Some new fixed point results on V - ψ -fuzzy contraction endowed with graph

Abstract

Recent researches in the development of metric space to V-fuzzy metric spaces encouraged us to write this paper. In this paper, we have defined a new contraction called V-ψ-fuzzy contraction in a V-fuzzy metric space (V, M, *) endowed with a graph. Here, we have extended the Banach Contraction Principle on V-fuzzy metric spaces in association with graph. Also, we have explained the characteristic of Graph as of partial ordering, which provides a way to find fixed point for n-tuples of a V-fuzzy metric space and an important application in the field of computer science.

Keywords

54H25

1. Introduction

A sequence of results in metric spaces make the fixed point theory very special and basic area of the researches in mathematics. Zadeh [8] was the first to introduce the fuzzy sets. Kramosil and Michlek [5] employed this concept in the metric spaces to introduce fuzzy metric spaces. George and Veeramani [1] modified the concept of fuzzy metric spaces given by [5]. Gregori and Sapna [9] extended the Banach fixed point theorem to fuzzy contractive mappings on complete fuzzy metric spaces. Jachymski [7] introduced the language of graph theory in replacement of partial orderings of metric spaces in terms of graph in a united way. Recently, Shukla [15] introduced the idea of G-fuzzy contraction in fuzzy metric spaces, which is a generalization of results of Jachymski [7] and Gregori and Sapena [9]. The fixed point results endowed with graph has been extended by many authors in literature [3 , 16–20]. Graph theory is also applicable for cyclic mapping [10] in different spaces.

In this paper, we have introduced a new contraction as V-ψ-fuzzy contraction using the concept of altering distance function, and we have generalized the result of Shukla [15] in V-fuzzy metric spaces. Some basic definitions and results are given below:

Definition 1.1. (Gregori and Sapena [9]). Let (K, M, *) be a fuzzy metric space. A mapping S : K → K is called t-uniformly continuous if for each r ∈ (0, 1) there exists s ∈ (0, 1) such that for each x, y ∈ K and t > 0, $[M (x, y, t) \geq 1 - s \Rightarrow M (Sx, Sy, t) \geq 1 - r] .$

Remark 1.1. The t-uniform continuity of S implies S is continuous for the topology deduced from M. For uniform structure in a fuzzy metric space, refer to [9].

Definition 1.2. (Gregori and Sapena [9]). Let (K, M, *) be a fuzzy metric space. A mapping S : K → K is called a fuzzy contractive mapping if there exists λ ∈ (0, 1) such that $\begin{matrix} \frac{1}{M (Sx, Sy, t)} - 1 \leq λ [\frac{1}{M (x, y, t)} - 1], \\ for each x, y \in K and t > 0 . \end{matrix}$ (1) (λ is called the contractive constant of S.)

Definition 1.3. ([4]) A binary operation * : [0.1] × [0, 1] → [0, 1] is called continuous t-norms if * satisfies following conditions:

* is commutative and associative;

* is continuous;

a * 1 = a, ∀ a ∈ [0, 1];

a * b ≤ c * d whenever a ≤ c and b ≤ d, for all a, b, c, d ∈ [0, 1].

Recently, Gupta and Kanwar [11] stamped the move of different generalizations to n-tuples as defined below:

Definition 1.4. ([11]) Let X be a non-empty set. A 3-tuple (X, V, *) is said to be a V-fuzzy metric space (denoted by VF-space), where * is a continuous t-norm and V is a fuzzy set on Xⁿ × (0, ∞) satisfying the following conditions for each t, s > 0,

V (x, x, x, …, x, y, t) >0, for all x, y ∈ X with x ≠ y;

V (x₁, x₁, x₁, …, x₁, x₂, t) ≥ V (x₁, x₂, x₃, …, x_n, t) ,

for all x₁, x₂, x₃, …, x_n ∈ X with x₂ ≠ x₃ ≠ ⋯ ≠ x_n;

V (x₁, x₂, x₃, …, x_n, t) =1 if x₁ = x₂ = x₃ = … = x_n;

V (x₁, x₂, x₃, …, x_n, t) = V (p (x₁, x₂, x₃, … x_n) , t), where p is a permutation function;

V (x₁, x₂, x₃, …, x_n-1, t + s) ≥ V (x₁, x₂, x₃, …, x_n-1, l, t) * V (l, l, l, …, l, x_n, s);

$lim_{t \to \infty} V (x_{1}, x_{2}, x_{3}, \dots, x_{n}, t) = 1$ ;

V (x₁, x₂, x₃, …, x_n) : (0, ∞) → [0, 1] is continuous.

