Generalized probabilistic G-contractions
- Seong-Hoon Cho^{1}Email author
Received: 10 August 2015
Accepted: 1 April 2016
Published: 12 April 2016
Abstract
In this paper, the notion of generalized probabilistic G-contractions in Menger probabilistic metric spaces endowed with a directed graph G is introduced and some new fixed point theorems for such mappings are established.
Keywords
fixed point coincidence point directed graph Menger probabilistic metric spaceMSC
47H10 54H251 Introduction and preliminaries
Ran and Reurings [1] gave a generalization of Banach contraction principle to partially ordered metric spaces. Since then, many authors obtained generalization and extension of the results of [2–7].
In particular, Ćirić et al. [3] extended the results of [1, 5, 6] to partially ordered Menger probabilistic metric spaces.
Samet et al. [8] introduced the notion of α-ψ-contractive type mappings and established some fixed point theorems for such mappings in complete metric spaces.
Cho [9] obtained a generalization of the results of [3] by introducing the concept of α-contractive type mappings in Menger probabilistic metric spaces.
Recently, Wu [10] obtained a generalization of the results of [3], and improved and extended the fixed point results of [4, 11, 12]. Also, Kamran et al. [13] introduced the notion of probabilistic G-contractions in Menger PM-spaces endowed with a graph G and obtained some fixed point results. Especially, they obtained the following result.
Theorem 1.1
Assume that there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(G)\). If either f is orbitally G-continuous or G is a C-graph, then f has a fixed point in \([x_{0}]_{\widetilde{G}}\).
Further if \((x,y)\in E(G)\) for any \(x,y\in M\), where \(M=\{x\in X:(x,fx)\in E(G)\}\), then f has a unique fixed point.
In this paper, we give some new fixed point theorems which are generalizations of the results of [3, 9, 10, 13], by introducing a concept of generalized probabilistic G-contractions in Menger PM-spaces with a directed graph \(G=(V(G),E(G))\) such that \(V(G)=X\) and \(\Omega\subset E(G)\).
We recall some definitions and results which will be needed in the sequel.
- (1)
f is nondecreasing and left-continuous;
- (2)
\(\sup\{f(t):t\in \mathbb {R}\}=1\);
- (3)
\(\inf\{f(t):t\in \mathbb {R}\}=0\).
We denote by D the set of all distribution functions.
Then \(\epsilon_{0} \in D\).
- (1)
\(\Delta(a,b)=\Delta(b,a)\) for all \(a,b\in[0,1]\);
- (2)
\(\Delta(\Delta(a,b),c)=\Delta(a, \Delta(b,c))\) for all \(a,b,c\in[0,1]\);
- (3)
\(\Delta(a,1)=a\) for all \(a\in[0,1]\);
- (4)
\(\Delta(a,b)\geq\Delta(c,d)\), whenever \(a\geq c\) and \(b\geq d\) for all \(a,b,c,d\in[0,1]\).
Then Δ is called a triangular norm (for short t-norm).
We denote \(\mathbb {N}\) by the set of all natural numbers.
A t-norm Δ is said to be of Hadžić-type [14] whenever the family of \(\{\Delta^{n}(t)\}_{n=1}^{\infty}\) is equicontinuous at \(t=1\).
- (1)for a t-norm Δ,$$ \mbox{it is of Had\v{z}i\'{c}-type}; $$(1.1)
- (2)
given \(\epsilon\in(0,1)\), there is a \(\delta\in(0,1)\) such that \(\Delta^{n}(x)>1-\epsilon\) for all \(n\in \mathbb {N}\), whenever \(x>1-\delta\).
- (PM1)
\(F_{x,y}(t)=\epsilon_{0}(t)\) for all \(t\in \mathbb {R}\) if and only if \(x=y\);
- (PM2)
\(F_{x,y}=F_{y,x}\) for all \(x,y\in X\);
- (PM3)
\(F_{x,y}(t+s)\geq\Delta(F_{x,z}(t),F_{z,y}(s))\) for all \(x, y, z \in X\) and all \(t,s\geq0\).
Then a 3-tuple \((X,F,\Delta)\) is called a Menger probabilistic metric space (briefly, Menger PM-space) [16, 17].
Let \((X,F,\Delta)\) be a Menger PM-space and ∈X, and let \(\epsilon >0\) and \(\lambda\in(0,1]\).
