STRONG CONVERGENCE THEOREMS FOR FIXED POINT PROBLEMS OF ASYMPTOTICALLY QUASI-𝜙-NONEXPANSIVE MAPPINGS IN THE INTERMEDIATE SENSE

Abstract

In this paper, we introduce a general iterative algorithm for asymptotically quasi-${\phi}$-nonexpansive mappings in the intermediate sense to have the strong convergence in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results announced by many authors.

1. Introduction

Let E be a real Banach space with the dual space E*. Let C be a nonempty closed convex subset of E. Let T : C → C be a nonlinear mapping. We denote by F(T) the set of fixed points of T.

A mapping T : C → C is said to be nonexpansive if

Three classical iteration processes are often used to approximate a fixed point of nonexpansive mapping. The first one is introduced by Halpern [3] and is defined as follows: Take an initial point x0 ∈ C arbitrarily and define {xn} recursively by

where is a sequence in the interval [0, 1]. The second iteration process is now known as Mann’s iteration process[6] which is defined as

where the initial point x1 is taken in C arbitrarily and the sequence is in the interval [0,1]. The third iteration process is referred to as Ishikawa’s iteration process [5] which is defined recursively by

where the initial point x1 is taken in C arbitrarily, are sequences in the interval [0, 1].

In general not much is known regarding the convergence of the iteration processes (1.1)-(1.3) unless the underlying space E has elegant properties which we briefly mention here.

Recently, Matsushita and Takahashi [7] proved strong convergence theorems for approximation of fixed points of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. More precisely, they proved the following theorem.

Theorem 1.1. Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive mapping from C into itself and let {αn} be a sequence of real numbers such that 0 ≤ αn < 1 and lim supn→∞ αn < 1. Suppose that {xn} is given by

where J is the duality mapping on E. If F(T) is nonempty, then {xn} converges strongly to ΠF(T)x0, where ΠF(T) is the generalized projection from C onto F(T).

In [4], Hao introduced the following iterative scheme for approximating a fixed point of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense in a reflexive, strictly convex and smooth Banach space: x0 ∈ E, C1 = C, x1 = ΠC1x0,

where ξn = max{0, supp∈F(T),x∈C(ϕ(p, Tnx) − ϕ(p, x))}.

Motivated by the fact above, the purpose of this paper is to prove a strong convergence theorem for finding a fixed point of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense in a reflexive, strictly convex and smooth Banach space, which has the Kadec-Klee property.

 

2. Preliminaries

Let E be a real Banach space and let E* be the dual space of E. The duality mapping J : E → 2E* is defined by

By Hahn-Banach theorem, J(x) is nonempty.

The modulus of convexity of E is the function δE : (0, 2] → [0, 1] defined by

E is said to be uniformly convex if ∀ε ∈ (0, 2], there exists a δ = δ(ε) > 0 such that for x, y ∈ E with ∥x∥ ≤ 1, ∥y∥ ≤ 1 and ∥x − y∥ ≥ ε, then Equivalently, E is uniformly convex if and only if δE(ε) > 0, ∀ε ∈ (0, 2]. E is strictly convex if for all x, y ∈ E, x ≠ y, ∥x∥ = ∥y∥ = 1, we have ∥λx+(1−λ)y∥ < 1, ∀λ ∈ (0, 1). The space E is said to be smooth if the limit

exists for all x, y ∈ S(E) = {z ∈ E : ∥z∥ = 1}. It is also said to be uniformly smooth if the limit exists uniformly in x, y ∈ S(E).

It is well known that if E is uniformly smooth, then J is norm-to-norm uniformly continuous on each bounded subset of E. If E is smooth, then J is single-valued.

Recall that a Banach space E has the Kadec-Klee property if for any sequence {xn} ⊂ E and x ∈ E with xn ⇀ x and ∥xn∥ → ∥x∥, then ∥xn − x∥ → 0 as n → ∞. It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.

In what follows, we always use ϕ : E × E → ℝ to denote the Lyapunov functional defined by

It follows from the definition of ϕ that

and

Following Alber [1], the generalized projection ΠC : E → C is defined by

The existence and uniqueness of the operator ΠC follows from the properties of the function ϕ(x, y) and strict monotonicity of mapping J (see [1,2,10]).

Lemma 2.1 ([1]). Let E be a reflexive, strictly convex and smooth Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:

Remark 2.1. If E is a real Hilbert space, then ϕ(x, y) = ∥x − y∥2 and ΠC is the metric projection PC of E onto C.

