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

Authors: Vishal Joshi; Deepak Singh; Adrian Petruşel
Journal: Journal of Function Spaces
 (2017)

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## Abstract

In this paper, we introduce some new classes of generalizedF-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.


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## Body

## 1. Introduction and Preliminaries

There are many extensions and generalizations of the metric space concept. In 1989, Bakhtin [1] introduced the notion of b-metric space, while Czerwik ([2, 3]) extensively used the concept of b-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 b-metric-like spaces, which extends the notions of partial metric spaces, b-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 F-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 F-contraction and proved certain fixed and common fixed point theorems. Recently, Secelean [13] described a large class of functions using the condition (F2′) instead of the condition (F2) in the definition of F-contraction presented by Wardowski [11]. Very recently, Piri and Kumam [14] improved the result of Secelean [13], by using the condition (F3′) instead of the condition (F3).In this paper, we consider the notions ofϕ-F-contraction and Suzuki-Berinde type F-contraction in the context of b-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,R, N, and R+≔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.Definition 1 (see [2]).
LetX be a nonempty set and s≥1 be a given real number. A function d:X×X→0,∞ is called a b-metric if, for all x,y,z∈X, the following conditions are satisfied: (b1)
d(x,y)=0 iff x=y;(b2)
d(x,y)=d(y,x);(b3)
d(x,y)≤s[d(x,z)+d(z,y)].
The pair(X,d) is called a b-metric space. The number s≥1 is called the coefficient of (X,d).Definition 2 (see [5]).
A functionσ:X×X→[0,∞) is called a metric-like if, for all x,y,z∈X, the following conditions are satisfied: (σ1)
σ(x,y)=0 implies x=y;(σ2)
σ(x,y)=σ(y,x);(σ3)
σ(x,y)≤σ(x,z)+σ(z,y).
The pair(X,σ) 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 b-metric-like space.Definition 3 (see [6]).
LetX be a nonempty set and s≥1 be a given real number. A function σb:X×X→[0,∞) is called a b-metric-like if, for all x,y,z∈X, the following conditions are satisfied: (σb1)
σb(x,y)=0 implies x=y;(σb2)
σb(x,y)=σb(y,x);(σb3)
σb(x,y)≤s[σb(x,z)+σb(z,y)].
The pair(X,σb) is called a b-metric-like space. The number s≥1 is called the coefficient of (X,σb).Example 4 (see [6]).
LetX=R+ and the mapping σb:X×X→R+ be defined by(1)σbx,y=max⁡x,y2, for all x,y∈X. Then (X,σb) is a b-metric-like space with the coefficient s=2>1, but it is neither a b-metric nor a metric-like space.Remark 5.
The class ofb-metric-like space (X,σb) is larger than the class of metric-like space, since a metric-like space is a special case of b-metric-like space (X,pb) when s=1. Also, the class of b-metric-like space (X,σb) is effectively larger than the class of b-metric space, since a b-metric space is a special case of a b-metric-like space (X,σb) when the self-distance σb(x,x)=0.Eachb-metric-like σb on X generalizes a topology τσb on X whose base is the family of open σb-balls B-σb(x,ϵ)={y∈X:|σb(x,y)-σb(x,x)|<ϵ} for all x∈X and ϵ>0.Definition 6 (see [6]).
A sequence{xn} in a b-metric-like space (X,σb) is said to be (1)
σb convergent to a point x∈X if σb(x,x)=limn→∞σb(x,xn);(2)
aσb-Cauchy sequence if limn,m→∞σb(xn,xm) exists (and is finite).Definition 7 (see [6]).
Ab-metric-like space (X,σb) is said to be σb-complete if every σb-Cauchy sequence {xn} in X, σb-converges to a point x∈X, such that(2)σbx,x=limn,m→∞⁡σbxn,xm=limn→∞⁡σbx,xn.Definition 8 (see [6]).
Suppose that(X,σb) is a b-metric-like space. A mapping T:X→X is said to be continuous at x∈X, if, for every ϵ>0, there exists δ>0 such that T(Bσb(x,δ))⊂Bσb(Tx,ϵ). We say that T is continuous on X if T is continuous at all x∈X.Lemma 9 (see [6]).
Let{yn} be a sequence in a b-metric-like space (X,σb) such that (3)σbyn,yn+1≤λσbyn-1,yn, for some λ, 0<λ<1/s, and each n∈N. Then {yn} is a Cauchy sequence in X and limn,m→∞σb(yn,ym)=0.Remark 10 (see [6]).
Let(X,σb) be a b-metric-like space with constant s≥1. Then it is clear that (4)σbsx,y=2σbx,y-σbx,x-σby,y satisfies σbs(x,x)=0, for all x∈X. So it is considered to be a b-metric induced by b-metric-like spaces.Remark 11 (see [15]).
Let(X,σb) be a b-metric-like space and let f:X→X be a continuous mapping. Then (5)limn→∞⁡σbxn,x=σbx,x⟹limn→∞⁡σbfx,fxn=σbx,x.Wardowski [11] introduced the F-contraction as follows.Definition 12.
LetF:R+→R be a mapping satisfying (F1)
F is strictly increasing, that is, for α,β∈R+ such that α<β implies F(α)<F(β);(F2)
for each sequence{αn} of positive numbers limn→∞αn=0 if and only if limn→∞F(αn)=-∞;(F3)
there existsk∈(0,1) such that limα→0+αkF(α)=0.
Denote the set of all functions satisfying (F1)–(F3) by I. In [13], Secelean changed the condition (F2) by an equivalent but a more simple condition (F2′). (F2′)
inf⁡F=-∞,
or also by(F2′′)
there exists a sequence{αn}n=1∞ of positive real numbers with limn→∞F(αn)=-∞.
Recently, Piri and Kumam [14] used the following condition (F3′) instead of (F3).(F3′)
F is continuous on (0,∞).
In our subsequent discussion, condition(F2′) is dropped out. Thus we utilize the functions F:R+→R which satisfy (F1) and (F3′). The class of all functions satisfying (F1) and (F3′) is denoted by ΔF.LetΦ be the set of functions ϕ:[0,∞)→[0,∞) such that(1)
ϕ is monotonic increasing, that is, t1≤t2⇒ϕ(t1)≤ϕ(t2);(2)
ϕ is continuous and ϕ(t)<t for each t>0.LetΨ denote the set of all continuous functions ψ:(0,∞)→(0,∞).Remark 13.
For recent interesting fixed point results forα-F-contractions, see [16–19].


