Fixed Point Results of Dynamic Process through -Contractions with Applications

Abstract

This article constitutes the new fixed point results of dynamic process D(ϒ, μ0) through FIC-integral contractions of the Ciric kind and investigates the said contraction to iterate a fixed point of set-valued mappings in the module of metric space. To do so, we use the dynamic process instead of the conventional Picard sequence. The main results are examined by tangible nontrivial examples which display the motivation for such investigation. The work is completed by giving an application to Liouville‐Caputo fractional differential equations.

1. Introduction and Preliminaries

In the recent past, the study of metric fixed point theory untied a portal to a new area of pure and applied mathematics, the fixed point theory and its application. This concept was prolonged by either extending metric space into its extensions or by modifying the structure of the contractions (see [1–7]). The most classical structure known as Banach contraction principle (or contraction) theorem was introduced by a Polish mathematician Banach in 1922 [8]. The applications of fixed points of Banach contraction mappings defined for different kinds of spaces is the guarantee of the existence and uniqueness of solutions of differential and integral type equations. The variety of these nonlinear problems imposes the search for more and better tools, which are recently very remarkable in the literature. One of such tools was recently conveyed by Wardowski [9], where the author originated a new class of contractive mapping called -contraction.

Nadler [10] using the idea of Pompeiu–Hausdorff metric discussed the Banach contraction mappings for set-valued functions rather than single-valued functions. Let be a metric space. For and , define the Pompeiu–Hausdorff metric induced by on as follows:for each , where denotes the set of all nonempty closed bounded subsets of and . An element is called a fixed point of a set-valued mapping, i.e., , then . Also, denote the family of nonempty compact subsets of by .

Some well-known results are related to this section and hereafter.

Lemma 1. Let and be nonempty proximal subsets of a metric space . If , then

Lemma 2 (see [11]). Let be a metric space and a sequence in such thatis not a Cauchy sequence. Then, there exists and subsequences of positive integers and , such that

Definition 1 (see [12]). Let be a multivalued mapping and be arbitrary and fixed. DefineEach element of is called a dynamic process of starting point . The dynamic process onward be written as .

Example 1 (see [12]). Let be a Banach space with a norm , . Let be such that, for every , is a collection of the functionsthat is,and let , , then the sequence is a dynamic process of with starting point .

A mapping is said to be -dynamic lower semicontinuous at , if for every dynamic process and for every subsequence of convergent to , we get . If is -dynamic lower semicontinuous at each , then is said to be -dynamic lower semicontinuous. If for every sequence and such that , we have , then is known as lower semicontinuous.

As of now, Branciari [5] generalized the second well-known contraction of Banach contraction mappings is determined, i.e., let be a metric space and a mapping such thatfor all , where , , and is the class of all functions which is Lebesgue integrable, summable on each compact subset of and for all . Then, has a fixed point.

The following lemmas are helpful for our main results. We shall also suppose that

Lemma 3 (see [6]). Let be a nonnegative sequence in such a way that . Then,

Lemma 4 (see [6]). Let be a nonnegative sequence. Then,

In 2012, Wardowski [9] initiated the term of -contraction and implemented on fixed point theorem related with -contraction. So, with the intent that, he generalizes contraction theorem which is a purely altered from many past results in the literature frame.

Definition 2 (see [9]). Let is called an -contraction on a metric space , if there exist and in such a way that, impliesFor each , where is the class of all functions such that  implies for all .  For each sequence of positive real numbers,  There is in such a way that .

From now, we present some well-defined examples of -contraction that are listed as follows:    

Owing to and (11), clearly, we conclude that every -contraction is a contractive mapping. Consequently, every -contraction is a continuous mapping (see more [13]).

The main purpose of this manuscript is to introduce the new concept of dynamic iterative process based on -integral contractions and prove some related multivalued fixed point results in the class of metric space. To approximate our main results by tangible examples are also determined. At the end, the work is completed by giving an application to Liouville–Caputo fractional differential equations.

2. Main Result

First, we give our main definition.

Definition 3. Let be a metric space, , and A set-valued map is said to be -integral contraction with respect to a dynamic process , if there exists such thatfor all , where

Remark 1. For the act of continuing our results, we consider only the dynamic processes satisfying the following structure:If the investigated process does not satisfy (15), then there is such thatandThen, we get which implies the existence of fixed point due to this consideration of dynamic process that satisfying (15) does not depreciate a generality of our approach.

Example 2. Let be defined by . Each set-valued -integral contraction on a metric space with respect to dynamic process assures thatUpon setting, we havefor all , , and . In view of the above observations, clearly, for such that , the following inequality also holds through that is, is a contraction.

