# Some Fixed Point Theorems inb-Metric Space Endowed with Graph

Authors: Maria Samreen; Tayyab Kamran; Naseer Shahzad
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We define some notions of contraction mappings inb-metric space endowed with a graph G and subsequently establish some fixed point results for such classes of contractions. According to the applications of our results, we obtain fixed point theorems for cyclic operators and an existence theorem for the solution of an integral equation.


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## Body

## 1. Introduction and Preliminaries

The study ofb-metric spaces was initiated in some works of Bakhtin, Heinonen, Bourbaki, and Czerwik [1–4]. Afterwards, several articles which deal with fixed point theorems for single-valued and multivalued functions in b-metric space appeared [1–8].Definition 1 (see [1, 4]).
LetX be a set, and let s≥1 be a given real number. A function d:X×X→ℝ+ is said to be a b-metric on X, and the pair (X,d) is called a b-metric space if, for all x,y,z∈X,  (d1)
d(x,y)=0 if and only if x=y,(d2)
d(x,y)=d(y,x),(d3)
d(x,z)≤s[d(x,y)+d(y,z)].Note that the class ofb-metric spaces contains the class of metric spaces.Example 2.
LetX:=lp(ℝ) with 0<p<1, where lp(ℝ):={x={xn}⊂ℝ:∑n=1∞‍|xn|p<∞}. Then, d(x,y)=(∑n=1∞‍|xn-yn|p)1/p is a b-metric on X with s=21/p.A sequence{xn} in a b-metric space X is said to be convergent if and only if there exists x∈X such that d(xn,x)→0 as n→∞. In this case, we write limn→∞xn=x. A sequence {xn} in a b-metric space X is said to be Cauchy if and only if d(xn,xm)→0 as m,n→∞. A b-metric space (X,d) is complete if every Cauchy sequence in X converges. In general, a b-metric is not continuous.The famous Banach contraction principle [9] infers that every contraction on a complete metric space has a unique fixed point. Recently, Jachymski [10] introduced the notion of Banach G-contraction to generalize Banach contraction principle as follows. Let (M,δ) be a metric space, let Δ be the diagonal of the Cartesian product M×M, and let G be a directed graph such that the set V(G) of its vertices coincides with M and the set E(G) of its edges contains all loops; that is, E(G)⊇Δ. Assume that G has no parallel edges. A mapping f:M→M is called a Banach G-contraction if (i) for  all x,y∈X  ((x,y)∈E(G)⇒(fx,fy)∈E(G)),  (ii)  ∃α, 0<α<1 such that for all x,y∈X,(x,y)∈E(G)⇒δ(fx,fy)≤αδ(x,y). A mapping f:M→M is known as Picard operator [11] if f has a unique fixed point x* and limn→∞fnx=x* for all x∈M.Various generalizations of Banach's principle have been obtained by weakening contractive conditions. In this context, Matkowski [12] introduced class of φ-contractions in metric fixed point theory, and subsequently further study was developed in this setting by different authors when underlying space was taken to be a partially ordered set (see, e.g., [13, 14]).Letφ:ℝ+→ℝ+. Consider the following properties:(i)φ
t1≤t2⇒φ(t1)≤φ(t2), for all t1,t2∈ℝ+,(ii)φ
φ(t)<t for t>0,(iii)φ
φ(0)=0,(iv)φ
limn→∞φn(t)=0 for all t≥0,(v)φ
∑n=0∞‍φn(t) converges for all t>0.It is easily seen that (i)φ and (iv)φ imply (ii)φ and (i)φ and (ii)φ imply (iii)φ.We recall that a functionφ satisfying (i)φ and (iv)φ is said to be a comparison function. A function φ satisfying (i)φ and (v)φ is known as (c)-comparison function.Any(c)-comparison function is a comparison function, but converse may not be true. For example, φ(t)=t/(1+t); t∈ℝ+ is a comparison function but not a (c)-comparison function. On the other hand, define φ(t)=t/2; 0≤t≤1 and φ(t)=t-(1/2); t>1, and then φ is a (c)-comparison function. For details on φ contractions, we refer the readers to [15, 16].Berinde [17] took further step to investigate φ contractions when the framework was taken to be a b-metric space, and for some technical reasons, he had to introduce the notion of b-comparison function; in particular, he obtained some estimations for rate of convergence [17]. For other related results, see also [5, 7, 17–21].Definition 3.
Lets≥1 be a fixed real number. A function φ:ℝ+→ℝ+ is known as b-comparison function if it satisfies (i)φ, and the following holds:(vi)φ
∑n=0∞‍snφn(t) converges for all t∈ℝ+.The concept ofb-comparison function coincides with comparison function when s=1. Let (X,d) be a b-metric space with coefficient s≥1, and then φ(t)=at; t∈ℝ+ with 0<a<(1/s) is a b-comparison function.