## 2. Fixed Point Results forϕ-F Contractive Mappings

We introduce the following concept.Definition 14.
Let(X,σb) be a b-metric-like space. A self-mapping T:X→X is said to be generalized ϕ-F contraction if F∈ΔF and(6)σbTx,Ty>0⟹ψσbx,y+FσbTx,Ty≤Fϕασbx,y+βσbx,Ty2s+γσby,Tx2s,for all x,y∈X, where α,β,γ∈[0,1] (not all zero simultaneously), such that α+β+γ≤1, ϕ∈Φ and ψ∈Ψ.For illustrating the above definition, the following example is presented.Example 15.
LetX=[0,25] and let the function σb:X×X→[0,∞) be defined by(7)σbx,y=max⁡x,y3/2, for all x,y∈X. It is obvious that (X,σb) is a complete b-metric-like space with s=2. Define the mapping T:X→X by (8)Tx=tan⁡120log⁡10+x+x. In order to verify the Condition (6) with α=0.6, β=0.2, and γ=0.2 such that α+β+γ≤1 and F(k)=log⁡k, for all k∈R+, we see that F∈ΔF. Let ϕ:[0,∞)→[0,∞) be given by ϕ(t)=5t/6 and functions ψ:(0,∞)→(0,∞) are defined by ψ(t)=1/t+10.
Here we note that(9)σbTx,Ty>0∀x,y∈X.Without loss of generality, assume that x≥y. Then, the following cases arise.Case 1.
Whenx≥y>Tx, calculating various terms appearing in the inequality (6), we conclude that left hand side of (6) comes out (10)ψσbx,y+FσbTx,Ty=1x3/2+10+log⁡tan120log⁡10+x+x3/2, and right hand side of (6) becomes (11)log⁡56αx3/2+β22x3/2+γ22y3/2. 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.Figure 1
Right hand side superimposes left hand side in Case1.Case 2.
Whenx>Tx≥y, with this assumption, evaluating the terms involved in Condition (6), we obtain the left hand side as (12)ψσbx,y+FσbTx,Ty=1x3/2+10+log⁡tan⁡120log⁡10+x+x3/2, and right hand side of (6) becomes (13)log56αx3/2+β22x3/2+γ22tan⁡120log10+x+x3/2.Figure 2 shows that right hand side expression is superimposing the left hand side expression, which validates our condition in this case.Figure 2
Right hand side is dominating left hand side in Case2.Thus all the hypothesis of Definition14 are fulfilled and therefore T is a ϕ-F contraction mapping.Our main result runs as follows.Theorem 16.
Let(X,σb) be a complete b-metric-like space and T be a continuous generalized ϕ-F contraction. If σb(Tx,Tx)≤σb(x,x), for all x∈X, then T has a unique fixed point in X.Proof.
Letx0 be an arbitrary point in X. Set Tx0=x1 and define a sequence {xn} in X by(14)xn+1=Txn,n∈N.If there exists n0∈N such that(15)σbxn0,xn0+1=0, then xn0 is the fixed point of T, which complete the proof.
Consequently, we supposeσb(xn,xn+1)>0, for all n∈N.
Then we have(16)0<σbxn,xn+1=σbTxn-1,Txn,∀n∈N,and by (6) we obtain(17)ψσbxn-1,xn+FσbTxn-1,Txn≤Fϕασbxn-1,xn+β2sσbxn-1,Txn+γ2sσbxn,Txn-1.Now, we claim that(18)σbxn,Txn<σbxn-1,Txn-1,∀n∈N.Suppose, on the contrary, that there exists n0∈N, such that(19)σbxn0,Txn0≥σbxn0-1,Txn0-1. Then, by (6), one gets (20)ψσbxn0-1,xn0+Fσbxn0,Txn0=ψσbxn0-1,xn0+FσbTxn0-1,Txn0≤Fϕασbxn0-1,xn0+β2sσbxn0-1,Txn0+γ2sσbxn0,Txn0-1≤Fϕασbxn0-1,xn0+β2ssσbxn0-1,xn0+sσbxn0,Txn0+γ2ssσbxn0,xn0-1+sσbxn0-1,Txn0-1=Fϕασbxn0-1,Txn0-1+β2σbxn0-1,Txn0-1+σbxn0,Txn0+γ2σbTxn0-1,xn0-1+σbxn0-1,Txn0-1≤Fϕασbxn0-1,Txn0-1+βσbxn0,Txn0+γσbxn0-1,Txn0-1.In view of the properties of Φ, Ψ, and (F1), we obtain that(21)σxn0,Txn0<ασbxn0-1,Txn0-1+βσbxn0,Txn0+γσbxn0-1,Txn0-1, which shows (22)σbxn0,Txn0<α+γ1-βσbxn0-1,Txn0-1; this implies (23)σbxn0,Txn0<σbxn0-1,Txn0-1,sinceα+γ1-β≤1. This is a contradiction and hence (18) holds; that is, σb(xn,Txn)<σb(xn-1,Txn-1). So {σb(xn,Txn)} is a decreasing sequence in R+ and is bounded below at 0; consequently it is convergent to some point, say p∈R+. Now we assert that p=0. On the contrary suppose p>0.
On the similar approach as discussed earlier, we conclude that(24)ψxn-1,xn+FσbTxn-1,Txn≤Fασbxn-1,Txn-1+βσbxn-1,Txn-1+γσbxn-1,Txn-1.Letting n→∞ and utilizing (F3′), we have(25)ψp+Fp≤Fα+β+γp. This is a contradiction, in view of the properties of Ψ and (F1) and the fact that α+β+γ≤1.
