Multivalued Problems, Orthogonal Mappings, and Fractional Integro-Differential Equation

Sharma, R. K.; Chandok, Sumit

doi:https://doi.org/10.1155/2020/6615478

Journal of Mathematics

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Abstract Introduction and Preliminaries Applications Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Recent Trends in Nonlinear and Variational Analysis

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Research Article | Open Access

Volume 2020 | Article ID 6615478 | https://doi.org/10.1155/2020/6615478

Multivalued Problems, Orthogonal Mappings, and Fractional Integro-Differential Equation

R. K. Sharma¹and Sumit Chandok¹

Academic Editor: Sun Young Cho

Received09 Oct 2020

Revised28 Oct 2020

Accepted31 Oct 2020

Published21 Nov 2020

Abstract

In this manuscript, we propose some sufficient conditions for the existence of solution for the multivalued orthogonal -contraction mappings in the framework of orthogonal metric spaces. As a consequence of results, we obtain some interesting results. Also as application of the results obtained, we investigate Ulam’s stability of fixed point problem and present a solution for the Caputo-type nonlinear fractional integro-differential equation. An example is also provided to illustrate the usability of the obtained results.

1. Introduction and Preliminaries

The theory of multivalued mappings has an important role in mathematics and allied sciences because of its many applications, for instance, in real and complex analysis as well as in optimal control problems. Over the years, this theory has increased its significance, and hence in the literature, there are many papers focusing on the discussion of abstract and practical problems involving multivalued mappings. As a matter of fact, amongst the various approaches utilized to develop this theory, one of the most interesting approaches is based on methods of fixed point theory.

Acknowledging the work of Nadler [1], Gordji et al. [2], and Wardowski [3–5], the aim of this paper is to introduce the notion of multivalued orthogonal -contraction mappings in the framework of orthogonal metric space and to establish some sufficient conditions for the existence of fixed points for such class of mappings. Many researchers [6–11] proved the existence of fixed points using the concept of -contraction introduced by Wardowski [3–5]. In 1974, Reich [12, 13] asked whether we can take into account nonempty closed and bounded set instead of nonempty compact set. Although a lot of fixed point theorists studied this problem, it has not been completely solved. There are some partial affirmative answers to this problem, for instance, Mizoguchi et al. [14] and Olgun et al. [15]. We provide a partial solution to Reich’s original problem using multivalued orthogonal -contraction mappings in the setting of orthogonal metric spaces. Also, as application of the interesting and new results obtained, we investigate Ulam’s stability of fixed point problem and present a solution for a Caputo-type nonlinear fractional integro-differential equation.

Recently, Gordji et al. [2] introduced the concept of an orthogonal set (briefly, O-set) and presented some fixed point theorems in orthogonal metric spaces.

Definition 1. Let and be a binary relation. If satisfies the following condition: there exists such that for all , , or for all , , then it is called an orthogonal set (briefly O-set). We denote this O-set by .

Example 1. Let . Define if there exists such that . It is easy to see that for all . Hence, is an O-set [2].

Example 2. Let be a metric space and be a Picard operator, that is, has a unique fixed point and for all . We define the binary relation on by ifThen, is an O-set [2].

Example 3. Let be an inner product space with the inner product . Define the binary relation on by if . It is easy to see that for all . Hence, is an O-set [2].
For more interesting examples for an O-set, see [2].

Definition 2. Let be an O-set. A sequence is called an orthogonal sequence (briefly, O-sequence) if for all , or for all .

Definition 3. Let be an orthogonal metric space ( is an O-set, and is a metric space). Then is said to be orthogonally continuous (or -continuous) at if, for each O-sequence in with , we have . Also, is said to be -continuous on if is -continuous for each .
It is easy to see that every continuous mapping is -continuous, but the converse is not true [2].

Definition 4. Let be an orthogonal set with the metric . Then is said to be orthogonally complete (briefly, O-complete) if every Cauchy O-sequence is convergent.
It is easy to see that every complete metric space is O-complete, but the converse is not true [2].

Definition 5. Let be an O-set. A mapping is said to be -preserving if , whenever . Also, is said to be weakly -preserving if or , whenever .
It is easy to see that every -preserving mapping is weakly -preserving. But the converse is not true [2].

