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# On Some Novel Fixed Point Results for Generalized -Contractions in -Metric-Like Spaces with Application

Kastriot Zoto1, Ilir Vardhami2, Dušan Bajović3, Zoran D. Mitrović3,*, Stojan Radenović4
1 Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra, 6001, Albania
2 Department of Mathematics, Faculty of Natural Sciences, University of Tirana, Tirana, 1001, Albania
3 Faculty of Electrical Engineering, University of Banja Luka, Banja Luka, 78000, Bosnia and Herzegovina
4 Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Beograd, 11120, Serbia
* Corresponding Author: Zoran D. Mitrović. Email:
(This article belongs to this Special Issue: Computational Aspects of Nonlinear Operator and Fixed Point Theory with Applications)

Computer Modeling in Engineering & Sciences 2023, 135(1), 673-686. https://doi.org/10.32604/cmes.2022.022878

Received 30 March 2022; Accepted 17 May 2022; Issue published 29 September 2022

### Abstract

The focus of our work is on the most recent results in fixed point theory related to contractive mappings. We describe variants of -contractions that expand, supplement and unify an important work widely discussed in the literature, based on existing classes of interpolative and -contractions. In particular, a large class of contractions in terms of and F for both linear and nonlinear contractions are defined in the framework of -metric-like spaces. The main result in our paper is that --weak contractions have a fixed point in -metric-like spaces if function F or the specified contraction is continuous. As an application of our results, we have shown the existence and uniqueness of solutions of some classes of nonlinear integral equations.

### Keywords

(ϕ, F)-contraction; (s, q, ϕ, F)-contraction; b-metric-like space; fixed point

1  Introduction

Fixed point theory has been studied for a long time. Its application relies on the existence of solutions to mathematical problems that are based on the contraction principle. An interesting generalization of the Banach contraction principle was given by Wardowski [1,2] using a different type of contraction called F-contraction and by Karapınar in [3] defining the type of interpolative contractions. These approaches have been extended by weakening the contractive conditions, removing some of the imposed conditions on the used mappings or relaxing axioms of the defined spaces. Starting from these aspects, many researchers have constructed new fixed point theorems in different types of spaces such as metric, b-metric and other generalized metric spaces, as cited in [331]. Nevertheless, in the papers of Younis et al. [4,5] the notion of Kanan mappings in the view of F-contraction in the setting of b-metric-like spaces has been expanded and an example related to electrical engineering has been given. In this paper, we introduce general types of (s,ϕ,F) and (s,q,ϕ,F)-contractions, which are variants of Wardowski contractions in the setting of b-metric-like spaces. Using these classes of contractive mappings, we establish unique fixed point theorems that unify and extend recent results on this topic.

2  Preliminaries

In this section, we list some well-known definitions and lemmas in terms of b-metric-like spaces.

Definition 2.1 [9]. Let V be a nonempty set and s1 be a given real number. A mapping b:V×V[0,+) is called a b-metric-like if for all γ,δ,νV these conditions are satisfied:

(i)b(γ,δ)=0impliesγ=δ;

(ii)b(γ,δ)=b(δ,γ);

(iii)b(γ,δ)s[b(γ,ν)+b(ν,δ)].

The pair (V,b) is called a b-metric-like space (in the sequel we use bm.l.s for short).

In a b-metric-like space (V,b), if γ,δV and b(γ,δ)=0, then γ=δ. However, the converse need not be true, and b(γ,γ) may be positive for γV.

Definition 2.2 [10]. Let (V,b) be a bm.l.s with parameter s1, {νn} be any sequence in V and νV. Then, the following applies:

(a)   The sequence {νn} is said to be convergent to ν if limn+b(νn,ν)=b(ν,ν);

(b)   The sequence {νn} is said to be a Cauchy sequence in (V,b) if limn,m+b(νn,νm) exists and is finite;

(c)   The pair (V,b) is called a complete bm.l.s if for every Cauchy sequence {νn}V, there exists νV such that limn,m+b(νn,νm)=limn+b(νn,ν)=b(ν,ν).