Definition 2.2. Let C be a nonempty closed convex subset of E and let T be a mapping from C into itself. A point p ∈ C is said to be an asymptotic fixed point of T if C contains a sequence {xn}, which converges weakly to p and limn→∞ ∥xn − Txn∥ = 0.

The set of asymptotic fixed points of T is denoted by

Definition 2.3. A mapping T : C → C is said to be

(1) relatively nonexpansive if and

for all x ∈ C and p ∈ F(T);

(2) quasi-ϕ-nonexpansive if F(T)≠ϕ and

for all x ∈ C and p ∈ F(T);

(3) asymptotically quasi-ϕ-nonexpansive if F(T)≠ϕ and there exists a sequence {kn} ⊂ [0,∞) with kn → 1 as n → ∞ such that

for all x ∈ C, p ∈ F(T) and n ≥ 1;

(4) asymptotically quasi-ϕ-nonexpansive in the intermediate sense if F(T)≠ϕ and

Put

Remark 2.2. From the definition, it is obvious that ξn → 0 as n → ∞ and

Remark 2.3. (1) It is easy to see that the class of quasi-ϕ-nonexpansive mappings contains the class of relatively nonexpansive mappings.

(2) The class of asymptotically quasi-ϕ-nonexpansive mappings is more general than the class of relatively asymptotically nonexpansive mappings.

(3) The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense in the framework.

Recall that T is said to be asymptotically regular on C if for any bounded subset K of C,

Definition 2.4. A mapping T : C → C is said to be closed if for any sequence {xn} ⊂ C with xn → x and Txn → y, Tn = y.

Lemma 2.5 ([4]). Let E be a reflexive, strictly convex and smooth Banach space such that both E and E* have the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let T : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Then F(T) is a closed convex subset of C.

 

3. Main results

Theorem 3.1. Let E be a reflexive, strictly convex and smooth Banach space such that both E and E* have the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let T : C → C be a closed, asymptotically regu-lar and asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Let {αn} be a sequence in [0, 1] and {βn} be a sequence in (0, 1) satisfying the following conditions:

Let {xn} be a sequence generated by

where ξn = max{0, supp∈F(T);x∈C(ϕ(p, Tnx) − ϕ(p, x))}, ΠCn+1 is the general-ized projection of E onto Cn+1. If F(T) is bounded in C, then {xn} converges strongly to ΠF(T)x1.

Proof. It follows from Lemma 2.2 that F(T) is a closed convex subset of C, so that ΠF(T)x is well defined for any x ∈ C.

We split the proof into six steps.

Step 1. We first show that Cn, n ≥ 1, is nonempty, closed and convex.

It is obvious that C1 = C is closed and convex. Suppose that Cn is closed and convex for some n ≥ 2. For z1, z2 ∈ Cn+1, we see that z1, z2 ∈ Cn. It follows that z = tz1 + (1 − t)z2 ∈ Cn, where t ∈ (0, 1). Notice that

and

These are equivalent to

and

Multiplying t and 1 − t on both sides of (3.2) and (3.3), respectively, we obtain that

That is,

Therefore, we have

This implies that Cn+1 is closed and convex for all n ≥ 1.

Step 2. We show that F(T) ⊂ Cn, ∀n ≥ 1.

For n = 1, we have F(T) ⊂ C1 = C. Now, assume that F(T) ⊂ Cn for some n ≥ 2. Put wn = J−1(βnJxn + (1−βn)JTnxn). For each x* ∈ F(T), we obtain from (2.2) and (2.3) that

and

Therefore, we have

So, x* ∈ Cn+1. It implies that F(T) ⊂ Cn+1.

Step 3. We prove that {xn} is bounded and limn→∞ ϕ(xn, x1) exists.

Since xn = ΠCnx1, we have from Lemma 2.1 that

Again, since F(T) ⊂ Cn, we have

It follows from Lemma 2.1 that for each u ∈ F(T) and for each n ≥ 1,

Therefore, {ϕ(xn, x1)} is bounded. By virtue of (2.1), {xn} is also bounded. Again, since xn = ΠCnx1, xn+1 = ΠCn+1x1 and xn+1 ∈ Cn+1 ⊂ Cn for all n ≥ 1, we have

This implies that {ϕ(xn, x1)} is nondecreasing and bounded. Hence, limn→∞ ϕ(xn, x1) exists.

