# A Generalization of Prešić Type Mappings in Metric-Like Spaces

Authors: Satish Shukla; Brian Fisher
Journal: Journal of Operators
 (2013)

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## Abstract

We generalize the result of Prešić in metric-like spaces by proving some common fixed point theorems for Prešić type mappings in metric-like spaces. An example is given which shows that the generalization is proper.


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## Body

## 1. Introduction and Preliminaries

Let(X,d) be any metric space, and let f:X→X be any mapping; then f is said to be a contraction on X if there exists λ∈[0,1) such that
(1)d(fx,fy)≤λd(x,y),∀x,y∈X.A pointx∈X is called a fixed point of f if fx=x. Banach [1] proved that every contraction on a complete metric space has a unique fixed point and this result is known as the Banach contraction principle. There are several generalizations of this famous principle. One such generalization is given by Prešić [2, 3]. When studying the convergence of some particular sequences, Prešić [2, 3] proved the following theorem.Theorem 1.
Let(X,d) be a complete metric space, k a positive integer, and f:Xk→X a mapping satisfying the following contractive type condition:
(2)d(f(x1,x2,…,xk),f(x2,x3,…,xk+1))≤∑i=1k‍qid(xi,xi+1),
for every x1,x2,…,xk+1∈X, where q1,q2,…,qk are nonnegative constants such that q1+q2+⋯+qk<1. Then there exists a unique point x∈X such that f(x,x,…,x)=x. Moreover, if x1,x2,…,xk are arbitrary points in X and for n∈ℕ,
(3)xn+k=f(xn,xn+1,…,xn+k-1),
then the sequence {xn} is convergent and limxn=f(limxn,limxn,…,limxn).Note that thek-step iterative sequence given by (3) represents a nonlinear difference equation. In view of Prešić theorem, it is obvious that if this sequence is convergent (which is ensured by the Prešić theorem) then the limit of the sequence is a fixed point of f. The result of Prešić is generalized by several authors, and some generalizations and applications of Prešić theorem can be seen in [4–15].On the other hand, Matthews [16] introduced the notion of a partial metric space as a part of the study of denotational semantics of a dataflow network. In this space, the usual metric is replaced by a partial metric with an interesting property that the self-distance of any point of space may not be zero. Further, Matthews showed that the Banach contraction principle is valid in a partial metric space and can be applied in program verifications. O'Neill [17] generalized the concept of a partial metric space a bit further by admitting negative distances. The partial metric defined by O'Neill is called the dualistic partial metric. Heckmann [18] generalized it by omitting the small self-distance axiom. The partial metric defined by Heckmann is called a weak partial metric.Recently, Amini-Harandi [19] generalized the partial metric spaces by introducing the metric-like spaces. Amini-Harandi introduced the notion of a σ-Cauchy sequence and completeness of metric-like spaces and proved some fixed point theorems in such spaces. In this paper, we prove some common fixed point theorems for Prešić type mappings in metric-like spaces. Our results generalize and extend the result of Prešić from complete spaces into σ-complete metric-like spaces. An example is given which shows that the generalization is proper.First we recall some definitions about partial metric and metric-like spaces.Definition 2 (see [16]).
A partial metric on a nonempty setX is a function p:X×X→ℝ+ (ℝ+ stands for nonnegative reals) such that for all x,y,z∈X:(p1)
x=y if and only if p(x,x)=p(x,y)=p(y,y);(p2)
p(x,x)≤p(x,y);(p3)
p(x,y)=p(y,x);(p4)