## 2. Main Results

Throughout this section, let(X,d) be a b-metric space with coefficient s≥1, and Δ is the diagonal of the Cartesian product X×X. G is a directed graph such that the set V(G) of its vertices coincides with X, and the set E(G) of its edges contains all loops; that is, E(G)⊇Δ. Assume that G has no parallel edges. We assign to each edge having vertices x and y a unique element d(x,y). Now, we introduce the following definition.Definition 4.
One says that a mappingf:X→X is a b-(φ,G) contraction if for all x,y∈X:
(1)(fx,fy)∈E(G)whenever(x,y)∈E(G);(2)d(fx,fy)≤φ(d(x,y))whenever(x,y)∈E(G),
where φ:ℝ+→ℝ+ is a comparison function.Remark 5.
Note that a BanachG-contraction is a b-(φ,G) contraction.Example 6.
Any constant mappingf:X→X is a b-(φ,G) contraction for any graph G with V(G)=X.Example 7.
Any self-mappingf on X is trivially a b-(φ,G1) contraction, where G1=(V(G),E(G))=(X,Δ).Example 8.
LetX=ℝ, and define d:X×X→ℝ by d(x,y)=|x-y|2. Then, d is a b-metric on X with s=2. Further, fx=x/2, for all x∈X. Then, f is a b-(φ,G0) contraction with φ(t)=t/4 and G0=(X,X×X). Note that d is not a metric on X.Definition 9.
Two sequences{xn} and {yn} in X are said to be equivalent if limn→∞d(xn,yn)=0, and if each of them is a Cauchy sequence, then they are called Cauchy equivalent.As a consequence of Definition9, we get the following lemma.Remark 10.
Let{xn} and {yn} be equivalent sequences in X. (i) If {xn} converges to x, then {yn} also converges to x and vice versa. (ii)  {yn} is a Cauchy sequence whenever {xn} is a Cauchy sequence and vice versa.Now, we recollect some preliminaries from graph theory which we need for the sequel. LetG=(V(G), and let E(G)) be a directed graph. By letter G~, we denote the undirected graph obtained from G by ignoring the direction of edges. If x and y are vertices in a graph G, then a path in G from x to y of length l is a sequence {xi}i=0l of l+1 vertices such that x0=x,xl=y, and (xi-1,xi)∈E(G) for i=1,…,l. A graph G is called connected if there is a path between any two vertices. G is weakly connected if G~ is connected. For a graph G such that E(G) is symmetric and x is a vertex in G, the subgraph Gx consisting of all edges and vertices which are contained in some path beginning at x is known as a component of G containing x. So that V(Gx)=[x]G~, where [x]G~ is the equivalence class of a relation R defined on V(G) by the rule: yRz if there is a path in G from y to z. Clearly, Gx is connected. A graph G is known as (C)-graph in X [22] if for any sequence {xn} in X with xn→x and (xn,xn+1)∈E(G) for n∈ℕ; then there exists a subsequence {xnk} of {xn} such that (xnk,x)∈E(G) for k∈ℕ.Proposition 11.
Letf:X→X be a b-(φ,G) contraction, where φ:ℝ+→ℝ+ is a comparison function; then  (i)
fis ab-(φ,G~)contraction and ab-(φ,G-1)contraction as well,(ii)
[x0]G~isf-invariant, andf|[x0]G~is ab-(φ,G~x)contraction provided thatx0∈Xis such thatfx0∈[x0]G~.Proof.
(i) By using (d2) (Definition1) it can be easily proved.
(ii) Letx∈[x0]G~. Then, there is a path x=z0,z1,…,zl=x0 between x and x0. Since f is a b-(φ,G) contraction, then (fzi-1,fzi)∈E(G) for all i=1,2,…,l. Thus, fx∈[fx0]G~=[x0]G~.
Suppose(x,y)∈E(G~x0). Then, (fx,fy)∈E(G), since f is a b-(φ,G) contraction. But [x0]G~ is f invariant, so we conclude that (fx,fy)∈E(G~x0). Condition (2) is satisfied automatically as G~x0 is a subgraph of G.From now on, we assume that coefficient ofb-comparison function is at least as large as the coefficient of b-metric s.Lemma 12.
Letf:X→X be a b-(φ,G) contraction, where φ:ℝ+→ℝ+ is a b-comparison function. Then, given any x∈X and y∈[x]G~, two sequences {fnx} and {fny} are equivalent.Proof.
