Coupled solutions for a bivariate weakly nonexpansive operator by iterations

We prove weak and strong convergence theorems for a double Krasnoselskij type iterative method to approximate coupled solutions of a bivariate nonexpansive operator F : C x C -->C, where C is a nonempty closed and convex subset of a Hilbert space. The new convergence theorems generalize, extend, improve and complement very important old and recent results in coupled fixed point theory. Some appropriate examples to illustrate our new results and their generalization are also given.


Introduction and preliminaries
Let X be a nonempty set. A pair (x, y) ∈ X × X is called a coupled fixed point of the mapping F : X × X → X if it is a solution of the system F (x, y) = x, F (y, x) = y.
The novelty of this paper is that it considered coupled fixed point problem in a partially ordered metric space for mixed monotone mapping F : X × X → X in conjunction with a contraction type condition of the form: In particular, the authors established three kinds of coupled fixed point results: 1) existence theorems (Theorems 2.1 and 2.2); 2) existence and uniqueness theorem (Theorem 2.4); and 3) theorems that ensure the equality of the coupled fixed point components (Theorems 2.5 and 2.6).
Theorem 1. ( [18], Theorem 2.1 and Theorem 2.6) Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that 1 (X, d) is a complete metric space. Let F : X × X → X be a continuous mapping having the mixed monotone property on X.
If F satisfies (1.1) and there exist x 0 , y 0 ∈ X such that x 0 ≤ F (x 0 , y 0 ) and y 0 ≥ F (y 0 , x 0 ) , then there exist x, y ∈ X such that x = F (x, y) and y = F (y, x) .
Suppose, additionally, that x 0 , y 0 ∈ X are comparable. Then for the coupled fixed point (x, y), we have x = y.
We note that in all the above mentioned cases, the main conclusion is drawn using ( [18], Theorem 2.6) which guarantees existence as well as equality of components of the coupled fixed point.
On the other hand, in almost all the papers dealing with study of coupled fixed points, no attention is paid to the constructive features of such a result, i.e., there is neither explicit mention of the method by which one could approximate that coupled fixed point, nor on the order of convergence and / or error estimates of the iteration processes involved.
All the above observations motivate for constructive study of coupled fixed points of a bivariate mapping F : X × X → X satisfying a weaker contractive condition of nonexpansiveness type and providing a constructive method to approximate these coupled fixed points which we generally meet in applications, i.e., when we have equality of the coupled fixed point components.
The only paper that considers asymptotically nonexpansive bivariate mappings and the existence of their coupled fixed points is due to Olaoluwa et al. [33]. No other attempt has been made to tackle this important problem. We find here coupled solutions for a bivariate weakly nonexpansive operator on Hilbert spaces through an iterative method. Since asymptotically nonexpansive (and nonexpansive) bivariate mappings are particular sub-classes of the weakly nonexpansive mappings considered in the present paper, therefore our results also generalize, improve and complement the corresponding results obtained in [33].
In order to illustrate the broader scope and novelty of our results, we present appropriate examples to delineate them from the existing coupled fixed point theorems in literature and indicate their potential use in applications.

