# Common Fixed Point Results for Generalizedαsp Contractive Mappings and Applications

Authors: Kastriot Zoto; Ilir Vardhami
Journal: Journal of Function Spaces
 (2018)

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## Abstract

A class ofg-αsp-admissible mappings and general type g-αsp contraction mappings on b-metric-like space are defined. Some fixed point results dealing with such a class of contractions are obtained. The generalized contractions considered in this work cover and unify many particular types of contractions. Finally, we present application to the existence of solutions for a system of integral equations by means of b-metric-like spaces.


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## Body

## 1. Introduction

The study of new classes of spaces and their basic properties are always favorite topics of interest. Recently some authors have introduced some generalizations of metric spaces in several ways and have studied fixed point problems in these spaces, as well as their applications. In this context, Matthews [1] introduced the notion of partial metric space where self-distance of an arbitrary point needs not be equal to zero. Hitzler and Seda [2] and Amini-Harandi [3] made further generalizations under the name of dislocated, respectively, metric-like space. Further, Shukla et al. introduced in [4] the notion of 0-σ complete metric space. This concept was further extended by Alghamdi et al. [5] under the name of b-metric-like spaces. This class of spaces has received significant attention lately.Also these generalizations have been associated with new and generalized classes of contractive mappings. In this direction Samet et al. [6] introduced the concept of α-admissible and α-ψ-contractive mappings, further extended to the (α,β)-contractive mappings, α,ψ,ϕ-contractive mappings, and f-α-admissible mappings. Many papers dealing with the above notions have been considered to prove fixed point results (e.g., see [7–25]).In this paper, following the above discussion, we introduce the notion ofg-αs-admissible mapping and also introduce the concepts of g-αsp,λ quasi-contractive mappings and rational (g-αsp,ψ,ϕ) contractive mappings in the larger framework of b-metric-like space. The considered contractive conditions not only generalize the known ones but also include and unify a huge number of existing results on the topic in the corresponding literature. Finally, we apply the given results to obtain existence of solutions of integral equations.