p(x,y)≤p(x,z)+p(z,y)-p(z,z).A partial metric space is a pair (X,p) such that X is a nonempty set and p is a partial metric on X. A sequence {xn} in (X,p) converges to a point x∈X if and only if p(x,x)=limn→∞p(xn,x). A sequence {xn} in (X,p) is called a p-Cauchy sequence if limn,m→∞p(xn,xm) exists and is finite. (X,p) is said to be complete if every p-Cauchy sequence {xn} in X converges to a point x∈X such that p(x,x)=limn,m→∞p(xn,xm).Definition 3 (see [19]).
A metric-like on a nonempty setX is a function σ:X×X→ℝ+ such that for all x,y,z∈X:(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).A metric-like space is a pair (X,σ) such that X is a nonempty set and σ is a metric-like on X. Note that a metric-like satisfies all the conditions of metric except that σ(x,x) may be positive for some x∈X. Each metric-like σ on X generates a topology τσ on X whose base is the family of open σ-balls
(4)Bσ(x,ϵ)={y∈X:|σ(x,y)-σ(x,x)|<ϵ},∀x∈X,ϵ>0.
A sequence{xn} in X converges to a point x∈X if and only if limn→∞σ(xn,x)=σ(x,x). A sequence {xn} is said to be σ-Cauchy if limn,m→∞σ(xn,xm) exists and is finite. The metric-like space (X,σ) is called complete if for each σ-Cauchy sequence {xn}, there exists x∈X such that
(5)limn→∞σ(xn,x)=σ(x,x)=limm,n→∞σ(xn,xm).Every partial metric space is a metric-like space but the converse may not be true.Example 4 (see [19]).
LetX={0,1} and σ:X×X→ℝ+ be defined by
(6)σ(x,y)={2,ifx=y=0,1,otherwise.
Then(X,σ) is a metric-like space, but it is not a partial metric space, as σ(0,0)≰σ(0,1).Example 5.
LetX=ℝ, k≥0, and σ:X×X→ℝ+ be defined by
(7)σ(x,y)={2k,ifx=y=0,k,otherwise.
Then(X,σ) is a metric-like space, but for k>0, it is not a partial metric space, as σ(0,0)≰σ(0,1).Example 6.
LetX=ℝ+ and σ:X×X→ℝ+ be defined by
(8)σ(x,y)={2x,ifx=y,max{x,y},otherwise.
Then(X,p) is a metric-like space, but it is not a partial metric space, as σ(1,1)=2≰σ(0,1)=1.For the following definition, we refer to [10] and the references therein.Definition 7.
LetX be a nonempty set, k a positive integer, and f:Xk→X a mapping. If f(x,x,…,x)=x, then x∈X is called a fixed point of f.Definition 8.
LetX be a nonempty set, k a positive integer, and f:Xk→X and g:X→X mappings.(a)
An elementx∈X is said to be a coincidence point of f and g if gx=f(x,…,x).(b)
Ifw=gx=f(x,…,x), then w is called a point of coincidence of f and g.(c)
Ifx=gx=f(x,…,x), then x is called a common fixed point of f and g.(d)
Mappingsf and g are said to be commuting if g(f(x,…,x))=f(gx,…,gx) for all x∈X.(e)
Mappingsf and g are said to be weakly compatible if they commute at their coincidence points.Let(X,σ) be a metric-like space, k a positive integer, and f:Xk→X a mapping. f is said to be Prešić type if
(9)σ(f(x1,x2,…,xk),f(x2,x3.…,xk+1))≤∑i=1k‍αiσ(xi,xi+1),
for all x1,x2,…,xk,xk+1∈X, where αi are nonnegative constants such that ∑i=1kαi<1.The following lemma shows that the self-distance of a point of coincidence of two maps of a metric-like space satisfying Prešić type contractive condition is minimum (zero), and it will be used in the sequel.Lemma 9.
Let(X,σ) be a metric-like space, k a positive integer, and f:Xk→X, g:X→X two mappings. Suppose that the following condition holds:
(10)σ(f(x1,x2,…,xk),f(x2,x3,…,xk+1))≤∑i=1k‍αiσ(gxi,gxi+1),
for all x1,x2,…,xk+1∈X, where α1,α2,…,αk are nonnegative constants such that α1+α2+⋯+αk<1. If f and g have a point of coincidence v∈X, then σ(v,v)=0.Proof.
Letv∈X be any point of coincidence of f and g. Then there exists u∈X such that f(u,u,…,u)=gu=v.
Now suppose thatσ(v,v)≠0. Then it follows from (10) that
(11)σ(v,v)=σ(f(u,u,…,u),f(u,u,…,u))≤α1σ(gu,gu)+α2σ(gu,gu)+⋯+αkσ(gu,gu)=[α1+α2+⋯+αk]σ(gu,gu)=[α1+α2+⋯+αk]σ(v,v)<σ(v,v)(asα1+α2+⋯+αk<1),
a contradiction. Therefore we must have σ(v,v)=0.Now we can state our main results.