Basic concepts about the graph are similar to those in [7].

Let (K, V, *) be a V-fuzzy metric space. Let Δ denote the diagonal of the cartesian product K × K. Consider a directed graph G such that the set V (G) of its vertices coincides with K, and the set E (G) of its edges contains all loops, i.e., E (G) ⊆ Δ. We assume that G has no parallel edges and the pair (V (G) , E (G)) is representing the graph G.

By G^-1 we denote the conversion of a graph G, i.e., the graph obtained from G by reversing the direction of edges. Thus we have $E (G^{- 1}) = {(x, y) \in K \times K : (y, x) \in E (G)} .$

The letter $\tilde{G}$ denotes the undirected graph obtained from G by ignoring the direction of edges. Actually, it will be more convenient for us to treat G as a directed graph for which the set of its edges is symmetric. Under this convention, $E (\tilde{G}) = E (G) \cup E (G^{- 1}) .$ (2)

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 $(x_{i})_{i = 0}^{l}$ of l + 1 vertices such that x₀ = x, x_l = y and (x_i-1, x_i) ∈ E (G) for i = 1, …, l. A graph G is called connected if there is a path between any two vertices of G. A graph G is weakly connected if $\tilde{G}$ is connected. For a graph G such that E (G) is symmetric and x is a vertex in G, the subgraph G_x consisting of all edges and vertices which are contained in some path beginning at x is called the component of G containing x. In this case V (G_x) = [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, G_x is connected. We recall here the result of Shukla [15], which gave us the idea to construct the problem in a V-fuzzy metric space.

Definition 1.5. ([15]). Let (K, M, *) be a fuzzy metric space and G be a graph. Two, sequences (x_n) _n∈N and (y_n) _n∈N in K are said to be Cauchy equivalent if each of them is a Cauchy sequence and $lim_{n \to \infty} M (x_{n}, y_{n}, t) = 1$ for all t > 0.

Definition 1.6. ([15]). Let (K, M, *) be a fuzzy metric space and G be a graph. The mapping S : K → K is said to be a G-fuzzy contraction if the following conditions hold:

(GF1) ((x, y) ∈ E (G) ⇒ (Sx, Sy) ∈ E (G)), for all x, y ∈ K i.e, S is edge-preserving;

(GF2) for all x, y ∈ K and t > 0, there exists λ ∈ (0, 1) such that $((x, y) \in E (G) \Rightarrow \frac{1}{M (Sx, Sy, t)} - 1 \leq λ [\frac{1}{M (x, y, t)} - 1]),$ where λ is called the contractive constant of S.

An obvious consequence for the symmetry of M (· , · , t) and (1.2) is the following remark.

Remark 1.2. ([15]) If S is a G-fuzzy contraction then it is both a G^-1-fuzzy contraction and a $\tilde{G}$ -fuzzy contraction.

Theorem 1.1. [[15]] Let (K, M, *) be a complete fuzzy metric space and G be a directed graph and let the 4-tuple (K, M, * , G) have the property (P_S). Let S : K → K be a G-fuzzy contraction and K_S = {x ∈ K : (x, Sx) ∈ E (G)}, then the following statements hold:

if x ∈ X_T, then $T |_{[x]_{\tilde{G}}}$ is a Picard operator;

if X_T≠ ∅ and G is weakly connected, then T is a Picard operator;

FixT≠ ∅ if and only if X_T≠ ∅;

If T ⊆ E (G), then T is a weakly Picard operator.

2. Main results

Now, we are ready to present our main result. First, we introduce the following definitions and lemma.

Definition 2.1. Let (K, V, *) be a V-fuzzy metric space and G be a graph. Two sequences (x_n) _n∈N and (y_n) _n∈N in K are said to be Cauchy equivalent if each of them is a Cauchy sequence and $lim_{n \to \infty} V (x_{n}, y_{n}, \dots, y_{n}, t) = 1$ for all t > 0.

