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
MSC
1 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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