COUPLED FIXED POINT THEOREMS FOR TWISTED (α,β )−ψ-EXPANSIVE MAPPINGS

In this paper, we introduce a new concept of cyclic and cyclic ordered α−ψ expansive mappings and investigate the existence of a fixed point for the mappings in this class. Further, we shall derive coupled fixed point theorems in complete metric spaces. The presented theorems generalize and improve many existing results in the literature. Moreover, some examples are given to illustrate our results.


INTRODUCTION
Fixed point theory is one of the most powerful and fruitful tools in nonlinear analysis, since it provides a simple proof for the existence and uniqueness of the solutions to various mathematical models. Its core subject is concerned with the conditions for the existence of one or more fixed points of a mapping f from a topological space X into itself. The Banach contraction principle [1] is one of the most versatile elementary results in fixed point theory. Moreover, being based on iteration process, it can be implemented on a computer to find the fixed point of a contractive mapping. This principle has many applications and was extended by several authors. Among them, we mention the α − ψ-contractive mapping, which was introduced by Samet et al. [2] via α-admissible mappings.
In this paper, we introduce a new concept of cyclic and cyclic ordered α − ψ expansive mappings and establish various fixed point theorems for such mappings in complete metric spaces.
The presented theorems extend, generalize and improve many existing results in the literature.
Some examples are given to illustrate our results.
For the sake of completeness, we recall some basic definitions and fundamental results.
Wang et al. [3] defined expansion mappings in the form of following theorem. Theorem 1.1. Let (X, d) be a complete metric space. If f : X → X is an onto mapping and there exists a constant k > 1 such that d( f x, f y) ≥ kd(x, y), for each x, y ∈ X. Then f has a unique fixed point in X.
In 2013, P. Salimi et al. [10] introduced the concept of twisted (α, β )-admissible mappings in the following form: Definition 1.6. Let (X, d) be a metric space and f : X → X be a twisted (α, β )-admissible mapping. Then f is said to be a holds for all x, y ∈ X, where ψ ∈ Ψ.
(iii) twisted (α, β ) − ψ-contractive mapping of type III, if there is p ≥ 1 such that Recently, Kang et al. [11] introduced the concept of twisted (α, β ) − ψ-expansive mappings in metric spaces as follows: Definition 1.7. Let (X, d) be a metric space and f : X → X be a twisted (α, β )-admissible mapping. Then f is said to be a holds for all x, y ∈ X, where ψ ∈ Ψ.
In what follows, we present the main results of Kang et al. [11].
Theorem 1.8. Let (X, d) be a complete metric space and f : X → X be a bijective, twisted (α, β )−ψ-expansive mapping of type I or type II or type III satisfying the following conditions: (iii) f is continuous.
Then f has a fixed point, that is, there exists z ∈ X such that f z = z.
In what follows, Kang et al. [11] proved that Theorem 1.8 still holds for f not necessarily continuous, assuming the following condition: Here, we give some suitable examples to illustrate the results given by Kang et al. [11].
Example 1.10. Let X = R be endowed with the usual metric We prove that Theorem 1.9 can be applied to f . Proof. Let α(x, y) ≥ 1 for x, y ∈ X. Then x ∈ [−1, 0] and y ∈ [0, 1], and so f −1 y ∈ [−1, 0] and Otherwise α(x, y)β (x, y) = 0 and (1) trivially holds. Then f is a twisted (α, β ) − ψ-expansive mapping of type I and, by Theorem 1.9, f has a fixed point. Clearly, 0 and −11 6 are two fixed points of f . Example 1.11. Let X, d, α and β be as in Example 1.9 and f : We prove that Theorem 1.9 can be applied to f .
Proof. Proceeding as in the proof of Example 1.10, we deduce that f −1 is a twisted (α, β )admissible mapping and that the conditions (i)and (ii) of Theorem 1.9 hold.
Clearly 0 and −5 2 are two fixed points of f . Example 1.12. Let X = [0, ∞) be endowed with the usual metric d(x, y) = |x − y|, for all x, y ∈ X and f : X → X be defined by We prove that Theorem 1.9 can be applied to f .
Proof. Proceeding as in the proof of Example 1.10, we deduce that f −1 is a twisted (α, β )admissible mapping and that the conditions (i) and (ii) of Theorem 1.9 hold. Moreover, if x, y ∈ (3) trivially holds. Then f is a twisted (α, β ) − ψ-expansive mapping of type III and, by Theorem 1.9, f has a fixed point. Clearly, 0 and 3 4 are two fixed points of f . To ensure the uniqueness of the fixed point in Theorems 1.8 and 1.9 Kang et al. [11] consider the following condition:

