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Existence Results for Integral Equations and Boundary Value Problems via Fixed Point Theorems for Generalized -Contractions in -Metric-Like Spaces
In this paper, we introduce some new classes of generalized -contractions and we establish certain fixed point results for such mappings in the setting of -metric-like spaces. Some examples will illustrate the results and the corresponding computer simulations are suggestive from the output point of view. A second purpose of the paper is to apply the abstract results in the study of the existence of a solution for an integral equation problem and for a boundary value problem related to a real life mathematical model, namely, the problem of conversion of solar energy to electrical energy. Our study is concluded with an open problem, related to an integrodifferential equation arising in the study of electrical and electronics circuit analysis.
1. Introduction and Preliminaries
There are many extensions and generalizations of the metric space concept. In 1989, Bakhtin  introduced the notion of -metric space, while Czerwik ([2, 3]) extensively used the concept of -metric space for proving fixed point theorems for single-valued and multivalued mappings. On the other hand, the concept of partial metric space was introduced by Matthews .
More recently, Amini-Harandi  generalized the concept of partial metric space by introducing the metric-like spaces. After that, in , Alghamdi et al. introduced -metric-like spaces, which extends the notions of partial metric spaces, -metric spaces, and metric-like spaces. There are many other types of generalized metric spaces (see [7, 8]), introduced by adapting and developing new metric axioms. These generalized metric spaces frequently appear to be metrizable and the contraction conditions may be conserved under various particular transforms. Hence, fixed point theory in such spaces may be an outcome of the fixed point theory in classical metric spaces. However, it is not true that all generalized fixed point results become obvious in this way. More specifically, these results are based on some contraction type conditions, and some of these conditions do not remain authentic when one considers the problem in the associated metric space; see, for example, the well-written papers [9, 10].
On the other hand, in 2012, Wardowski  introduced a new contraction mapping, called -contraction, and proved a fixed point result as a generalization of the Banach contraction principle. After this, Abbas et al.  generalized the idea of -contraction and proved certain fixed and common fixed point theorems. Recently, Secelean  described a large class of functions using the condition instead of the condition in the definition of -contraction presented by Wardowski . Very recently, Piri and Kumam  improved the result of Secelean , by using the condition instead of the condition .
In this paper, we consider the notions of --contraction and Suzuki-Berinde type -contraction in the context of -metric-like spaces in order to prove certain fixed point results. Some illustrative examples are considered, which validate the hypothesis of proved results. Moreover, some applications to integral equations and a boundary value problem related to a mathematical model of conversion of solar energy to electrical energy are also given. Finally, an open problem is also suggested for the utilization of our results to some engineering problems.
In this paper, , , and will denote the set of all real numbers, natural numbers, and the set of all real nonnegative numbers, respectively.
For the beginning, some necessary definitions and fundamental results, which will be used in the sequel, are presented here.
Definition 1 (see ). Let be a nonempty set and be a given real number. A function is called a -metric if, for all , the following conditions are satisfied: () iff ;();(). The pair is called a -metric space. The number is called the coefficient of .
Definition 2 (see ). A function is called a metric-like if, for all , the following conditions are satisfied: (σ1) implies ;(σ2);(σ3). The pair is called a metric-like space.
Definition 3 (see ). Let be a nonempty set and be a given real number. A function is called a -metric-like if, for all , the following conditions are satisfied: implies ;;. The pair is called a -metric-like space. The number is called the coefficient of .
Example 4 (see ). Let and the mapping be defined by for all Then is a -metric-like space with the coefficient , but it is neither a -metric nor a metric-like space.
Remark 5. The class of -metric-like space is larger than the class of metric-like space, since a metric-like space is a special case of -metric-like space when . Also, the class of -metric-like space is effectively larger than the class of -metric space, since a -metric space is a special case of a -metric-like space when the self-distance .
Each -metric-like on generalizes a topology on whose base is the family of open -balls for all and
Definition 6 (see ). A sequence in a -metric-like space is said to be (1) convergent to a point if ;(2)a -Cauchy sequence if exists (and is finite).