Step 4. Next, we prove that where is some point in C.

Now, since {xn} is bounded and the space E is reflexive, we may assume that there exists a subsequence {xni} of {xn} such that Since Cn is closed and convex, it is easy to see that for each n ≥ 1. This implies that

On the other hand, it follows from the weak lower semicontinuity of the norm that

which implies that as ni → ∞. Hence, as ni → ∞. In view of the Kadec Klee property of E, we see that as ni → ∞. If there exists another subsequence such that we have

which implies This shows that

Step 5. Now we prove that

Since and limn→∞ ϕ(xn, x1) exists, we see that

Hence, we have

Since xn+1 ∈ Cn+1, xn → and αn → 0, it follows from (3.1) and Remark 2.2 that

as n → ∞. This implies that

Therefore we obtain

and so

This shows that {Jyn} is bounded. Since E is reflexive, E* is reflexive. Without loss of generality, we can assume that In view of reflexivity of E, we see that J(E) = E*. Hence, there exists y ∈ E such that This implies that J(yn) ⇀ Jy. And

Taking lim infn→∞ for both sides of (3.7), we have from (3.4) that

which shows that and so

It follows from (3.6) and the Kadec-Klee property of E* that Since J−1 is norm-weak-continuous, we have

It follows from (3.5),(3.8) and the Kadec-Klee property of E that we have

On the other hand, since {xn} is bounded and T is asymptotically quasi-ϕ-nonexapnsive in the intermediate sense, for any given p ∈ F(T), we have from (2.3) that

This implies that {Tnxn} is bounded. Since

it implies that {wn} is also bounded. From (3.1), we have

It follows from (3.9) that as n → ∞. Since J−1 is norm-weaklycontinuous, this implies that

as n → ∞. Note that

This together with (3.10) shows that

as n → ∞. Since we have Since

By condition (ii) and (3.11), we have that

Since J−1 is norm-weakly-continuous, this implies that

It follows from (3.12) that

This together with (3.13) and the Kadec-Klee property of E shows that

as n → ∞. Again, by the asymptotic regularity of T, we have

as n → ∞. That is, It follows from the closedness of T that

Step 6. Finally, we prove that

Let w = ΠF(T)x1. Since w ∈ F(T) ⊂ Cn and xn = ΠCnx1, we have

This implies that

From the definition of and (3.14), we see that This completes the proof.

Remark 3.1. If we take αn = 0 for all n ∈ ℕ, then the iterative scheme (3.1) reduces to following scheme:

where

which is (1.2) and an improvement to (1.1).

In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following.

Corollary 3.2. Let E be a Hilbert space. Let C be a nonempty closed convex subset of H. Let T : C → C be a closed, asymptotically regular and asymptot-ically quasi-ϕ-nonexpansive mapping in the intermediate sense. Let {αn} be a sequence in [0, 1] and {βn} be a sequence in (0, 1) satisfying the following condi-tions:

Let {xn} be a sequence generated by

where

is the metric projection from E onto Cn+1. If F(T) is bounded in C, then {xn} converges strongly to PF(T)x1.

Proof. . If E is a Hilbert space, then J = I (the identity mapping) and ϕ(x, y) = ∥x − y∥2. We can obtain the desired conclusion easily from Theorem 3.1. This completes the proof.

If T is quasi-ϕ- nonexpansive, then Theorem 3.1 is reduced to the following without involving boundedness of F(T) and asymptotically regularity on C.

Corollary 3.3. Let E be a reflexive, strictly convex and smooth Banach space such that both E and E* have the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let T : C → C be a closed, quasi-ϕ-nonexpansive mapping with F(T) ≠ ϕ. Let {αn} be a sequence in [0, 1] and {βn} be a sequence in (0, 1) satisfying the following conditions:

Let {xn} be a sequence generated by

where is the generalized projection of E onto Cn+1. Then {xn} converges strongly to ΠF(T)x1.

Remark 3.2. (1) By Remark 3.1, Theorem 3.1 extends Theorem 2.1 of Hao [4].

(2) Theorem 3.1 generelized Theorem 3.1 of Matsushita and Takahashi [7] in the following respects:

(3) Corollary 3.1 generalized and improves Corollary 2.5 of Hao [4], Theorem 3.4 of Nakajo and Takahashi [8] and Theorem 2.1 of Su and Qin [9] in the following aspects: