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

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. In this paper, we introduce some new classes of generalized F-contractions and we establish certain fixed point results for such mappings in the setting of b-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.


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 -metriclike 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 (2  ) instead of the condition (2) 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 (3  ) instead of the condition (3).
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 2 Journal of Function Spaces 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, R, N, and R + fl [0, ∞) 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.
The pair (, ) is called a -metric space.The number  ≥ 1 is called the coefficient of (, ).
In the following definition, Alghamdi et al. [6] extended Definition 2 in order to introduce the new notion of -metriclike space.
Definition 3 (see [6]).Let  be a nonempty set and  ≥ 1 be a given real number.A function   :  ×  → [0, ∞) is called a -metric-like if, for all , ,  ∈ , the following conditions are satisfied: The pair (,   ) is called a -metric-like space.The number  ≥ 1 is called the coefficient of (,   ).
Example 4 (see [6]).Let  = R + and the mapping   :  ×  → R + be defined by for all ,  ∈ .Then (,   ) is a -metric-like space with the coefficient  = 2 > 1, 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  = 1.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 selfdistance   (, ) = 0.
Remark 10 (see [6]).Let (,   ) be a -metric-like space with constant  ≥ 1.Then it is clear that satisfies    (, ) = 0, for all  ∈ .So it is considered to be a -metric induced by -metric-like spaces.
Denote the set of all functions satisfying (1)-(3) by I.In [13], Secelean changed the condition (2) by an equivalent but a more simple condition (2  ).
In our subsequent discussion, condition (2  ) is dropped out.Thus we utilize the functions  : R + → R which satisfy (1) and (3  ).The class of all functions satisfying (1) and (3  ) is denoted by Δ  .

Fixed Point Results for 𝜙-𝐹 Contractive Mappings
We introduce the following concept.
For illustrating the above definition, the following example is presented.
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  0 be an arbitrary point in .Set  0 =  1 and define a sequence {  } in  by If there exists  0 ∈ N such that then   0 is the fixed point of , which complete the proof.
Then we have and by ( 6) we obtain Now, we claim that Suppose, on the contrary, that there exists  0 ∈ N, such that Then, by ( 6), one gets In view of the properties of Φ, Ψ, and (1), we obtain that which shows this implies This is a contradiction and hence (18) holds; that is,   (  ,   ) <   ( −1 ,  −1 ).So {  (  ,   )} is a decreasing sequence in R + and is bounded below at 0; consequently it is convergent to some point, say  ∈ R + .Now we assert that  = 0. On the contrary suppose  > 0.
On the similar approach as discussed earlier, we conclude that Letting  → ∞ and utilizing (3  ), we have This is a contradiction, in view of the properties of Ψ and (1) and the fact that  +  +  ≤ 1.
In order to show the uniqueness of fixed point, suppose V is another fixed point such that  ̸ = V.Then we have This is a contradiction, in view of (1) and Ψ.Thus we have  = V.Hence  has a unique fixed point.This completes the proof.
In order to illustrate our result, we present the following example.
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 as By 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  = 0.0477177.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.

Results via Suzuki-Berinde Type 𝐹-Contractions
Berinde initiated some new mappings, called weak contraction mappings in a metric space [20][21][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.
Theorem 21.Let (,   ) be a complete -metric-like space and  :  →  be a continuous Suzuki-Berinde type  contraction.Then  has a unique fixed point in .
Letting  → ∞ in (52), we have which is a contradiction in view of (1) and the properties of .Thus we have  = 0. Consequently, we have lim Now we will show that {  } is a Cauchy sequence.
Case 2. When  = 1, 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  = 0 in Theorem 21, then fixed point theorem for Suzuki-type generalized -contraction in the setting of metric-like spaces is obtained.( Then  has a unique fixed point.with  ≥ 0. Then  has a unique fixed point in .
Next, we present an example which substantiates the hypothesis of Theorem 21.
Case 1.If  ≥  >  ≥ , then values of terms appearing in (45) are evaluated as follows: (, ) =  2 ;   (, ) =  2 ; (, ) =  2 ;   (, ) =  2 ; (, ) =  2 ; 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  = 1, which validates condition (45).
Case 2. When  >  >  > , then the same conclusion will be obtained as in Case 1.
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  = 0, which is indeed unique, as demonstrated by Figure 7.

Applications
then () 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: (      The following example demonstrates the validity of hypothesis of Theorem 25. Example 26.Consider the subsequent integral equation in  = ([0, 1], R).
For obtaining the existence of solution of integral equation (76), we will show that  is a fixed point of , that is,  = , where We notice that the integral equation ( 76) is a particular case of (69), in which () = /( + 1), (, ) = /( + 1), and (, ()) = 1/2(1 + ()).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 , V ∈ R, we get The approximate solution of the integral equation ( 76) is represented geometrically by Figure 8.
Utilizing the obtained approximate solution and (77), one can get Subsequent 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.

Application to Conversion 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 by where  : [0, 1] ×  + →  is a continuous function.

Journal of Function Spaces 13
The above problem is equivalent to the integral equation where (, ) is Green's function, given by Here  > 0 is a constant obtained by the values of  and , mentioned in (82).
Let  = ([0, 1],  + ) be the set of all nonnegative continuous real functions defined on [0, 1].We endow  with the -metric-like Here, we notice that the function  :  + →  defined by () = log(), for each  ∈ ([0, 1],  + ) and for  > 0, is in Δ  .Consequently all the conditions of Corollary 19 are satisfied by operator  with  = /2 with 0 <  ≤ 1. 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:  () =  () + ∫    (, ,  () ,   ()) . (92) It is an open question, whether the existence of solution of aforementioned integrodifferential equation can be established from our results, proved in this article.

Figure 1 :
Figure 1: Right hand side superimposes left hand side in Case 1.

Figure 2 :
Figure 2: Right hand side is dominating left hand side in Case 2.

Figure 3 :
Figure 3: Right hand side function superimposes left hand side function.

Figure 4 :
Figure 4: Fixed point of the mapping .

Figure 6 :
Figure 6: Plot showing domination of right hand side function over left hand side function.

Figure 7 :
Figure 7: Plot showing Fixed point of .

Figure 10 :
Figure 10: Error between solution and integral equation.