Nonexpansive bivariate operators
In this paper, we define the concept of nonexpansiveness for bivariate mappings as follows.
Definition 1. Let X be a normed linear space and C be a subset of for all x, y, u, v ∈ C, where a, b ≥ 0 and a + b ≤ 1.
A similar but stronger concept has been introduced in [33].
Definition 2. ( [33]) Let X be a normed linear space and C be a subset of X. A mapping F : for all x, y, u, v ∈ C.
Note that our condition (2.1) is more general than (2.2): any nonexpansive mapping F is weakly nonexpansive but the converse is not true, in general, as shown below.
Observe that for F in this example, the double sequence {(x n , y n )} n≥0 , defined by the Picard-type iteration with x 0 , y 0 ∈ X, is convergent but its limit is not the coupled fixed point of F (except for the case x 0 = y 0 ); a fact which follows immediately from the expressions of x n and y n : It is important to note that Opoitsev [36] was the first who studied coupled fixed points of bivariate mappings (see also [34], [35]) where a double Picard-type iteration sequence {(x n , y n )} n≥0 of the form (2.4) was used.
In order to state our main results, we need some concepts and results, adapted from the case of mono-variate operators to the case of bivariate operators.
The concept of demicompact operator has been introduced by Petryshyn [39] (see also [19] and [40]) for a mapping T : C → H, where C is a subset of a Hilbert space H. For the bivariate case it is adapted as follows.
has the property that whenever {u n } and {v n } are bounded sequences in C with the property that {F (u n , v n ) − u n } and {F (v n , u n ) − v n } converge strongly, then there exists a subsequence {(u n k , v n k )} of {(u n , v n )} such that u n k → u and v n k → v strongly.
We need the following version of the well known Browder-Gohde-Kirk fixed point theorem (see, for example, Theorem 3.1 in [7]), stated here in the Hilbert space setting.
Theorem 2. Let C be a bounded, closed and convex subset of a Hilbert space H and let F : C × C → C be a (weakly) nonexpansive operator. Then F has at least one coupled fixed point in C.
Proof. Let T : C → C be given by T (x) = F (x, x), x ∈ C. By the (weakly) nonexpansiveness property of F , we obtain the nonexpansiveness of T and hence by Browder-Gohde-Kirk fixed point theorem, it follows that F ix (T ) = ∅.
Remark 1. Theorem 2 shows that F has at least one (coupled) fixed point of the form (x, x) ∈ C ×C, but in general, for a bivariate mapping F it is also possible to have coupled fixed points (x, y) with unequal components, i.e., such that x = y, as shown by the following example.
We close this section by stating another auxiliary result that will be needed.

Main results
The main result of this paper is the following strong convergence theorem for a double Krasnoselskij-type algorithm associated with bivariate weakly nonexpansive operators on Hilbert spaces. Theorem 3. Let C be a bounded, closed and convex subset of a Hilbert space H and let F : C × C → C be weakly nonexpansive and demicompact operator. Then the set of coupled fixed points of F is nonempty and the double iterative algorithm {(x n , x n )} ∞ n=0 given by x 0 in C and
Proof. By Theorem 2, F has at least one coupled fixed point with equal components, (x, x) ∈ C × C. We first show that the sequence {x n − F (x n , x n )} n ∈ N converges strongly to zero.
Indeed, by using Lemma 1, On the other hand, (3.3), and weak nonexpansiveness of F and the fact that F (x, x) = x, it follows that for any real number a we have (3.4) If we choose now a nonzero a such that a 2 ≤ λ(1 − λ), then from the last inequality we obtain (3.5) (we used the Cauchy-Schwarz inequality,