Definition 6. (see [3, 5]). Let be a mapping satisfying the following: (F1) For all , implies (F2) For every sequence in , we have if and only if (F3) There exists a number such that If , then using (F1), we have [5, 11]. Inspired by the work of Wardowski [3–5], we denote be the family of all the functions satisfying (F1) and (F3) We denote be the family of all the functions satisfying (F1), (F3), and (F4) for all with Here, (left limit at ) and (right limit at ) for all . From mathematical analysis, the following is true for all :

Example 4. Let functions defined as follows:(1), for all .(2), for all .(3), for all .Then .
Let be a metric space and be a Hausdorff–Pompeiu metric induced by metric on a set . Denote the family of all nonempty subsets of , the family of all nonempty, and closed and bounded subsets of and the family of all nonempty compact subsets of . defined by, for every :where .

2. Multivalued Results

In this section, we establish some results on the existence of fixed point for weak orthogonal multivalued contraction mappings using conditions of Wardowski [3–5].

Now, we define the following orthogonal relation between two nonempty subsets of an orthogonal set.

Definition 7. Let and be two nonempty subsets of an orthogonal set . The set is orthogonal to set is denoted by and defined as follows: , if for every and , .
It is easy to observe the following results.

Lemma 1. Let be an orthogonal metric space, and . Then there exists such that .

Lemma 2. Let be an orthogonal metric space, and , . Then there exists such that .
Now, we are ready to present our first result.

Theorem 1. Let be an O-complete orthogonal metric space and be a multivalued mapping on . Assume that the following conditions are satisfied:(i)There exists such that or (ii)For all , implies (iii)If is an orthogonal sequence in such that , then or for all (iv)If , there exists such that for all with satisfying the following: Then has at least a fixed point.

Proof. By assumption (i), there exists such that or . By assumption (ii), we get ; that is, there exists such that or . If , then is a fixed point of . Suppose that . Since is compact, . As , using (F1), we have . Therefore, using (iv), we getContinuing this process inductively, we can construct an orthogonal sequence in such that , for all . Thus we have or for all .
If for some , then is a fixed point of .
So, we may assume that for all . Since is closed, we have , for all . Also . So using (F1), we have . Further from (iv) and for every , we haveHence from the strictly increasing property of , we get . We know that , . Therefore, the sequence is strictly decreasing sequence. Suppose that , for some .
Furthermore, for all , we haveTaking in (7), we get , which is contradiction, and hence . By (F3), there exists such thatUsing (6), we getFrom (9), the following holds for all :Letting in (10), we get . Hence there exists such that for all . So, we have all for all :Now, we have to show that is a Cauchy orthogonal sequence. Consider such that . Using the triangle inequality and (11), we haveBy the convergence of series, , passing to limit , we get .
This shows that is a Cauchy orthogonal sequence. Since is O-complete, there exists such that .
Now, we claim that . Assume the contrary that . Hence there exists such that , . Therefore, further by our assumption, or , and using (iv), we getNow using strict increasing property of and , we get . Taking , we get . Hence, the result is obtained.
Here it should be noted that in Theorem 1, is compact for all . Now, we have the following result in which we give a partial answer to Reich’s problem for a closed and bounded set.

Theorem 2. Let be an O-complete orthogonal metric space and be a multivalued mapping on . Assume that the following conditions are satisfied:(i)There exists such that or (ii)For all , implies (iii)If is an orthogonal sequence in such that , then or for all .(iv)If , there exists such that for all with satisfying the following: Then has at least a fixed point.

Proof. Let . Since is nonempty for all , by assumption (i), we can choose such that or . If , then is a fixed point of . Let . Then since is closed. Since , then from (F1), we getUsing (iv), we getFrom (F4), we get . So from (16), we haveBy assumption (ii), we get . Continuing this process, we construct an orthogonal sequence in such that for all . Thus we have or for all .
If for some , then is a fixed point of , and so the proof is completed.
So, we may assume that for all . Since is closed, we have , for all . Also , and from (F1), we get .
Furthermore, using (iv), we haveSince . Therefore, using (18), we getSo from (19), we can get a sequence in such that there exists and for all . Now, proceeding on the same lines of Theorem 1, we get the result.

3. Consequences

In this section, we give some interesting consequences of the results proved in the previous section.

The following result is an immediate consequence of Theorem 1.

Corollary 1. Let be an O-complete orthogonal metric space and . Assume that the following conditions are satisfied:(i)There exists such that or (ii)For all , implies (iii)If is an orthogonal sequence in such that , then or for all .(iv)There exists some , such that for all with , , either of the following contractive conditions hold: Then has at least a fixed point in each of these cases.