Definition 2.2. Let (K, V, *) be a V-fuzzy metric space and G be a graph. The mapping S : K → K is said to be V-ψ-fuzzy contraction if the following conditions hold:

For all x, y ∈ K

(GF1) (x, y) ∈ E (G) ⇒ (Sx, Sy) ∈ E (G), i.e., S is edge-preserving.

(GF2) For all t > 0, $\begin{matrix} (x, y) \in E (G) & \Rightarrow & \frac{1}{V (Sx, Sy, Sy, \dots, Sy, φ (t)} - 1 \\ \leq & ψ [\frac{1}{V (x, y, y, \dots, y, φ (t))} - 1], \end{matrix}$ where ψ : [0, 1] → [0, 1] is such that ψ is continuous, ψ (0) =0 and ψⁿ (a_n) →0, whenever a_n → 0 as n→ ∞.

Now we start with lemma below:

Lemma 2.1. Let S : K → K be a V-ψ-fuzzy contraction, then given x ∈ K and $y \in [x]_{\tilde{G}}$ , $lim_{n \to \infty} V (S^{n} x, S^{n} y, S^{n} y, \dots, S^{n} y, φ (t)) = 1$ for all t > 0 .

Proof. Let x ∈ K and $y \in [x]_{\tilde{G}}$ . Then there exists a path $(x_{i})_{i = 0}^{m}$ in $\tilde{G}$ from x to y, where x = x₀, y = x_m and (x_i, x_i-1) ∈ E (G) for i = 1, 2, …, m.

By Remark 1.2, S is a $\tilde{V}$ -ψ-fuzzy contraction.

Therefore by (GF1) we have,

$(S^{n} x_{i}, S^{n} x_{i - 1}) \in E (\tilde{G})$ and by (GF2), for i = 1, 2, …, m and t > 0, we have $\begin{matrix} [\frac{1}{V (S^{n} x_{i - 1}, S^{n} x_{i}, \dots S^{n} x_{i}, φ (t))} - 1] \\ = [\frac{1}{V ({SS}^{n - 1} x_{i - 1}, {SS}^{n - 1} x_{i}, \dots {SS}^{n - 1} x_{i}, φ (t))} - 1] . \end{matrix}$ Again, V (Sx, Sy, Sy, …, Sy, φ (t)) >0 implies V (Sx, Sy, Sy, …, Sy, φ (t/c)) >0,

therefore $\begin{matrix} [\frac{1}{V ({SS}^{n - 1} x_{i - 1}, {SS}^{n - 1} x_{i}, \dots {SS}^{n - 1} x_{i}, φ (t))} - 1] \\ \leq ψ [\frac{1}{V (S^{n - 1} x_{i - 1}, S^{n - 1} x_{i}, \dots S^{n - 1} x_{i}, φ (t / c))} - 1], \end{matrix}$ this implies, $\begin{matrix} ψ [\frac{1}{V ({SS}^{n - 2} x_{i - 1}, {SS}^{n - 2} x_{i}, \dots {SS}^{n - 2} x_{i}, φ (t))} - 1] \\ \leq ψ^{2} (\frac{1}{V (S^{n - 2} x_{i - 1}, S^{n - 2} x_{i}, \dots S^{n - 1} x_{i}, φ (t / c^{2}))} - 1) . \end{matrix}$

Continuing like this, we have $\begin{matrix} [\frac{1}{V (S^{n} x_{i - 1}, S^{n} x_{i}, \dots S^{n} x_{i}, φ (t))} - 1] \\ \leq ψ^{n - 1} [\frac{1}{V (x_{i - 1}, x_{i}, \dots x_{i}, φ (t / c^{n}))} - 1] . \end{matrix}$ (2.1)