So if (1.3) holds, then Menger space \((X,F,\Delta)\) is a Hausdorff topological space in the \((\epsilon,\lambda)\)-topology (see [18, 21]).
Remark 1.1
- (1)
\(\{x_{n}\}\) is convergent to x (we write \(\lim_{n\to\infty}x_{n}=x\)) if and only if, given \(\epsilon>0\) and \(\lambda \in(0,1)\), there exists \(n_{0}\in \mathbb {N}\) such that \(F_{x_{n},x}(\epsilon)>1-\lambda\), for all \(n\geq n_{0}\).
- (2)
\(\{x_{n}\}\) is a Cauchy sequence if and only if, given \(\epsilon>0\) and \(\lambda\in(0,1)\), there exists \(n_{0}\in \mathbb {N}\) such that \(F_{x_{n},x_{m}}(\epsilon)>1-\lambda\), for all \(m>n\geq n_{0}\).
- (3)
\((X,F,\Delta)\) is complete if and only if each Cauchy sequence in X is convergent to some point in X.
Example 1.1
Then \((X,F,\Delta_{m})\) is a Menger PM-space (see [18]).
Remark 1.2
If \((X,d)\) is complete, then \((X,F,\Delta_{m})\) is complete. In fact, let \(\{x_{n}\}\) be any Cauchy sequence in \((X,F,\Delta_{m})\).
Hence, \(\{x_{n}\}\) is a Cauchy sequence in \((X,d)\). Since \((X,d)\) is complete, there exists \(x_{*}\in X\) such that \(\lim_{n\to\infty }d(x_{n},x_{*})=0\).
Note that \(\Phi\subset\Phi_{w}\).
Fang [23] gave the corrected version of Theorem 12 of [11] by introducing the notion of right-locally monotone functions as follows: \(\phi:[0,\infty) \to[0,\infty)\) is right-locally monotone if and only if \(\forall t\geq0\), \(\exists\delta>0\) s.t. it is monotone on \([t,t+\delta)\).
Lemma 1.1
[23]
- (1)If a right-locally monotone function \(\phi:[0,\infty) \to [0,\infty)\) satisfiesthen \(\phi\in\Phi\).$$\phi(0)=0,\qquad \phi(t)< t \quad\textit{and}\quad \lim_{r\to t^{+}}\inf \phi(r)< t\quad \textit{for all } t>0, $$
- (2)If a function \(\phi:[0,\infty) \to[0,\infty)\) satisfiesthen \(\phi\in\Phi_{w}\).$$\phi(t)< t \quad\textit{and}\quad \lim_{r\to t^{+}}\sup\phi(r)< t \quad \textit{for all } t>0, $$
- (3)If a function \(\alpha:[0,\infty) \to[0,1)\) is piecewise monotone andthen \(\phi\in\Phi\).$$\phi(t)=\alpha(t)t\quad \textit{for all } t\geq0, $$
Lemma 1.2
[23]
If \(\phi\in\Phi_{w}\), then \(\forall t>0\), \(\exists r\geq t\) s.t. \(\phi(r)< t\).
Lemma 1.3
[23]
Lemma 1.4
[18]
Let \((X,F,\Delta)\) be a Menger PM-space and \(x,y\in X\), where Δ is continuous. Suppose that \(\{x_{n}\}\) is a sequence of points in X. If \(\lim_{n\to\infty}x_{n}=x\), then \(\lim_{n\to\infty}\inf F_{x_{n},y}(t)=F_{x,y}(t)\) for all \(t>0\).
Lemma 1.5
Proof
By induction, we show that (1.5) holds.
Let \(m=n+1\).
Thus, (1.5) holds for \(m=n+1\).
Hence, (1.5) holds for all \(m\geq n+1\). □
Lemma 1.6
[24]
Then \((X,F,\Delta_{m})\) is a Menger PM-space, which is called a Menger PM-space induced by the metric d.
Remark 1.3
Let \((X,d)\) be a metric space. Suppose that \((X,F,\Delta_{m})\) is a Menger PM-space induced by d.
- (1)
If \(f:X\to X\) is continuous in \((X,d)\), then it is continuous in \((X,F,\Delta_{m})\).
- (2)
If a sequence \(\{x_{n}\}\) is convergent to a point x in \((X,d)\), then it is convergent to x in \((X,F,\Delta_{m})\).