## 2. Preliminaries

Definition 1 (see [8]).
LetX be a nonempty set and s≥1 be a given real number. A mapping d:X×X→0,∞ is called a b-metric if, for all x,y,z∈X, the following conditions are satisfied: (i)
dx,y=0 if and only if x=y;(ii)
d(x,y)=d(y,x);(iii)
d(x,y)≤sd(x,z)+d(z,y).The pair(X,d) is called a b-metric space with parameter s.Definition 2 (see [26]).
LetX be a nonempty set. A mapping σ:X×X→0,∞ is called metric-like if, for all x,y,z∈X, the following conditions are satisfied: (i)
σ(x,y)=0 implies x=y;(ii)
σ(x,y)=σ(y,x);(iii)
σ(x,y)≤σ(x,z)+σ(z,y).The pair(X,σ) is called a metric-like space.Definition 3 (see [5]).
LetX be a nonempty set and s≥1 be a given real number. A mapping σb:X×X→0,∞ is called b-metric-like if, for all x,y,z∈X, the following conditions are satisfied: (i)
σb(x,y)=0 implies x=y;(ii)
σb(x,y)=σb(y,x);(iii)
σb(x,y)≤sσb(x,z)+σb(z,y).The pair(X,σb) is called a b-metric-like space.In ab-metric-like space (X,σb), if x,y∈X and σb(x,y)=0, then x=y, but the converse needs not be true, and σb(x,x) may be positive for x∈X.Example 4 (see [18]).
LetX=0,+∞ and m>1 be a constant. Define a function σb:X2→0,∞ by σb(x,y)=(x+y)m or σb(x,y)=max{x,y}m. Then X,σb is a b-metric-like space with parameter s=2m-1. Clearly, (X,σb) is neither a b-metric, nor metric-like, nor partial b-metric space.Definition 5 (see [5]).
Let(X,σb) be a b-metric-like space with parameter s, and let {xn} be any sequence in X and x∈X. Then (1)
The sequence{xn} is said to be convergent to x if limn→∞σb(xn,x)=σb(x,x);(2)
The sequence{xn} is said to be a Cauchy sequence in (X,σb) if limn,m→∞σb(xn,xm) exists and is finite;(3)
(X,σb) is said to be a complete b-metric-like space if, for every Cauchy sequence {xn} in X, there exists an x∈X such that limn,m→∞σb(xn,xm)=limn→∞σb(xn,x)=σb(x,x).The limit of a sequence in ab-metric-like space need not be unique.Proposition 6 (see [5]).
Let(X,σb) be a b-metric-like space with parameter s, and let {xn} be any sequence in X with x∈X such that limn→∞σb(xn,x)=0. Then (i)
x is unique,(ii)
σb(x,y)/s≤limn→∞σb(xn,y)≤sσb(x,y) for all y∈X.In 2012, Samet et al. [6] introduced the class of α-admissible mappings.Definition 7.
LetX be a nonempty set, f:X→X, and α:X×X→R+ be mappings. We say that f is an α-admissible mapping if α(x,y)≥1 implies that α(fx,fy)≥1, for all x,y∈X.In 2014 a new notion ofg-α-admissible mapping was introduced by Rosa and Vetro [27].Definition 8.
For a nonempty setX, let f,g:X→X and α:X×X→0,+∞ be mappings. The mapping f is called g-α-admissible if, for all x,y∈X such that α(gx,gy)≥1, we have α(fx,fy)≥1.Definition 9 (see [28]).
Let(X,σb) be a b-metric-like space with parameter s≥1, and let α:X×X→0,∞ be a function and arbitrary constants q, p such that q≥1 and p≥1. A self-mapping f:X→X is αqsp-admissible if α(x,y)≥qsp implies α(fx,fy)≥qsp, for all x,y∈X.Examples 3.3 and 3.4 in [28] illustrate Definition 9.Lemma 10 (see [9]).
Let(X,σb) be a b-metric-like space with parameter s≥1 and f:X→X be a given mapping. Suppose that f is continuous at u∈X. Then for all sequence {xn} in X such that xn→u, we have fxn→fu; that is,(1)limn→∞σbfxn,fu=σbfu,fu.Lemma 11.
Let{xn} be a sequence in a b-metric-like space (X,σb) with parameter s≥1, such that (2)σbxn,xn+1≤λσbxn-1,xn∀n∈N,for some λ, where 0≤λ<1/s. Then (1)
limn→∞σb(xn,xn+1)=0,(2)
{xn} is a Cauchy sequence in (X,σb) and limn,m→∞σb(xn,xm)=0.Proof.
For the proof of the lemma, one can use the following clear inequalities:(3)σbxn+1,xn+2≤λσbxn,xn+1≤λ2σbxn-1,xn≤⋯≤λn+1σbx0,x1,σbxm,xn≤sσbxm,xm+1+s2σbxm+1,xm+2+⋯+sn-m-1σbxn-2,xn-1+sn-mσbxn-1,xn,where m,n∈N and n>m.Lemma 12 (see [5]).
Let(X,σb) be a b-metric-like space with parameter s≥1 and suppose that {xn} and {yn} are σb-convergent to x and y, respectively. Then we have (4)1s2σbx,y-1sσbx,x-σby,y≤liminfn→∞σbxn,yn≤limsupn→∞σbxn,yn≤sσbx,x+s2σby,y+s2σbx,y.In particular, if σb(x,y)=0, then we have limn→∞σb(xn,yn)=0.
Moreover, for eachz∈X, we have (5)1sσbx,z-σbx,x≤liminfn→∞σbxn,z≤limsupn→∞σbxn,z≤sσbx,z+sσbx,x. In particular, if σb(x,x)=0, then (6)1sσbx,z≤liminfn→∞σbxn,z≤limsupn→∞σbxn,z≤sσbx,z.The following result is useful.Lemma 13 (see [28]).
Let(X,σb) be a b-metric-like space with parameter s≥1. Then (1)
Ifσb(x,y)=0, then σb(x,x)=σb(y,y)=0;(2)
If{xn} is a sequence such that limn→∞σb(xn,xn+1)=0, then we have(7)limn→∞σbxn,xn=limn→∞σbxn+1,xn+1=0;(3)
Ifx≠y, then σb(x,y)>0.Lemma 14 (see [29]).
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and let {xn} be a sequence such that (8)limn→∞σbxn,xn+1=0. If for the sequence {xn}, limn,m→∞σb(xn,xm)≠0, then there exist ε>0 and sequences {m(k)} and {n(k)} of positive integers with nk>mk>k, such that (9)ε≤σbxmk,xnk≤εs,σbxmk,xnk-1<ε,εs2≤limsupk→∞σbxmk-1,xnk-1≤εs,εs≤limsupk→∞σbxnk-1,xmk≤ε,εs≤limsupk→∞σbxmk-1,xnk≤εs2.