Letx∈X, and let y∈[x]G~; then there exists a path {xi}i=0l in G~ from x to y with x0=x, xl=y, and (xi-1,xi)∈E(G~). From Proposition 11, f is a b-(φ,G~) contraction. So,
(3)(fnxi-1,fnxi)∈E(G~)impliesd(fnxi-1,fnxi)≤φ(d(fn-1xi-1,fn-1xi))
for all n∈ℕ and i=0,1,2,…,l. Hence,
(4)d(fnxi-1,fnxi)≤φn(d(xi-1,xi))∀n∈ℕ,i=0,1,2,…,l.
We observe that {fnxi}i=0l is a path in G~ from fnx to fny. Using (d3) Definition 1 and (4), we have
(5)d(fnx,fny)≤∑i=1l‍sid(fnxi-1,fnxi)≤∑i=1l‍siφn(d(xi-1,xi)).
Letting n→∞, we obtain d(fnx,fny)→0.Proposition 13.
Letf be a b-(φ,G) contraction, where φ:ℝ+→ℝ+ is a b-comparison function. Suppose that there is z0 in X such that fz0∈[z0]G~. Then, {fnz0} is a Cauchy sequence in X.Proof.
Sincefz0∈[z0]G~, let {yi}i=0r be a path from z0 to fz0. Then using the same arguments as in Lemma 12, we arrive at
(6)d(fnz0,fn+1z0)≤∑i=1r‍siφn(d(yi-1,yi)),∀n∈ℕ.
Let m>n≥1, and then from above inequality; it follows for p≥1(7)d(fnz0,fn+pz0)≤sd(fnz0,fn+1z0)+s2d(fn+1z0,fn+2z0)+⋯+spd(fn+p-1z0,fn+pz0)≤1sn-1[∑j=nn+p-1‍sjd(fjz0,fj+1z0)]≤1sn-1[∑i=1r‍si∑j=nn+p-1‍sjφj(d(yi-1,yi))].
Denoting for each i=1,2,…,r(8)Sni=∑k=0n‍skφk(d(yi-1,yi)),n≥1,
relation (7) becomes
(9)d(fnz0,fn+pz0)≤1sn-1[∑i=1r‍si[Sn+p-1i-Sn-1i]],
since φ is a b-comparison function, so that for each i=1,2,…,r,
(10)∑k=0∞‍skφk(d(yi-1,yi))<∞.
Then, corresponding to each i, there is a real number Si such that
(11)limn→∞Sni=Si.
In view of (11), relation (9) gives d(fnz0,fn+pz0)→0 as n→∞. Which shows that {fnz0} is a Cauchy sequence in X.Definition 14.
Letf:X→X, and let y∈X, and the sequence {fny} in X is such that fny→x* with (fny,fn+1y)∈E(G) for n∈ℕ. One says that a graph G is (Cf)-graph if there exists a subsequence {fnky} and a natural number p such that (fnky,x*)∈E(G) for all k≥p [23]. One says that a graph G is (Hf)-graph if fny∈[x*]G~ for n≥1; then r(fny,x*)→0(asn→∞), where r(fny,x*)=∑i=1M‍sid(zi-1,zi); {zi}i=0M is a path from fny to x* in G~.Obviously every(C)-graph is a (Cf)-graph for any self-mapping f on X, but converse may not hold as shown in the following.Example 15.
LetX=[0,1] with respect to b-metric d(x,y)=|x-y|2. Consider a graph G consisting of V(G)=X and E(G)={(n/(n+1), (n+1)/(n+2)):n∈ℕ}∪{(x/2n,x/2n+1):n∈ℕ, x∈[0,1]}∪{((x/22n), 0):n∈ℕ, x∈[0,1]}. Note that G is not a (C)-graph as n/(n+1)→1. Define f:X→X as fx=x/2. Then, G is a (Cf)-graph, since fnx=x/2n→0.Example 16.
LetX={1/n:n∈ℕ}∪{0}∪ℕ with respect to b-metric d(x,y)=|x-y|2, and let I be identity map on X. Consider a graph G2 consisting of V(G2)=X and
(12)E(G2)={(1n,1n+1),(1n+1,n),(n,0),(15n,0);n∈ℕ},
since xn=1/n→0 as n→∞. We note that G2 is a (CI)-graph, but r(xn,0)=2|(1/n)-(1/(n+1))|2+22|(1/(n+1))-n|2+22n2↛0 as n→∞. Thus, G2 is not an (HI)-graph.Example 17.