So, we must havep=0; that is,(26)limn→∞σbxn,Txn=0.Now we shall prove that {xn} is a Cauchy sequence. In fact, we will establish that(27)limn,m→∞σbxn,xm=0.On the contrary, suppose that there exists ϵ>0, and two sequences {pn} and {qn} on natural number such that p(n)>q(n)>n,(28)σbxpn,xqn≥ϵ,σbxpn-1,xqn<ϵ,∀n∈N.From the triangle inequality, one can obtain(29)σbxpn-1,xqn-1<sσbxpn-1,xqn+σbxqn,xqn-1<sσbxqn,xqn-1+sϵ=sσbTxqn-1,xqn-1+sϵ,∀n∈N.From (26), there exists N1∈N, such that(30)σbxpn-1,Txpn-1<ϵ,σbxqn-1,Txqn-1<ϵ,∀n≥N1,which, together with (29), shows(31)σbxpn-1,xqn-1<2ϵs,∀n≥N1;therefore(32)Fσbxpn-1,xqn-1<F2sϵ,∀n≥N1.From (28), we get(33)ϵ≤σbxpn,xqn=σbTxpn-1,Txqn-1. Employing (6), one acquires (34)ψσbxpn-1,xqn-1+FσbTxpn-1,Txqn-1≤Fϕασbxpn-1,xqn-1+β2sσbxpn-1,Txqn-1+γ2sσbxqn-1,Txpn-1≤Fϕασbxpn-1,xqn-1+β2σbxpn-1,xqn-1+σbxqn-1,Txqn-1+γ2σxqn-1,xpn-1+σbxpn-1,Txpn-1,∀n≥N1.In view of (30), (31), and (32), we get (35)ψ2sϵ+FσbTxpn-1,Txqn-1<Fα2sϵ+β22sϵ+ϵ+γ22sϵ+ϵ⟹σbTxpn-1,Txqn-1<α2sϵ+β22sϵ+ϵ+γ22sϵ+ϵ.This amounts to say that limn→∞σb(Txp(n)-1,Txq(n)-1)=0. Hence limn→∞σb(xp(n),xq(n))=0, which is a contradiction with (28). This validates (27). Therefore {xn} is a Cauchy sequence in X. Since (X,σb) is complete, there exists w∈X, such that(36)σbw,w=limn→∞σbxn,w=limn,m→∞σbxn,xm=0. Since T is continuous, we get(37)σbTw,Tw=limn→∞σbTxn,Tw=limn→∞σbxn+1,Tw=0.Due to the fact that σ(Tw,Tw)≤σ(w,w), from above, we have limn→∞σb(xn,Tw)=0. Since σb(w,Tw)≤sσb(w,xn)+σb(xn,Tw), letting n→∞, we obtain that σb(w,Tw)=0. Thus w=Tw and so T has a fixed point.
In order to show the uniqueness of fixed point, supposev is another fixed point such that w≠v. Then we have (38)σbTw,Tu>0⟹Fσbw,v=FσbTw,Tv≤Fϕασbw,v+βσbw,Tv2s+γσbv,Tw2s-ψσbw,v≤Fα+β2s+γ2sσbw,v-ψσbw,v.This is a contradiction, in view of (F1) and Ψ. Thus we have w=v. Hence T has a unique fixed point. This completes the proof.In order to illustrate our result, we present the following example.Example 17.
LetX=[0.0477177,∞) and let the function σb:X×X→[0,∞) be defined by σb(x,y)=x2+y2+|x-y|2, for all x,y∈X. It is obvious that (X,σb) is a complete b-metric-like space with s=2. Let the mapping T:X→X be defined by (39)Tx=ex2/1+x20+ex2/1+x. We verify the condition (6) with α=0.8, β=0.1, and γ=0.1 (clearly α+β+γ≤1) and F(p)=log⁡p+p, for all p∈R+. Notice that F∈ΔF and σb(Tx,Ty)>0, for all x,y∈X. Consider ϕ,ψ:[0,∞)→[0,∞) given by ϕ(t)=t/2 and ψ(t)=1/(t2+10).Various terms involved in the inequality (6) are calculated as follows: (40)σbTx,Ty=ex2/1+x20+ex2/1+x2+ey2/1+y20+ey2/1+y2+ex2/1+x20+ex2/1+x-ey2/1+y20+ey2/1+y2;σbx,y=x2+y2+x-y2;σbx,Ty=x2+ey2/1+y20+ey2/1+y2+x-ey2/1+y20+ey2/1+y2;σby,Tx=y2+ex2/1+x20+ex2/1+x2+y-ex2/1+x20+ex2/1+x2.Utilizing aforementioned values, the left hand side of (6) becomes (41)1x2+y2+x-y22+10+log⁡ex2/1+x20+ex2/1+x2+ey2/1+y20+ey2/1+y2+ex2/1+x20+ex2/1+x-ey2/1+y20+ey2/1+y2+ex2/1+x20+ex2/1+x2+ey2/1+y20+ey2/1+y2+ex2/1+x20+ex2/1+x-ey2/1+y20+ey2/1+y2,and the right hand side is obtained as(42)log⁡12αx2+y2+x-y2+β4x2+ey2/1+y20+ey2/1+y2+x-ey2/1+y20+ey2/1+y2+γ4y2+ex2/1+x20+ex2/1+x2+y-ex2/1+x20+ex2/1+x2+12αx2+y2+x-y2+β4x2+ey2/1+y20+ey2/1+y2+x-ey2/1+y20+ey2/1+y2+γ4y2+ex2/1+x20+ex2/1+x2+y-ex2/1+x20+ex2/1+x2.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.Figure 3
Right hand side function superimposes left hand side function.Furthermore,T is continuous and we also have that σb(Tx,Tx)≤σb(x,x) for all x∈X.Thus, all the conditions of Theorem16 are satisfied and, consequently, the mapping T has a unique fixed point as x=0.0477177. This is also demonstrated by Figure 4, where the mapping T and the first diagonal are represented.Figure 4
Fixed point of the mappingT.If we chooseψ(t)=τ in Theorem 16, then the following corollary is obtained.Corollary 18.
Let(X,σb) be a b-metric-like space and T:X→X be a continuous self-mapping. If there exist τ>0 and F∈ΔF such that for all x,y∈X,(43)σbTx,Ty>0⟹τ+FσbTx,Ty≤Fϕασbx,y+βσbx,Ty2s+γσby,Tx2s,where α,β,γ∈[0,1] (not all zero simultaneously) such that α+β+γ≤1 and ϕ∈Φ. Then T has a unique fixed point.Replacingβ=γ=0 in Corollary 18, subsequent result is obtained.Corollary 19.
Let(X,σb) be a b-metric-like space and T:X→X be a continuous self-mapping. If there exist τ>0 and F∈ΔF such that for all x,y∈X,(44)σbTx,Ty>0⟹τ+FσbTx,Ty≤Fϕασbx,y,where α∈(0,1] and ϕ∈Φ. Then T has a unique fixed point.


