Generalized Interpolative Contractions and an Application

Nazam, Muhammad; Aydi, Hassen; Hussain, Aftab

doi:https://doi.org/10.1155/2021/6461477

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Fixed Points and Computational Optimization

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Volume 2021 | Article ID 6461477 | https://doi.org/10.1155/2021/6461477

Generalized Interpolative Contractions and an Application

Muhammad Nazam,¹Hassen Aydi,^2,3and Aftab Hussain⁴

Academic Editor: Sun Young Cho

Received01 Oct 2021

Accepted29 Oct 2021

Published15 Nov 2021

Abstract

In this study, we introduce a property (P) and the generalized interpolative contractions of types I, II, III, and IV. We investigate certain conditions for the existence of fixed points of generalized interpolative contractions. We derive several new results from the main theorems. As an application, we resolve the Urysohn integral equation.

1. Introduction

Fixed-point theory is an outstanding example of a central principle with multiple implementations. In diverse areas, such as differential equations and artificial intelligence, it has always been a significant theoretical method. Furthermore, the development of accurate and efficient techniques for computing fixed points has significantly increased the concept’s utility for applications, making fixed-point methods a major tool in the arsenal of the applied mathematician. The key element in the metric fixed-point theory is the Banach contraction principle (BCP). It states that every contraction, in the complete metric space, admits a unique fixed point. This principle has been generalized by many ways (see [1]). Recently, Gordji et al. [2] presented a new generalization of the BCP by defining the notion of orthogonal sets and hence orthogonal metric spaces. They presented an example supporting the fact that their main theorem is a real generalization of the BCP. Baghani et al. [3] extended the work of [2] to -contractions. Chandok et al. [4] extended the results given in [3] to multivalued -contractions.

On the contrary, Karapinar [5] introduced interpolative contractions and presented a method to obtain fixed points of such contractions. Karapinar et al. [6–9], in subsequent papers, investigated Rus–Reich–Ćirić-type interpolative contractions, Hardy–Rogers-type interpolative contractions, Rus–Reich–Ćirić-type -interpolative contractions, and Boyd–Wong- and Matkowski-type interpolative contractions to ensure the existence of fixed points in variant (generalized) metric spaces. Gautam et al. [10] presented some fixed-point results for Chatterjea and cyclic Chatterjea interpolative contractions in complete quasi-partial -metric spaces. Debnath et al. [11] proved some fixed-point theorems for Rus–Reich–Ćirić- and Hardy–Rogers-type interpolative contractions in -metric spaces.

Boyd–Wong [12] generalized the well-known Banach contraction principle (BCP) [13] by introducing a control function , verifying the below conditions for each :(1)(2)

The related result of Boyd–Wong [12] is as follows.

Theorem 1. Let be a self-mapping on a complete metric space so thatwhere verifies (1)-(2). Then, has a unique fixed point in (say, ) and the sequence is convergent to , for each .

It is noted that Theorem 1 is an improvement of main results of Rakotch [14] and Browder [15]. The Boyd–Wong idea has been generalized by Matkowski [16], Samet et al. [17], Karapinar et al. [18], Pasicki [19], and Proinov [20], respectively. Recently, Nazam et al. [21] introduced several conditions on the newly introduced functions to generalize and improve the results in [12, 16–20].

The Banach contraction principle (BCP) and its generalization (GBCP) have been extensively applied to show the existence of solutions to various mathematical models. For instant, in [22–27], authors have applied GBCP to show the existence of solution to a matrix equation:where (set of positive definite matrices) and are arbitrary matrices for each and are entries of block matrices given by

Consider the system of fractional differential equations:under boundary conditions,where denotes CFD of order defined by

The existence of solutions of the above system has been shown in [21] by using GBCP. In [28], authors have employed the GBCP for the existence of solutions to a system of integral equations:for all , , and , where is a continuous function, are lower semicontinuous operators, and .

In this paper, motivated by the interpolation notion of contractions and the applications of GBCP, we investigate different conditions on the functions to show the existence of fixed points of generalized interpolative contractions (a new GBCP) of type I, II, III, and IV and hence, we apply GBCP of type I to resolve the Urysohn integral equation.

2. Preliminaries

Before stating our main results, we need to define some basic notions for better understanding of readers.

Definition 1 (see [2]). Let be a binary relation defined on a nonempty set (i.e., ) verifying the property (O). Then, is called an orthogonal set (in short, O-set):

Example 1. Let be the set of integers. Consider if and only if . Then, is an O-set. Indeed, for each .

Definition 2 (see [2]). A sequence is said to be an O-sequence if either or , for all .