## 3. Main Results

We begin this section with the following definition.Definition 15.
Let(X,σb) be a b-metric-like space with parameter s≥1, and let f,g:X→X and α:X×X→0,∞ be given mappings and arbitrary constant p such that p≥2. The mapping f:X→X is g-αsp-admissible if, for all x,y∈X,α(gx,gy)≥sp implies α(fx,fy)≥sp.Remark 16.
Takings=1 in definition we obtain an g-α-admissible mapping defined in [27]. Taking g=Ix as the identity mapping on X, we deduce the definition of f-αsp-admissible mapping as in [28]. For s=1 and g=Ix the definition reduces to the definition of an α-admissible mapping in a metric space [6].
In the sequel, according to [28] we shall consider the following properties in case of q=1.Let(X,σb) be a complete b-metric-like space with parameter s≥1 and α:X×X→0,∞ be a function. Then(Hsp)
If{xn} is a sequence in X such that gxn→gx∈X as n→∞ and α(gxn,gxn+1)≥sp, then there exists a subsequence {gxnk} of {gxn} such that α(gxnk,gx)≥sp for all k∈N.(Usp)
For allu,v∈C(f,g), we have α(gu,gv)≥sp or α(gv,gu)≥sp, where C(f,g) denotes the set of all coincidence points of f and g.Example 17.
LetX=0,+∞. We define the mappings f,g:X→X by fx=ex;gx=1/2x2 for all x∈X, and α:X×X→0,∞ by (10)αx,y=s2x,y∈0,10otherwise.
Thenf is g-αsp-admissible.Extending the well known definition of quasi-contraction from Cirić, we introduce the notion of a generalized(αsp-g,λ)-quasi-contraction in the setting of a b-metric-like space.Definition 18.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f,g:X→X be given mappings. We say that f is a generalized (αsp-g,λ)-quasi-contraction if f is a g-αsp-admissible mapping satisfying(11)αgx,gyσbfx,fy≤λmax⁡σbgx,gy,σbgx,fx,σbgy,fy,σbgx,fy,σbgy,fx,σbgx,gx,σbgy,gyfor all x,y∈X and λ∈0,1/2.Remark 19.
If we takeα(gx,gy)=s2 (p=2), the definition reduces to the definition of an s-λ quasi-contraction. If we take s=1, the definition reduces to the λ-quasi-contraction in the setting of metric spaces.Theorem 20.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f,g be self-mappings on X such that f(X)⊂g(X), f(X) or g(X) is a closed subset of X, and α:X×X→0,+∞ is a given mapping. Suppose that the following conditions are satisfied: (i)
f is g-αsp-admissible mapping;(ii)
f is a generalized (g-αsp,λ) contractive mapping;(iii)
there existsx0∈X such that α(gx0,fx0)≥sp;(iv)
propertiesHsp and Usp are satisfied.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Proof.
By hypothesis (iii), there exists anx0∈X such that α(gx0,fx0)≥sp. We define a sequence {xn} and {yn} in X by yn=fxn=gxn+1 for all n∈N. If yn=yn+1 for some n∈N, then gxn+1=yn=yn+1=fxn+1 and f and g have a point of coincidence. Without loss of generality, one can suppose that yn≠yn+1 (by Lemma 13; i.e., σb(yn,yn+1)>0) for each n∈N.
Sincef is g-αsp-admissible mapping, we have (12)αgx0,gx1=αgx0,fx0≥sp,αgx1,gx2=αfx0,fx1≥sp,αgx2,gx3=αfx1,fx2≥sp.Hence, by induction, we get α(yn,yn+1)=α(gxn,gxn+1)≥sp for all n∈N.
By condition (11), we have(13)spσbyn,yn+1=spσbfxn,fxn+1≤αgxn,gxn+1σbfxn,fxn+1≤λmax⁡σbgxn,gxn+1,σbgxn,fxn,σbgxn+1,fxn+1,σbgxn,fxn+1,σbgxn+1,fxn,σbgxn,gxn,σbgxn+1,gxn+1=λmax⁡σbyn-1,yn,σbyn-1,yn,σbyn,yn+1,σbyn-1,yn+1,σbyn,yn,σbyn-1,yn-1,σbyn,yn≤λmax⁡σbyn-1,yn,σbyn-1,yn,σbyn,yn+1,sσbyn-1,yn+σbyn,yn+1,2sσbyn-1,yn,2sσbyn-1,yn,2sσbyn-1,yn=λmax⁡σbyn-1,yn,σbyn-1,yn,σbyn,yn+1,sσbyn-1,yn+σbyn,yn+1,2sσbyn-1,yn.If σb(yn-1,yn)<σb(yn,yn+1) for some n∈N, then, from inequality (13), we have (14)σbyn,yn+1≤2λsp-1σbyn,yn+1, a contradiction since 2λ/sp-1<1.