## 3. Results via Suzuki-Berinde TypeF-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 F-contraction, Suzuki-Berinde type F-contractive mapping is defined in the framework of b-metric-like spaces.Definition 20.
Let(X,σb) be a b-metric-like space with s≥1. A self-mapping T:X→X is said to be Suzuki-Berinde type F-contraction, if there exists F∈ΔF such that, for all x,y∈X with x≠y,(45)12sσbx,Tx<σbx,y⟹Fs2σbTx,Ty≤F1sMx,y-ψMx,y+LNx,y,where(46)Mx,y=max⁡σbx,y,σbx,Tx,σby,Ty,σbx,Ty+σby,Tx4s,Nx,y=minσbsx,Ty,σbsy,Tx,σbx,Tx,σby,Tywith L≥0 and ψ∈Ψ.Theorem 21.
Let(X,σb) be a complete b-metric-like space and T:X→X be a continuous Suzuki-Berinde type F contraction. Then T has a unique fixed point in X.Proof.
Letx0∈X be any arbitrary point. We construct a sequence {xn} in X in such a way that xn=Txn-1 for all n∈N.
Suppose thatσb(xn0,xn0+1)=σ(xn0,Txn0)=0, for some n0≥0. Then one can get xn0=xn0+1=Txn0 and then xn0 is a required fixed point. So, we are done in this case. Thus, from now on we assume that σ(xn,Txn)>0, for all n∈N. Consequently, we have (47)12sσbxn,Txn<σbxn,Txn=σbxn,xn+1,∀n∈N. Then by the Definition 20 with x=xn and y=xn+1, we have(48)Fs2σbTxn,Txn+1≤F1sMxn,xn+1-ψMxn,xn+1+LNxn,xn+1.Notice that(49)σbxn,xn+2+σbxn+1,xn+14s≤sσbxn,xn+1+3σbxn+1,xn+24s≤max⁡σbxn,xn+1,σbxn+1,xn+2.Thus, we have (50)Mxn,xn+1=maxσbxn,xn+1,σbxn+1,xn+2. If M(xn,xn+1)=σb(xn+1,xn+2) then, from (48), we have(51)Fs2σbxn+1,xn+2≤F1sσbxn+1,xn+2-ψσbxn+1,xn+2.This leads to a contradiction, in view of (F1) and the hypothesis of ψ. Then we arrive at(52)Fs2σbxn+1,xn+2≤F1sσbxn,xn+1-ψσbxn,xn+1≤F1sσbxn,xn+1.Thus, from (52) and (F1), we get that(53)s2σbxn+1,xn+2<1sσbxn,xn+1,or equivalently(54)σbxn,xn+1<σbxn-1,xn,∀n∈N, as s≥1. Therefore {σb(xn,xn+1)}n∈N is a nonnegative decreasing sequence of real numbers and is bounded below at 0, consequently convergent to some point p∈R+; now we claim that p=0. Let us suppose that p>0.
Lettingn→∞ in (52), we have(55)Fs2p≤F1sp-ψp, which is a contradiction in view of (F1) and the properties of ψ. Thus we have p=0.
Consequently, we have(56)limn→∞σbxn,Txn=limn→∞σbxn,xn+1=0. Now we will show that {xn} is a Cauchy sequence.
Case 1. When s>1, rewrite (53) as (57)σbxn,xn+1<1s3σbxn-1,xn. Since 1/s3>0 and 1/s3<1/s, as s>1, then, by Lemma 9, the sequence {xn} is a Cauchy sequence.
Case 2. When s=1, following the same approach as in Theorem 16 and utilizing the condition (45), it is easy to show that {xn} 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 Theorem21.If we setL=0 in Theorem 21, then fixed point theorem for Suzuki-type generalized F-contraction in the setting of b-metric-like spaces is obtained.Corollary 22.
Let(X,σb) be a b-metric-like space with s≥1; let T:X→X be a continuous mapping. If there exist F∈ΔF and ψ∈Ψ such that for all x,y∈X with x≠y, (58)12sσbx,Tx<σbx,y⟹Fs2σbTx,Ty≤F1sMx,y-ψMx,y,where(59)Mx,y=max⁡σbx,y,σbx,Tx,σby,Ty,σbx,Ty+σby,Tx4s. Then T has a unique fixed point.If we chooseψ(t)=τ>0, then Berinde-Wardowski type fixed point result in the framework of b-metric-like spaces is acquired.Corollary 23.
Let(X,σb) be a b-metric-like space with s≥1 and let T:X→X be a continuous mapping. If there exists F∈ΔF such that for all x,y∈X with x≠y, (60)σbTx,Ty>0⟹τ+Fs2σbTx,Ty≤F1sMx,y+LNx,y,where(61)Mx,y=maxσbx,y,σbx,Tx,σby,Ty,σbx,Ty+σby,Tx4s,Nx,y=minσbsx,Ty,σbsy,Tx,σbx,Tx,σby,Tywith L≥0. Then T has a unique fixed point in X.Next, we present an example which substantiates the hypothesis of Theorem21.Example 24.
LetX=[0,2] and let the function σb:X×X→[0,∞) be defined by σb(x,y)=max{x,y}2, for all x,y∈X. It is obvious that (X,σb) is a complete b-metric-like space with s=2. Define the mapping T:X→X by Tx=(1/20)(log(1+x3)+x3).In order to show thatT is a Suzuki-Berinde type F-contraction mapping, we first verify the condition (45) with F(t)=log⁡t, for all t∈R+. Clearly F∈ΔF. Let ψ:(0,∞)→(0,∞) be given by ψ(t)=t/100.Consider(62)12sσbx,Tx=x24≤x2≤σbx,y;∀x,y∈X. Next, we discuss subsequent possible cases for x,y∈X.Case 1.
Ifx≥y>Tx≥Ty, then values of terms appearing in (45) are evaluated as follows: (63)σbTx,Ty=2400log⁡1+x3+x32;σbx,y=x2;σbx,Tx=x2;σby,Ty=y2;σbx,Ty=x2;σby,Tx=y2;σbsx,Ty=x2-Ty2;σbsy,Tx=y2-Tx2.Employing aforementioned values to the left hand side of (45), we get (64)Fs2σbTx,Ty=log4400log1+x3+x32, and right hand side of (45) is obtained as (65)F1sMx,y-ψMx,y+LNx,y=log⁡x22-x2100+Ly2-1400log⁡1+x3+x32.It is very easy to verify that (66)Fs2σbTx,Ty≤F1sMx,y-ψMx,y+LNx,y,forL≥0,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 L=1, which validates condition (45).Figure 5
Plot of inequality for verification of Condition (45).Case 2.
Whenx>Tx>y>Ty, then the same conclusion will be obtained as in Case 1.Case 3.
Ify>x>Ty>Tx, then after calculating the terms involved in (45), the left hand side comes out (67)Fs2σbTx,Ty=log4400log1+y3+y32 and right hand side of (45) is obtained as (68)F1sMx,y-ψMx,y+LNx,y=logy22-y2100+Lx2-1400log1+y3+y32. By Figure 6, it is clear that Condition (45) is satisfied for all x,y∈X with y>x>Ty>Tx.Figure 6
Plot showing domination of right hand side function over left hand side function.Same result will be obtained wheny>Ty>x>Tx.Moreover, the mappingT is continuous. Then all the conditions of Theorem 21 are satisfied and hence T has a fixed point x=0, which is indeed unique, as demonstrated by Figure 7.Figure 7
Plot showing Fixed point ofTx.