Definition 2.3 [10]. Let (V,b) be a bm.l.s with parameter s1 and f be a self-mapping on V. We say that the function f is continuous if and only if limn+b(fνn,fν)=b(fν,fν), for each sequence {νn}V, which satisfies limn+b(νn,ν)=b(ν,ν).

Note that in a bm.l.s with parameter s1, if limn,m+b(νn,νm)=0 then the limit of the sequence {νn} is unique if it exists.

Lemma 2.1 [11,12]. Let (V,b) be a complete bm.l.s with parameter s1 and {νn} be a sequence such that b(νn,νn+1)λb(νn1,νn), for all nN, where λ[0,1). Then {vn} is a b-Cauchy sequence such that limn,m+b(νn,νm)=0.

Lemma 2.2. [9]. Let (V,b) be a bm.l.s with parameter s1 and suppose that {νn} converges to a ν and b(ν,ν)=0. Then

s1b(v,z)liminfn+b(νn,z)limsupn+b(νn,z)sb(ν,z),

for all zV.

Lemma 2.3. [9]. Let (V,b), be a bm.l.s with parameter s1. Then, the following applies:

(a)   If b(γ,δ)=0, then b(γ,γ)=b(δ,δ)=0;

(b)   If {νn} is a sequence such that limn+b(νn,νn+1)=0, then we have

limn+b(νn,νn)=limn+b(νn+1,νn+1)=0;

(c)   If γδ, then b(γ,δ)>0.

Lemma 2.4. [13]. Let (V,b) be a complete bm.l.s with parameter s1. Let {νn}V be a sequence such that limn+b(νn,νn+1)=0. If for sequence {νn} holds limn,m+b(νn,νm)0, then there exist ε>0 and sequences {mk}k=1+ and {nk}k=1+ of natural numbers with nk>mk>k, such that

b(νmk,νnk)ε,b(νmk,νnk1)<ε,

εs2limsupk+,b(νmk1,νnk1)εs,

εslimsupk+,b(νnk1,νmk)εs2and

εslimsupk+,b(νmk1,νnk)εs2.

3  Results

We begin the main section with a definition that is an expanding outlook of Wardowski type (ϕ,F)-contractions in the frame of a generalized metric space such as bm.l.s.

Definition 3.1. Let (V,b) be a bm.l.s with parameter s1 and f be a self-mapping on V. We say that f is a (s,q,ϕ,F)-contraction if there exist the functions F:(0,+)R and ϕ:(0,+)(0,+) such that

(a)   F is strictly increasing;

(b)   liminfrt+ϕ(r)>0 for all t>0;

(c)   For all ν,δV with fνfδ, and for some q>0

ϕ(b(ν,δ))+F(sqb(fν,fδ))F(b(ν,δ)).(1)

Remark 3.1. In the above definitions property F and conditions (1) yield

b(fν,fδ)sqb(fν,fδ)<b(ν,δ).

The continuity of the mapping f follows from the inequality b(fν,fδ)<b(ν,δ).

Remark 3.2.

•   Our definition generalizes the previous definitions given in [1416]. It contains a reduced number of conditions compared with the previous definitions.

•   The definition of (s,q,F)-contraction is an immediate consequence of Definition 2.1, if we take ϕ:(0,+)(0,+) to be a constant function.

•   If s=1 we get the definition of Wardowski in [1,2] in the case of metric spaces.

•   For s=1 the definition is valid in the framework of a metric space.

The following is the first fixed point theorem for (s,q,ϕ,F)-contraction type mapping.

Theorem 3.1. Let (V,b) be a complete bm.l.s with parameter s1. If f is a (s,q,ϕ,F)-contraction on V, then the function f has a unique fixed point in V.

Proof. Let be ν0V and the Picard iterative sequence {νn} defined by νn+1=f(νn) for n{0,1,2,}. The proof is clear in the case that there exists n0N, with νn0+1=νn0. So, we will assume that νn+1νn, which implies fνnfνn1 and b(fνn,fνn1)>0, for all nN{0}. Using inequality (1), we have

F(sqb(νn,νn+1))ϕ(b(νn1,νn))+F(sqb(νn,νn+1))=ϕ(b(νn1,νn))+F(sqb(fνn1,fνn))F(b(νn1,νn)).(2)

Further from inequality (2), we get

sqb(νn,νn+1)<b(νn1,νn),

which implies

b(νn,νn+1)<1sqb(νn1,νn).