Definition 3 (see [2]). The O-set endowed with a metric is called an O-metric space (in short, OMS) denoted by .

Definition 4 (see [2]). The O-sequence is said to be O-Cauchy if . If each O-Cauchy sequence converges in , then is called O-complete.

Remark 1. Each complete metric space is O-complete, but the converse is not true in general (see [2], for details).

Lemma 1. Let be an OMS and be an O-sequence, verifying . If the sequence is not Cauchy, then there are , , and such that

The proof of this lemma has the same arguments that are given in [20]. We omit details.

Definition 5. Let be a self-mapping. An element is said to be a fixed point of if .

Definition 6 (see [3]). Let be an OMS and be a binary relation. is called -regular if, for each sequence so that for each and as , we have either , or , for all .

Definition 7 (see [2]). A mapping is said to be asymptotically regular at a point of ifIf is asymptotically regular at each point in , then it is named as an asymptotically regular mapping.

In this section, we initiate the notion of -interpolative contractions. We consider various conditions on control functions to ensure the existence of fixed points of -interpolative contractions. In the following, we develop the strategy towards main results.

Let .

Definition 8. A mapping is said to be strictly -admissible if , for all , with and otherwise.

Example 2. Let , and we define the relation byThen, is O-set. Define byThen, is -admissible.

Definition 9. Let and be a binary relation. Such is called -preserving if, for each and such that or , there is such that or .

Example 3. Let , and we define the relation byThen, is an O-set. We define byThen, is -preserving. Indeed, for , there is such that either or , and then, there is such that either or .
Let be a metric space. For a mapping and positive real numbers , we define the mappings byIt is important to note that, despite , some exponents are negative; for example, if , , and , then . If any one of goes to , then . Moreover, we have the following interesting facts about the exponents that can be proved by using basic algebraic tools:The following observations are essential for the proofs of main theorems.

Observation 1. The following inequality holds for all and :

Proof. We note that the equality holds for . We can assume that ; then, , . Let so that , . Define the function byThis implies thatSince (otherwise ), we have . This implies that ; hence, , that is, .

Observation 2. Let . For any nonempty set , we define the mapping byThen, the pair is a metric space.

Definition 10. Let be a metric space. A mapping is said to have property if, for any real number , it satisfies the following inequality:

Example 4. Let and consider the metric defined by for all . The mapping defined by , for all and , satisfies the property . Indeed,

Example 5. Every identity mapping satisfies the property . The constant mapping does not satisfy the property . The mapping is defined by for all which satisfies the property only for .

Example 6. Let . The mapping defined by for all satisfies the property . In fact, the mapping defined by , for all , for , satisfies the property .

Example 7. Let . The mapping defined by for all satisfies the property .

Example 8. Let . The mapping defined by for all satisfies the property .

Remark 2. The proof of Theorem 2 depends largely on the use of either “Observations 1 and 2” or “Property .”
We proceed with the property .

Definition 11. Let be an OMS. A mapping is said to be a -interpolative fractional contraction of types I, II, III, and IV, for , respectively, if there exist a strictly -admissible mapping and , for , and , for , such thatfor all and .
If either or or in -interpolative fractional contraction of type I, we receive the recently announced -contraction by Proinov [20] which provided .
We also note that, for and , for all , , contraction (24) () can be written as follows:and then, we haveThis represents a general version of the contraction introduced by Wardowski [29], and if either or or and , then type I represents an -contraction [29].

Remark 3. It is very important to note that the set of self-mappings satisfying property and contraction (24) is not empty. For example, the mappings , for all , and , for all , satisfy both the property and contraction (24) with and , for all , where .
In the next result, we give a set of conditions that guarantee the existence of a fixed point of a self-mapping .

Theorem 2. Let be an -regular O-complete metric space (in short, OCMS). Let be an -preserving mapping verifying (24) for and property . Suppose the relation is transitive and the functions are so that(i)For each , there is such that or (ii) are nondecreasing and , for all (iii), for all (iv)Then, admits a fixed point in .

