Fixed point theorems for $\alpha$--contractive mappings of Meir--Keeler type and applications

In this paper, we introduce the notion of $\alpha$--contractive mapping of Meir--Keeler type in complete metric spaces and prove new theorems which assure the existence, uniqueness and iterative approximation of the fixed point for this type of contraction. The presented theorems extend, generalize and improve several existing results in literature. To validate our results, we establish the existence and uniqueness of solution to a class of third order two point boundary value problems.

Then T has a unique fixed point x * ∈ X and T n x → x * (as n → ∞) for every x ∈ X, where T n denotes the n-th order iterate of T .
In another direction, Ran and Reurings [10] extended Banach's contraction principle to the setting of ordered metric spaces and obtained some interesting applications to matrix equations. Later on, the results of Ran and Reurings were extended and generalized by many authors (e.g., [1-4, 6, 8, 9, 11-13] and the references therein). In particular, Harjani et al. [5] unified these two directions by studying the fixed points of Meir-Keeler type contractions in ordered metric spaces.
In this context, the aim of this paper is to unify the concepts of Meir-Keeler contraction [7] and α − ψ-contractive type mapping [14] and establish some new fixed point theorems in complete metric spaces for such mappings. Several consequences of our results are presented in Section 3. We validate our results with an application to the study of the existence and uniqueness of solutions for a class of third order two point boundary value problems.

Preliminaries
Throughout this paper, let N denote the set of all non-negative integers, Z the set of all integers and R the set of all real numbers. We start by introducing the concept of α-contractive mapping of Meir-Keeler type. Subsequently, we prove some lemmas useful later.
In what follows, let (X, d) be a metric space, T : X → X and α : X × X → [0, +∞), if not stated otherwise.

Lemma 2.1. If T is an α-contractive mapping of Meir-Keeler type, then
α(x, y)d(T x, T y) < d(x, y) for all x, y ∈ X with x y.

Lemma 2.2.
Assume that T is α-admissible and α-contractive of Meir-Keeler type. Let x, y ∈ X such that α(x, y) ≥ 1. Then α(T n x, T n y) ≥ 1 for all n ∈ N, the sequence {d(T n x, T n y)} is nonincreasing, and d(T n x, T n y) → 0 (as n → ∞).
Then α is N-transitive, but not necessarily transitive (see, also, Corollary 3.7). Definition 2.8. Let x, y ∈ X. A vector ζ = (z 0 , z 1 , . . . , z n ) ∈ X n+1 is called an α-chain (of order n) from x to y if z 0 = x, z n = y and, for every i ∈ {1, 2, . . . , n}, Definition 2.9. We say that X is α-connected if for every x, y ∈ X with x y, there exists an α-chain from x to y.

Existence and uniqueness of fixed points
Now, we are ready to present and prove the first main result of the paper.
Then T has a fixed point, that is, there exists x * ∈ X such that T x * = x * .
Proof. Define the sequence {x n } in X by x n+1 = T x n for all n ∈ N; equivalently, and d(x n , x n+1 ) → 0 as n → +∞.
Finally, let m, n ≥ k + 1 and assume that q(x n ) ≤ q(x m ) without any loss of generality. Then, by the triangle inequality, (6) and (15), it follows that Concluding, {x n } is a Cauchy sequence in the complete metric space (X, d), hence convergent to some x * ∈ X. Moreover, {x n } is a (T, α)-orbital sequence by (4), hence, by (A3), there exists a subsequence {x n(k) } of {x n } such that T x n(k) → T x * as k → +∞. But T x n(k) = x n(k)+1 → x * as k → +∞, hence T x * = x * by the uniqueness of the limit, which concludes the proof.
In the next theorem, we replace the continuity of the mapping T by a regularity condition over the metric space (X, d). Proof. Following the proof of Theorem 2.1, we only have to prove that x * is a fixed point of T . Since {x n } is a (T, α)-orbital sequence, then, by (A4), there exists a subsequence {x n(k) } of {x n } such that Next, using Lemma 2.1, we get (with equality when x n(k) = x * ). As x n(k) → x * , we obtain that x n(k)+1 = T x n(k) → T x * . As {x n(k)+1 } is a subsequence of {x n } and x n → x * we have x n(k)+1 → x * . Now, the uniqueness of the limit gives us T x * = x * and the proof is complete.
To assure the uniqueness of the fixed point, we will consider the following additional assumption.
This is the purpose of the next theorem.