In view of Lemma 2.1, the corresponding Picard sequence {νn} with the initial point ν0 is a Cauchy sequence such that limn,m+b(νn.νn)=0. Since (V,b) is a complete b-metric-like space, we conclude that there exists νV such that

limn+b(νn,ν)=b(ν,ν)=limn,m+b(νn,νm)=0.(3)

According to the (1), it follows:

F(sqb(fν,fνn))ϕ(b(ν,νn))+F(sqb(fν,fνn))F(b(ν,νn)),

that from property of F we get

sqb(fν,fνn)b(ν,νn).(4)

From triangular property and (4), we have

b(fν,fν)2sb(fν,fνn)2sqb(fν,fνn)2b(ν,νn).

Since f is continuous and using (3), (4) we obtain

b(fν,fν)=limn+b(fνn,fν)limn+2b(νn,ν)=2b(ν,ν)=0.(5)

Since b(ν,fν)s[b(ν,fνn)+b(fν,fνn)], as n+ we obtain that b(ν,fν)=0. Thus fν=ν and so f has a fixed point. Also from (3), we have b(ν,ν)=0. To prove the uniqueness of the fixed point, suppose that uV is another different fixed point. From uν follows fufν, then

F(sqb(u,ν))=F(sqb(fu,fν))ϕ(b(u,ν))+F(sqb(fu,fν))F(b(u,ν)),

which implies

b(u,ν)<1sqb(u,ν).

Previous inequality is a contradiction, so b(u,ν)=0 and the fixed point is unique.

Corollary 3.1. Let (V,b) be a bm.l.s with parameter s1 and f be a self-mapping on V. If there exist an increasing function F:(0,+)R and a positive constant τ such that

τ+F(sqb(fν,fδ))F(b(ν,δ))(6)

for all ν,δV with fνfδ, and for some q>0, then f has a unique fixed point in V.

Proof. Inequality (1) implies (6) if we set ϕ(r)=τ>0.

Example 3.1. Let V=[0,+) and b(x,y)=x2+y2+|xy|2, for all x,yV. It is clear that b is a b-metric-like on V, with parameter s=2 and (V,b) is complete. Also, b is not a metric-like nor b-metric (nor a metric on V). Consider the self-mapping f:VV by fx=ln(1+x)5. For all x,yV and constant q=2, we have

s2b(fx,fy)=4(f2x+f2y+|fxfx|2)=4((ln(x+1)5)2+(ln(y+1)5)2+|ln(x+1)5ln(y+1)5|2)4[x225+y225+|x5y5|2]=425[x2+y2+|xy|2]15b(x,y).

Taking the logarithms in the above inequality and fixing τ=ln5 and the function F(t)=lnt then the conditions of Corollary 3.1 are satisfied and clearly x=0 is a unique fixed point of f.

With the aim of expanding the initiated Definition 2.1 and starting a result that includes Theorem 3.1 and its respective corollaries, we will use a class of implicit relations, which makes simultaneously effective enormous literature on this topic.

Let Γ4 be the set of all continuous functions g:[0,+)4[0,+) satisfying

(a)   g is non-decreasing with respect to each variable:

(b)   g(t,t,t,t)t for t[0,+).

Definition 3.2. Let (V,b) be a bm.l.s with parameter s1 and f:VV be a self mapping. We say that f is generalized (s,q,ϕ,F)-g-weak contraction, if there exist functions F:(0,+)R, ϕ:(0,+)(0,+) and gΓ4 such that

(a) F is strictly increasing;

(b) liminfrt+ϕ(r)>0 for all t>0;

(c)ϕ(b(x,y))+F(sqb(fx,fy))F(g(b(x,y),b(x,fx),b(y,fy),b(x,fy)+b(y,fx)4s))(7)

for all x,yV with fxfy, and for some q1.

Remark 3.3.

•   The above definition reduces to a generalized (s,q,F)-g-weak contraction by setting ϕ:(0,+)(0,+) to be a constant function ϕ(r)=τ>0.