Hence, for alln∈N, σb(yn,yn+1)≤σb(yn-1,yn), and also by inequality (13), we get(15)σbyn,yn+1≤2λsp-1σbyn-1,yn=cσbyn-1,yn,where 0≤c=2λ/sp-1<1/s.
Then, by Lemma11, we have(16)limn→∞σbyn,yn+1=0,and the sequence {yn} is a Cauchy sequence, and limn,m→∞σb(yn,ym)=0. By completeness of (X,σb) the sequence {yn}={fxn}={gxn+1} converges to a point u. By hypothesis, since g(X) is closed, then u∈g(X). Therefore, there exists z∈X such that u=gz. That is(17)0=limn,m→∞σbyn,ym=limn→∞σbyn,gz=limn→∞σbfxn,gz=limn→∞σbgxn+1,gz=σbgz,gz.
Since propertyHsp is satisfied, there exists a subsequence {ynk} of {yn} such that α(ynk-1,gz)≥sp (that is α(gxnk,gz)≥sp) for all k∈N. If fz≠gz, applying contractive condition (ii) of theorem, with x=xnk and y=u, we obtain(18)spσbfxnk,fz≤αgxnk,gzσbfxnk,fz≤λmax⁡σbgxnk,gz,σbgxnk,fxnk,σbgz,fz,σbgxnk,fz,σbgz,fxnk,σbgxnk,gxnk,σbgz,gz=λmax⁡σbynk-1,gz,σbynk-1,ynk,σbgz,fz,σbynk-1,fz,σbgz,ynk,σbynk-1,ynk-1,σbgz,gz.
Taking the upper limit ask→∞ in the above inequality, and using (16), (17), and Lemmas 12 and 13, we have(19)sp-1σbgz,fz=sp1sσbgz,fz≤2λsσbgz,fz. Hence, from above we get σb(gz,fz)=0; that is, fz=gz and fz=gz=u is a point of coincidence of f and g. By using condition (11) and property Usp one can be convinced that the point of coincidence is unique and by weak compatibility of f and g it follows that z is a unique common fixed point.Example 21.
LetX=0,+∞ and σb(x,y)=(x+y)2; for all x,y∈X is a b-metric-like space with parameter s=2. Define the mappings f and g on X by (20)
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.It is clear that f(X)⊆g(X). For x,y∈X such that α(gx,gy)≥s2; then gx,gy∈0,1/2 and this implies that x,y∈0,1. By definitions we have fx,fy∈0,1/2 and α(fx,fy)≥s2; that is, f is g-αsp-admissible mapping;
Forx,y∈0,1, we have (21)αgx,gyσbfx,fy=s2σbfx,fy=4σb18x,18y=418x+18y2=41612x+12y2=14σbgx,gy≤λmax⁡σbgx,gy,σbgx,fx,σbgy,fy,σbgx,fy,σbgy,fx,σbgx,gx,σbgy,gy, where 1/4≤λ<1/2.Obviously the other assumptions of theorem can be verified andx=0 is the unique common fixed point of f and g.Theorem 22.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f,g be self-mappings on X such that f(X)⊂g(X),f(X) or g(X) is a closed subset of X, and α:X×X→0,+∞ a given mapping. Suppose that the following conditions are satisfied: (i)
f is g-αsp-admissible mapping;(ii)
for allx,y∈X, and constants ai≥0, i=1,…,5,(22)αgx,gyσbfx,fy≤a1σbgx,gy+a2σbgx,fx+a3σbgy,fy+a4σbgx,fy+a5σbgy,fx,wherea1+a2+a3+a4+a5<1/2;(iii)
there existsx0∈X such that α(gx0,fx0)≥sp;(iv)
propertiesHsp and Usp are satisfied.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Proof.
This theorem can be considered as a corollary of Theorem20, since, for all x,y∈X, inequality (22) is a special case of (11).If we considerαgx,gy=s2 (where p=2) in Theorem 20, then we deduce the following corollary.Corollary 23.
Let(X,σb) be a complete b-metric-like space with parameter s≥1, and f, g be self-mappings on X such that f(X)⊂g(X),f(X) or g(X) is a closed subset of X and satisfy the condition(23)s2σbfx,fy≤λmax⁡σbgx,gy,σbgx,fx,σbgy,fy,σbgx,fy,σbgy,fx,σbgx,gx,σbgy,gy for all x,y∈X and λ∈0,1/2.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Remark 24.