- (3)
If \((X,d)\) is complete, then \((X,F,\Delta_{m})\) is complete.
Lemma 1.7
[25]
If X is a nonempty set and \(h:X\to X\) is a function, then there exists \(Y \subset X\) such that \(h(Y)=h(X)\) and \(h:Y\to X\) is one-to-one.
Let X be a nonempty set, and let \(\Omega=\{(x,x):x\in X\}\) the diagonal of the Cartesian product \(X\times X\).
- (1)
the set \(V(G)\) of its vertices coincides with X, i.e. \(V(G)=X\);
- (2)
the set \(E(G)\) of its edges contains all loops, i.e. \(\Omega\subset E(G)\).
If G has no parallel edges, then we can identify G with the pair \((V(G), E(G))\).
Let \(G=(V(G), E(G))\) be a directed graph.
Note that \(G^{-1}=(V(G), E(G^{-1}))\).
Then p is called a path in G from x to y of length N.
Denote \(\Xi(G)\) by the family of all path in G.
If, for any \(x,y\in V(G)\), there is a path \(p\in\Xi(G)\) from x to y, then the graph G called connected. A graph G is called weakly connected, whenever G̃ is connected.
Let G be a graph such that \(E(G)\) is symmetric and \(x\in V(G)\).
Then the relation ℜ is an equivalence relation on \(V(G)\), and \([x]_{G}=V(G_{x})\), where \([x]_{G}\) is the equivalence class of \(x\in V(G)\).
Note that the component \(G_{x}\) of G containing x is connected.
For the details of the graph theory, we refer to [26].
Let \((X,F,\Delta)\) be a Menger PM-space, and let \(G=(V(G),E(G))\) be a directed graph such that \(V(G)=X\) and \(\Omega\subset E(G)\).
Then the graph G is said to be a C-graph if and only if, for any sequence \(\{x_{n}\}\subset X\) with \(\lim_{n\to\infty }x_{n}=x_{*}\in X\), there exist a subsequence \(\{x_{n_{k}}\}\) of \(\{ x_{n}\}\) and an \(N\in \mathbb {N}\) such that \((x_{n_{k}},x_{*})\in E(G)\) (resp. \((x_{*},x_{n_{k}})\in E(G)\)) for all \(k \geq N\) whenever \((x_{n},x_{n+1})\in E(G)\) (resp. \((x_{n+1},x_{n})\in E(G)\)) for all \(n\in \mathbb {N}\).
The following definitions are in [13].
- (1)f is continuous if and only if, for any \(x\in X\) and a sequence \(\{x_{n}\}\subset X\) with \(\lim_{n\to\infty}x_{n}=x\),$$\lim_{n\to\infty}fx_{n}=fx. $$
- (2)f is G-continuous if and only if, for any \(x\in X\) and a sequence \(\{x_{n}\}\subset X\) with \(\lim_{n\to\infty}x_{n}=x\) and \((x_{n},x_{n+1})\in E(G)\) for all \(n\in \mathbb {N}\),$$\lim_{n\to\infty}fx_{n}=fx. $$
- (3)f is orbitally continuous if and only if, for all \(x,y\in X\) and any sequence \(\{k_{n}\}\subset \mathbb {N}\) with \(\lim_{n\to \infty}f^{k_{n}}x=y\),$$\lim_{n\to\infty}ff^{k_{n}}x=fy. $$
- (4)f is orbitally G-continuous if and only if, for all \(x,y\in X\) and any sequence \(\{k_{n}\}\subset \mathbb {N}\) with \(\lim_{n\to\infty}f^{k_{n}}x=y\) and \((f^{k_{n}}x,f^{k_{n}+1}x)\in E(G) \) for all \(k\in \mathbb {N}\),$$\lim_{n\to\infty}ff^{k_{n}}x=fy. $$
2 Main results
- (1)
f preserves edges of G, i.e. \((x,y)\in E(G) \Longrightarrow(fx,fy)\in E(G)\);
- (2)there exists \(\phi\in\Phi_{w}\) such thatfor all \(x,y\in X\) with \((x,y)\in E(G)\) and all \(t>0\).$$ F_{fx,fy}\bigl(\phi(t)\bigr) \geq\min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\} $$(2.1)
Theorem 2.1
Let \((X,F,\Delta)\) be complete. Suppose that a map \(f:X\to X\) is a generalized probabilistic G-contraction. Assume that there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(G)\). If either f is orbitally G-continuous or Δ is a continuous t-norm and G is a C-graph, then f has a fixed point in \([x_{0}]_{\widetilde{G}}\).
Further if \((x,y)\in E(G)\) for any \(x,y\in M\), where \(M=\{x\in X:(x,fx)\in E(G)\}\), then f has a unique fixed point.
Proof
Let \(x_{0}\in X\) be such that \((x_{0},fx_{0})\in E(G)\). Let \(x_{n}=f^{n}x_{0}\) for all \(n\in \mathbb {N}\cup\{0\}\).
If there exists \(n_{0}\in \mathbb {N}\) such that \(x_{n_{0}}=x_{n_{0}+1}\), then \(x_{n_{0}}=x_{n_{0}+1}=fx_{n_{0}}\), and so \(x_{n_{0}}\) is a fixed point of f.
Then the above path is in G̃. Hence, \(x_{n_{0}}=x_{n_{0}+1}\in[x_{0}]_{\widetilde{G}}\).
Hence, the proof is finished.
Assume that \(x_{n-1}\neq x_{n}\) for all \(n\in \mathbb {N}\).
As in the proof of Lemma 1.4, we have \(\phi(t)>0\) for all \(t>0\).
Next, we show that \(\{x_{n}\}\) is a Cauchy sequence.
Let \(\epsilon\in(0,1)\) be given.
If f is orbitally G-continuous, then \(\lim_{n\to\infty }x_{n}=fx_{*}\). Hence, \(x_{*}=fx_{*}\).
Suppose that Δ is continuous and G is C-graph.
Suppose that \((x,y)\in E(G)\) for any \(x,y\in M\).
Let \(x_{*}\) and \(y_{*}\) be two fixed point of f.
Then \(x_{*},y_{*}\in M\). By assumption, \((x_{*},y_{*})\in E(G)\).
Example 2.1
Let \(X=[0,\infty)\), and let \(d(x,y)=| x-y |\) for all \(x,y\in X\).
Then \((X,F,\Delta_{m})\) is a complete Menger PM-space.
Then \(\phi\in\Phi_{w}\) and \(\phi(t)\geq{1\over 2}t\) for all \(t\geq0\).
Further assume that X is endowed with a graph G consisting of \(V(G)=X\) and \(E(G)=\{(x,y)\in X\times X:y \preceq x\}\).
Obviously, f preserves edges, and it is orbitally G-continuous. If \(x_{0}=0\), then \((x_{0}, fx_{0})=(0,0)\in E(G)\).
Thus, (2.1) is satisfied. Hence, all the conditions of Theorem 2.1 are satisfied and f has a fixed point \(x_{*}=0\in [0]_{\widetilde{G}}\). Furthermore, \(M=\{0\}\) and the fixed point is unique.
Remark 2.1
Note that in Theorem 2.1 the assumption of orbitally G-continuity can be replaced by orbitally continuity, G-continuity or continuity.
Remark 2.2
Theorem 2.1 is a generalization of Theorem 3.1 in [23] to the case of a Menger PM-space endowed with a graph.
Corollary 2.2
- (1)
f preserves edges of G;
- (2)there exists \(\phi\in\Phi\) such thatfor all \(x,y\in X\) with \((x,y)\in E(G)\) and all \(t>0\).$$F_{fx,fy}\bigl(\phi(t)\bigr) \geq\min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\} $$
Remark 2.3
- (1)
- (2)
In Corollary 2.2, let \(\phi(s)=ks\) for all \(s\geq 0\), where \(k\in(0,1)\). If G is a graph such that \(V(G)=X\) and \(E(G)=\{(x,y)\in X\times X:\alpha(x,y)\geq1\}\), where \(\alpha:X\times X \to[0,\infty)\) is a function, then Corollary 2.2 reduces to Theorem 2.1 of [9].
- (3)
If G is a graph such that \(V(G)=X\) and \(E(G)=\{(x,y)\in X\times X:x\preceq y\}\), where ⪯ is a partial order on X, then Corollary 2.2 become to Theorem 2.1 of [10].
Corollary 2.3
Let \((X,F,\Delta)\) be complete. Suppose that a map \(f:X\to X\) is generalized probabilistic G-contraction. Assume that either f is continuous or Δ is a continuous t-norm and G is a C-graph.
Then f has a fixed point in \([x_{0}]_{\widetilde{G}}\) for some \(x_{0}\in Q\) if and only if \(Q\neq\emptyset\), where \(Q=\{x\in X:(x,fx)\in E(\widetilde{G})\}\). Further if, for any \(x,y\in Q\), \((x,y)\in E(\widetilde{G})\) then f has a unique fixed point.
Proof
If f has a fixed point in \([x_{0}]_{\widetilde{G}}\), say \(x_{*}\), then \((x_{*},fx_{*})=(x_{*},x_{*})\in\Omega\subset E(\widetilde{G})\). Thus, \(Q\neq\emptyset\).
Suppose that \(Q\neq\emptyset\).
Then there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(\widetilde{G})\).
We have two cases: \((x_{0},fx_{0})\in E(G) \) or \((x_{0},fx_{0})\in E(G^{-1})\).
If \((x_{0},fx_{0})\in E(G) \), then following Theorem 2.1 f has a fixed point in \([x_{0}]_{\widetilde{G}}\).
Assume that \((x_{0},fx_{0})\in E(G^{-1})\).
Then \((fx_{0},x_{0})\in E(G)\). Since f is preserves edges of G, \((f^{n+1}x_{0},f^{n}x_{0})\in E(G)\) for all \(n\in \mathbb {N}\cup\{0\}\).
In the same way as the proof of Theorem 2.1 with condition (PM2), we deduce that f has a fixed point in \([x_{0}]_{\widetilde{G}}\).
Suppose that, for any \(x,y\in Q\), \((x,y)\in E(\widetilde{G})\).
Let \(x_{*}\) and \(y_{*}\) be two fixed points of f.
Then \(x_{*},y_{*}\in Q\). By assumption, \((x_{*},y_{*})\in E(\widetilde{G})\).
Let \((x_{*},y_{*})\in E(G^{-1})\), then \((y_{*},x_{*})\in E(G)\).
Remark 2.4
If \(\phi\in\Phi\) and G is a graph such that \(V(G)=X\) and \(E(G)=\{ (x,y)\in X\times X:{x\preceq y}\}\), where ⪯ is a partial order on X, then Corollary 2.3 reduces to Theorem 2.2 of [10].
In the following result, we can drop continuity of the t-norm Δ.
Corollary 2.4
Assume that there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(G)\). If either f is orbitally G-continuous or G is a C-graph, then f has a fixed point in \([x_{0}]_{\widetilde{G}}\).
Further if \((x,y)\in E(G)\) for any \(x,y\in M\), where \(M=\{x\in X:(x,fx)\in E(G)\}\), then f has a unique fixed point.
Proof
Let \(x_{0}\in X\) be such that \((x_{0},fx_{0})\in E(G)\), and let \(x_{n}=f^{n}x_{0}\) for all \(n\in \mathbb {N}\cup\{0\}\).
Note that (2.6) to be satisfied implies that (2.1) is satisfied.
If f is orbitally G-continuous, then \(\lim_{n\to\infty }x_{n}=fx_{*}\), and so \(x_{*}=fx_{*}\).
Assume that G is a C-graph.
Since \(\phi\in\Phi_{w}\), for each \(t>0\), there exists \(r\geq t\) such that \(\phi(r)< t\).
Since \(\lim_{n\to\infty}a_{n}=1\) and \(\Delta(t,t) \) is continuous at \(t=1\), \(\lim_{n\to\infty}\Delta(a_{n},a_{n})=\Delta(1,1)=1\). Hence, from (2.7) we have \(F_{x_{*},fx_{*}}(t)=1\) for all \(t>0\), and so \(x_{*}=fx_{*}\). □
Remark 2.5
Corollary 2.4 is a generalization of Theorem 3.1 in [23] to the case of a Menger PM-space endowed with a graph.
Theorem 2.5
- (1)
\(f(X) \subset h(X)\);
- (2)
\(h(X)\) is closed;
- (3)
\((hx,hy)\in E(G)\) implies \((fx,fy)\in E(G)\);
- (4)
there exists \(x_{0}\in X\) such that \((hx_{0},fx_{0})\in E(G)\);
- (5)there exists \(\phi\in\Phi_{w}\) such thatfor all \(x,y\in X\) with \((hx,hy)\in E(G)\) and all \(t>0\);$$ F_{fx,fy}\bigl(\phi(t)\bigr) \geq\min\bigl\{ F_{hx,hy}(t),F_{hx,fx}(t),F_{hy,fy}(t)\bigr\} $$(2.8)
- (6)
if \(\{x_{n}\}\) is a sequence in X such that \((hx_{n},hx_{n+1})\in E(G)\) for all \(n\in \mathbb {N}\cup\{0\}\) and \(\lim_{n\to\infty}hx_{n}=hu\) for some \(u\in X\), then \((hx_{n},hu)\in E(G)\) for all \(n\in \mathbb {N}\cup\{0\}\).
Then f and h have a coincidence point in X. Further if f and h commute at their coincidence points and \((hu,hhu)\in E(G)\), then f and h have a common fixed point in X.
Proof
By Lemma 1.7, there exists \(Y\subset X\) such that \(h(Y)=h(X)\) and \(h:Y\to X\) is one-to-one. Define a mapping \(U:h(Y) \to h(Y)\) by \(U(hx)=fx\). Since \(h:Y\to X\) is one-to-one, U is well defined.
By (3), \((hx,hy)\in E(G)\) implies \((U(hx),U(hy))\in E(G)\).
Suppose that f and h commute at their coincidence points and \((hu,hhu)\in E(G)\). Let \(w=hu=fu\). Then \(fw=fhu=hfu=hw\), and \((hu,hw)=(hu,hhu)\in E(G)\).
By Lemma 1.2, \(w=fw\). Hence \(w=fw=hw\). Thus, w is a common fixed point of f and h. □
Remark 2.6
Theorem 2.5 is a generalization of Theorem 3.4 of [3]. If we have \(\phi(s)=ks\) for all \(s\geq0\), where \(k\in(0,1)\), and \(V(G)=X\) and \(E(G)=\{(x,y):x\leq y\}\), where ≤ is a partial order on X, then Theorem 2.5 reduces to Theorem 3.4 of [3].
Theorem 2.6
Suppose that f preserves edges, and assume that there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(G)\), where \(f=f_{0}f_{1}\). If either f is orbitally G-continuous or Δ is a continuous t-norm and G is a C-graph, then f has a fixed point in \([x_{0}]_{\widetilde{G}}\).
Further if \((x,y)\in E(G)\) for any \(x,y\in M\), where \(M=\{x\in X:(x,fx)\in E(G)\}\), then \(f_{0}\) and \(f_{1}\) have a common fixed point whenever \(f_{0}\) is commutative with \(f_{1}\).
Proof
Suppose that \((x,y)\in E(G)\) for any \(x,y\in M\).
Then from Theorem 2.1 f has a unique fixed point.
Since \(f_{0}\) is commutative with \(f_{1}\) and \(fx_{*}=x_{*}\), \(ff_{0}x_{*}=f_{0}(f_{1}f_{0}x_{*}) =f_{0}(f_{0}f_{1}x_{*})=f_{0}fx_{*}=f_{0}x_{*}\). Similarly, we obtain \(ff_{1}x_{*}=f_{1}x_{*}\). From the uniqueness of fixed point of f, we have \(x_{*}=f_{0}x_{*}=f_{1}x_{*}\). □
Example 2.2
Then \((X,F,\Delta_{m})\) is a complete Menger PM-space.
Then \(\phi\in\Phi_{w}\) and \(\phi(t)\geq{1\over 2}t\) for all \(t\geq0\).
Further assume that X is endowed with a graph G consisting of \(V(G)=X\) and \(E(G)=\{(x,y)\in X\times X:y \preceq x\}\).
Obviously, G is a C-graph.
Obviously, f preserves edges.
Let \((x,y)\in E(G)\).
We consider the following three cases:
Thus, (2.10) is satisfied.
For \(x_{0}=4\), \((x_{0},fx_{0})=(4,{1\over 6})\in E(G)\). Hence, all the conditions of Theorem 2.6 are satisfied and f has a fixed point \(x_{*}=0\in[x_{0}]_{\widetilde{G}}\).