LetX={1/n:n∈ℕ}∪{5/n:n∈ℕ}∪{0} with respect to b-metric d(x,y)=|x-y|2, and let I be identity map on X. Consider a graph G3 consisting of V(G3)=X and
(13)E(G3)={(1n,1n+1),(1n+1,5n),(5n,0);n∈ℕ},
since xn=1/n→0 as n→∞. Clearly G3 is not a (CI)-graph, but it is easy to verify that G3 is an (HI)-graph.Above examples show that for a givenf notions, (Cf)-graph and (Hf)-graph are independent even if f is identity map.Theorem 18.
Let(X,d) be a complete b-metric space, and let f be a b-(φ,G) contraction, where φ is b-comparison function. Assume that d is continuous and there is z0 in X for which (z0,fz0) is an edge in G~. Then, the following assertions hold.  (1)
IfG is a (Cf)-graph, then f has a unique fixed point t∈[z0]G~, and for any y∈[z0]G~, fny→t. Further, if G is weakly connected, then f is Picard operator.(2)
IfG is weakly connected (Hf)-graph, then f has a unique fixed point t∈X and for any y∈X, fny→t.Proof.
(1) It follows from Proposition13 that {fnz0} is a Cauchy sequence in X. Since X is complete, there exists t∈X such that fnz0→t. Since (fnz0,fn+1z0)∈E(G), for all n∈ℕ, and G is a (Cf) graphs, there exists a subsequence {fnkz0} of {fnz0} and p∈ℕ such that (fnkz0,t)∈E(G) for all k≥p. Observe that (z0,fz0,f2z0,…,fn1z0,…,fnpz0,t) is a path in G~. Therefore, t∈[z0]G~. From (2), we get
(14)d(fnk+1z0,ft)≤φ(d(fnkz0,t))<d(fnkz0,t)∀k≥n0.
Letting k→∞, we obtain limk→∞fnk+1z0=ft, as d is continuous. Since {fnkz0} is a subsequence of {fnz0}, we conclude that ft=t. Finally, if y∈[z0]G~, it follows from Lemma 12 that fny→t.
(2) LetG be weakly connected (Hf)-graph. From Proposition 13, fnz0→t∈X, and then r(fnz0,t)→0 as n→∞. Now, for each n∈ℕ, let {yin};i=0,1,…,Mn be a path from fnz0 to t with y0=t and yMnn=fnz0 in G~; then
(15)d(t,ft)≤s[d(t,fn+1z0)+d(fn+1z0,ft)]≤s[d(t,fn+1z0)+∑i=1Mn‍sid(fyi-1n,fyin)]≤s[d(t,fn+1z0)+∑i=1Mn‍siφ(d(yi-1n,yin))]<s[d(t,fn+1z0)+∑i=1Mn‍sid(yi-1n,yin)]=s[d(t,fn+1z0)+r(fnz0,t)].
Letting n→∞, the above inequality yields ft=t. Let y∈[z0]G~:=X be arbitrary; then from Lemma 12 and Remark 10, it is easily seen that fny→t.The following example shows that the condition of(Cf) or (Hf)-graph in the hypothesis of Theorem 18 can not be dropped.Example 19.
LetX=[0,1], let d(x,y)=|x-y|2, and let fx=x/2 for all x∈(0,1] and f0=1/2. Then, (X,d) is a complete b-metric space with s=2. Further, d is continuous, and f is a b-(φ,G1) contraction (with φ(t)=t/4), where V(G1)=X and E(G1)={(x,y)∈(0,1]×(0,1]; x≥y}∪{(0,0),(0,1)}. Note that G1 is weakly connected, but f has no fixed point in [z0]G1~=X. Observe that G1 is not a (Cf)-graph because the sequence fnx=x/2n→0 for x∈(0,1] and (fnx,fn+1x)∈E(G1);n∈ℕ, but it does not contain any subsequence such that (xnk,0)∈E(G1). Also, we note that for any fixed x∈(0,1], r(fnx,0)=2[|x/2n-1|2+|1-0|2]↛0 as n→∞.Definition 20.
Let(X,d) be a b-metric space. A mapping f:X→X is called orbitally continuous if for all x,y∈X and any sequence {kn}n∈ℕ of positive integers, fknx→y implies f(fknx)→fy as n→∞. A mapping f:X→X is called orbitally G-continuous if for all x,y∈X and any sequence {kn}n∈ℕ of positive integers, fknx→y and (fknx,fkn+1x)∈E(G) for all n∈ℕ imply f(fknx)→fy.Theorem 21.
Let(X,d) be a complete b-metric space, and let f be a b-(φ,G) contraction, where φ is a b-comparison function. Assume that d is continuous, f is orbitally G-continuous, and there is z0 in X for which (z0,fz0) is an edge in G. Then, f has a fixed point t∈X. Moreover, for any y∈[z0]G~, fny→t.Proof.