Theorem 2.3. If adding (A5) to the hypotheses of Theorem 2.1 (or Theorem 2.2), then x * is the unique fixed point of T and T n
it follows by Lemma 2.2 and the symmetry of d, that Now, since z 0 = x * is a fixed point of T , it follows that T n (z 0 ) = x * for all n, which finally leads to T n z i → x * (as n → +∞) for all i ∈ {1, 2, . . . , n}, using (16); hence, T n x → x * (as n → +∞). In particular, if x is another fixed point of T , it follows that x = x * which is a contradiction, and the proof is concluded.

Some corollaries
In this section, we will derive some corollaries from our previous theorems.

Coupled fixed point theorems for bivariate α-contractive mappings of Meir-Keeler type on complete metric spaces
The theorems obtained in the previous section allow us to derive some coupled fixed point results in complete metric spaces. First, let us recall the following definitions.

Definition 3.1 ([4]).
Let X be a nonempty set and F : X × X → X be a given mapping. A pair (x, y) ∈ X × X is called a coupled fixed point of F if F(x, y) = x and F(y, x) = y.
Also, x ∈ X is called a fixed point of F if (x, x) is a coupled fixed point, i.e., F(x, x) = x.

Definition 3.2 ([11])
. Let X be a nonempty set, and F, G : X × X → X. The symmetric composition (or, the scomposition for short) of A and B is defined by

Remark 3.1 ([11]
). The s-composition is an associative law. Also, the projection mapping is the identity element with respect to the s-composition (i.e., F * P X = P X * F = F for all F : X × X → X).
Consequently, for any F : X × X → X one can define the functional powers (i.e., the iterates) of F with respect to the s-composition by We have the following result.
Suppose that 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. Consider
Then, clearly, (X × X, D) is a complete metric space. Also, let T : X × X → X × X be defined by which concludes our argument.
We claim next that T is a β-contractive mapping of Meir-Keeler type (with respect to D). Indeed, let ε > 0 and let δ(ε) > 0 for which (17) (17). These two inequalities lead straight to which proves our claim.
Concluding, all the hypotheses of Theorem 2.1 applied to the metric space (X × X, D), the mapping T and the function β are satisfied, hence T has a fixed point (x * , y * ) ∈ X × X, meaning that (x * , y * ) is a coupled fixed point of F. The proof is now complete.
Proof. Using the notations in the proof of Corollary 3.1, it easily follows by (B4) that (X × X, D) is β-regular, hence (T, β)-regular. By following the proof of Corollary 3.1, the conclusion follows by Theorem 2.2 applied to the metric space (X × X, D), the mapping T and the function β.
For the uniqueness of the coupled fixed point, we consider the following assumption. Proof. We use the notations in the proof of Corollary 3.1. Then, by Theorem 2.3, it follows that (x * , y * ) is the unique fixed point of T , hence the unique coupled fixed point of F. Since (y * , x * ) is also a coupled fixed point of F, then (x * , y * ) = (y * , x * ), hence x * = y * , meaning also that x * is the unique fixed point of F. Since T n (x, y) = (F n (x, y), F n (y, x)) for all n ∈ N and x, y ∈ X, the proof is complete.

Fixed point theorems for R-contractive mappings of Meir-Keeler type on a metric space endowed with a Ntransitive binary relation
The notions and results in Section 2 easily translate to the setting of metric spaces endowed with a N-transitive binary relation.
In what follows, let (X, d) be a metric space, R be a binary relation over X and T : X → X. We first start with some terminology that is symmetrical to that in Section 2.

Definition 3.3.
We say that T is a R-contractive mapping of Meir-Keeler type (with respect to d) if for all ε > 0, there exists δ(ε) > 0 such that

Definition 3.4. We say that T is R-preserving if
x, y ∈ X : xRy ⇒ T xRT y. Definition 3.5. We say that a sequence {x n } in X is (T, R)-orbital if x n = T n x 0 and x n Rx n+1 for all n ∈ N. Definition 3.6. We say that T is R-orbitally continuous if for every (T, R)-orbital sequence {x n } in X such that x n → x ∈ X as n → +∞, there exists a subsequence {x n(k) } of {x n } such that T x n(k) → T x as k → +∞.