## 2. Main Results

The following theorem extends and generalizes the result of Prešić in metric-like spaces.Theorem 10.
Let(X,σ) be a metric-like space, k a positive integer, and f:Xk→X, g:X→X two mappings such that f(Xk)⊂g(X) and g(X) is a complete subspace of X. Suppose that the following condition holds:
(12)σ(f(x1,x2,…,xk),f(x2,x3,…,xk+1))≤∑i=1k‍αiσ(gxi,gxi+1),
for every x1,x2,…,xk+1∈X, where α1,α2,…,αk are nonnegative constants such that α1+α2+⋯+αk<1. Then f and g have a unique point of coincidence v∈X and σ(v,v)=0. Moreover, if f and g are weakly compatible, then v is the unique common fixed point of f and g.Proof.
Letx1,x2,…,xk∈X be arbitrary points. As f(Xk)⊂g(X), we can define a sequence {yn} in X by yn=gxn for n=1,2,…,k and yn+k=gxn+k=f(xn,xn+1,…,xn+k-1) for n=1,2,….
For simplicity putσn=σ(yn,yn+1) for n=1,2,… and μ=max{σ1/θ,σ2/θ2,…,σk/θk}, where θ=[∑i=1kαi]1/k<1.
By mathematical induction, we will show that(13)σn≤μθn∀n∈ℕ.
According to the definition of μ, it is clear that (13) is true for n=1,2,…,k. Let the following k inequalities:
(14)σn≤μθn,σn+1≤μθn+1,…,σn+k-1≤μθn+k-1,
be the induction hypothesis.
Now using (12), we obtain
(15)σn+k=σ(yn+k,yn+k+1)=σ(f(xn,xn+1,…,xn+k-1),f(xn+1,xn+2,…,xn+k))≤α1σ(gxn,gxn+1)+α2σ(gxn+1,gxn+2)+⋯+αkσ(gxn+k-1,gxn+k)=α1σ(yn,yn+1)+α2σ(yn+1,yn+2)+⋯+αkσ(yn+k-1,yn+k)=α1σn+α2σn+1+⋯+αkσn+k-1≤α1μθn+α2μθn+1+⋯+αkμθn+k-1≤α1μθn+α2μθn+⋯+αkμθn(asθ=[∑i=1k‍αi]1/k<1)=[α1+α2+⋯+αk]μθn=[∑i=1k‍αi]μθn=μθn+k.
Thus, the inductive proof of (13) is complete.
Now letn,m∈ℕ with m>n. Then from (13), we have
(16)σ(yn,ym)≤σ(yn,yn+1)+σ(yn+1,yn+2)+⋯+σ(ym-1,ym)=σn+σn+1+⋯+σm-1≤μθn+μθn+1+μθn+2+⋯=μθn[1+θ+θ2+⋯]=μθn1-θ.
As, θ<1, it follows from the previous inequality that
(17)limn,m→∞σ(yn,ym)=0.
Thus, {yn}={gxn} is a σ-Cauchy sequence in (X,σ). By the completeness of g(X), there exists u,v∈X such that v=gu and
(18)limn→∞σ(yn,v)=limm,n→∞σ(yn,ym)=σ(v,v)=0.
We will now show thatv is a point of coincidence of f and g. For any n∈ℕ we have
(19)σ(gu,f(u,u,…,u))≤σ(gu,yn+k)+σ(yn+k,f(u,u,…,u))≤σ(v,yn+k)+σ(f(xn,…,xn+k-1),f(u,u,…,u))≤σ(v,yn+k)+σ(f(xn,…,xn+k-1),f(xn+1,…,xn+k-1,u))+σ(f(xn+1,…,xn+k-1,u),f(xn+2,…,xn+k-1,u,u))+⋯+σ(f(xn+k-1,u,…,u),f(u,u,…,u)).
Using (12) and writing σn=σ(yn,yn+1)=σ(gxn,gxn+1), gu=v in the previous inequality we obtain
(20)σ(v,f(u,u,…,u))≤σ(v,yn+k)+α1σn+⋯+αk-1σn+k-2+αkσ(yn+k-1,v)+α1σn+1+⋯+αk-2σn+k-2+αk-1σ(yn+k-1,v)+αkσ(v,v)+⋯+α1σ(yn+k-1,v)+[α2+⋯+αk]σ(v,v).
