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: (σ1) implies ;(σ2);(σ3). 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.
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.
Example 24. 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 show that is a Suzuki-Berinde type -contraction mapping, we first verify the condition (45) with , for all . Clearly . Let be given by .
Consider Next, we discuss subsequent possible cases for .
Case 1. If , then values of terms appearing in (45) are evaluated as follows: Employing aforementioned values to the left hand side of (45), we get and right hand side of (45) is obtained as It is very easy to verify that which is pictorially justified by Figure 5, in which we see that the surface showing right hand side expression is dominating the surface representing left hand side expression for , which validates condition (45).
Case 2. When , then the same conclusion will be obtained as in Case .
Case 3. If , then after calculating the terms involved in (45), the left hand side comes out and right hand side of (45) is obtained as By Figure 6, it is clear that Condition (45) is satisfied for all with .
Same result will be obtained when .
Moreover, the mapping is continuous. Then all the conditions of Theorem 21 are satisfied and hence has a fixed point , which is indeed unique, as demonstrated by Figure 7.
4. Applications
4.1. An Application to Integral Equations
In this section, we obtain the solution of the subsequent integral equation for an unknown function :where is a nondecreasing function, is a nonincreasing continuous function, and is an nondecreasing continuous function. Let be given continuous function.
Let be the set of real continuous functions on and let be given byOne can easily see that is a complete -metric-like space (in view of Remark 5, since it a complete -metric space). Let the mapping be defined bythen is a solution of (69) if and only if it is a fixed point of . Now, we prove the following theorem to show the existence of solution of integral equation.
Theorem 25. Assume that the following assumptions hold:(1) (2)For all , the following inequality holds:Then the integral equation (69) has a solution.
Proof. By the conditions - and taking into account the integral equation (69), we have that is,Consequently, by passing to logarithms, we get , and this turns into , for , , , and . Thus, all the conditions of Corollary 19 are satisfied. Hence, we conclude that has a unique fixed point in which yields the fact that integral equation (69) has a unique solution which belongs to .
The following example demonstrates the validity of hypothesis of Theorem 25.
Example 26. Consider the subsequent integral equation in .For obtaining the existence of solution of integral equation (76), we will show that is a fixed point of , that is, , whereWe notice that the integral equation (76) is a particular case of (69), in which , , and .
Indeed, the functions , , and are continuous. Moreover, the function is nondecreasing with respect to ; the active variable under integral and function is nonincreasing for which is considered to be a nondecreasing function. Thus the assumptions with respect to functions are satisfied. Further, for all , we get for . Therefore, condition (2) of Theorem 25 is fulfilled. For condition (1), we have Thus, condition (1) is fulfilled.
Subsequently, we conclude that all the conditions of Theorem 25 are satisfied. Hence, the integral equation (76) has a solution in . Moreover, the approximate solution of the integral equation (76) isThe approximate solution of the integral equation (76) is represented geometrically by Figure 8.
Utilizing the obtained approximate solution and (77), one can getSubsequent is the plot of , mentioned in (81). By Figures 8 and 9 one can easily deduce that the plot of approximate solution with green surface almost coincides with the plot of with purple surface. This shows that approximate solution mentioned in (80) is a fixed point of (76) and hence is a solution of the integral equation (76). Also the error between the approximate solution and the value of is given by Figure 10.
4.2. Application to Conversion of Solar Energy to Electrical Energy
Solar panels currently are being produced and marketed in mass to counteract the dependency humans have on the less forgiving fossil fuels. In 2007, 18.8 trillion kilowatt hours of electricity were produced globally [23]. In comparison, the sunlight received on the Earth’s surface in one hour is enough to power the entire world for a year [24]. The question is, how do those radiant warm rays of light become electricity? With a basic understanding of how light is transformed into electricity, a mathematical model can be presented of the electric current in an RLC parallel circuit [25], also known as a “tuning” circuit. Such problems mathematically modeled as a Cauchy problem attached to differential equation are represented bywhere is a continuous function.
The above problem is equivalent to the integral equationwhere is Green’s function, given by Here is a constant obtained by the values of and , mentioned in (82).
Let be the set of all nonnegative continuous real functions defined on . We endow with the -metric-likeThen clearly is a complete -metric-like space with .
Consider the self-map , defined by It is obvious that is a solution of (83) if and only if is a fixed point of .
The existence of a fixed point of is obtained in our next theorem.
Theorem 27. Consider the Cauchy problem (82). Suppose there exists such that , for all , , where .
Then the integral equation (83) has a solution.
Proof. Already note that is a complete -metric-like space, where is given by (85).
Next, for all such that , we have Since as , this implies that This amounts to say thatConsequently, by passing to logarithms, one can obtain Or Here, we notice that the function defined by , for each and for , is in . Consequently all the conditions of Corollary 19 are satisfied by operator with with . Consequently mapping has a fixed point which is the solution of integral equation (83) and hence the equation which represented the conversion of solar energy to electric energy has a solution.
Remark 28. We also notice that Theorem 16 can be utilized to study the existence of the solution of following real time problems:(i)Solution of equation generated by the motion of pendulum;(ii)Problems related to simple harmonic motion;(iii)Solution of equations of vibrations.
Open Problem. For further applications of the presented results, an open problem is suggested as follows.
In electrical and electronics circuits analysis, the following integrodifferential equation appears: It is an open question, whether the existence of solution of aforementioned integrodifferential equation can be established from our results, proved in this article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.