## 4. Applications

### 4.1. An Application to Integral Equations

In this section, we obtain the solution of the subsequent integral equation for an unknown functionu:(69)ut=pt+∫abGt,zfz,uzdz,t∈a,b,where u:R+→R is a nondecreasing function, f:[a,b]×R+→R is a nonincreasing continuous function, and G:[a,b]×[a,b]→[0,∞) is an nondecreasing continuous function. Let p:[a,b]→R be given continuous function.LetX be the set C[a,b] of real continuous functions on [a,b] and let σb:X×X→[0,∞) be given by(70)σbu,v=maxa≤t≤but-vt2.One can easily see that (X,σb) is a complete b-metric-like space (in view of Remark 5, since it a complete b-metric space). Let the mapping T:X→X be defined by(71)Tut=pt+∫abGt,zfz,uzdz,t∈a,b;then u(t) is a solution of (69) if and only if it is a fixed point of T. Now, we prove the following theorem to show the existence of solution of integral equation.Theorem 25.
Assume that the following assumptions hold:(1)
(72)maxa≤t≤b∫abGt,z2dz≤1b-a;(2)
For allx,y∈R+, the following inequality holds:(73)fz,x-fz,y2≤12x-y2e-τ.Then the integral equation (69) has a solution.Proof.
By the conditions(1)-(2) and taking into account the integral equation (69), we have (74)σbTu1,Tu2=maxa≤t≤bTu1t-Tu2t2=maxa≤t≤bpt+∫abGt,zfz,u1zdz-pt+∫abGt,zfz,u2zdz2=maxa≤t≤b∫abGt,zfz,u1z-Gt,zfz,u2zdz2≤maxa≤t≤b∫abGt,z2dz·∫abfz,u1z-fz,u2z2dz=maxa≤t≤b∫abGt,z2dz·∫abfz,u1z-fz,u2z2dz≤1b-a·∫ab12u1z-u2z2e-τdz≤12b-a∫abmaxa≤t≤bu1t-u2t2e-τdz=maxa≤t≤b12u1t-u2t2e-τ=12σbu1,u2e-τ;that is,(75)σbTu1,Tu2≤σbu1,u2e-τ2.Consequently, by passing to logarithms, we get τ+log⁡(σb(Tu1,Tu2))≤log⁡(σb(u1,u2)/2), and this turns into τ+F(σb(Tu1,Tu2))≤F(ϕ(σb(u1,u2))), for F(t)=log⁡t, t>0, α=1, and ϕ(t)=1/2. Thus, all the conditions of Corollary 19 are satisfied. Hence, we conclude that T has a unique fixed point u∗ in X which yields the fact that integral equation (69) has a unique solution which belongs to X=C[a,b].The following example demonstrates the validity of hypothesis of Theorem25.Example 26.
Consider the subsequent integral equation inX=C([0,1],R).(76)ut=tt+1+12∫01st+111+usds;t∈0,1.
For obtaining the existence of solution of integral equation (76), we will show that u is a fixed point of T, that is, Tu=u, where(77)Tut=tt+1+12∫01st+111+usds;t∈0,1.We notice that the integral equation (76) is a particular case of (69), in which p(t)=t/(t+1), G(t,s)=s/(t+1), and f(s,u(s))=1/21+us.
Indeed, the functionsp, G, and f are continuous. Moreover, the function G is nondecreasing with respect to s; the active variable under integral and function f is nonincreasing for u(s) which is considered to be a nondecreasing function. Thus the assumptions with respect to functions are satisfied. Further, for all u,v∈R, we get (78)0≤fs,u-fs,v2≤121+u-121+v2≤14v-u2≤12v-u2e-0.2≤12v-u2e-τ,for τ=0.2. Therefore, condition (2) of Theorem 25 is fulfilled. For condition (1), we have (79)maxa≤t≤b∫01Gt,s2ds=max0≤t≤1∫01s2t+12ds=max0≤t≤113t+12≤1. Thus, condition (1) is fulfilled.
Subsequently, we conclude that all the conditions of Theorem25 are satisfied. Hence, the integral equation (76) has a solution in X=C([0,1],R). Moreover, the approximate solution of the integral equation (76) is(80)ut=2t+0.33562t+1.The approximate solution of the integral equation (76) is represented geometrically by Figure 8.
Utilizing the obtained approximate solution and (77), one can get(81)Tut=tt+1+12t+1∫012ss+14s+2.3356ds;t∈0,1.
Subsequent is the plot ofTu(t), 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 Tu(t) 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 u(t) is given by Figure 10.Figure 8
Geometrical representation of approximate solution of (76).Figure 9
Plot ofTu(t) represented by (81).Figure 10
Error between solution and integral equation.


