CQ-Type Algorithm for Reckoning Best Proximity Points of EP-Operators

: We introduce a new class of non-self mappings by means of a condition which is called the (EP)-condition. This class includes proximal generalized nonexpansive mappings. It is shown that the existence of best proximity points for (EP)-mappings is equivalent to the existence of an approximate best proximity point sequence generated by a three-step iterative process. We also construct a CQ-type algorithm which generates a strongly convergent sequence to the best proximity point for a given (EP)-mapping.


Introduction
The Banach contraction principle, which is the central result of the metric fixed point theory, has for decades been a source of inspiration for many authors. It states that any contraction mapping acting on a complete metric space has a unique fixed point, which is the limit of a sequence obtained by successive iterations of the given mapping. The attempts to extend this fundamental result have generated an impressive amount of scientific papers, as well as new areas of research. For instance, the theory of nonexpansive mappings, which naturally generalizes contraction mappings, has been a central topic during the last five decades. Fundamental existence results for nonexpansive mappings have been obtained by Kirk [1], Browder [2], and Göhde [3]. Later on, even wider classes of mappings were proposed and studied (see for instance Suzuki [4], García-Falset et al. [5]). At the same time, besides Picard's iteration used for contractions, some authors have introduced other iteration schemes (such as Mann [6] and Ishikawa [7]). This was in part due to the fact that Picard's iterative sequence for nonexpansive mappings does not necessarily converge. For more recently introduced iterative schemes, one can see Noor [8], Agrawal et al. [9], Abbas and Nazir, [10], Sintunavarat and Pitea [11], Thakur et al. [12][13][14], etc.
Another natural extension is to consider non-self mappings between two disjoint sets instead of mappings of a set into itself. In this setting, however, there is no point asking for fixed points, but instead one looks for best proximity points. More precisely, let T : X → Y be a mapping between two subsets X and Y of a metric space E. A best proximity point x ∈ X is a point such that d (x, Tx) is minimal. The interest for this type of problem was ignited by Fan [15]. Later on, authors such as Reich [16], Seghal and Singh [17], Naraghirad [18], and others have picked up on this subject and extended Fan's result in multiple ways.
The results presented in this paper relate to the above mentioned context as follows. Firstly, we consider the iterative process introduced by Thakur et al. [12] (which we shall call henceforth Proof. To prove that X 0 is a closed set, take a sequence {x n } ⊂ X 0 , converging in the norm to some point p ∈ X. As {x n } ⊂ X 0 , one can associate a sequence {y n } ⊂ Y 0 , such that x n − y n = d (X, Y) for all n. On the other hand, the P-property implies that x n − x m = y n − y m , for all n and m. Thus, {y n } ⊂ Y 0 is a Cauchy sequence, which converges to some q ∈ Y, since Y is a closed set. Using now the inequality The set X 0 is bounded since X is bounded.
To prove the convexity of the set X 0 , take x 1 , x 2 ∈ X 0 and α ∈ [0, 1]. There exist y 1 , y 2 ∈ Y 0 such that From the convexity of the set Y we get Thus X 0 is convex. The proof for Y 0 is similar.
A Banach space E is called uniformly convex (see for instance [24]) if, for each ε ∈ (0, 2], there exists δ > 0 such that for x, y ∈ E, Let C be a nonempty closed convex subset of a Banach space E. Given a bounded sequence {x n } ⊂ E, setting, for a given x ∈ E, one defines the asymptotic radius and, respectively, the asymptotic center In a uniformly convex Banach space the asymptotic center of a bounded sequence consists of a single element [25]. In a paper published in 2011, García-Falset et al. introduced a new class of mappings satisfying the so-called condition (E) defined as follows.

Definition 2 ([5]
). Let C be a nonempty subset of a Banach space (E, · ). We say that a mapping T : C → E satisfies the condition (E µ ) if there exists µ ≥ 1 such that for all x, y ∈ C, A mapping T is said to satisfy the condition (E) whenever it satisfies (E µ ) for some µ ≥ 1.
This condition is weaker than Suzuki's condition (C) for generalized nonexpansive mappings, a fact which follows from [4] Lemma 7. Recently Thakur et al. [12] have introduced a new iterative process, whose convergence to best proximity points of maps which satisfy the condition (E) we shall study. The iterative process, for a mapping satisfying the condition (E), is as follows.
for all n ≥ 1, where {α n } and {β n } are sequences in (0, 1). The following lemma is the counterpart of Lemma 3.1 from [12], but for mappings satisfying the condition (E). We shall denote the set of fixed points of a mapping T by F(T).

