a strongly subadditive map. Suppose that the following assumptions hold:(B1)For each , there is such that , for all , where is defined in (B2)There is such that and , for each , where M comes from strong subadditivity of Then, admits a unique fixed point in . Moreover, for each , .

Proof. It is sufficient to take ; then, by Theorem 1, has a unique fixed point in . Also, for each , .
Now, a natural question arises. If in Theorem 2, one replaces the sequence by , then under what conditions we have the case thatIn the following, we propose a positive answer to this question.

Theorem 3. Consider the assumptions of Theorem 2 and let be the unique fixed point of . Then, , where is defined by .

Proof. Let . By strong subadditivity of and conditions (B1) and (B2), we havewhere , M comes from strong subadditivity of , and . We get thatThus,Now, let . Since , there is so that for , . Thus,Therefore,Since is arbitrary, . Also, by Theorem 2, . Thus, .

Theorem 4. Suppose is a map such that all the conditions of Theorem 1 are satisfied for ( is Nth iteration of for some positive integer N). Then, admits a unique fixed point.

Proof. By Theorem 1, has a unique fixed point . However,so is also fixed point of . Since the fixed point of is unique, one writes that . Also, if , then . By uniqueness, we obtain that .
Now, we present an extension of Theorem 1.
Let denote the class of functions such that and for every (where ).

Theorem 5. Let be a strongly subadditive map. Suppose that the following assumptions hold:(C1)For all , there is so that , for each (C2)There is so that and for each , , where M comes from strong subadditivity of Then, admits a unique fixed point and , for each .

Proof. Fix and let for . The proof is divided in two steps. Step 1. We claim that . By conditions (C1) and (C2) and strong subadditivity of , the sequence is monotonously decreasing and is bounded below. Hence, . Suppose that . Using the strong subadditivity of H and condition (C2), we have Taking , we get (by (C2)). Since , this implies that , that is, a contradiction, and so step 1 is completed. Step 2. We shall prove that is a Cauchy sequence. Suppose that . By triangle inequality, strong subadditivity of , and condition (C2), we haveUnder the assumption that , Step 1 now implies thatTherefore,However, since , this implies that , which is again a contradiction. Therefore, Step 2 is established.
Now, let . Since is a Cauchy sequence in , which is complete, we have . The continuity of yields that . Now, we shall show that is the unique fixed point. Let be another fixed point of . So, , and this completes the proof.

4. An Application to Nonlinear Integrodifferential Equations

Consider the following assumptions:(i)For all ,(ii)Let . For each , there is such that for each satisfying and for each .(iii)For each , there is such that(iv)For each , there is such that

Theorem 6. Let be a Banach space. Suppose that the assumptions hold. Then, the initial value problem (1) has a unique solution in , for which .

Proof. We shall prove that the operator , given asadmits a unique fixed point, which is a solution of (1). For this, by condition , we havefor all . Hence, G is subadditive. Let . By condition , for each , there is such thatfor each satisfying , so G is strongly subadditive. By condition , for each , there is so that . Also, by condition , we conclude that for each , there is such that , for which . Therefore, all conditions of Theorem 1 hold, and so (1) has a unique solution.
Now, an example is provided in order to make use of our result.

Example 2. Let () be the map as in Example A.1. (given later in Appendix section). Take and . We consider the following initial value problem:For all and , we define in (1):and the norm on . Then, is a Banach space.(i)For all and , we have(ii)For every , we have with and for each satisfying .(iii)For every , we haveAgain by , we haveTherefore,with . Also, . Thus, all conditions of Theorem 6 hold, and so the initial value problem (26) possesses a unique solution in .

5. Conclusion

We established some fixed-point theorems for strongly subadditive maps. We applied the obtained results to ensure the existence and uniqueness of a solution for a system of nonlinear integrodifferential equations. Moreover, some related examples have been presented in support of the given results.

Appendix

The following examples are in support of Theorem 1.

Example A.1. Given and with , take given asThen, is subadditive and strongly subadditive. Here, 0 is the unique fixed point of .
For this, since , it follows thatTherefore, is subadditive. The continuity of is clear. To show that is strongly subadditive: given any , we should find and so that when . It suffices to ensure thatis bounded in some neighborhood of . Let and . Note thatFirst, we consider the case . Then,Now, we show that is bounded from above for z in a neighborhood of 1. Using power series expansion of , we getThese inequalities imply thatFor , we have andIf , then , and it follows that . Thus, . Now, suppose that or 1. Choose so that if , then . Since , we get for . After including the case , we haveAlso,(A1)For any , there is so that (A2)For any , we haveChoosing , and . Thus, all conditions of Theorem 1 hold and is the unique fixed point of . Moreover, for each , .

Example A.2. Let () be as in Example A.1. and be given asClearly, is a subadditive and strongly subadditive map.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

Nabil Mlaiki would like to thank Prince Sultan University for funding this work through the research group “Nonlinear Analysis Methods in Applied Mathematics” (NAMAM) (group no. RG-DES-2017-01-17).

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Copyright © 2019 Hüseyin Işık et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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