Theorem 1. Let be a complete metric space, and be a set-valued -integral contraction with respect to the dynamic process . Assume that

Proof. In view of , if there exists such that , then the existence of a fixed point is obvious. Therefore, if we let , then for every . Using (15) and by Lemma 1, one writesMoreover, since is compact, we obtain such that . Using (21), we haveIn view of the above observations, is decreasing and hence convergent. We now show that In the light of (D1), there exist and such that for all . So, we haveLet us set for and from (24), we see that . By means of (), we haveAlso, in the light of , there is such thatFurthermore, from (24), we can write for all Taking limit as in (27) and using (26), we haveLet us perceive that, from (28), there is such that for all . We haveNow, in order to show that is a Cauchy sequence, we consider , such that . From (29) and by virtue of metric condition, we haveIn the light of (30) and view of convergence of series , we see that . Hence, is Cauchy sequence in . Furthermore, for the completeness of there is such that . Since is compact, then we have . By Lemma 1, one writesSo, and . Suppose, on the contrary, . Then, there exist and subsequence of such that for each (otherwise, there is such that for every , which yields that ). By contractive condition, one writesUpon letting in (32),which is a contradiction. On the other hand, we see that the mapping is -dynamic lower semicontinuous, we haveand by virtue of closedness of implies that which means that is a fixed point of .

Remark 2. If in Theorem 1, instead of the contractive condition (13), we assume the following conditionwhere andfor all , , then there exists a fixed point of the mapping with the assumptions (D1) and (D2) on Theorem 1.

Corollary 1. Let be a complete metric space, , , , and . Assume that there exists such that

for all , , where

Then, there exists a fixed point of the mapping with the assumptions (D1) and (D2) on Theorem 1.

Proof. If we choose , the proof follows from Theorem 1.

Corollary 2. Let be a complete metric space, , , , and . Assume that there exists such thatfor all , , where

Then, there exists a fixed point of the mapping with the assumptions (D1) and (D2) on Theorem 1.

Proof. If we choose , the proof follows from Theorem 1.
The direct consequence of Theorem 1 for single-valued maps is the following.

Corollary 3. Let be a complete metric space, , , , and . Assume that there exists such that impliesfor all and for each . Suppose also that a mapping is -dynamic lower semicontinuous. Then, has a fixed point.

Corollary 4. Let be a complete metric space, , , and . Assume that there exists such that impliesfor all and for each . Suppose also that a mapping is lower semicontinuous. Then, has a fixed point.

Example 3. Let with the usual metric, constitutes a complete metric space. Consider a mapping by , and byDefine dynamic iterative process : a sequence is given by for all with initial point and such that

Continuing the above iterative process, we see thatis a dynamic iterative process of starting from the point . Setting for all and For and , we haveand soHence, all the required hypotheses of Theorem 1 are satisfied and hence 0 is a fixed point of .

3. An Application

In this frame of study, we deal with some new aspects of Liouville–Caputo fractional differential equations in module of complete metric space. Several earlier developments on fixed point theory and its applications involving fractional calculus can be found in [14].

Define the Liouville–Caputo fractional differential equations based on order bywhere , , , and the collection represents positive real number and represents the Gamma function. Let be the space of all continuous real-valued functions on And, complete metric space be given by

Now, consider the following fractional differential equations and its integral boundary valued problem:where , andwhere , and be a continuous function. Let be defined byfor and . Now, we start the main result of this section.

Theorem 2. Let be a continuous function, nondecreasing on second variable and there is a nonconstant function such that and implieswhere and

Then, equations (49) and (50) has at least one solution on .

Proof. For every and owing to operator , one writes
Upon setting, we see that
In the light of above observation, we have
which implies thatBy above virtue, we haveFurthermore, by contractive condition (13) upon setting of for all and , we havefor all , and for each given such that . Thus, all the required hypotheses of Theorem 1 are satisfied, and hence equations (49) and (50) has at least one solution on .

Example 4. Based upon the Liouville–Caputo fractional differential equations based on order . Let us consider the following integral boundary-value problem:andwhere , , and . So, the above setting is an example of equations (49) and (50). Hence, here is clearly the pair of equations (57) and (58) has at least one solution.

4. Conclusions

In this paper, we have investigated the preexisting results of fixed point for set-valued mappings rather than the conventional mappings. Based upon a Wardowski integral and with a nonnegative Lebesque integrable mapping, we have transformed the conventional theorems of fixed point into the module of . Instead of the traditional Picard sequence, the dynamic process is adopted to iterate the fixed point. Afterwards, the results have been explained by rendering concrete examples, and some foremost corollaries have been deduced from the prime results. Also, we provide illustrative applications to Liouville–Caputo fractional differential equations.

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 no conflicts of interest.

Authors’ Contributions

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