### 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 by(82)d2Idt2+RLdIdt=Kt,It;I0=0,I′0=0,where K:[0,1]×R+→R is a continuous function.The above problem is equivalent to the integral equation(83)ut=∫0tGt,sKs,Isds,t∈0,1,where G(t,s) is Green’s function, given by(84)Gt,s=t-seτt-s,0≤s≤t≤1;0,0≤t≤s≤1. Here τ>0 is a constant obtained by the values of R and L, mentioned in (82).LetX=C([0,1],R+) be the set of all nonnegative continuous real functions defined on [0,1]. We endow X with the b-metric-like(85)σbI1,I2=supt∈0,1I1t+I2t2e-2τt,∀I1,I2∈Xwithτ>0.Then clearly (X,σb) is a complete b-metric-like space with s=2.Consider the self-mapA:X×X, defined by(86)AIt=∫0tGt,sKs,Isds,t∈0,1. It is obvious that I∗ is a solution of (83) if and only if I∗ is a fixed point of A.The existence of a fixed point ofA is obtained in our next theorem.Theorem 27.
Consider the Cauchy problem (82). Suppose there exists τ>0 such that Ks,I1+Ks,I2≤e-τ/2(αI1+I2/2), for all s∈0,1, I1,I2∈R+, where 0<α≤1.
Then the integral equation (83) has a solution.Proof.
Already note that(C([0,1],R+),σb) is a complete b-metric-like space, where σb(I1,I2) is given by (85).
Next, for allI1,I2∈X such that FI1(t)≠FI2(t), we have (87)AI1t+AI2t2≤∫0tGt,sKs,I1sds+∫0tGt,sKs,I2sds2≤∫0tGt,sKs,I1s+Ks,I2sds2≤∫0tGt,se-τ/2αI1s+I2s2ds2=∫0tGt,se-τ/2αI1s+I2s2e-τseτsds2≤e-τα2σbI1,I2∫0teτst-seτt-sds2=e-τα2σbI1,I2e2τt∫0tt-sds2=e-τα2σbI1,I2e2τtt44≤e-τα2e2τtσbI1,I2.Since t4/4≤1 as t∈[0,1], this implies that (88)AI1t+AI2t2e-2τt≤e-τα2σbI1,I2.This amounts to say that(89)σbAI1,AI2≤e-τα2σbI1,I2.
Consequently, by passing to logarithms, one can obtain(90)log⁡σbAI1,AI2≤log⁡e-τα2σbI1,I2. Or (91)τ+log⁡σbAI1,AI2≤log⁡ϕασbI1,I2. Here, we notice that the function F:R+→R defined by F(w)=log(w), for each w∈C([0,1],R+) and for τ>0, is in ΔF. Consequently all the conditions of Corollary 19 are satisfied by operator A with ϕ=t/2 with 0<α≤1. Consequently mapping A 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 Theorem16 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=pt+∫atft,s,Is,I′sds. It is an open question, whether the existence of solution of aforementioned integrodifferential equation can be established from our results, proved in this article.


## 4.1. An Application to Integral Equations

In this section, we obtain the solution of the subsequent integral equation for an unknown functionu:(69)ut=pt+∫abGt,zfz,uzdz,t∈a,b,where u:R+→R is a nondecreasing function, f:[a,b]×R+→R is a nonincreasing continuous function, and G:[a,b]×[a,b]→[0,∞) is an nondecreasing continuous function. Let p:[a,b]→R be given continuous function.LetX be the set C[a,b] of real continuous functions on [a,b] and let σb:X×X→[0,∞) be given by(70)σbu,v=maxa≤t≤but-vt2.One can easily see that (X,σb) is a complete b-metric-like space (in view of Remark 5, since it a complete b-metric space). Let the mapping T:X→X be defined by(71)Tut=pt+∫abGt,zfz,uzdz,t∈a,b;then u(t) is a solution of (69) if and only if it is a fixed point of T. Now, we prove the following theorem to show the existence of solution of integral equation.Theorem 25.