Lemma 2.
Let C be a nonempty closed convex subset of a Banach space (E, · ), and let T : C → C be a mapping satisfying the condition (E) such that F(T) = ∅. For arbitrary chosen x 1 ∈ C, let the sequence {x n } be generated by the iterative process Equation (1). Then lim n→∞ x n − p exists for any p ∈ F(T).
Proof. Let p ∈ F(T). As the mapping T satisfies condition (E), we have for any x ∈ C. Applying Equation (2) and using the triangle axiom, one has Similarly, using Equation (3), we get Now Equations (2) and (4) yield which means that the sequence { x n − p } is bounded and nonincreasing for any p ∈ F(T). Thus, the limit lim n→∞ x n − p exists.
The following theorem is an extension of Theorem 3.2 from [12] to the class of mappings satisfying condition (E). It is worth to compare it with Theorems 2 and 3 from [5]. We shall need the following technical lemma.

Lemma 3 ([26]
). Suppose (E, · ) is a uniformly convex Banach space and {t n } is a sequence bounded away from 0 and 1, i.e., 0 < b ≤ t n ≤ c < 1 for all n ≥ 1. Let {x n } and {y n } be two sequences in E such that lim sup n→∞ x n ≤ a, lim sup n→∞ y n ≤ a and lim sup n→∞ t n x n + (1 − t n ) y n = a hold for some a ≥ 0.
Then lim n→∞ x n − y n = 0. Theorem 1. Let C be a nonempty closed convex subset of a uniformly convex Banach space E and let T : C → C be a mapping satisfying condition (E). Given a point x 1 ∈ C, let the sequence {x n }, n ≥ 1, be generated by the iterative process Equation (1) On the other hand, using Equations (2) and (5), together with the properties of the norm, we get Thus, Whereas from Equation (2) we have that z n − p ≤ x n − p and thus It follows lim Thus, the conditions of Lemma 3 are satisfied yielding lim n→∞ Tx n − x n = 0.
Conversely, assume that {x n } is bounded and lim n→∞ Tx n − x n = 0. Take a point p ∈ A (C, {X n }). Using the fact that the mapping T satisfies the condition (E), we have which means that T p lies in A (C, {X n }). On the other hand, since E is uniformly convex, A (C, {X n }) is a singleton and hence T p = p. Corollary 1. Let C be a nonempty compact convex subset of a uniformly convex Banach space and let {x n } and T be as in Theorem 1. If F (T) = ∅, then the sequence {x n } converges strongly to a fixed point of T.
Proof. If F (T) = ∅, then, according to Theorem 1 lim n→∞ x n − Tx n = 0. As C is assumed to be compact, the sequence {x n } has a convergent subsequence x n k to some point p ∈ C. Since the mapping T satisfies the condition (E), for all n ≥ 1 and some µ ≥ 1, we have The uniqueness of the limit implies that x n k converges strongly to T p, meaning that T p ∈ F (T). On the other hand, according to Lemma 2, the limit lim n→∞ x n − p exists which completes the proof.

Best Proximity Point Problem for (EP)-Mappings
Let X and Y be two convex subsets in a Banach space. A non-self mapping T : X → Y is called nonexpansive if Tx − Ty ≤ x − y , for all x, y ∈ X.
Gabeleh [19] introduced a condition on mappings which is weaker than nonexpansiveness and which resembles Suzuki's condition (C), but in the context of non-self mappings.

Definition 3 ([19]
). Let (X, Y) be a pair of of nonempty subsets of a Banach space. A mapping T : X → Y is said to be proximal generalized nonexpansive if and only if for all x, y, u, v ∈ X such that u − Tx = d (X, Y) = v − Ty , The above definition can be widened by taking some λ ∈ (0, 1) instead of 1/2. Next we introduce a new condition on non-self mappings which can be seen as the analogue of the condition (E) introduced by García-Falset et al. [5] and which involves the metric projection.