Definition 7 (see ). A -metric-like space is said to be -complete if every -Cauchy sequence in , -converges to a point , such that
Definition 8 (see ). Suppose that is a -metric-like space. A mapping is said to be continuous at , if, for every , there exists such that . We say that is continuous on if is continuous at all .
Lemma 9 (see ). Let be a sequence in a -metric-like space such that for some , , and each . Then is a Cauchy sequence in and
Remark 10 (see ). Let be a -metric-like space with constant . Then it is clear that satisfies , for all . So it is considered to be a -metric induced by -metric-like spaces.
Remark 11 (see ). Let be a -metric-like space and let be a continuous mapping. Then
Wardowski  introduced the -contraction as follows.
Definition 12. Let be a mapping satisfying () is strictly increasing, that is, for such that implies ;()for each sequence of positive numbers if and only if ;()there exists such that .Denote the set of all functions satisfying ()–() by . In , Secelean changed the condition by an equivalent but a more simple condition . (), or also by()there exists a sequence of positive real numbers with . Recently, Piri and Kumam  used the following condition instead of .() is continuous on . In our subsequent discussion, condition is dropped out. Thus we utilize the functions which satisfy and . The class of all functions satisfying and is denoted by .
Let be the set of functions such that(1) is monotonic increasing, that is, ;(2) is continuous and for each .
Let denote the set of all continuous functions .
2. Fixed Point Results for - Contractive Mappings
We introduce the following concept.
Definition 14. Let be a -metric-like space. A self-mapping is said to be generalized - contraction if andfor all , where (not all zero simultaneously), such that , and .
For illustrating the above definition, the following example is presented.
Example 15. Let and let the function be defined by for all . It is obvious that is a complete -metric-like space with . Define the mapping by In order to verify the Condition (6) with , , and such that and , for all , we see that . Let be given by and functions are defined by .
Here we note that Without loss of generality, assume that . Then, the following cases arise.
Case 1. When , calculating various terms appearing in the inequality (6), we conclude that left hand side of (6) comes out and right hand side of (6) becomes It is evident from Figure 1 that the surface representing right hand side is dominating the surface representing left hand side. This concludes that, in this case, the condition (6) is verified.
Case 2. When , with this assumption, evaluating the terms involved in Condition (6), we obtain the left hand side as and right hand side of (6) becomes Figure 2 shows that right hand side expression is superimposing the left hand side expression, which validates our condition in this case.
Thus all the hypothesis of Definition 14 are fulfilled and therefore is a - contraction mapping.
Our main result runs as follows.
Theorem 16. Let be a complete -metric-like space and be a continuous generalized -F contraction. If , for all , then has a unique fixed point in .
Proof. Let be an arbitrary point in . Set and define a sequence in byIf there exists such that then is the fixed point of , which complete the proof.
Consequently, we suppose , for all .
Then we haveand by (6) we obtainNow, we claim thatSuppose, on the contrary, that there exists , such that Then, by (6), one gets In view of the properties of , , and , we obtain that which shows this implies This is a contradiction and hence (18) holds; that is, . So is a decreasing sequence in and is bounded below at ; consequently it is convergent to some point, say . Now we assert that . On the contrary suppose .
On the similar approach as discussed earlier, we conclude thatLetting and utilizing , we have This is a contradiction, in view of the properties of and and the fact that
So, we must have ; that is,Now we shall prove that is a Cauchy sequence. In fact, we will establish thatOn the contrary, suppose that there exists , and two sequences and on natural number such that ,From the triangle inequality, one can obtainFrom (26), there exists , such thatwhich, together with (29), showsthereforeFrom (28), we get Employing (6), one acquires In view of (30), (31), and (32), we get This amounts to say that . Hence , which is a contradiction with (28). This validates (27). Therefore is a Cauchy sequence in . Since is complete, there exists , such that Since is continuous, we getDue to the fact that , from above, we have . Since , letting , we obtain that . Thus and so has a fixed point.