Theorem20 generalizes Theorem 18 in [30]. For α(x,y)=s2 and, for each x,y∈X, Theorems 20 and 22 reduce to Theorems 3.2 and 3.13 of [31]. In Theorem 22, by choosing the constants ai in certain manner, we obtain, as particular cases, certain classes of g-αsp types of classical contractions (such as Kannan, Chatterjea, Reich, and Zamfirescu contractions).Contraction-type mappings have been generalized in several directions. A series of generalizations start with Samet et al. [6] with the concept of α-admissible mappings and α-ψ-contractive mapping. Later many authors used these classes of mappings under weakly and generalized weakly contractivity conditions and discussed fixed point results in various spaces.The contractivity conditions considered in the second part of this section are constructed via auxiliary functions defined with the familiesΨ,Φ, and S, respectively:ψ:0,∞→0,∞ is an increasing and continuous function;ϕ:0,∞→0,∞ is continuous function and ϕ(t)<ψ(t) for all t>0;β:0,∞→0,1 satisfying the condition β(tn)→1asn→∞ implies that tn→0 as n→∞.Let(X,σb) be a b-metric-like space with parameter s≥1. For two self-mappings f,g:X→X, we define the set N(x,y) with the involvement of rational terms.(24)Nx,y=max⁡σbgx,gy,σbgx,fx,σbgy,fy,σbgx,fy+σbgy,fx4s,σbgx,fxσbgy,fy1+σbfx,fy,σbgy,fy1+σbgx,fx1+σbgx,gyfor all x,y∈X.We introduce now the notion of rational(g-αsp,ψ,ϕ) contraction in the setting of b-metric-like spaces.Definition 25.
Let(X,σb) be a b-metric-like space with parameter s≥1 and f,g:X→X two self-mappings. Also, let α:X×X→0,∞ and p≥2. We say that f is called a generalized (g-αsp,ψ,ϕ) contractive mapping, if there exist ψ∈Ψ,ϕ∈Φ such that(25)ψαgx,gyσbfx,fy≤ϕNx,yfor all x,y∈X with α(x,y)≥sp, where N(x,y) is defined as in (24).Remark 26.
( 1 ) If we take g=IX as the identity mapping on X, then we obtain the definition of (αsp-ψ,ϕ) contractive mapping as in [28].
( 2 ) Taking s=1 in the Definition 25, we obtain (g-α-ψ,ϕ)-contractive mappings.
( 3 ) For s=1 and g=IX the definition reduces to the definition of an (α-ψ,ϕ)-contractive mapping.
( 4 ) The definition reduces to a (ψ,ϕ,s-g)-contractive mapping if we take α(gx,gy)=sp.
( 5 ) The definition reduces to an g-αsp-ϕ contractive mapping if we take ψ(t)=t.Theorem 27.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f, g be self-mappings on X such that f(X)⊂g(X),f(X) or g(X) is a closed subset of X, and α:X×X→0,+∞ a given mapping. Suppose that the following conditions are satisfied: (i)
f is g-αsp-admissible mapping;(ii)
f is a generalized (g-αsp-ψ,ϕ) contractive mapping;(iii)
there existsx0∈X such that α(gx0,fx0)≥sp;(iv)
propertiesHsp and Usp are satisfied.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Proof.
From the similar arguments as in proof of Theorem20, we construct the sequences {xn} and {yn} in X by yn=fxn=gxn+1 for all n∈N. Supposing that yn≠yn+1 (which by Lemma 13 implies σb(yn,yn+1)>0) for each n∈N, we get(26)αyn,yn+1=αgxn,gxn+1≥sp.
By (26) and condition (25) we have(27)ψspσbyn,yn+1=ψspσbfxn,fxn+1≤ψαgxn,gxn+1σbfxn,fxn+1≤ϕNxn,xn+1,where(28)Nxn,xn+1=maxσbgxn,gxn+1,σbgxn,fxn,σbgxn+1,fxn+1,σbgxn,fxn+1+σbgxn+1,fxn4s,σbgxn,fxnσbgxn+1,fxn+11+σbfxn,fxn+1,σbgxn+1,fxn+11+σbgxn,fxn1+σbgxn,gxn+1=max⁡σbyn-1,yn,σbyn-1,yn,σbyn,yn+1,σbyn-1,yn+1+σbyn,yn4s,σbyn-1,yn,σbyn,yn+11+σbyn,yn+1,σbyn,yn+11+σbyn-1,yn1+σbyn-1,yn≤max⁡σbyn-1,yn,σbyn,yn+1,sσbyn-1,yn+σbyn,yn+1+2sσbyn-1,yn4s.If we assume that, for some n∈N,(29)σbyn-1,yn<σbyn,yn+1, then, from inequality (28), we get(30)Nxn,xn+1≤σbyn,yn+1.Again, by (30) and using condition (27) and property of ψ, we obtain(31)σbyn,yn+1≤Nxn,xn+1. Hence as a result we have(32)Nxn,xn+1=σbyn,yn+1.Then by (27) using (32) we obtain(33)ψspσbyn,yn+1≤ϕσbyn,yn+1 which gives a contradiction, since we have assumed that σb(xn,xn+1)>0 and property ϕ(t)<ψ(t) for all t>0. So σb(yn,yn+1)≤σb(yn-1,yn), for