Proof. Step 1: simplification of :Step 2: by (i), for an arbitrary , there is such that or . It is assumed that is an -preserving mapping, so there is such that or , and then, there is such that or . In general, there is such that or for all. Hence, , for all . Note that if , then is a fixed point of , for all . We assume that , for all . Thus, , for each (otherwise, , for some ). Let , for all . By the first part of (ii) and (24) ( i = 1), we haveIn view of second part of (ii), we writeSince is nondecreasing, one gets , for each . This shows that the sequence is decreasing, so there is such that . If , by (29), one obtainsThis contradicts (iii), so , i.e., is an asymptotically regular mapping.
Step 3: we claim that is a Cauchy sequence. If not, then, by Lemma 1, there are and of and such that (9) and (10) hold. By (9), we infer that . Since , for all , by transitivity of , we have and hence, for all . Letting and in (24) (), we have, for each ,We note thatIf , we haveBy (9), we have , and (33) impliesIt is a contradiction to (iv), so is a Cauchy sequence in the OCMS ; hence, there is so that as , and the -regularity of yields that or . Thus, . We claim that . Assume that for infinitely many values of . By (24) (),By the first part of (ii), we get . Applying limit , we obtain . This implies that ; hence, .

Next result gives an idea on conditions ensuring the existence of fixed points of verifying (24) ().

Theorem 3. Let be an -regular OCMS. Let be an -preserving mapping verifying (24) () and property . Assume the relation is transitive and the functions are such that(i)For each , there is such that or (ii), for all (iii)(iv)If and are converging to the same limit and is strictly decreasing, then (v), for all (vi), for all Then, possesses a fixed point in .

Proof. Note that we need (i)-(iv) to show that is an asymptotically regular. Condition (v) is needed to establish that is Cauchy and (vi) is useful to ensure that the mapping has a fixed point.
By (i), for an arbitrary , there is so that or . Since is -preserving, there is so that or , and then, so that or . In general, there is in order that or , for all . Hence, . Note that if , then is a fixed point of . Suppose that , for all . Thus, (otherwise ). Since , by (ii) and (24) (), we writeInequality (36) shows that is strictly decreasing. If it is not bounded below, in view of (iii), we get . This implies thatThus, ; otherwise, we have(i.e., a contradiction to (iii)). If it is bounded below, then is a convergent sequence, and by (36), also converges and both have the same limit. Thus, by (iv), one gets . Hence, is asymptotically regular.
Now, we claim that is a Cauchy sequence. If is not a Cauchy sequence, so, by Lemma 1, there exist and and such that (9) and (10) hold. By (9), we infer that . Since , for all , so, by transitivity of , we have , and hence, for all . Letting and in (24), one writes, for all ,We note thatIf , we haveBy (9), we have and (41) impliesIt contradicts (v), so is a Cauchy sequence in the OCMS . Hence, there is in order that as .
To show that , we have two cases: Case 1: if , for some , then, since taking limit on both sides, we have . This implies ; thus, . Case 2: if, for all , , then by -regularity of , we find or , so . By (24) (), one writesBy taking and , one writesTake . Note that and as . Applying limits on (45), we haveThis contradicts (vi) if . Thus, we have , i.e., , that is, is a fixed point of .

Remark 4. Observe thatThe next two results address the -interpolative fractional contractions of types II and III.

Theorem 4. Let be an -regular OCMS. Let be an -preserving mapping verifying (24) for and property . Suppose the relation is transitive, and the functions are so that(i)For each , there is such that or (ii) are nondecreasing and , for all (iii), for all (iv)Then, has a fixed point in .

Proof. Keeping in view the simplifications for and given in Remark 4 with the fact that and following the proof of Theorem 2, we assert that admits a fixed point in . If , then we have a contradiction to the definition of function .

Theorem 5. Let be an -regular OCMS. Let be an -preserving mapping verifying (24) () and property . Assume the relation is transitive, and the functions are so that(i)For each , there is such that or (ii), for all (iii)(iv)If and are converging to the same limit and is strictly decreasing, then (v), for all (vi), for all Then, possesses a fixed point in .

Proof. Keeping in view the simplifications for and given in Remark 4 with the fact that and following the proof of Theorem 2, we assert that admits a fixed point in . If , then we have a contradiction to the definition of function .

The next two results address the -interpolative fractional contraction of type IV.

Theorem 6. Let be an -regular OCMS. Let be an -preserving mapping verifying (24) for with and property . Suppose the relation is transitive and the functions are so that(i)For each , there is such that or (ii) are nondecreasing and , for all (iii), for all (iv)Then, has a fixed point in .

Proof. Keeping in view the simplifications for given in Remark 4 and following the proof of Theorem 4, we assert that admits a fixed point in .

Theorem 7. Let be an -regular OCMS. Let be an -preserving mapping verifying (24) () with and property . Assume the relation is transitive and the functions are so that(i)For each , there is such that or (ii), for all (iii)(iv)If and are converging to the same limit and is strictly decreasing, then (v), for all (vi), for all Then, possesses a fixed point in .