•   Fixing the parameter s=1 we get the definition of (ϕ,F)-g-weak contraction in the setting of metric and metric-like spaces.

•   Fixing s=1 and ϕ(r)=τ>0 we get the definition of (F-g)-weak contraction in the setting of metric and metric-like spaces.

Theorem 3.2. Let (V,b) be a bm.l.s with parameter s1 and the self mapping f:VV be a generalized (s,q,ϕ,F)-g-weak contraction. If f or F is continuous, then f has a unique fixed point in V.

Proof. Let u0V be arbitrary and construct the Picard iterative sequence {un} as un+1=f(un) for n{0,1,2,}. The proof is clear in the case that there exists n0N, with un0+1=un0. Therefore, we assume that un+1un, which means fumfun1 or b(fun,fun1)>0 for all nN{0}. Using (7) for x=un, y=un1 we have

ϕ(b(un,un+1))+F(sqb(un,un+1))=ϕ(b(fun1,fun))+F(sqb(fun1,fun))F(g(b(un1,un),b(un1,fun1),b(un,fun),b(un1,fun)+b(un,fun1)4s))=F(g(b(un1,un),b(un1,un),b(un,un+1),b(un1,un+1)+b(un,un)4s))F(g(b(un1,un),b(un1,un),b(un,un+1),sb(un1,un)+sb(un,un+1)+2sb(un1,un)4s))=F(g(b(un1,un),b(un1,un),b(un,un+1),b(un,un+1)+3b(un1,un)4)).(8)

If we assume that b(un1,un)b(un,un+1), then inequality (8) yields

ϕ(b(un,un+1))+F(sqb(un,un+1))F(g(b(un,un+1),b(un,un+1),b(un,un+1),b(un,un+1)))F(b(un,un+1)),

for all nN. So, we obtain

F(sqb(un,un+1)F(b(un,un+1))ϕ(b(un,un+1))<F(b(un,un+1)),

which is a contradiction. Therefore

b(un,un+1)<b(un1,un),

for all nN. Thus, the sequence {b(un1,un)} is decreasing and bounded below. Consequently, there exists l0 such that b(un1,un)l as n+. If l>0, then by taking the limit in (8) we get

ϕ(l)+F(sql)F(l),

which is a contradiction. Therefore, we conclude that l=0 and

limn+b(un1,un)=0.(9)

Next, we show that limn,mb(un,um)=0. Suppose the opposite, limn,mb(un,um)>0. Then by Lemma 2.4, there exist ε>0 and sequences {mk} and {nk} of positive integers, with nk>mk>k, such that

b(umk,unk)ε,b(umk,unk1)<ε,εs2limsupk+,b(umk1,unk1)εs,εslimsupk+,b(unk1,umk)εs2andεslimsupk+,b(umk1,unk)εs2.

From condition (7), we get

ϕ(b(umk,unk))+F(sqb(umk,unk))=ϕ(b(umk,unk))+F(sqb(fumk1,funk1))F(g(b(umk1,unk1),b(umk1,fumk1),b(unk1,funk1),b(umk1,funk1)+b(unk1,fumk1)4s))=F(g(b(umk1,unk1),b(umk1,umk),b(unk1,unk),b(umk1,unk)+b(unk1,umk)4s))(10)

Taking the upper limit in (10) as k+ and using Lemma 2.3, Lemma 2.4 and (9), we get

liminfn+,ϕ(b(umk,unk))+F(sqε)liminfn+,ϕ(b(umk,unk))+F(limsupn+sqb(umk,unk))F(limsupn+g(b(umk1,unk1),b(umk1,umk),b(unk1,unk),b(umk1,unk)+b(unk1,umk)4s))F(g(ε,0,0,ε2s))F(εs).