Now we can find a strictly decreasing sequence (a_n) _n∈N of positive number such that $\sum_{i = 1}^{\infty} a_{i} = 1$ and then using (2.1), we have $\begin{matrix} V (S^{n} x, S^{n} y, \dots, S^{n} y, φ (t)) \\ = V (S^{n} x_{0}, S^{n} x_{m}, \dots, S^{n} x_{m}, \sum_{i = 1}^{\infty} a_{i} φ (t)) \\ \geq V (S^{n} x_{0}, S^{n} x_{m}, \dots, S^{n} x_{m}, \sum_{i = 1}^{m} a_{i} φ (t)) \\ \geq \prod_{i = 1}^{m} V (S^{n} x_{i - 1}, S^{n} x_{i}, \dots, S^{n} x_{i}, a_{i} φ (t)) \\ \geq \prod_{i = 1}^{m} [\frac{1}{1 + ψ^{n} [\frac{1}{V (x_{i - 1}, x_{i}, \dots, x_{i}, φ (t / c^{n}))} - 1]}] . \end{matrix}$ Letting n→ ∞, φ (t/cⁿ)→ ∞ and by the definition, ψⁿ (r_i) →0 as r_i → 0, gives $lim_{n \to \infty} V (S^{n} x, S^{n} y, \dots, S^{n} y, φ (t)) = 1, for all t > 0 .$ It proves Sⁿx and Sⁿy are equivalent.□

Next we prove the convergence towards fixed point for iterative sequences in a V-fuzzy metric space with connected graph.

Theorem 2.1. The following statements are equivalent:

G is weakly connected.

For any $\tilde{V}$ -ψ-fuzzy contraction S : K → K and for given x, y ∈ K, the sequences (Sⁿx) _n∈N and (Sⁿy) _n∈N are Cauchy equivalent.

For any $\tilde{V}$ -ψ-contraction S : K → K, card(FixS) ≤1.

Proof. (i)⇒(ii).

Let G is weakly connected and S be a V-ψ-fuzzy contraction.

Also, we render for weakly connected graph that $[x]_{\tilde{G}} = K$ and therefore $S^{p} x \in [x]_{\bar{G}}$ for all p ∈ N.

By Lemma 2.1, (Sⁿx) _n∈N is a Cauchy sequence. Therefore, $lim_{n \to \infty} V (S^{n} x, S^{n} y, \dots, S^{n} y, φ (t)) = 1$ , for all t > 0.

Also, y ∈ K implies $y \in [x]_{\bar{G}}$ . As (Sⁿx) _n∈N is a Cauchy sequence, therefore (Sⁿy) _n∈N is also a Cauchy sequence.

Hence the sequences (Sⁿx) _n∈N and (Sⁿy) _n∈N are Cauchy equivalent.

(ii)⇒(iii)

Let x, y ∈ FixS, where S is V-ψ-fuzzy contraction. Since x, y ∈ Fix (Sⁿ), V (x, y, …, y, t) = V (Sⁿx, Sⁿy, …, Sⁿy, t). So as from (ii) (Sⁿx) _n∈N and (Sⁿy) _n∈N are Cauchy equivalent.

Therefore x = y.

(iii)⇒(i) Suppose (iii) holds, but G is not weakly connected.

i.e. $\tilde{G}$ is disconnected. Let u ∈ K, then the sets $[u]_{\tilde{G}}$ and $K | [u]_{[\tilde{G}]}$ are non-empty.

Let $v \in K | [u]_{[\tilde{G}]}$ and define a mapping S : K → K by $Sx = {\begin{matrix} u & if x \in [u]_{[\tilde{G}]} \\ v & if x \in K | [u]_{[\tilde{G}]} . \end{matrix}$ Clearly, FixS = {u, v}.

Now, we will prove that S is a V-ψ-fuzzy contraction.

If (x, y) ∈ E (G) then $[x]_{[\tilde{G}]} = [y]_{[\tilde{G}]}$ , so either $x, y \in [u]_{[\tilde{G}]}$ or $x, y \in K / [u]_{[\tilde{G}]}$ .

In both the cases, we have Sx = Sy and so (Sx, Sy) ∈ E (G) (since E (G) ⊇ Δ) and GF1 is satisfied. Also, V (Sx, Sy, …, Sy, φ (t)) =1 for all t > 0, so GF2 is satisfied.

Hence S is V-ψ-fuzzy contraction and card(FixS) =2 > 1.

A contradiction prove that V is weakly connected.