Remark 3.2. Clearly, if T is continuous, then T is R-orbitally continuous (for any R).
Definition 3.7. We say that (X, d) is (T, R)-regular if for every (T, R)-orbital sequence {x n } in X such that x n → x ∈ X as n → +∞, there exists a subsequence {x n(k) } of {x n } such that x n(k) Rx for all k. Definition 3.8. We say that (X, d) is R-regular if for every sequence {x n } in X such that x n → x ∈ X as n → +∞ and x n Rx n+1 for all n, there exists a subsequence {x n(k) } of {x n } such that x n(k) Rx for all k.

Remark 3.3.
Clearly, if (X, d) is R-regular, then it is also (T, R)-regular (for any T ). Definition 3.9. Let N ∈ N. We say that R is N-transitive (on X) if In particular, for N = 1 we recover the usual transitivity property. Definition 3.10. Let x, y ∈ X. A vector ζ = (z 0 , z 1 , . . . , z n ) ∈ X n+1 is called a R-chain (of order n) from x to y if z 0 = x, z n = y and z i−1 Rz i or z i Rz i−1 for every i ∈ {1, 2, . . . , n}.
Definition 3.11. We say that X is R-connected if for every x, y ∈ X with x y, there exists a R-chain from x to y.
The main results in Section 2 translate to the setting of metric spaces endowed with an arbitrary binary relation as follows. Proof. Define the mapping α : The conclusions then follows directly from Theorems 2.1, 2.2 and 2.3.
The following result is a consequence of Corollary 3.4 for bivariate R-contractive mappings of Meir-Keeler type. Corollary 3.6. Let (X, d) be a complete metric space, R a N-transitive binary relation over X (for some N ∈ N \ {0}), and F : X × X → X such that for every ε > 0 there exists δ(ε) > 0 for which: Suppose that (E1) for all x, y, u, v ∈ X, xRy, vRu =⇒ F(x, y)RF(u, v); (E2) there exists (x 0 , y 0 ) ∈ X × X such that If either (E3) F is continuous, or (E4) for every sequence {(x n , y n )} in X × X such that x n → x ∈ X, y n → y ∈ X as n → +∞, and x n Rx n+1 , y n+1 Ry n for all n ∈ N, there exists a subsequence {(x n(k) , y n(k) )} such that x n(k) Rx, yRy n(k) for all k ∈ N, then F has a coupled fixed point (x * , y * ) ∈ X × X. Additionally, if (E5) X is R-connected, then x * = y * , (x * , x * ) is the unique coupled fixed point of F, x * is the unique fixed point of F and F n (x, y) → x * as n → ∞ for all x, y ∈ X.

Fixed point results for cyclic contractive mappings of Meir-Keeler type
In this section, we obtain some fixed point results for cyclic α-contractions of Meir-Keeler type. We start by recalling the result obtained by Kirk, Srinivasan and Veeramani in [6] for cyclic contractive mappings.

Then N i=1 A i is non-empty and T has a unique fixed point in
The aim of our next result is to weaken the contraction condition (F2) by considering the following condition of Meir-Keeler type: (F3) for every ε > 0, there exists δ(ε) > 0 such that Then N i=1 A i is non-empty and T has a fixed point We check that the conditions in Theorem 2.2 are satisfied for the complete metric space (Y, d), the mappings α and T .
It follows that there exist i, j ∈ {1, . . . , N} such that note that j − i − 1 + N ≥ 0, and we conclude that the subsequence x n(k) satisfies hence α(x n(k) , x) ≥ 1 for all k, which proves our claim. Now, all the conditions in Theorem 2.2 (for (Y, d), α and T ) are satisfied, hence there exists a fixed point Now, the rest of the conclusion follows by Theorem 2.3.

Some consequences in ordered metric spaces
Clearly, the initial result of Meir and Keeler (Theorem 1.1) follows as a particular case of our Theorems 2.2 and 2.3, by simply choosing α(x, y) = 1 for all x, y ∈ X. In what follows, we will also show that several fixed point and coupled fixed point results in ordered metric spaces can be easily deduced (and improved) from our theorems.