Letting n→∞ and using (18) in the previous inequality, we obtain
(21)σ(v,f(u,u,…,u))=0,that isf(u,u,…,u)=v=gu.
Therefore, v is a point of coincidence of f and g.
We will now show that it is unique. Suppose thatv′ is another point of coincidence of f and g. Then there exists u′∈X such that f(u′,u′,…,u′)=gu′=v′. By Lemma 9, we have σ(v′,v′)=0. Now it follows from (12) that
(22)σ(v,v′)=σ(f(u,u,…,u),f(u′,u′,…,u′))≤σ(f(u,…,u),f(u,…,u,u′))+σ(f(u,…,u,u′),f(u,…,u,u′,u′))+⋯+σ(f(u,u′…,u′),f(u′,u′,…,u′))≤α1σ(gu,gu)+⋯+αk-1σ(gu,gu)+αkσ(gu,gu′)+α1σ(gu,gu)+⋯+σk-2σ(gu,gu)+αk-1σ(gu,gu′)+αkσ(gu′,gu′)+⋯+α1σ(gu,gu′)+α2σ(gu′,gu′)+⋯+αkσ(gu′,gu′).
As σ(gu,gu)=σ(v,v)=0=σ(v′,v′)=σ(gu′,gu′), it follows from previous inequality that
(23)σ(v,v′)≤[α1+α2+⋯+αk]p(gu,gu′)=[α1+α2+⋯+αk]p(v,v′)<p(v,v′)(asα1+α2+⋯+αk<1),
a contradiction. Therefore, we must have σ(v,v′)=0, that is, v=v′. Thus the point of coincidence of f and g is unique.
Now suppose thatf and g are weakly compatible. Put w=gv. Then w=gv=gf(u,u,…,u)=f(gu,gu,…,gu)=f(v,v,…,v), so w is a point of coincidence of f and g. However, by the uniqueness of v, we have w=v=gv=f(v,v,…,v). Thus, v is the unique common fixed point of f and g.Takingf=IX (i.e., the identity mapping of X) we obtain following fixed point result for Prešić type mapping in a metric-like space.Corollary 11.
Let(X,σ) be a complete metric-like space, k a positive integer, and f:Xk→X a Prešić type mapping on X. Then f has a unique fixed point v∈X and σ(v,v)=0.The following example shows that Corollary11 is a proper generalization of Theorem 1.Example 12.
LetX=[0,1], and define σ:X×X→ℝ+ by
(24)σ(x,y)={2x,ifx=y,max{x,y},otherwise.
Then,(X,σ) is a complete metric-like space. Note that (X,σ) is not a partial metric space, as σ(1,1)≰σ(0,1).
Definef:X2→X by
(25)f(x,y)={116,ifx=y=1,x+y8,otherwise.
Note thatf is not a Prešić type mapping in the usual metric space (X,d), where d(x,y)=|x-y| for all x,y∈X. Indeed, for the points x1=x2=1,x3=9/10 we have
(26)d(f(x1,x2),f(x2,x3))=d(116,1980)=740,
and α1d(1,1)+α2d(1,9/10)=(1/10)α2. Therefore, the condition (2) of Theorem 1 is not satisfied with α1+α2<1. Thus the result of Prešić is not applicable here.
On the other hand, it is easy to see thatf satisfies all the conditions of Corollary 11, with α1=α2∈[1/8,1) and f(0,0)=0; that is, 0 is the unique fixed point of f.



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