Corollary 2.1. Let (K, V, *) be a complete V-fuzzy metric space. Then the following statements are equivalent

V is weakly connected;

for any V-fuzzy contraction S : K → K, there exists x^* ∈ K such that $lim_{x \to \infty} S^{n} x = x^{*}$ for all x ∈ K.

Proposition 2.1. [[15]] Assume that S : K → K is V-ψ-fuzzy contraction such that for some x₀ ∈ K, $S x_{0} \in [x_{0}]_{[\tilde{G}]}$ . Let ${\tilde{G}}_{x_{0}}$ be the component of $\tilde{G}$ containing x₀. Then $[x_{0}]_{[\tilde{G}]}$ is S-invariant and $S |_{[x_{0}] \tilde{G}}$ is a ${\tilde{G}}_{x_{0}}$ -ψ-fuzzy contraction. Moreover, it if $x, y \in [x_{0}]_{\tilde{G}}$ , then the sequences (Sⁿx) _n∈N and (Sⁿy) _n∈N are Cauchy equivalent.

Theorem 2.2. Let (K, V, *) be a complete V-fuzzy metric space and G be a directed graph and let the 4-tuple (K, V, * , G) have the property (P_S). Let S : K → K be a V-ψ-fuzzy contraction and K_S = {x ∈ K : (x, Sx) ∈ E (G)}, then the following statements hold:

If x ∈ K_S, then S|_{[x]_G} is a Picard operator;

If K_S ≠ φ and G is weakly connected, then S is a Picard operator;

FixS ≠ φ If and only if K_S ≠ φ;

If S ⊆ E (G), then S is a weakly Picard operator.

Proof. Let x ∈ K_S. By definition of K_S, (x, Sx) ∈ E (G) and so we have $Sx \in [x]_{\tilde{G}}$ .□

By Proposition 2.1, we have $S : [x]_{\tilde{G}} \to [x]_{\tilde{G}}$ and also S is V-ψ-fuzzy contraction.

Consider $y \in {\tilde{G}}_{x}$ then (Sⁿx) _n∈N and (Sⁿy) _n∈N are Cauchy equivalent and so (Sⁿx) _n∈N is a Cauchy sequence. By completeness of K, sequence (Sⁿx) must be convergent. So, there exists x^* ∈ K such that $\begin{matrix} lim_{n \to \infty} V (S^{n} x, x^{*}, \dots, x^{*}, φ (t)) = 1, for all t > 0 . \end{matrix}$ (2.2) Since (x, Sx) ∈ E (G) implies $(x, Sx) \in E (\tilde{G})$ and so by (GF1) we have, $(S^{n} x, S^{n + 1} x) \in E (G), for all n \in N .$ Now, by property (P_S), there exists a subsequence (S^{m
_n}x) _n∈N such that (S^{m
_n}x, x^*) ∈ E (G) for all n ∈ N.

Hence (x, Sx, Sx², …, S^{m
_n}x, x^*) is a path in G and so in $\tilde{G}$ . Therefore, $x^{*} \in [x]_{\tilde{G}}$ . Using (GF2) we have, $\begin{matrix} \frac{1}{V (S^{m_{n + 1}} x, {Sx}^{*}, \dots, {Sx}^{*}, φ (t))} - 1 \\ \leq ψ [\frac{1}{V (S^{m_{n}} x, x^{*}, \dots, x^{*}, φ (t))} - 1], \end{matrix}$ for all t > 0. Let ϵ > 0 be given, then by virtue of property of φ, we can find t > 0 such that φ (ct) < ϵ and using above inequality we have, $\begin{matrix} V (x^{*}, {Sx}^{*}, \dots, ϵ) & \geq V (x^{*}, S^{m_{n + 1}} x, \dots, S^{m_{n + 1}} x, \frac{ϵ}{2}) \\ * V (S^{m_{n + 1}}, {Sx}^{*}, \dots, {Sx}^{*}, \frac{ϵ}{2}) \\ \geq V (x^{*}, S^{k_{n} + 1}, \dots, S^{k_{n} + 1}, \frac{ϵ}{2}) \\ * [\frac{1}{1 + ψ [\frac{1}{V (S^{k_{n}} x, x^{*}, \dots, x^{*}, φ (t))} - 1]}] . \end{matrix}$ Letting n→ ∞ and using (2.2), we have $V (x^{*}, {Sx}^{*}, \dots, {Sx}^{*}, φ (t)) = 1, for all t > 0 .$ Thus, Sx^* = x^*, i.e. $x^{*} \in [x]_{\tilde{G}}$ is a fixed point of S. Hence by Theorem 2.1 and definition of Picard operator, $S |_{[x]_{\tilde{G}}}$ is a Picard operator.