Fixed point results in ordered metric spaces
Let X be a nonempty set. Recall that a binary relation over X is called a partial order if it is reflexive, transitive and anti-symmetric. If is a partial order over X, then x, y ∈ X are called comparable (subject to ) if x y or y x. Also, X is called -connected if for every x, y ∈ X, there exist z 0 , z 1 , . . . , z n ∈ X such that z 0 = x, z n = y and z i−1 , z i are comparable for every i ∈ {1, 2, . . . , n}.
In [5], Harjani et al. obtained several fixed point results in partially ordered sets for mappings satisfying some contraction condition of Meir-Keeler type. The main results in [5] for the case of nondecreasing mappings can be summarized as follows.
Theorem 4.1 ([5]). Let (X, d) be a complete metric space, a partial order over X and T : X → X such that for all ε > 0 there exists δ(ε) > 0 for which: As it can be easily seen, this result follows straight from Corollary 3.5, with R being the partial order . Moreover, (G5) can be replaced by the weaker assumption: Also, if x * is the unique fixed point of T , then T n (x) → x * (as n → ∞) for every x ∈ X. This follows by Corollary 3.5 and its an extension of the conclusion in Theorem 4.1.

Coupled fixed point results in ordered metric spaces
In [13], Samet studied the coupled fixed points of mixed strict monotone mappings that satisfied a contraction condition of Meir-Keeler type, thereby extending the previous work of Bhaskar and Lakshmikantham [4]. In what follows we present an extension of the results of Samet [13]; in this direction, we do not require that the mixed monotone property be strict and we also weaken other assumptions. We also improve the conclusion.
First, recall the following definition:

Definition 4.1 ([4]).
Let (X, ) be a partially ordered set. A mapping F : X × X → X is said to have the mixed monotone property if Our extension of the main results in [13] follows straight from Corollary 3.6, with R being the partial order , and can be stated as follows.
Theorem 4.2. Let (X, d) be a complete metric space, a partial order over X and F : X × X → X such that for every ε > 0 there exists δ(ε) > 0 for which: Suppose that: (H1) F has the mixed monotone property; (H2) there exist x 0 , y 0 ∈ X such that x 0 F(x 0 , y 0 ) and y 0 F(y 0 , x 0 ).

If either
(H3) F is continuous, or (H4) (X, d, ) has the following property: if {x n } is a nondecreasing (respectively, nonincreasing) sequence in X such that x n → x, then x n x (respectively, x n x) for all n, then F has a coupled fixed point (x * , y * ) ∈ X × X. In addition, if (H5) X is -connected, then x * = y * , (x * , x * ) is the unique coupled fixed point of F, x * is the unique fixed point of F and F n (x, y) → x * as n → ∞ for all x, y ∈ X.

Application to a third order two point boundary value problem
We study the existence and uniqueness of solution to the third order differential equation where f ∈ C([0, 1] × R, R), with the boundary value conditions This problem is equivalent to finding a solution x ∈ C([0, 1], R) to the integral equation Clearly, G(t, s) ≥ 0 for all t, s ∈ [0, 1]. Also, we can verify easily that for all t ∈ [0, 1].
It is well known that (X, d) is a complete metric space. Define the mapping T : X → X by It is easy to observe that α is N-transitive by (J1), T is α-admissible by (J3) and α(x 0 , T x 0 ) ≥ 1 by (J4). Also, it follows in a standard fashion that T is continuous, hence we omit this proof. Now, using (J2), (22) and the fact that ϕ is nondecreasing, it follows that for all x, y ∈ X with α(x, y) ≥ 1:  , y)) for all x, y ∈ X with α(x, y) ≥ 1.
This clearly leads to α(x, y)d (T x, T y) ≤ ϕ(d(x, y)) for all x, y ∈ X.
Let x, y ∈ X with ε ≤ d(x, y) < ε + δ(ε). Then, by (23) and (24), it follows that α(x, y)d (T x, T y) ≤ ϕ(d(x, y)) < ε; hence, we conclude that T is α-contractive mapping of Meir-Keeler type. Now, we can apply Theorem 2.1 and obtain the existence of a fixed point of T , hence the existence of a solution to (20)-(21). In addition, (J5) ensures that X is α-connected and the uniqueness of the solution follows by Theorem 2.3. The proof is now complete. ξ(x(t), z(t)) ≥ 0, inf 1] ξ(y(t), z(t)) ≥ 0. while all the other conditions and conclusions remain unchanged. In this case, the proof follows similarly, by letting ξ : R 2 → R be defined by ξ (a, b) = a − b (a, b ∈ R).