Assume that the following assumptions hold:(1)
(72)maxa≤t≤b∫abGt,z2dz≤1b-a;(2)
For allx,y∈R+, the following inequality holds:(73)fz,x-fz,y2≤12x-y2e-τ.Then the integral equation (69) has a solution.Proof.
By the conditions(1)-(2) and taking into account the integral equation (69), we have (74)σbTu1,Tu2=maxa≤t≤bTu1t-Tu2t2=maxa≤t≤bpt+∫abGt,zfz,u1zdz-pt+∫abGt,zfz,u2zdz2=maxa≤t≤b∫abGt,zfz,u1z-Gt,zfz,u2zdz2≤maxa≤t≤b∫abGt,z2dz·∫abfz,u1z-fz,u2z2dz=maxa≤t≤b∫abGt,z2dz·∫abfz,u1z-fz,u2z2dz≤1b-a·∫ab12u1z-u2z2e-τdz≤12b-a∫abmaxa≤t≤bu1t-u2t2e-τdz=maxa≤t≤b12u1t-u2t2e-τ=12σbu1,u2e-τ;that is,(75)σbTu1,Tu2≤σbu1,u2e-τ2.Consequently, by passing to logarithms, we get τ+log⁡(σb(Tu1,Tu2))≤log⁡(σb(u1,u2)/2), and this turns into τ+F(σb(Tu1,Tu2))≤F(ϕ(σb(u1,u2))), for F(t)=log⁡t, t>0, α=1, and ϕ(t)=1/2. Thus, all the conditions of Corollary 19 are satisfied. Hence, we conclude that T has a unique fixed point u∗ in X which yields the fact that integral equation (69) has a unique solution which belongs to X=C[a,b].The following example demonstrates the validity of hypothesis of Theorem25.Example 26.
Consider the subsequent integral equation inX=C([0,1],R).(76)ut=tt+1+12∫01st+111+usds;t∈0,1.
For obtaining the existence of solution of integral equation (76), we will show that u is a fixed point of T, that is, Tu=u, where(77)Tut=tt+1+12∫01st+111+usds;t∈0,1.We notice that the integral equation (76) is a particular case of (69), in which p(t)=t/(t+1), G(t,s)=s/(t+1), and f(s,u(s))=1/21+us.
Indeed, the functionsp, G, and f are continuous. Moreover, the function G is nondecreasing with respect to s; the active variable under integral and function f is nonincreasing for u(s) which is considered to be a nondecreasing function. Thus the assumptions with respect to functions are satisfied. Further, for all u,v∈R, we get (78)0≤fs,u-fs,v2≤121+u-121+v2≤14v-u2≤12v-u2e-0.2≤12v-u2e-τ,for τ=0.2. Therefore, condition (2) of Theorem 25 is fulfilled. For condition (1), we have (79)maxa≤t≤b∫01Gt,s2ds=max0≤t≤1∫01s2t+12ds=max0≤t≤113t+12≤1. Thus, condition (1) is fulfilled.
Subsequently, we conclude that all the conditions of Theorem25 are satisfied. Hence, the integral equation (76) has a solution in X=C([0,1],R). Moreover, the approximate solution of the integral equation (76) is(80)ut=2t+0.33562t+1.The approximate solution of the integral equation (76) is represented geometrically by Figure 8.
Utilizing the obtained approximate solution and (77), one can get(81)Tut=tt+1+12t+1∫012ss+14s+2.3356ds;t∈0,1.
Subsequent is the plot ofTu(t), 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 Tu(t) 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 u(t) is given by Figure 10.Figure 8
Geometrical representation of approximate solution of (76).Figure 9
Plot ofTu(t) represented by (81).Figure 10
Error between solution and integral equation.