Hence, the acquired inequality

liminfn+,ϕ(b(umk,unk))+F(εsq)<F(εs),

is a contradiction since ε>0. So limn,mb(un,um)=0, and the sequence {un} is a Cauchy sequence in the complete b-metric-like space (V,b). Thus, there exists uV, such that

limn+b(un,u)=b(u,u)=limn,m+b(un,um)=0.(11)

Let n0N such that un+1fu for all nn0 and ufu. Now using condition (7) and property F, we have

ϕ(b(un,u))+F(sqb(un+1,fu))=ϕ(b(un,u))+F(sqb(fun,fu))F(g(b(un,u),b(un,fun),b(u,fu),b(un,fu)+b(u,fun)4s))=F(g(b(un,u),b(un,un+1),b(u,fu),b(un,fu)+b(u,un+1)4s)),

which implies

ϕ(b(un,fu))+sqb(un+1,fu)<g(b(un,u),b(un,un+1),b(u,fu),b(u,un+1)2s).(12)

Taking the upper limit in (12), and using Lemma 2.1 and result (9), it follows that

liminfn+ϕ(b(un,fu))+sq1b(u,fu)=sq1sb(u,fu)<g(0,0,b(u,fu),0)b(u,fu).(13)

Since q1, the inequality (13) implies b(u,fu)=0 and therefore fu=u. Thus, u is a fixed point and

0=b(u,fu)=b(u,u).(14)

Let u and v be two fixed points of f, where fu=u and fv=v. Since uv, it implies fufv. By (7) we have

ϕ(u,v)+F(sqb(u,v))=ϕ(u,v)+F(sqb(fu,fv))F(g(b(u,v),b(u,fu),b(v,fv),b(u,fv)+b(v,fu)4s))=F(g(b(u,v),b(u,u),b(v,v),b(u,v)+b(v,u)4s))=F(g(b(u,v),b(u,u),b(v,v),b(u,v)2s))=F(g(b(u,v),0,0,b(u,v)2s))F(g(b(u,v),b(u,v),b(u,v),b(u,v)))F(b(u,v)).(15)

Since this is a contradiction, it implies b(u,v)=0. Therefore, u=v and the fixed point is unique.

Theorem 3.3. Let (V,b) be bm.l.s with parameter s1 and f:VV be a continuous self-mapping. Assume that there exist the functions F:(0,+)R, gΓ4 and the constant τ>0 such that

(a)   F is strictly increasing;

(b)   τ+F(sqb(fx,fy))F(g(b(x,y),b(x,fx),b(y,fy),b(x,fy)+b(y,fx)4s)) for all x,yV with fxfy, for some q1.

Then f has a unique fixed point in V.

Proof. The proof follows from Theorem 3.2. by setting ϕ(r)=τ.

Corollary 3.2. Let (V,b) be a bm.l.s with parameter s1 and f:VV be a continuous self-mapping. Assume that there exist the functions F:(0,+)R, ϕ:(0,+)(0,+) such that:

(a)   F is strictly increasing;

(b)   liminfrt+ϕ(r)>0 for all t>0;

(c)  ϕ(b(x,y))+F(sqb(fx,fy))F(max(b(x,y),b(x,fx),b(y,fy),b(x,fy)+b(y,fx)4s))(16)

for all x,yV with fxfy, and for some q1. Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking gΓ4 as g(t1,t2,t3,t4)=max{t1,t2,t3,t4}.

Corollary 3.3. Let (V,b) be a bm.l.s with parameter s1 and f:VV be a continuous self-mapping. Assume that there exist functions F:(0,+)R, ϕ:(0,+)(0,+) such that:

(a)   F is strictly increasing;

(b)   liminfrt+ϕ(r)>0 for all t>0;

(c)  ϕ(b(x,y))+F(sqb(fx,fy))F(max(a1b(x,y)+a2b(x,fx)+a3b(y,fy)+a4b(x,fy)+b(y,fx)4s))(17)

for all x,yV with fxfy, and for some q1. Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2. by taking gΓ4 as g(t1,t2,t3,t4)=a1t1+a2t2+a3t3+a4t4 with 0<a1+a2+a3+a4<1.

Recently, many authors have studied new types of contractions known as interpolative contractions and hybrid contractions. The reader can refer to [3,11,1721]. The rest of the paper deals with this type of contractions extended in the setting of b-metric-like spaces, which can be obtained from our results as a certain special cases.