Definition 4.
Let (X, Y) be a pair of of nonempty subsets of a Banach space (E, · ) such that X 0 = ∅ and denote by P X 0 : E → X 0 the metric projection operator onto X 0 . A mapping T : X → Y is said to satisfy the condition (EP) if and only if

Proposition 1. Any proximal generalized nonexpansive mapping satisfies the condition (EP).
Proof. From Definition 3 it is clear that u, v ∈ X 0 (and hence X 0 = ∅) and that Tx ∈ Y 0 . Also, from the definition of the metric projection we have u = P X 0 Tx and v = P X 0 Ty. For any λ ∈ (0, 1) we have Since the mapping T is proximal generalized nonexpansive, it follows that On the other hand, the triangle inequality together the inequality Equation (11), yield which means that the condition (EP) is satisfied for µ = 2.
Next, we adapt the iterative process Equation (1) for the case of non-self mappings using the metric projection as follows.
It is clear from Lemma 1 that the set X 0 is convex. Also, since the iterative process Equation (12) involves the metric projection onto X 0 and convex combinations of elements from X 0 , it is clear that {x n } ⊂ X 0 .
The notion of approximate fixed point sequence has a natural extension in the context of best proximity point problem.

Definition 5 ([19]
). Let (X, Y) be a pair of nonempty sets of a Banach space and T : X → Y be a non-self mapping. A sequence {x n } ⊂ X is said to be an approximate best proximity point sequence for T if and only if lim n→∞ x n − Tx n = d (X, Y) .

Theorem 2.
Let (X, Y) be a pair of nonempty subsets of a Banach space E, where the pair has the P-property, X is convex, Y is closed and convex, and X 0 = ∅. Suppose the mapping T : X → Y satisfies the condition (EP) with T (X 0 ) ⊆ Y 0 and let {x n } be the sequence generated by the iterative process (12). Then, the mapping T has a best proximity point if and only if {x n } is bounded and lim n→∞ x n − Tx n = d (X, Y).
Proof. According to Lemma 1 the set X 0 is closed and convex. If p is a best proximity point for the mapping T, then p is a fixed point for the mapping P X 0 T : X 0 → X 0 , i.e., F P X 0 T = ∅. Thus, according to Theorem 1, the sequence {x n } is bounded and lim n→∞ x n − P X 0 Tx n = 0. Also, since Conversely, suppose that lim n→∞ x n − Tx n = d (X, Y). Using this fact while passing to the limit in Equation (13) gives lim n→∞ x n − P X 0 Tx n = 0. Since by assumption the sequence {x n } is bounded, according to Theorem 1, there exists p ∈ X 0 such that P X 0 T p = p, which means that p − T p = d (X, Y).

Corollary 2.
Let (X, Y), T, and {x n } be as in Theorem 2 and suppose additionally that X is compact. If F(P X 0 T) = ∅ , then the sequence {x n } generated by the iterative process (12) converges strongly to a best proximity point of T.
Proof. Since X is compact, the sequence {x n } has a subsequence x n k converging strongly to some point z ∈ X. Also, since F(P X 0 T) = ∅, we have that lim n→∞ x n − P X 0 Tx n = 0. Letting k → ∞ in the relation we obtain that x n k converges strongly to P X 0 Tz and by the uniqueness of the limit we have z = P X 0 Tz, i.e., z ∈ F(P X 0 T). Applying now Lemma 2 yields the conclusion.

Strong Convergence via a CQ-Type Algorithm
In this section we introduce an algorithm which is a hybrid between the iterative process (12) and the CQ algorithm introduced by Nakajo and Takahashi [20]. The main outcome is the strong convergence of the resulting sequence. Before dealing with the main result, let us establish the following preliminaries.
Let H be a real Hilbert and denote the inner product by ·, · and, respectively, the norm by · . Let X and Y be nonempty closed and convex subsets of H. Given a mapping T : X → Y, we denote the set of its best proximity points by X T , i.e., Clearly X T ⊆ X 0 (for details, one can see [27]). For a sequence {x n } ⊂ X let where denotes the weak convergence, be the weak ω-limit set.

Lemma 4 ([28]
). Let K be a closed and convex subset of a real Hilbert space H and let P K be the metric projection from H onto K. Then, given x ∈ H and z ∈ K, for all y ∈ K.

