Strong convergence theorems for total quasi-ϕ-asymptotically nonexpansive multi-valued mappings in Banach spaces

The main purpose of this article is to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and prove the strong convergence theorem in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. In order to get the theorems, the hybrid algorithms are presented and are used to approximate the fixed point. The results presented in this article improve and extend some recent results announced by some authors.

AMS (MOS) Subject Classification: 47J06, 47J25.

Throughout this article, we always assume that X is a real Banach space with the dual X* and J : X → 2^X is the normalized duality mapping defined by

In the sequal, we use F(T) to denote the set of fixed points of a mapping T, and use ℛ and ℛ⁺ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively. We denote by x_n → x and x_n ⇀ x the strong convergence and weak convergence of a sequence {x_n}, respectively.

A Banach space X is said to be strictly convex if for all x, y ∈ U = {z ∈ X : ||z|| = 1} with x ≠ y. X is said to be uniformly convex if, for each ϵ ∈ (0, 2], there exists δ > 0 such that for all x, y ∈ U with ||x - y|| ≥ ϵ. X is said to be smooth if the limit

exists for all x, y ∈ U. X is said to be uniformly smooth if the above limit is attained uniformly in x, y ∈ U.

Remark 1.1. The following basic properties of a Banach space X can be found in Cioranescu [1].

(i) If X is uniformly smooth, then J is uniformly continuous on each bounded subset of X;

(ii) If X is a reflexive and strictly convex Banach space, then J^-1 is norm-weak-continuous;

(iii) If X is a smooth, strictly convex and reflexive Banach space, then J is single-valued, one-to-one and onto;

(iv) A Banach space X is uniformly smooth if and only if X* is uniformly convex;

(v) Each uniformly convex Banach space X has the Kadec-Klee property, i.e., for any sequence {x_n} ⊂ X, if x_n ⇀ x ∈ X and ||x_n|| → ||x||, then x_n → x.

Let X be a smooth Banach space. We always use ϕ : X × X → ℛ⁺ to denote the Lyapunov functional defined by

It is obvious from the definition of the function ϕ that

Following Alber [2], the generalized projection Π_C : X → C is defined by

Lemma 1.2. [2] Let X be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of X. Then the following conclusions hold:

(a) ϕ(x, Π_Cy) + ϕ(Π_Cy, y) ≤ ϕ(x, y) for all x ∈ C and y ∈ X;

(b) If x ∈ X and z ∈ C, then

(c) For x, y ∈ X, ϕ(x, y) = 0 if and only if x = y.

Let X be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of X and T : C → C be a mapping. A point p ∈ C is said to be an asymptotic fixed point of T if there exists a sequence {x_n} ⊂ C such that x_n ⇀ p and ||x_n - Tx_n|| → 0. We denoted the set of all asymptotic fixed points of T by .

Definition 1.3. (1) A mapping T : C → C is said to be relatively nonexpansive [3] if and

(2) A mapping T : C → C is said to be closed if, for any sequence {x_n} ⊂ C with x_n → x and Tx_n → y, then Tx = y.

Definition 1.4. (1) A mapping T : C → C is said to be quasi-ϕ-nonexpansive if and

(2) A mapping T : C → C is said to be quasi-ϕ-asymptotically nonexpansive if and there exists a real sequence {k_n} ⊂ [1, ∞) with k_n → 1 such that

(3) A mapping T : C → C is said to be total quasi-ϕ-asymptotically nonexpansive if and there exist nonnegative real sequences {ν_n}, {μ_n} with ν_n → 0, μ_n → 0 (as n → ∞) and a strictly increasing continuous function ζ : ℛ⁺ → ℛ⁺ with ζ(0) = 0 such that for all x ∈ C, p ∈ F(T)

Remark 1.5. From the definitions, it is easy to know that

(1) Taking ζ(t) = t, t ≥ 0, ν_n = k_n - 1 and μ_n = 0, then ν_n → 0(as n → ∞) and (1.3) can be rewritten as

This implies that the class of total quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-asymptotically nonexpansive mappings as a subclass, but the converse is not true.