Theorem 3.4. Let (V,b) be a bm.l.s with parameter s1 and f:VV be a continuous self-mapping. Assume that there exist the functions F:(0,+)R, ϕ:(0,+)(0,+) such that

(a)   F is strictly increasing;

(b)   liminfrt+ϕ(r)>0 for all t>0;

(c)  ϕ(b(x,y))+F(sqb(fx,fy))F([a1(b(x,y))p+a2(b(x,fx))p+a3(b(y,fy))p+a4(b(x,fy)+b(y,fx)4s)p]1p)(18)

for all x,yV with fxfy, and for some q1. Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking gΓ4 as

g(t1,t2,t3,t4)=[a1t1p+a2t2p+a3t3p+a4t4p]1p,p>0,

where 0<a1+a2+a3+a4<1.

Theorem 3.5. Let (V,b) be a bm.l.s with parameter s1 and f:VV be a continuous self-mapping. Assume that there exist the functions F:(0,+)R, ϕ:(0,+)(0,+) such that

(a)   F is strictly increasing;

(b)   liminfrt+ϕ(r)>0 for all t>0;

(c)  ϕ(b(x,y))+F(sqb(fx,fy))F([max{(b(x,y))p,(b(x,fx))p,(b(y,fy))p,(b(x,fy)+b(y,fx)4s)p}]1p)(19)

for all x,yV with fxfy, and for some q1.

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking gΓ4 as

g(t1,t2,t3,t4)=[max{t1p,t2p,t3p,t4p}]1p,p>0.

Theorem 3.6. Let (V,b) be a bm.l.s with parameter s1 and f:VV be a continuous self-mapping. Assume that there exist the functions F:(0,+)R, ϕ:(0,+)(0,+) such that:

(a)   F is strictly increasing;

(b)   liminfrt+ϕ(r)>0 for all t>0;

(c)  (b(x,y))+F(sqb(fx,fy))F((b(x,y))a1(b(x,fx))a2(b(y,fy))a3(b(x,fy)+b(y,fx)4s)1(a1+a2+a3))(20)

for all x,yV with fxfy, and for some q1.

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking gΓ4 as

g(t1,t2,t3,t4)=t1a1t2a2t3a3t41(a1+a2+a3),

where a1,a2,a3(0,1) and a1+a2+a3<1.

Theorem 3.7. Let (V,b) be a bm.l.s with parameter s1 and f:VV be a continuous self-mapping. Assume that there exist the functions F:(0,+)R, ϕ:(0,+)(0,+) and λ(0,1) such that

(a)   F is strictly increasing;

(b)   liminfrt+ϕ(r)>0 for all t>0;

(c)  ϕ(b(x,y))+F(sqb(fx,fy))F([λmax{(b(x,y))p,(b(x,fx))p,(b(y,fy))p,(b(x,fy)+b(y,fx)4s)p}]1p)(21)

for all x,yV with fxfy, and for some q1.

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking gΓ4 as

g(t1,t2,t3,t4)=[λmax{t1p,t2p,t3p,t4p}]1p,p>0,λ(0,1).

Corollary 3.4. Let (V,b) be a bm.l.s with parameter s1 and f:VV be a continuous self-mapping. Assume that there exist the functions F:(0,+)R, ϕ:(0,+)(0,+) such that

(a) F is strictly increasing;

(b) liminfrt+ϕ(r)>0 for all t>0;

(c)ϕ(b(x,y))+F(sqb(fx,fy))F((b(x,fx))a1(b(y,fy))1a1)(22)

for all x,yVF(fix(f)) with fxfy, for some q1.

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking gΓ4 as g(t1,t2,t3,t4)=t2at31a, where a(0,1).

Remark 3.4.

•   Varieties of further results can be obtained by extending the set Γ4 to Γ5, Γ6, Γ7, etc.

•   Many significant fixed point theorems that were established for types of interpolative and hybrid contractive conditions essentially belong to the class of generalized (ϕ,s,q,F)-g-contractions.

4  Application

The study of the existence, nonexistence and uniqueness of the solution of differential and integral equations, plays a fundamental role in the research on nonlinear analysis and engineering mathematics. One of the main tools developed in this area consists of the application of a fixed point method.