all n∈N. Hence, the sequence of nonnegative numbers {σb(yn,yn+1)} is nonincreasing. Thus it converges to a nonnegative number, say r≥0. That is limn→∞σb(yn,yn+1)=r and also limn→∞σb(yn,yn+1)=limn→∞N(xn,xn+1)=r. If r>0, then letting n→+∞ in (27) we get ψ(spr)≤ϕ(r) which implies r=0; that is,(34)limn→∞σbyn,yn+1=limn→∞Nxn,xn+1=0.Now for the sequence {yn} we shall show that limn,m→∞σb(yn,ym)=0. Suppose, on the contrary, that limn,m→∞σb(yn,ym)≠0. Then, using Lemma 14, we get that there exists ε>0 for which we can find subsequences {ymk} and {ynk} of yn, with nk>mk>k, such that the following hold:(35)σbymk,ynk≥ε,σbymk,ynk-1<ε;εs2≤limsupk→∞σbymk-1,ynk-1≤εs;εs≤limsupk→∞σbynk-1,ymk≤ε;εs≤limsupk→∞σbymk-1,ynk≤εs2 From the definition of N(x,y), we have(36)Nxmk-1,xnk-1=Nxmk,xnk=Ngxmk,gxnk=max⁡σbgxmk,gxnk,σbgxmk,fxmk,σbgxnk,fxnk,σbgxmk,fxnk+σbgxnk,fxmk4s,σbgxmk,fxmkσbgxnk,fxnk1+σbfxmk,fxnk,σbgxnk,fxnk1+σbgxmk,fxmk1+σbgxmk,gxnk=max⁡σbymk-1,ynk-1,σbymk-1,ymk,σbynk-1,ynk,σbymk-1,ynk+σbynk-1,ymk4sσbymk-1,ymkσbynk-1,ynk1+σbymk,ynk,σbynk-1,ynk1+σbymk-1,ymk1+σbymk-1,ynk-1.Taking the upper limit as k→∞ in (36), and using (34), (35), we get(37)limsupk→∞Nxmk-1,xnk-1≤maxεs,0,0,εs2+ε4s,0,0=εs.Applying (25) with x=xnk and y=xmk, we obtain(38)ψσbymk,ynk=ψσbfxmk,fxnk≤ψspσbfxmk,fxnk≤ψαgxmk,gxnkσbfxmk,fxnk≤ϕNxmk,xnk.Taking the upper limit in (38), using (35) and (37), we obtain (39)ψεs≤ψεsp-1=ψspεs≤ψsplimsupk→∞σbfxmk,fxnk≤ϕlimsupk→∞Nxmk-1,xnk-1≤ϕεs, which implies that ε=0, a contradiction with ε>0. Hence, limn,m→∞σb(yn,ym)=0; that is, {yn} is a Cauchy sequence in X. From the completeness of (X,σb), there exists u∈X such that(40)limn→∞σbyn,u=limn→∞σbfxn,u=limn→∞σbgxn+1,u=limn,m→∞σbyn,ym=σbu,u=0.By hypothesis, since g(X) is closed, by (40), u∈g(X). Therefore, there exists z∈X such that u=gz. And (40) can be written as (41)0=limn,m→∞σbyn,ym=limn→∞σbyn,gz=limn→∞σbfxn,gz=limn→∞σbgxn+1,gz=σbgz,gz. Since property Hsp is satisfied there exists a subsequence ynk of yn such that α(ynk-1,gz)≥sp (that is α(gxnk,gz)≥sp) for all k∈N. If fz≠gz, applying contractive condition (26), with x=xnk and y=z, we obtain(42)ψspσbynk,fz=ψspσbfxnk,fz≤ψαgxnk,gzσbfxnk,fz≤ϕNxnk,z=ϕNgxnk,gz,where(43)Nxnk,z=Ngxnk,gz=max⁡σbgxnk,gz,σbgxnk,fxnk,σbgz,fz,σbgxnk,fz+σbgz,fxnk4s,σbgxnk,fxnkσbgz,fz1+σbfxnk,fzσbgz,fz1+σbgxnk,fxnk1+σbgxnk,gz=max⁡σbynk-1,gz,σbynk-1,ynk,σbgz,fz,σbynk-1,fz+σbgz,ynk4s,σbynk-1,ynkσbgz,fz1+σbynk,fz,σbgz,fz1+σbynk-1,ynk1+σbynk-1,gz.Taking the upper limit in (43) and using Lemma 13 and (40), we obtain(44)limsupn→∞Nxnk,u≤max0,0,σbgz,fz,sσbgz,fz+04s,0,σbgz,fz1+01+0=σbgz,fz.Taking the upper limit as k→∞ in (42), and using (44) and Lemma 13, we obtain(45)ψsp-1σbgz,fz=ψsp1sσbgz,fz≤ψsplimsupk→∞σbfxnk,fz≤ϕlimsupk→∞Nxnk,z≤ϕσbgz,fz.
In view of property ofψ,ϕ from (45), we get σb(gz,fz)=0 which implies that fz=gz. Hence, u=fz=gz is a point of coincidence for f and g. Similarly as in Theorem 20 by using condition (26) and property Usp and weak compatibility it can be shown that z is a unique common fixed point.By takingϕ(t)=ψ(t)-φ(t), where φ∈Ψ, in Theorem 27, we obtain the following result.Corollary 28.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f, g be self-mappings on X such that f(X)⊂g(X),f(X) or g(X) is a closed subset of X, and α:X×X→0,+∞ a given mapping. Suppose that the following conditions are satisfied: (i)
f is g-αsp-admissible mapping;(ii)
there exist functionsψ,φ∈Ψ such that(46)ψαgx,gyσbfx,fy≤ψNx,y-φNx,y;(iii)
there existsx0∈X such that α(gx0,fx0)≥sp;(iv)