Corollary 2.7
Suppose that f preserves edges, and assume that there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(G)\), where \(f=f_{0}f_{1}\). If f is orbitally G-continuous or G is a C-graph, then f has a fixed point in \([x_{0}]_{\widetilde{G}}\).
Further if \((x,y)\in E(G)\) for any \(x,y\in M\), where \(M=\{x\in X:(x,fx)\in E(G)\}\), then \(f_{0}\) and \(f_{1}\) have a common fixed point whenever \(f_{0}\) is commutative with \(f_{1}\).
Proof
Suppose that \((x,y)\in E(G)\) for any \(x,y\in M\).
Then from Corollary 2.4 f has a unique fixed point.
Since \(f_{0}\) is commutative with \(f_{1}\), as in the proof of Theorem 2.6 we have \(x_{*}= f_{0}x_{*}= f_{1}x_{*}\). □
Remark 2.7
Corollary 2.7 is a generalization of Corollary 2.1 of [23] to the case of Menger PM-space endowed with a graph.
Corollary 2.8
- (1)
\((x,y)\in E(G)\) implies \((fx,fy)\in E(G)\);
- (2)there exists \(\phi\in\Phi_{w}\) such thatfor all \(x,y\in X\) with \((x,y)\in E(G)\), where ϕ is nondecreasing;$$\begin{aligned} &d(fx,fy) \\ &\quad\leq\phi\bigl(\max\bigl\{ d(x,y),d(x,fx),d(y,fy)\bigr\} \bigr) \end{aligned}$$(2.13)
- (3)
there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(G)\);
- (4a)
f is continuous, or
- (4b)
if \(\{x_{n}\}\) is a sequence in X such that \(\lim_{n\to \infty}x_{n}=x_{*}\in X\) and \((x_{n},x_{n+1})\in E(G)\) for all \(n\in \mathbb {N}\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\} \) such that \((x_{n_{k}},x_{*})\in E(G)\) for all \(k\in \mathbb {N}\).
Then f has a fixed point in \([x_{0}]_{\widetilde{G}}\).
Proof
Suppose that equality holds in (2.13) and \(x\neq fx\) for all \(x\in X\).
Thus, if equality holds in (2.13), then f has a fixed point.
Assume that equality is not satisfied in (2.13).
Let \((X,F, \Delta_{m})\) be the induced Menger PM-space by \((X,d)\).
By Lemma 1.6, \((X,F, \Delta_{m})\) is complete. By Remark 1.3, (4a) implies f is continuous in \((X,F, \Delta_{m})\), and (4b) implies G is C-graph.
We show that (2.1) is satisfied.
Hence, (2.1) is satisfied. By Theorem 2.1 and Remark 2.1, f has a fixed point in \([x_{0}]_{\widetilde{G}}\). □
Corollary 2.9
Let \((X,d)\) be a complete metric space, and let \(G=(V(G),E(G))\) be a directed graph satisfying \(V(G)=X\) and \(\Omega\subset E(G)\). Let \(f:X\to X\) be a map.
- (1)
\((x,y)\in E(G)\) implies \((fx,fy)\in E(G)\);
- (2)there exists \(\phi\in\Phi_{w}\) such thatfor all \(x,y\in X\) with \((x,y)\in E(G)\), where ϕ is nondecreasing;$$d(fx,fy)\leq\phi\bigl(d(x,y)\bigr) $$
- (3)
there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(G)\);
- (4)
either f is continuous or if \(\{x_{n}\}\) is a sequence in X such that \(\lim_{n\to\infty}x_{n}=x_{*}\in X\) and \((x_{n},x_{n+1})\in E(G)\) for all \(n\in \mathbb {N}\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \((x_{n_{k}},x_{*})\in E(G)\) for all \(k\in \mathbb {N}\).
Then f has a fixed point in \([x_{0}]_{\widetilde{G}}\).
Remark 2.8
Corollary 2.9 is a generalization of the results of [5]. If we have a graph G such that \(V(G)=X\) and \(E(G)=\{(x,y)\in X\times X:x\preceq y\}\), where ⪯ is a partial order on X, and \(\phi (s)=ks\) for all \(s\geq0\), where \(k\in[0,1)\), then Corollary 2.9 reduces to Theorem 2.1 and Theorem 2.2 of [5].
Declarations
Acknowledgement
The author thanks the editor and the referees for their useful comments and suggestions.
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Authors’ Affiliations
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