Lemma 5 ([28]
). Let K be a closed and convex subset of a real Hilbert space H. Let {x n } be a sequence in H and let x ∈ H. Let q = P K x. If {x n } is such that w ω ({x n }) ⊂ K and satisfies the condition A Banach space (E, · ) is said to have the Opial property if, for every sequence {x n } ⊂ E such that x n z, the inequality lim inf n→∞ x n − z < lim inf n→∞ x n − y holds whenever y = z. It is worth mentioning that any Hilbert space has the Opial property (for a proof, please see [29]).
Lemma 6 (Theorem 1, [5]). Let C be a nonempty subset of a Banach space E and let T : C → E be a given mapping. If a) there exists a sequence {x n } ⊂ C such that x n − Tx n → 0 and z n z, b) T satisfies the condition (E) on C, c) (E, · ) has the Opial property, then Tz = z.
Consider now the following algorithm: x 0 ∈ X 0 arbitrary, where {α n } and {β n } are real sequences bounded away from 0 and 1. Clearly the projection is well defined since the set X 0 is closed and convex, according to Lemma 1.
Theorem 3. Let (X, Y) be a pair of nonempty closed and convex subsets of a real Hilbert space, and suppose the pair has the P-property. Let T : X → Y be a mapping which satisfies the condition (EP) such that X T is a nonempty convex subset of X 0 . Then, the sequence {x n }, generated by the algorithm (12), converges to a best proximity point. In particular, it converges to p, where p = P X T (x 0 ). Moreover, the same holds true for the sequences {w n }, {y n } and {z n }.
Proof. Let x 0 ∈ X 0 . Clearly the sets Q n and C n respectively, are closed and convex subsets of X. Let us prove that X T ⊂ C n ∩ Q n . Let z ∈ X T . Clearly, d (z, Tz) = d (X, Y), i.e., d z, P X 0 Tz = 0. Keeping in mind that the mapping T satisfies the condition (EP), we have Similarly, we get the inequality and, respectively, Hence, z ∈ C n , i.e., X T ⊂ C n .
The inclusion X T ⊂ Q n follows by induction. Indeed, it is clear from the definition that Q 0 = X 0 and that X T ⊂ X 0 , respectively. Assume X T ⊂ Q n . As C n and Q n are closed and convex sets, for x n+1 = P C n ∩Q n (x 0 ), according to Lemma 4, one has x n+1 − z, x n+1 − x 0 ≤ 0 for all z ∈ C n ∩ Q n . Using again the definition of the set Q n and noticing that X T ⊂ C n ∩ Q n yields X T ⊂ Q n+1 , which completes the induction.
Let p = P X T (x 0 ). Since X T ⊂ C n ∩ Q n and x n+1 = P C n ∩Q n (x 0 ), we have which also means that the sequence {x n } is bounded. Since x n+1 ∈ Q n , we obtain On the other hand, the triangle axiom and the definition of C n yield and thus z n − x n → 0 for n → ∞.
Noticing that z n − x n = β n P X 0 Tx n − x n , it follows that P X 0 Tx n − x n → 0 as n → ∞, since the sequence {β n } is bounded away from 0 and 1.
Consider now the mapping P X 0 T : X 0 → X 0 , which clearly satisfies the condition (E). The set of its fixed points is the set X T . Recalling that any Hilbert space has the Opial property, while Applying Lemma 6, yields the inclusion w ω ({x n }) ⊂ X T . This fact, together with inequality Equation (18), according to Lemma 5, provides the strong convergence of the sequence {x n } to the point p = P X T (x 0 ).
Turning now to the strong convergence of the other sequences, we have w n − x n ≤ w n − x n+1 + x n+1 − x n ≤ x n − x n+1 + x n+1 − x n and thus w n − x n → 0. Similarly, one obtains y n − x n → 0. Lastly, the strong convergence of the sequences {w n }, {y n }, and {z n } follow by taking n → ∞ in the inequalities w n − p ≤ w n − x n + x n − p , y n − p ≤ y n − x n + x n − p , z n − p ≤ z n − x n + x n − p .

Conclusions
The starting point of our study in this paper has two main ingredients. One of them is the iterative process introduced by Thakur et al. [12], for Suzuki generalized nonexpansive mappings. The other is a class of mappings satisfying the condition (E), introduced by García-Falset et al. and which is even larger. We firstly extended the main results from [12] to the case of mappings satisfying condition (E). Afterwards, we have progressed to the setting of best proximity point problem, which is a generalization of the fixed point problem, by introducing a new class of non-self mappings. These generalize the class of proximal generalized nonexpansive mappings introduced by Gabeleh [19]. We have also adapted the iterative process from [12] to the setting of non-self mappings, using the metric projection, and have studied the convergence of the resulting iterative sequence. In the last part, we have constructed a CQ-type algorithm [20] for the iterative process under consideration and have proved the strong convergence of the resulting sequence to a best proximity point for mappings satisfying the condition (EP).