To prove part (B), let us assume that K_S ≠ φ and G is weakly connected then $[x]_{\tilde{G}} = K$ for all x ∈ K. Hence by the proof of part (A), S is a Picard operator.

To prove part (C), it is given that FixS ≠ φ then there must be some x ∈ FixS implies Sx = x. Also E (G) ⊇ Δ, so we have (x, Su) ∈ E (G). So x ∈ K_S and FixS ⊂ K_S ≠ φ. Then by part (A), $S |_{[x]_{\tilde{G}}}$ is a Picard operator and therefore FixS ≠ φ.

To prove part (D), consider S ⊆ E (G), then (x, Sx) ∈ E (G) for all x ∈ K and so K = K_S. From (A) it is obvious that S is a Picard operator and every Picard operator is a weakly Picard operator. Hence (D) is proved.

Corollary 2.2. Let (K, V, *) be a complete V-fuzzy metric space and ⪯ be a partial order defined an K. Let S : K → K be a non-decreasing mapping (i.e. x ⪯ y ⇒ S_x ⪯ S_y) such that $\begin{matrix} x & ⪯ & y \Rightarrow \frac{1}{V (Sx, Sy, \dots, t)} - 1 \\ \leq & ψ [\frac{1}{V (x, y, \dots, t)} - 1] . \end{matrix}$ Assume that the following conditions holds:

If there is a non-decreasing sequence (x_n) _n∈N in K which converges to point x ∈ K and x_n+1 ≼ x_n for all n ∈ N, then x_n ≼ x or x_n ⪯ x_n for all n ∈ N. If there exists x₀ ∈ K such that x₀ ≼ Sx₀ or Sx₀ ≼ x₀, then S has a fixed point in K.

Proof. Let G be the graph defined by V (G) = K and $E (G) = {(x, y) \in K \times K : x ⪯ y \lor y ⪯ x} .$ Since S is nondecreasing, therefore (GF1) holds, and by using the given contractive condition, (GF2) also holds. Thus, S is a V-ψ-fuzzy contraction. Also, x_n ≼ x or x ≼ x_n for all n ∈ N implies property (P_S) is satisfied and the assumption (x₀, Sx₀) ∈ E (G) gives x₀ ∈ K_S. Therefore by statement(A) of Theorem 2.2, $S |_{[x_{0}]_{\tilde{G}}}$ is a Picard operator and so has a fixed point in S|_[x₀].□

Now, we apply our result on Kirk’s [10] for cyclic contractions. Let X be a nonempty set, m be a positive integer, C_i, i = 1, 2, . . . . m are nonempty subsets of X and $T : ⋃_{i = 1}^{m} C_{i} \to ⋃_{i = 1}^{m} C_{i}$ be a mapping then $B = ⋃_{i = 1}^{m} C_{i}$ is said to be cyclic representation of B with respect to T if T (C₁) ⊂ C₂, T (C₂) ⊂ C₃, …, T (C_m) ⊂ C₁ and then T is called a cyclic operator.

Corollary 2.3. Let (K, V, *) be a complete V-fuzzy metric space, m be a positive integer, C_i, i = 1, 2, . . . . m be closed subsets of X and $B = ⋃_{i = 1}^{m} C_{i}$ be a cyclic representation of B with respect to S. Suppose C_m+i = C_i for all i ∈ N, x ∈ C_i, y ∈ C_i+1 and following contractive conditions satisfied: $\begin{matrix} \frac{1}{V (Sx, Sy, \dots, t)} - 1 \\ \leq ψ [\frac{1}{V (x, y, \dots, t)} - 1] \end{matrix}$ then S has unique fixed point in $x^{*} \in ⋂_{i = 1}^{m} C_{i}$ .

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