It follows from Proposition13 that {fnz0} is a Cauchy sequence in (X,d). Since X is complete, there exists t∈X such that limn→∞fnz0=t. Since (fnz0,fn+1z0)∈E(G) for all n∈ℕ, f is orbitally G-continuous. Therefore, continuity of d implies that ft=t. Let y∈[z0]G~ be arbitrary; then it follows from Lemma 12 that limn→∞fny=t.Slightly strengthening the continuity condition onf, our next theorem deals with the graph G which may fail to have the property that there is z0 in X for which (z0,fz0) is an edge in G.Theorem 22.
Let(X,d) be a complete b-metric space, and let f be a b-(φ,G) contraction, where φ is a b-comparison function. Assume that d is continuous, f is orbitally continuous, and there is z0 in X for which fz0∈[z0]G~. Then, for any y∈[z0]G~, fny→t∈X, where t is a fixed point of f.Proof.
It follows from Proposition13 that {fnz0} is a Cauchy sequence in X. Since X is complete, there exists t∈X such that fnz0→t. Since f is orbitally continuous, then limn→∞ffnz0=ft which yields ft=t. Let y∈[z0]G~ be arbitrary; then from Lemma 12, limn→∞fny=t.Remark 23.
In addition to the hypothesis of Theorems21 and 22, if we assume that G is weakly connected, then f will become Picard operator [11] on X.Remark 24.
Theorem18 generalizes/extends claims 40 and 50 of [10, Theorem 3.2] and [7, Theorem 4(1)]. Theorem 21 generalizes claims 20 and 30 of [10, Theorem 3.3]. Theorem 22 generalizes claims 20 and 30 of [10, Theorem 3.4], and thus generalizes extends results of Nieto and Rodríguez-López [24, Theorems 2.1 and 2.3], Petrusel and Rus [11, Theorem 4.3], and Ran and Reurings [25, Theorem 2.1]. We mention here that Theorem 18 can not be improved using comparison function instead of b-comparison function (see, Gwóźdź-Łukawska and Jachymski [26, Example 2]).We observe that Theorem22 can be used to extend famous fixed point theorem of Edelstein to the case of b-metric space. We need to define notion of ϵ-chainable property for b-metric space.Definition 25.
Ab-metric space (X,d) is said to be ϵ-chainable, for some ϵ>0, if for x,y∈X there exist xi∈X;i=0,1,2,…,l with x0=x,xl=y such that d(xi-1,xi)<ϵ for i=1,2,…,l.Corollary 26.
Let(X,d) be a complete ϵ-chainable b-metric space. Assume that d is continuous, and there exists a b-comparison function φ:ℝ+→ℝ+ such that f:X→X satisfying,
(16)d(x,y)<ϵimpliesd(fx,fy)≤φ(d(x,y)),
for all x,y∈X. Then, f is a Picard operator.Proof.
Consider a graphG consisting of V(G)∶=X and (x,y)∈E(G) if and only if d(x,y)<ϵ. Since X is ϵ-chainable, G is weakly connected. Let (x,y)∈E(G), and from (16), we have
(17)d(fx,fy)≤φ(d(x,y))<d(x,y)<ϵ.
Then, d(fx,fy)∈E(G). Therefore, in view of (16), f is b-(φ,G) contraction. Further, (16) implies that f is continuous. Now, the conclusion follows by using Theorem 22.Now, we establish a fixed point theorem using a general contractive condition.Theorem 27.
Let(X,d) be a complete b-metric space, and let G be a (Cf)-graph in X×X such that V(G)=X and f:X→X is an edge-preserving mapping. Assume that d is continuous and there exist α,β,γ≥0 with sα+(s+1)β+s(s+1)γ<1 and for all (x,y)∈E(G)(18)d(fx,fy)≤αd(x,y)+β[d(x,fx)+d(y,fy)]+γ[d(x,fy)+d(y,fx)].
If there is z0 in X for which (z0,fz0) is an edge in G, then f has a fixed point in [z0]G~.Proof.
Sincef is edge-preserving, then (fnz0,fn+1z0)∈E(G) for all n∈ℕ. From (18) and using (d3), it follows that
(19)d(fnz0,fn+1z0)≤αd(fn-1z0,fnz0)+β[d(fn-1z0,fnz0)+d(fnz0,fn+1z0)]+γs[d(fn-1z0,fnz0)+d(fnz0,fn+1z0)].
On rearranging,
(20)d(fnz0,fn+1z0)≤[α+β+γs1-β-γs]d(fn-1z0,fnz0).