CYCLIC RESULTS
In this section, we show how is possible to apply the results of Kang et al. [11] for proving, in a natural way, some analogous fixed point results involving a cyclic mapping. Definition 2.1. Let (X, d) be a metric space and A, B be two non-empty and closed subsets holds for all x ∈ A and y ∈ B, where ψ ∈ Ψ.
(iii) cyclic α − ψ-expansive mapping of type III, if there is p ≥ 1 such that holds for all x ∈ A and y ∈ B, where ψ ∈ Ψ. Now, we prove the following result for a continuous cyclic mapping.
Theorem 2.2. Let (X, d) be a complete metric space and A, B be two non-empty and closed subsets of X such that α : X × X → [0, ∞) and f : A ∪ B → A ∪ B be a bijective, continuous and generalized cyclic α − ψ-expansive mapping of type I or type II or type III. If there exists Also for cyclic α − ψ-expansive mappings, we can omit the continuity condition as is shown in the following theorem: Theorem 2.3. Let (X, d) be a complete metric space and A, B be two non-empty and closed subsets of X such that α : X × X → [0, ∞) and f : A ∪ B → A ∪ B be a bijective and cyclic α − ψexpansive mapping of type I or type II or type III. Also suppose that the following conditions hold: Let {x n } be a sequence in Y such that α(x 2n , x 2n+1 ) ≥ 1 and β (x 2n , x 2n+1 ) ≥ 1 for all n ∈ N∪{0} and x n → x as n → ∞, then x 2n ∈ A and x 2n+1 ∈ B. Now, since B is closed, then x ∈ B and hence We deduce that all the hypotheses of Theorem 1.9 are satisfied with X = Y and hence f has a fixed point.

CYCLIC ORDERED RESULTS
By using the similar arguments to those presented in the previous section, we are able to obtain results in the setting of ordered complete metric spaces. and holds for all x ∈ A and y ∈ B with x y, where ψ ∈ Ψ.
(ii) cyclic ordered α − ψ-expansive mapping of type II, if there is 0 < p ≤ 1 such that holds for all x ∈ A and y ∈ B with x y, where ψ ∈ Ψ.
(iii) cyclic ordered α − ψ-expansive mapping of type III, if there is p ≥ 1 such that holds for all x ∈ A and y ∈ B with x y, where ψ ∈ Ψ.  (3)) holds for all x, y ∈ Y . Let β (x, y) ≥ 1 for x, y ∈ X, then x ∈ A and y ∈ B with x y. It follows that f −1 x ∈ B and f −1 y ∈ A with f −1 y f −1 x, since f is decreasing. Therefore β ( f −1 y, f −1 x) ≥ 1, implies that, f −1 is a twisted (α, β )-admissible map- Then all the conditions of Theorem 1.8 are satisfied with X = Y and f has a fixed point in A ∪ B, say z. Since z ∈ A implies z = f −1 z ∈ B and z ∈ B implies z = f −1 z ∈ A, then z ∈ A ∩ B. Theorem 3.3. Let (X, d, ) be an ordered complete metric space and A, B be two non-empty and closed subsets of X. Let α : X × X → [0, ∞) and f : A ∪ B → A ∪ B be a bijective, cyclic ordered α − ψ-expansive mapping of type I or type II or type III.Also suppose that the following conditions hold: α(x 2n , x 2n+1 ) ≥ 1 and β (x 2n , x 2n+1 ) ≥ 1 for all n ∈ N ∪ {0} and x n → x as n → ∞, then x 2 n ∈ A and x 2n+1 ∈ B with x 2 n x 2n+1 . Since B is closed and by (iii), we deduce that x ∈ B and x 2 n x, We deduce that all the hypotheses of Theorem 1.9 are satisfied with X = Y and hence f has a fixed point.

COUPLED FIXED POINT
Now, we shall show that the coupled fixed point theorems in complete metric spaces can also be derived from these results. Before proving the result, we recall the following definition due to Bhaskar and Lakshmikantham [12]. Definition 4.1. Let f : X × X → X be a given mapping. We say that (x, y) ∈ X × X is a coupled fixed point of F if F(x, y) = x and F(y, x) = y.
Moreover, from the condition (ii) of the hypotheses of the theorem, we find that there exists So, we have transformed the problem to the complete metric space (Y, ρ). Therefore, all the hypotheses of Theorem 1. Theorem 4.4. Let (X, d) be a complete metric space and F : X × X → X be a given bijective mapping. Suppose that there exists ψ ∈ Ψ and functions α, β : for all (x, y), (u, v) ∈ X × X and 0 < p ≤ 1.
Suppose also that (i), (ii) and (iii) of Theorem 4.3 are satisfied.
Then F has a coupled fixed point, that is, there exists (x * , y * ) ∈ X × X such that x * = F(x * , y * ) and y * = F(y * , x * ).
Proof. For the proof of our result, we consider the mapping f given by (4) as a bijective mapping such that Also, consider the complete metric space (Y, ρ), where Y = X × X and From (12), we have Also, we define the functions η 1 , η 2 : Y ×Y → [0, ∞) as given by (8) and (9).
Moreover, from the condition (ii) of the hypotheses of the theorem, we find that there exists Theorem 4.5. Let (X, d) be a complete metric space and F : X × X → X be a given bijective mapping. Suppose that there exists ψ ∈ Ψ and functions α, β : for all (x, y), (u, v) ∈ X × X and p ≥ 1.
Suppose also that (i), (ii) and (iii) of Theorem 4.3 are satisfied.
Then F has a coupled fixed point, that is, there exists (x * , y * ) ∈ X × X such that x * = F(x * , y * ) and y * = F(y * , x * ).
Proof. For the proof of our result, we consider the mapping f given by (4) as a bijective mapping such that Also, consider the complete metric space (Y, ρ), where Y = X × X and From (16), we have Also, we define the functions η 1 , η 2 : Y ×Y → [0, ∞) as given by (8) and (9).