So, by (3.5) we get
(3.6) By (3.5) we deduce that { x n − x } is a decreasing sequence of non negative real numbers, hence it is convergent. By the inequality (3.5), we also have This shows that x n − F (x n , x n ) → 0 (strongly) and so it follows by demicompactness of F that there exists a subsequence {x n k } ⊂ C and a point q ∈ C such that lim k→∞ x n k = q.
As F is nonexpansive, so it is continuous. This implies lim k→∞ F (x n k , x n k ) = F (q, q).
By (3.7), 0 = lim k→∞ (x n k − F (x n k , x n k )) = q − F (q, q), which shows that (q, q) is a coupled fixed point of F .
Using now the inequality (3.6), with x = q, we deduce that the sequence of nonnegative real numbers { x n − q } n≥0 is nonincreasing, hence convergent.
Since its subsequence { x n k − q } k≥0 converges to 0, it follows that the sequence { x n − q } n≥0 itself converges to 0, that is, the sequence {(x n , x n )} converges strongly to (q, q), as n → ∞.
We now introduce the concept of demicompactness at a point for a bivariate operator (adapted from the original definition of Petryshyn [39]).
Remark 3. Clearly, if F is demicompact on C, then it is demicompact at 0 but the converse is not true.
The demicompactness of F on the whole C in Theorem 3 may be weakened to the demicompactness at 0. Theorem 4. Let H be a Hilbert space, C a closed, bounded and convex subset of H, and F : C × C → C a weakly nonexpansive mapping such that F is demicompact at 0.
Then the Krasnoselkij-type double sequence {(x n , x n )} ∞ n=0 given by x 0 in C and (3.1) converges (strongly) to a coupled fixed point of F .
Proof. Note that in the proof of Theorem 3, we actually used the demicompactness of F at 0, so the arguments used there can be applied here.
Remark 4. The conclusion of Theorem 4 remains true if instead of the demicompactness of F at 0, we suppose I − F (x, x) maps closed sets in C into closed sets of H (see [39]). If in Theorems 3 and 4, we remove the demicompactness assumption, then (see [7]), the Krasnoselskij iteration does no longer converge strongly, in general, but it could converge (at least) weakly to a fixed point, as shown in the next theorem, which extends Theorem 3.3 in [7].
Denote by F ix (F ), the set of all coupled fixed points of F with equal components, i.e., F ix (F ) = {p ∈ C : F (p, p) = p}. given by x 0 in C and x n+1 = (1 − λ)x n + λF (x n , x n ), n ≥ 0, (3.8) converges weakly to p, for any λ ∈ (0, 1). ), converges weakly to a certain p 0 , then p 0 is a fixed point of T (and hence of F ) and therefore p 0 = p. Suppose that {x n j } ∞ j=0 does not converge weakly to p. As F is weakly nonexpansive, so we have which shows that T is nonexpansive and hence we get: Using the arguments in the proof of Theorem 2, it follows x n j − T x n j → 0, as n → ∞, and so the last inequality implies that As in the proof of Theorem 2, we have which shows, together with x n j ⇀ p 0 (as j → ∞), that On the other hand, we have Since C is bounded, the sequence is bounded, too, and so by the relations (3.9) -(3.11) we get Remark 5.
The assumption F ix (F ) = {(p, p)} in Theorem 5 may be removed to obtain the following more general result (similar to Theorem 3.4 in [7]).
Theorem 6. Let C be a bounded, closed and convex subset of a Hilbert space and F : C × C → C be weakly nonexpansive operator. Then the Krasnoselskij algorithm {x n } ∞ n=0 given by x 0 in C and x n+1 = (1 − λ)x n + λF (x n , x n ), n ≥ 0, converges weakly to a coupled fixed point of F .
Proof. We essentially follow the steps and arguments of the proof of Theorem 3.4 in [7]. For each p ∈ F ix (F ) and each n, we have, as in the proof of Theorem 2, which shows that the function g(p) = lim n→∞ x n − p is well defined and is a lower semicontinuous convex function on the nonempty convex set is closed, convex, and, hence, weakly compact. Therefore . Moreover, F 0 contains exactly one point. Indeed, since F 0 is convex and closed, for p 0 , p 1 ∈ F 0 , and p λ = (1 − λ)p 0 + λp 1 , Since p 1 − x n → d 0 and p 0 − x n → d 0 , the later relation implies that Now, in order to show that x n = F n (x 0 , x 0 ) ⇀ p 0 , it suffices to assume that x n j ⇀ p for an infinite subsequence and then prove that p = p 0 . By the arguments in the proof of Theorem 3, p ∈ F ix (F ). Considering the definition of g and the fact that x n j → p, we have which means that p = p 0 .

Conclusions and further study
Example 3. Let X = R (with the usual metric), C = [−1, 1]. Define bivariate function F : Then F satisfies (2.1) and is demicompact. Hence, all the assumptions of Theorem 2 are satisfied. It is easy to see that F possesses a unique coupled fixed point, (0, 0), and the Krasnoselskij-type iteration algorithm (3.1) yields the sequence Since −1 < 1 − 2λ < 1, it follows that (x n , x n ) converges to (0, 0) as n → ∞, for any initial value x 0 . This shows that, for weakly nonexpansive mappings, by using a Krasnoselskij-type iteration we can reach the convergence, while, by means of Picard-type iterations, this cannot be obtained, in general. Indeed, in this case, the Picard-type iteration (u n , u n ) associated with F is given by u n+1 = −u n , n ≥ 0, which is not convergent (except for the case u 0 = 0).
Finally, let us note that the double sequence {(x n , y n )}, defined for each component by a formula of the form (3.1) with F (x n , y n ) and F (y n , x n ), respectively, instead of F (x n , x n ), in the case of the function F in Example 1 will be given by This also indicates that it is not necessary to consider only the case of a double sequence with equal components {(x n , x n )} in Theorems 3-6 (but the proof of a convergence theorem for such an iterative method will be essentially different from the one given in this paper).
To conclude this paper, we note that, for the general case of a weakly nonexpansive bivariate mapping F , the Picard-type iteration process (2.4) does not generally converge or, even if it converges, its limit is not a coupled fixed point of F , but the Krasnoselskij type iteration process always converges to a coupled fixed point of F .
A similar approach for other contractive conditions and algorithms existing in literature (see [7], [20]) will be considered in our future work.