Repeating iteratively, we have
(21)d(fnz0,fn+1z0)≤[α+β+γs1-β-γs]nd(z0,fz0).
For m>n≥1 and using (d3) Definition 1, we have
(22)d(fnz0,fmz0)≤sd(fnz0,fn+1z0)+s2d(fn+1z0,fn+2z0)+⋯+sm-nd(fm-1z0,fmz0)≤1sn-1d(z0,fz0)∑j=nm-1‍sj[α+β+γs1-β-γs]j(using(21))<1sn-1d(z0,fz0)∑j=n∞‍sj[α+β+γs1-β-γs]j.
Since s[(α+β+γs)/(1-β-γs)]<1, then {fnz0} is a Cauchy sequence in X. By completeness of X, the sequence {fnz0} converges to t∈X. Since G is a (Cf)-graph, there exists a subsequence {fnkz0} and a natural number p such that (fnkz0,t)∈E(G) for all k≥p. Using (18) for all k≥p, we have
(23)d(fnk+1z0,ft)≤αd(fnkz0,t)+β[d(fnkz0,fnk+1z0)+d(t,ft)]+γ[d(fnkz0,ft)+d(t,fnk+1z0)].
Since the b-metric d is continuous and β+γ<1, letting k→∞ inequality (23) yields ft=t. Also note that (z0,fz0,f2z0,…,fn1z0,…,fnp,t) is a path in G and hence in G~, therefore t∈[z0]G~.We note that Theorem27 does not guarantee the uniqueness of fixed point, but this can be accomplished under some assumptions as in the following theorem.Theorem 28.
In addition to the hypothesis of Theorem27, one further assumes that if α+2sβ+2γ<1 for the same set of α,β,γ≥0 and for any two fixed points t1,t2∃z∈X such that (t1,z) and (t2,z)∈E(G). Then, f has a unique fixed point.Proof.
Lett1 and t2 be two fixed points of f, and then there exists z∈X such that (t1,z),(t2,z)∈E(G). By induction, we have (t1,fnz),(t2,fnz)∈E(G) for all n=0,1,…. From (18), we have
(24)d(t1,fnz)≤αd(t1,fn-1z)+βd(fn-1z,fnz)+γ[d(t1,fnz)+d(fn-1z,t1)]≤αd(t1,fn-1z)+βs[d(fn-1z,t1)+d(t1,fnz)]+γ[d(t1,fnz)+d(fn-1z,t1)].
On rearranging,
(25)d(t1,fnz)≤[α+sβ+γ1-sβ-γ]d(t1,fn-1z),∀n=1,2,….
Continuing recursively, (25) gives
(26)d(t1,fnz)≤[α+sβ+γ1-sβ-γ]nd(t1,z).
Since[(α+sβ+γ)/(1-sβ-γ)]<1, then limn→∞d(t1,fnz)=0. Similarly, one can show that limn→∞d(t2,fnz)=0, which by using (d3) Definition 1 infers that d(t1,t2)=0.Suppose that(X,⪯) is a partially ordered set. Consider graph G2 consisting of E(G2)={(x,y)∈X×X:x⪯yory⪯x}, and V(G2) coincides with X. We note that if a self-mapping f is monotone with respect to the order, then, for graph G2, it is obvious that f is edge-preserving, or equivalently we can say that f maps comparable elements onto comparable elements.Following corollaries are the direct consequences of Theorem28.Corollary 29.
Let(X,d) be a complete metric space, where X is partially ordered set with respect to ⪯. Let f:X→X be nondecreasing (or nonincreasing) with respect to ⪯. Assume that there exists α,β,γ≥0 with α+2β+2γ<1 such that,
(27)d(fx,fy)≤αd(x,y)+β[d(x,fx)+d(y,fy)]+γ[d(x,fy)+d(y,fx)],
for all comparable x,y∈X. If the following conditions hold:  (i)
there existsx0∈X such that x0⪯fx0,(ii)
for nondecreasing (or nonincreasing) sequence{xn}→x∈X, there exists a subsequence {xnk} such that xnk⪯x, for  all  k.Then, f has a fixed point. Moreover, if for  all x,y∈X there exists z∈X such that x⪯z and y⪯z, then the fixed point is unique.Corollary 30.