In order to show the uniqueness of fixed point, suppose is another fixed point such that . Then we have This is a contradiction, in view of and . Thus we have . Hence has a unique fixed point. This completes the proof.
In order to illustrate our result, we present the following example.
Example 17. Let and let the function be defined by , for all . It is obvious that is a complete -metric-like space with . Let the mapping be defined by We verify the condition (6) with , , and (clearly ) and , for all . Notice that and , for all Consider given by and .
Various terms involved in the inequality (6) are calculated as follows: Utilizing aforementioned values, the left hand side of (6) becomes and the right hand side is obtained asBy Figure 3 it is obvious that the surface representing right hand side function is dominating the surface representing left hand side function. So, the condition (6) is verified.
Furthermore, is continuous and we also have that for all .
Thus, all the conditions of Theorem 16 are satisfied and, consequently, the mapping has a unique fixed point as . This is also demonstrated by Figure 4, where the mapping and the first diagonal are represented.
If we choose in Theorem 16, then the following corollary is obtained.
Corollary 18. Let be a -metric-like space and be a continuous self-mapping. If there exist and such that for all ,where (not all zero simultaneously) such that and . Then has a unique fixed point.
Replacing in Corollary 18, subsequent result is obtained.
Corollary 19. Let be a -metric-like space and be a continuous self-mapping. If there exist and such that for all ,where and . Then has a unique fixed point.
3. Results via Suzuki-Berinde Type -Contractions
Berinde initiated some new mappings, called weak contraction mappings in a metric space [20–22]. He demonstrated that Banach’s, Kannan’s, and Chatterjea’s mappings are weak contractions. Afterward, a lot of generalizations of these results in several spaces appeared in the literature. Berinde type weak contractions are usually called almost contractions. Clubbing the ideas of Berinde, Suzuki and the notion of -contraction, Suzuki-Berinde type -contractive mapping is defined in the framework of -metric-like spaces.
Definition 20. Let be a -metric-like space with . A self-mapping is said to be Suzuki-Berinde type -contraction, if there exists such that, for all with ,wherewith and .
Theorem 21. Let be a complete -metric-like space and be a continuous Suzuki-Berinde type contraction. Then has a unique fixed point in .
Proof. Let be any arbitrary point. We construct a sequence in in such a way that for all .
Suppose that , for some . Then one can get and then is a required fixed point. So, we are done in this case. Thus, from now on we assume that , for all . Consequently, we have Then by the Definition 20 with and , we haveNotice thatThus, we have If then, from (48), we haveThis leads to a contradiction, in view of and the hypothesis of . Then we arrive atThus, from (52) and , we get thator equivalently as Therefore is a nonnegative decreasing sequence of real numbers and is bounded below at , consequently convergent to some point ; now we claim that . Let us suppose that .
Letting in (52), we have which is a contradiction in view of and the properties of . Thus we have .
Consequently, we have Now we will show that is a Cauchy sequence.
Case 1. When , rewrite (53) as Since and , as , then, by Lemma 9, the sequence is a Cauchy sequence.
Case 2. When , following the same approach as in Theorem 16 and utilizing the condition (45), it is easy to show that is a Cauchy sequence in this case. Also the rest of the proof can be obtained with the similar approach as in Theorem 16.
We now discuss the following consequences of Theorem 21.
If we set in Theorem 21, then fixed point theorem for Suzuki-type generalized -contraction in the setting of -metric-like spaces is obtained.
Corollary 22. Let be a -metric-like space with ; let be a continuous mapping. If there exist and such that for all with , where Then has a unique fixed point.
If we choose , then Berinde-Wardowski type fixed point result in the framework of -metric-like spaces is acquired.
Corollary 23. Let be a -metric-like space with and let be a continuous mapping. If there exists such that for all with , wherewith . Then has a unique fixed point in .
Next, we present an example which substantiates the hypothesis of Theorem 21.