## 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 by(82)d2Idt2+RLdIdt=Kt,It;I0=0,I′0=0,where K:[0,1]×R+→R is a continuous function.The above problem is equivalent to the integral equation(83)ut=∫0tGt,sKs,Isds,t∈0,1,where G(t,s) is Green’s function, given by(84)Gt,s=t-seτt-s,0≤s≤t≤1;0,0≤t≤s≤1. Here τ>0 is a constant obtained by the values of R and L, mentioned in (82).LetX=C([0,1],R+) be the set of all nonnegative continuous real functions defined on [0,1]. We endow X with the b-metric-like(85)σbI1,I2=supt∈0,1I1t+I2t2e-2τt,∀I1,I2∈Xwithτ>0.Then clearly (X,σb) is a complete b-metric-like space with s=2.Consider the self-mapA:X×X, defined by(86)AIt=∫0tGt,sKs,Isds,t∈0,1. It is obvious that I∗ is a solution of (83) if and only if I∗ is a fixed point of A.The existence of a fixed point ofA is obtained in our next theorem.Theorem 27.
Consider the Cauchy problem (82). Suppose there exists τ>0 such that Ks,I1+Ks,I2≤e-τ/2(αI1+I2/2), for all s∈0,1, I1,I2∈R+, where 0<α≤1.
Then the integral equation (83) has a solution.Proof.
Already note that(C([0,1],R+),σb) is a complete b-metric-like space, where σb(I1,I2) is given by (85).
Next, for allI1,I2∈X such that FI1(t)≠FI2(t), we have (87)AI1t+AI2t2≤∫0tGt,sKs,I1sds+∫0tGt,sKs,I2sds2≤∫0tGt,sKs,I1s+Ks,I2sds2≤∫0tGt,se-τ/2αI1s+I2s2ds2=∫0tGt,se-τ/2αI1s+I2s2e-τseτsds2≤e-τα2σbI1,I2∫0teτst-seτt-sds2=e-τα2σbI1,I2e2τt∫0tt-sds2=e-τα2σbI1,I2e2τtt44≤e-τα2e2τtσbI1,I2.Since t4/4≤1 as t∈[0,1], this implies that (88)AI1t+AI2t2e-2τt≤e-τα2σbI1,I2.This amounts to say that(89)σbAI1,AI2≤e-τα2σbI1,I2.
Consequently, by passing to logarithms, one can obtain(90)log⁡σbAI1,AI2≤log⁡e-τα2σbI1,I2. Or (91)τ+log⁡σbAI1,AI2≤log⁡ϕασbI1,I2. Here, we notice that the function F:R+→R defined by F(w)=log(w), for each w∈C([0,1],R+) and for τ>0, is in ΔF. Consequently all the conditions of Corollary 19 are satisfied by operator A with ϕ=t/2 with 0<α≤1. Consequently mapping A 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 Theorem16 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=pt+∫atft,s,Is,I′sds. It is an open question, whether the existence of solution of aforementioned integrodifferential equation can be established from our results, proved in this article.



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*Source: 1649864-2017-11-19.xml*