propertiesHsp and Usp are satisfied.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Theorem 29.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f, g be self-mappings on X such that f(X)⊂g(X),f(X) or g(X) is a closed subset of X, and α:X×X→0,+∞ a given mapping. Suppose that the following conditions are satisfied: (i)
f is g-αsp-admissible mapping;(ii)
there exist functionsψ∈Ψ,ϕ∈Φ,β∈S such that(47)ψαgx,gyσbfx,fy≤βNx,yϕNx,y;(iii)
there existsx0∈X such that αgx0,fx0≥sp;(iv)
propertiesHsp and Usp are satisfied.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Proof.
This corollary is a special case of Theorem27 since inequality (47) implies inequality (25).By takingψ(t)=t and ϕ(t)=β(t)t, where β∈S in Theorem 27, we obtain the following result.Corollary 30.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f, g be self-mappings on X such that f(X)⊂g(X),f(X) or g(X) is a closed subset of X, and α:X×X→0,+∞. Suppose that the following conditions are satisfied: (i)
f is a g-αsp-admissible mapping;(ii)
there exists functionβ∈S such that(48)αgx,gyσbfx,fy≤βNx,yNx,y;(iii)
there exists anx0∈X such that αgx0,fx0≥sp;(iv)
propertiesHsp and Usp are satisfied.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.If we takeψ(t)=t in Theorem 27, then we get the following result.Corollary 31.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f, g be self-mappings on X such that f(X)⊂g(X),f(X) or g(X) is a closed subset of X, and α:X×X→0,+∞. Suppose that the following conditions are satisfied: (i)
f is a g-αsp-admissible mapping;(ii)
there exist functionϕ∈Ψ such that(49)αgx,gyσbfx,fy≤ϕNx,y;(iii)
there exists anx0∈X such that α(gx0,fx0)≥sp;(iv)
propertiesHsp and Usp are satisfied.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Corollary 32.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f, g be self-mappings on X such that f(X)⊂g(X),f(X) or g(X) is a closed subset of X, and α:X×X→0,+∞. Suppose that the following conditions are satisfied: (i)
f is a g-αsp-admissible mapping;(ii)
there exist functionsψ,φ∈Ψ such that(50)αgx,gyσbfx,fy≤Nx,y-φNx,y;(iii)
there exists anx0∈X such that αgx0,fx0≥sp;(iv)
propertiesHsp and Usp are satisfied.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Proof.
It follows from Corollary28 by taking ψ(t)=t.Corollary 33.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f, g be self-mappings on X such that f(X)⊂g(X),f(X) or g(X) is a closed subset of X, and α:X×X→0,+∞. Suppose that the following conditions are satisfied: (i)
f is a g-αsp-admissible mapping;(ii)
there exists functionψ∈Ψ such that(51)ψαgx,gyσbfx,fy≤λψNx,y;for allx,y∈X, where Nx,y is defined as in (32) and 0<λ<1;(iii)
there exists anx0∈X such that αgx0,fx0≥sp;(iv)
propertiesHsp and Usp are satisfied.
Thenf and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Proof.
In Theorem27 take ϕ(t)=λψ(t), where 0<λ<1.Corollary 34.
Let(X,σb) be a complete b-metric-like space with parameter s≥1 and f,g be two self-maps of X and ψ∈Ψ such that they satisfy the condition(52)ψspσbfx,fy≤λψNx,yfor all x,y∈X, where N(x,y) is defined as in (32) and 0<λ<1. Then f and g have a unique common fixed point in X.Proof.
In Theorem27 take ϕ(t)=λψ(t), where 0<λ<1 and α(gx,gy)=sp.Remark 35.
Our results generalize, extend, and improve the results appearing in the literature [3, 7, 10, 12, 16, 21, 26, 28, 29, 31–34]. It is clear that several more corresponding results can be derived by our main theorems by choosing constant p and the mappings ψ,ϕ, in a suitable way.