Let us study the existence of solution for the nonlinear integral equation

v(t)=λ10tG1(t,ρ)H1(ρ,v(ρ))dρ+λ20kG2(t,ρ)H2(ρ,v(ρ))dρ;t,k[0,1],(23)

where λi are positive constants and functions Gi:[0,1]×[0,1]R+,Hi:[0,1]×RR for i=1,2 are given.

Let V=C([0,1]) be the set of real continuous functions defined on [0, 1] endowed with the b-metric-like

b(v,u)=supρ[0,1]|v(ρ)+u(ρ)|m   for all   v,uV,mN.(24)

It is obvious that (V,b) is a complete b-metric-like space with parameter s=2m1.

Consider the mapping f:VV by

fv(t)=λ10tG1(t,ρ)H1(ρ,v(ρ))dρ+λ20kG2(t,ρ)H2(ρ,v(ρ))dρ;

for all vC[0,1] and t,k[0,1].

Theorem 4.1. Consider the integral Eq. (1) via the following assertions:

i.   The mapping f:VV is continuous;

ii.  Hi:[0,1]×RR are continuous and there exist constants Ai satisfying

Hi(ρ,v(ρ))+Hi(ρ,u(ρ))Ai|v(ρ)+u(ρ)|

for i=1,2 and t,ρ,k[0,1];

iii. The constants λi,Ai and functions Gi, for i=1,2 satisfy condition

0<λ1A10tG1(t,ρ)dρ+λ2A20kG2(t,ρ)<1sq+1m

for t,k(0,1) and q1. Then the integral Eq. (23) has a unique solution v(t)V.

Proof. For all t[0,1], and v,uV we have

sqσb(fv(t),fu(t))=sq|fv(t)+fu(t)|m=sq|λ10tG1(t,ρ)H1(ρ,v(ρ))dρ+λ20kG2(t,ρ)H2(ρ,v(ρ))dρ+λ10tG1(t,ρ)H1(ρ,u(ρ))dρ+λ20kG2(t,ρ)H2(ρ,u(ρ))dρ|m=sq|λ10tG1(t,ρ)(H1(ρ,v(ρ))+H1(ρ,u(ρ)))dρ+λ20kG2(t,ρ)(H2(ρ,v(ρ))+H2(ρ,u(ρ)))dρ|msq|λ10tG1(t,ρ)A1(|v(ρ)+u(ρ)|)dρ+λ20kG2(t,ρ)A2(|v(ρ)+u(ρ)|)dρ|m=sq|λ10tG1(t,ρ)A1(|v(ρ)+u(ρ)|m)1mdρ+λ20kG2(t,ρ)A2(|v(ρ)+u(ρ)|m)1mdρ|msq|λ10tG1(t,ρ)A1(b(v,u))1mdρ+λ20kG2(t,ρ)A2(b(v,u))1mdρ|m=sq|λ1(b(v,u))1m0tA1G1(t,ρ)dρ+λ2(b(v,u))1m0kA2G2(t,ρ)dρ|m=sq|(b(v,u))1m(A1λ10tG1(t,ρ)dρ+A2λ20kG2(t,ρ)dρ)|msq|1sq+1m(b(v,u))1m|m=b(v,u)s.(25)

Hence, by taking logarithms in inequality (25) we get

lns+ln(sqb(fv,fu))ln(b(v,u)).

Further, fixing F(ζ)=ln(ζ), τ=lns and taking gΓ4 as g(t1,t2,t3,t4)=t1 we obtain

τ+F(sqb(fv,fu))F(g(b(v,u),b(v,fv),b(u,fu),b(v,fu)+b(u,fv)2s)).

Therefore, f is a (s,q,F)-g-contraction on V and all conditions of Theorem 3.3 are satisfied. Thus, v(t) is the unique fixed point of f, i.e., the solution of the integral Eq. (23).

5  Conclusion

The Definitions 2.1 and 3.2 not only a large class of contractions in terms of ϕ,s,q,g and F in the metric, b-metric, metric-like, partial metric, but also have a unifying power for both linear and nonlinear contractions in the framework of b-metric-like spaces.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The author declares that they have no conflicts of interest to report regarding the present study.

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