Proof. As each functions , , and , where , is strictly increasing on , so the proof immediately follows from Theorem 1.
As a consequence of Theorem 1, we have the following result for single-valued mapping by replacing condition (iii) with is -continuous.

Corollary 2. Let be an O-complete orthogonal metric space and . Assume that the following conditions are satisfied:(i)There exists some , such that for all with , : where .(ii)There exists such that or .(iii)For all , implies (iv) is -continuousThen, has a fixed point.

Proof. Here, we can choose as a multivalued mapping by considering is a singleton set for every . Arguing on the same lines of Theorem 1, we consider is a Cauchy orthogonal sequence and . As is -continuous, we havei.e., is a fixed point of .
As a consequence of Corollary 2, we have the following result by taking , .

Corollary 3. Let be an O-complete orthogonal metric space and . Assume that the following conditions are satisfied:(i)There exists some , such that for all with , : where .(ii)There exists such that or .(iii)For all , implies .(iv) is -continuous.Then has a fixed point.

4. Illustration

In this section, we illustrate an example which shows that is a multivalued orthogonal mapping and satisfies the condition (iv) of Theorem 1, but it is not multivalued orthogonal contraction (, for with ).

Example 5. Let and be a mapping defined by for all .
Define a relation on by if and only if .
Thus is an O-complete orthogonal metric space. Now, we define a mapping byWe claim that is a multivalued orthogonal mapping satisfying condition (iv) of Theorem 1 with respect to and . To see this, we have the following cases.
First, we observe that for all , if and only if and or . Case 1. For and , we have Case 2. For , we getThis shows that satisfies (iv) of Theorem 1. Hence, has a fixed point.
On the contrary, is not multivalued orthogonal contraction (, ), as

5. Applications

In this section, we present the Ulam stability and solve a nonlinear fractional differential-type equation using Corollary 3.

5.1. Ulam Stability

The Ulam [16, 17] stability has attracted attention of several authors in fixed point theory [18]. On orthogonal metric space , , we investigate the fixed point equation:and the inequality (for ):

Equation (28) is called the Ulam stable if it satisfies the following condition: (A) There is a constant , for each , and for every solution of the inequality (29), there is a solution for equation (28) such that

Theorem 3. Under the hypothesis of Corollary 3, the fixed point equation (28) is Ulam stable.

Proof. On account of Corollary 3, we guarantee a unique such that , that is, forms a solution of (28). Let and be an -solution, that is,We haveHence, , where . Therefore, equation (28) is Ulam stable.

5.2. Application to Nonlinear Fractional Integro-Differential Equation

Here, we give a solution for a Caputo-type nonlinear fractional integro-differential equation. For more details on fractional calculus, see [19–25] and references cited therein.

The Caputo derivative of a continuous mapping (order ) is given bywhere represents the gamma function and refers to the integer part of the positive real number .

In this section, we examine the nonlinear fractional integro-differential equation of the Caputo type:where , , and is a continuous function (for more details, see [20]).

We consider with supremum norm . So is a Banach space.

The space endowed with the metric defined as and define an orthogonal relation if and only if , for all . Then is an orthogonal metric space.

Clearly, a solution of equation (34) is a fixed point of the integral equation:

Theorem 4. Assume that is a continuous function satisfyingfor each , for some and for all . Then the fractional differential equation (34) with given boundary conditions has a solution.

Proof. The space endowed with the metric defined as , for all . Define an orthogonal relation if and only if , for all . Then is an orthogonal metric space. Define as in (35). So is -continuous. First, we show that is -preserving, let for all . Now, from (35). we havewhich implies that .

Now, we have to show that satisfies (i) of Corollary 2 for . For all , , we havefor all . Therefore, the condition (i) of Corollary 2 holds. Accordingly, all axioms of Corollary 2 are verified, and has a fixed point. The Caputo-type nonlinear fractional differential equation (34) possesses a solution is yielded.

6. Conclusions

In this manuscript, we prove some existence results for the multivalued orthogonal mappings using the conditions (F1) and (F2) of Wardowski’s and obtain the stability of a fixed point problem and a solution for the Caputo-type nonlinear fractional differential equation.

Now, we have an open question, whether we can obtain Theorems 1 and 2 with condition (F1) only of Wardowski in the setting of orthogonal metric space?

Data Availability

No data are used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no known conflicts of financial interest or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors are grateful to the AISTDF-DST, India CRD/2018/00017.

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Copyright © 2020 R. K. Sharma and Sumit Chandok. 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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