Proof. Keeping in view the simplifications for given in Remark 4 and following the proof of Theorem 5, we assert that admits a fixed point in .

4. The Generality of the Main Results

Let us define , for all , in any one of Theorems 2 and 3, we receive a general version of the interpolative Boyd–Wong fixed-point theorem proved in [9], and defining in Theorem 2, we receive the following result (interpolative fractional version of Wardowski fixed-point theorem with only monotonicity condition on ).

Corollary 1. Let be a complete metric space. Let be a mapping so thatwhere is nondecreasing and . Then, there is a fixed point of in .

If we define in Theorem 2, we get an interpolative fractional version of fixed-point theorem presented in [4].

Corollary 2. Let be a complete metric space. Let be a mapping so thatwhere is nondecreasing and . Then, has a fixed point in .

We receive the following interpolative fractional version of Moradi theorem [30] if we take in Theorem 2.

Corollary 3. Let be an -regular OCMS. Let be an -preserving mapping so thatwhere(i) is an upper semicontinuous function with , for all (ii) is nondecreasing

Assume that, for each , there is such that or . Then, has a unique fixed point in .

Defining and in Corollary 3, we have the next result.

Corollary 4. Let be an -regular and OCMS. Let be an -preserving mapping so thatwhere is nondecreasing. Assume that, for each , there is such that or . Then, has a fixed point in .

Observe that Corollary 4 is an improvement of Jleli–Samet fixed-point theorem [31] and the results of Li and Jiang [32] and Ahmad et al. [33].

An improvement of Skof fixed-point theorem [34] may be stated by putting in Theorem 2, for , with either or or .

Corollary 5. Let be an -regular OCMS. Let be an -preserving mapping so thatwhere is nondecreasing and . Assume that, for each , there is so that or . Then, has a unique fixed point in .

5. The Existence of the Solution to Urysohn Integral Equation (UIE)

In this section, we will apply Theorem 2 for the existence of the unique solution to UIE:

This integral equation encapsulates both Volterra integral equation (VIE) and Fredholm integral equation (FIE), depending on the region of integration (IR). If IR , where is fixed, then UIE is VIE, and for IR , where are fixed, UIE is FIE. In the literature, one can find many approaches to find a unique solution to UIE (see [35–39] and references therein). We are interested to use a fixed-point technique for this purpose. The fixed-point technique is simple and elegant to show the existence of a unique solution to further mathematical models.

Let IR be a set of finite measure and .

Define the norm by

An equivalent norm can be defined as follows:

Then, is a Banach space. Let . The metric associated to norm is given by , for all . Define an orthogonal relation on by

Then, is an OCMS (see Theorem 4.1 in [3]). Let be defined by

Then, is a strictly -admissible mapping. Put . Let (A1) The kernel satisfies Carathéodory conditions and (A2) The function is continuous and bounded on . (A3) There exists a positive constant such that (A4) For any , there is such that or . (A5) There exists a nonnegative and measurable function such that and integrable over IR with for all and with .

Theorem 8. Suppose that the mappings and mentioned above satisfy conditions (A1)–(A5); then, the UIE (53) has a unique solution.

Proof. Define the mapping , in accordance with the abovementioned notations, byThe operator is -preserving: let with ; then, . Since, for almost every IR,this implies that . Thus, .
Self-operator: conditions (A1) and (A3) imply that is continuous and compact mapping from to (see Lemma 3 in [35]).
By (A4), for any , there is such that or , and using the fact that is -preserving, we have with or , for all . We will check the contractive condition (24) of Theorem 2 in the next lines. By (A5) and Holder inequality, we haveThis implies, by integrating with respect to ,Thus, we haveThis implies thatThat is,Taking on both sides and defining with , , we haveThe defined and satisfy remaining conditions of Theorem 2. Hence, by Theorem 2, the operator has a unique fixed point. This means that the UIE (53) has a unique solution.

6. Conclusion

The interpolative contractions are broad enough to include well-known contractions. The presented theorems provide a general criterion for the existence of a unique fixed point of interpolative contraction mappings. Fixed-point methodology is used to investigate the presence of a solution to a UIE.

Data Availability

Data sharing is not applicable to this article as no dataset was generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors equally conceived the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Journal of Mathematics

Fixed Points and Computational Optimization

Generalized Interpolative Contractions and an Application

Abstract

1. Introduction

2. Preliminaries

3. -Interpolative Contractions and Related Fixed-Point Results

4. The Generality of the Main Results

5. The Existence of the Solution to Urysohn Integral Equation (UIE)

6. Conclusion

Data Availability

Conflicts of Interest

Authors’ Contributions

References

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