## 4. Application

In this section we will use Theorem27 to show that there is a solution to the following system of integral equations:(53)xt=∫0tKr,xrdr,xt=∫0txrdr.Let X=C0,T,R be the set of real continuous functions defined on 0,T for T>0. We define a b-metric-like σb:X×X→0,∞ by (54)σbx,y=maxt∈0,Txt+ytmfor all x,y∈X, where m>1.It is evident thatX,σb,sm-1 is a complete b-metric-like space.Consider the mappingsf,g:X→X by(55)fxt=∫0tKr,xrdr,gxt=∫0txrdr, and let ζ:R×R→R be a given function.Theorem 36.
Consider the system of integral equations (53) and suppose that the following assertions hold: (i)
K:0,T×R→R+ (that is Kr,x(r)≥0, ∫0t‍x(r)dr≥0) is continuous;(ii)
ifKr,x(r)=x(r) for all r∈0,T, then we have (56)Kr,∫0txωdω=∫0tKω,xωdω∀r∈0,T;(iii)
there existsx0∈X such that ζgx0(t),fx0(t)≥0 for all t∈0,T;(iv)
for allt∈[0,T] and x,y∈X, (57)ζgxt,gyt≥0impliesthatζfxt,fyt≥0;(v)
propertiesHsp and Usp are satisfied;(vi)
there is a continuous functionγ:0,T→R+ such that(58)Kr,xr+Kr,yr≤γrxr+yrfor allr∈0,T, x,y∈X;(vii)
there exist constantL∈0,1, p≥2 such that for all r∈0,T(59)supt∈0,T∫0Tγrdr≤Lspm.
Then the system of integral equations (53) has a unique solution in x∈X.Proof.
We define a functionα:X×X→0,∞ by(60)αx,y=spifξxt,yt≥00otherwise. It is easy to see that the function f is g-αsp-admissible. By condition (ii) f and g are weakly compatible.
Letx,y∈X=C0,T,R be such that αgx,gy≥sp, that is, ζgx(t),gy(t)≥0 for all t∈0,T, then from conditions (iv), (vi), and (vii), for all t∈0,T, we have (61)σbfxt,fyt=fxt+fytm=∫0tKr,xrdr+∫0tKr,yrdrm≤∫0tKr,xrdr+∫0tKr,yrdrm=∫0tKr,xr+Kr,yrdrm≤∫0tγrxr+yrdrm≤supt∈0,Tγrm∫0txr+yrdrm≤supt∈0,Tγrmgxt+gytm≤Lsp1/mmσbgxt,gyt=Lspσbgxt,gyt≤LspNxt,yt, which implies that(62)αgx,gyσbfxt,fyt≤LNxt,yt.Therefore, taking ψ(x)=x and ϕ(x)=Lx, where L∈0,1, and in view of assertion (v) all of the conditions of Theorem 27 are satisfied, and, as a result, the mappings f and g have a unique common fixed point in X, which is a solution of the system of integral equations in (53).


## 5. Conclusions

In this manuscript, we defined new rational contraction using a larger class ofαsp-admissible mappings and auxiliaries functions in the framework of b-metric-like spaces. The presented main theorems of the paper cover and unify a huge number of published results and also complement the previous work on the topic in the related literature.



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