Journal of Function Spaces

Volume 2017 (2017), Article ID 1649864, 14 pages

https://doi.org/10.1155/2017/1649864

## Existence Results for Integral Equations and Boundary Value Problems via Fixed Point Theorems for Generalized -Contractions in -Metric-Like Spaces

^{1}Department of Mathematics, Jabalpur Engineering College, Jabalpur, India^{2}Department of Applied Sciences, NITTTR, Ministry of HRD, Govt. of India, Bhopal 462002, India^{3}Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

Correspondence should be addressed to Adrian Petruşel

Received 25 June 2017; Accepted 19 September 2017; Published 19 November 2017

Academic Editor: Ahmad S. Al-Rawashdeh

Copyright © 2017 Vishal Joshi 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.

#### Abstract

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 [1] 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 [4].

More recently, Amini-Harandi [5] generalized the concept of partial metric space by introducing the metric-like spaces. After that, in [6], 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 [11] 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. [12] generalized the idea of -contraction and proved certain fixed and common fixed point theorems. Recently, Secelean [13] described a large class of functions using the condition instead of the condition in the definition of -contraction presented by Wardowski [11]. Very recently, Piri and Kumam [14] improved the result of Secelean [13], 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 [2]). *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 [5]). *A function is called a metric-like if, for all , the following conditions are satisfied: (*σ*1) implies ;(*σ*2);(*σ*3). The pair is called a metric-like space.

In the following definition, Alghamdi et al. [6] extended Definition 2 in order to introduce the new notion of -metric-like space.

*Definition 3 (see [6]). *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 [6]). *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 [6]). *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 [6]). *A -metric-like space is said to be -complete if every -Cauchy sequence in , -converges to a point , such that

*Definition 8 (see [6]). *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 [6]). *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 [6]). *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 [15]). *Let be a -metric-like space and let be a continuous mapping. Then

Wardowski [11] 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 [13], 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 [14] 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 .

*Remark 13. *For recent interesting fixed point results for --contractions, see [16–19].

#### 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.