Let(X,d) be a complete metric space, where X is partially ordered set with respect to ⪯. Let f:X→X be nondecreasing (or nonincreasing) with respect to ⪯. Assume that there exists a constant 0<c<1/2 such that
(28)d(fx,fy)≤c[d(x,fx)+d(y,fy)],
for  all comparable x,y∈X. If the following conditions hold:  (i)
there existsx0∈X such that x0⪯fx0,(ii)
for nondecreasing (or nonincreasing) sequence{xn}→x∈X, there exists a subsequence {xnk} such that xnk⪯x, for  all  k.Then, f has a fixed point. Moreover, if for all x,y∈X there exists z∈X such that x⪯z and y⪯z, then the fixed point is unique.Corollary 31.
Let(X,d) be a complete metric space, where X is partially ordered set with respect to ⪯. Let f:X→X be nondecreasing (or nonincreasing) with respect to ⪯. Assume that there exists a constant 0<c<1/2 such that
(29)d(fx,fy)≤c[d(x,fy)+d(y,fx)],
for all comparable x,y∈X. If the following conditions hold:  (i)
there existsx0∈X such that x0⪯fx0,(ii)
for nondecreasing (or nonincreasing) sequence{xn}→x∈X, there exists a subsequence {xnk} such that xnk⪯x, for  all k.Then, f has a fixed point. Moreover, if for all x,y∈X there exists z∈X such that x⪯z and y⪯z, then the fixed point is unique.Remark 32.
We note that in Theorem27, the condition “there is z0 in X for which (z0,fz0) is an edge in G” yields fnz0→t, where t∈X is a fixed point of f. Consider a graph G:=(X,X×X). For such graph under the assumptions of Theorems 27 and 28, it infers that f is a Picard operator. Thus, many standard fixed point theorems can be easily deduced from Theorem 28 as follows.Corollary 33 (Hardy and Rogers [27]).
Let(X,d) be a complete metric space, and let f:X→X. Suppose that there exists constants α,β,γ≥0 such that
(30)d(fx,fy)≤αd(x,y)+β[d(x,fx)+d(y,fy)]+γ[d(x,fy)+d(y,fx)],
for all x,y∈X, where α+2β+2γ<1; then f has a unique fixed point in X.Corollary 34 (Kannan [28]).
Let(X,d) be a complete metric space, and let f:X→X. Suppose that there exists a constants c such that
(31)d(fx,fy)≤c[d(x,fx)+d(y,fy)],
for all x,y∈X, where 0<c<1/2; then f is Picard operator.Corollary 35 (Chatterjea [29]).
Let(X,d) be a complete metric space, and let f:X→X. Suppose that there exists a constants c such that
(32)d(fx,fy)≤c[d(x,fy)+d(y,fx)],
for all x,y∈X, where 0<c<1/2; then f is Picard operator.


## 3. Applications

LetX be a nonempty set, let m be a positive integer, {Xi}i=1m be nonempty closed subsets of X, and let f:⋃i=1mXi→⋃i=1mXi be an operator. Then, X:=⋃i=1mXi is known as cyclic representation of X w.r.t. f if
(33)f(X1)⊂X2,…,f(Xm-1)⊂Xm,f(Xm)⊂X1,
and operator f is known as cyclic operator [30].Theorem 36.
Let(X,d) be complete b-metric space such that d is continuous functional on X×X. Let m be positive integer, let {Xi}i=1m be nonempty closed subsets of X, let Y:=⋃i=1mXi, φ:ℝ+→ℝ+ be a b-comparison function, and let f:Y→Y. Further, suppose that  (i)
⋃i=1mXi is cyclic representation of  Y w.r.t. f,(ii)
d(fx,fy)≤φ(d(x,y)) whenever x∈Xi,y∈Xi+1, where Xm+1=X1.Then, f has a unique fixed point t∈⋂i=1mXi and fny→t for any y∈⋃i=1mXi.Proof.
We note that(Y,d) is complete b-metric space. Let us consider a graph G consisting of V(G):=Y and E(G):=Δ∪{(x,y)∈Y×Y:x∈Xi,y∈Xi+1;i=1,…,m}. By (i), it follows that f preserves edges. Now, let fnx→x* in Y such that (fnx,fn+1x)∈E(G) for all n≥1; then in view of (33), sequence {fnx} has infinitely many terms in each Xi so that one can easily extract a subsequence of {fnx} converging to x* in each Xi.  Since Xi's are closed, then x*∈⋂i=1mXi.  Now, it is easy to form a subsequence {fnkx} in some Xj, j∈{1,…,m} such that (fnkx,x*)∈E(G) for k≥1, and it indicates that G is weakly connected (Cf)-graph, and thus conclusion follows from Theorem 18.Remark 37 (see [31, Theorem  2.1(1)]).
It can be deduced from Theorem36 if (X,d) is a metric space.On the same lines as in proof of Theorem36, we obtain the following consequence of Theorem 28.Theorem 38.
Let(X,d) be a complete b-metric space such that d is continuous functional on X×X. Let m be positive integer, let {Xi}i=1m be nonempty closed subsets of X, let Y:=⋃i=1mXi, and let f:Y→Y. Further, suppose that  (i)
⋃i=1mXi is cyclic representation of Y w.r.t. f,(ii)
there existα,β,γ≥0 with sα+s(s+1)β+s(s+1)γ<1 such that
(34)d(fx,fy)≤αd(x,y)+β[d(x,fx)+d(y,fy)]+γ[d(x,fy)+d(y,fx)],
wheneverx∈Xi,y∈Xi+1, where Xm+1=X1.
Then,f has a fixed point t∈⋂i=1mXi.Remark 39.
[32, Theorem 7] and [33, Theorem 3.1] can be deduced from Theorem 38, but it does not ensure f to be a Picard operator.Remark 40.
We note that in proof [32, Theorem 7], the author's argument to prove that is G (as assumed in proof of Theorem 36), a (C)-graph is valid only if the terms of sequence {xn} are Picard iterations; otherwise it is void. For example, let Y=⋃i=13Xi where X1={1/n:nisodd}∪{0}, X2={1/n:niseven}∪{0}, X3={0}, and define f:Y→Y as fx=x/2;x∈Y∖X2 and fx=0;x∈X2. We see that (33) is satisfied and 1/n→0, but X3 does not contain infinitely many terms of {xn}.  In the following, we give the corrected argument to prove that G is a (C)-graph.Letxn→x in X such that (xn,xn+1)∈E(G) for all n≥1. Keeping in mind construction of G, there exists at least one pair of closed sets {Xj,Xj+1} for some j∈{1,2,…,m} such that both sets contain infinitely many terms of sequence {xn}, since Xi's are closed so that x∈Xj∩Xj+1 for some j∈{1,…,m}, and thus one can easily extract a subsequence such that (xnk,x)∈E(G) holds for k≥1.Now, we establish an existence theorem for the solution of an integral equation as a consequence of our Theorem18.Theorem 41.
Consider the integral equation(35)x(t)=∫ab‍k(t,s,x(s))ds+g(t),t∈[a,b],
where k:[a,b]×[a,b]×ℝn→ℝn and g:[a,b]→ℝn is continuous. Assume that(i)
k(t,s,·):ℝn→ℝn is nondecreasing for each t,s∈[a,b],(ii)
there exists a(c)-comparison function φ:ℝ+→ℝ+ and a continuous function p:[a,b] × [a,b]→ℝ+ such that |k(t,s,x(s))-k(t,s,y(s))|≤p(t,s)φ(|x(s)-y(s)|) for each t,s∈[a,b] and x≤y (i.e., x(t)≤y(t) for all t∈[a,b]),(iii)
supt∈[a,b]∫ab‍p(t,s)ds≤1,(iv)
there existsx0∈C([a,b],ℝn) such that x0(t)≤∫ab‍k(t,s,x0(s))ds+g(t) for all t∈[a,b].Then, the integral equation (35) has a unique solution in the set {x∈C([a,b],ℝn):x(t)≤x0(t) or x(t)≥x0(t) , for all t∈[a,b]}.Proof.
Let(X,∥·∥∞), where X=C([a,b],ℝn), and define a mapping T:C([a,b],ℝn)→C([a,b],ℝn) by
(36)Tx(t)=∫ab‍k(t,s,x(s))ds+g(t),t∈[a,b].
Consider a graph G consisting of V(G):=X and E(G)={(x,y)∈X×X:x(t)≤y(t) for all t∈[a,b]}. From property (i), we observe that the mapping T is nondecreasing, thus T preserves edges. Furthermore, G is a (C)-graph; that is, for every nondecreasing sequence {xn}⊂X which converges to some z∈X; then xn(t)≤z(t) for all t∈[a,b]. Now, for every x,y∈X with (x,y)∈E(G), we have
(37)|Tx(t)-Ty(t)|≤∫ab‍|k(t,s,x(s))-k(t,s,y(s))|ds≤∫ab‍p(t,s)φ(|x(s)-y(s)|)ds≤φ(∥x-y∥∞)∫ab‍p(t,s)ds.
Hence, d(Tx,Ty)≤φ(d(x,y)). From (iv), we have (x0,Tx0)∈E(G), so that [x0]G~={x∈C([a,b],ℝn):x(t)≤x0(t) or x(t)≥x0(t) for all t∈[a,b]}. The conclusion follows from Theorem 18.Note that Theorem41 specifies region of solution which invokes the novelty of our result.



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*Source: 967132-2013-10-08.xml*