(2) The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true.

(3) The class of quasi-ϕ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.

Let C be a nonempty closed convex subset of a Banach space X. Let N(C) be the family of nonempty subsets of C.

Definition 1.6. (1) A multi-valued mapping T : C → N(C) is said to be relatively nonexpansive [3] if and

(2) A multi-valued mapping T : C → N(C) is said to be closed if, for any sequence {x_n} ⊂ C with x_n → x and w_n ∈ T(x_n) with w_n → y, then y ∈ Tx.

Definition 1.7. (1) A multi-valued mapping T : C → N(C) is said to be quasi-ϕ-nonexpansive if and

(2) A multi-valued mapping T : C → N(C) is said to be quasi-ϕ-asymptotically non-expansive if and there exists a real sequence {k_n} ⊂ [1, ∞) with k_n → 1 such that

(3) A multi-valued mapping T : C → N(C) is said to be total quasi-ϕ-asymptotically nonexpansive if and there exist nonnegative real sequences {ν_n}, {μ_n} with ν_n → 0, μ_n → 0(as n → ∞) and a strictly increasing continuous function ζ : ℛ⁺ → ℛ⁺ with ζ(0) = 0 such that for all x ∈ C, p ∈ F(T)

(4) A total quasi-ϕ-asymptotically nonexpansive multi-valued mapping T : C → N(C) is said to be uniformly L-Lipschitz continuous if there exists a constant L > 0 such that

Remark 1.8. From the definitions, it is easy to know that

(1) Taking ζ(t) = t, t ≥ 0, ν_n = k_n - 1 and μ_n = 0, then ν_n → 0 (as n → ∞) and (1.6) can be rewritten as

This implies that the class of total quasi-ϕ-asymptotically nonexpansive multi-valued mappings contains properly the class of quasi-ϕ-asymptotically nonexpansive multi-valued mappings as a subclass, but the converse is not true.

(2) The class of quasi-ϕ-asymptotically nonexpansive multi-valued mappings contains properly the class of quasi-ϕ-nonexpansive multi-valued mappings as a subclass, but the converse is not true.

(3) The class of quasi-ϕ-nonexpansive multi-valued mappings contains properly the class of relatively nonexpansive multi-valued mappings as a subclass, but the converse is not true.

In 2005, Matsushita and Takahashi [3] proved weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space X. In 2008, Plubtieng and Ungchittrakool [4] proved the strong convergence theorems to approximate a fixed point of two relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space X. In 2010, Chang et al. [5] obtained the strong convergence theorem for an infinite family of quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space X with Kadec-Klee property. In 2011, Chang et al. [6] proved some approximation theorems of common fixed points for countable families of total quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space X with Kadec-Klee property. In 2011, Homaeipour and Razani [7] proved weak and strong convergence theorems for a single relatively nonexpansive multi-valued mapping in a uniformly convex and uniformly smooth Banach space X.

Motivated and inspired by the researches going on in this direction, the purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multivalued mapping which contains many kinds of mappings as its special cases, and then by using the hybrid iterative algorithm to prove some strong convergence theorems in uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the article improve and extend some recent results announced by some authors.

Lemma 2.1. [6] Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed convex set of X. Let {x_n} and {y_n} be two sequences in C such that x_n → p and ϕ(x_n, y_n) → 0, where ϕ is the function defined by (1.1), then y_n → p.

Lemma 2.2. Let X and C be as in Lemma 2.1. Let T : C → N(C) be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences {ν_n}, {μ_n} and a strictly increasing continuous function ζ : ℛ⁺ → ℛ⁺ such that ν_n → 0, μ_n → 0(as n → ∞) and ζ(0) = 0. If μ₁ = 0, then the fixed point set F(T) is a closed and convex subset of C.

Proof. Let {x_n} be a sequence in F(T) with x_n → p (as n → ∞), we prove that p ∈ F(T). In fact, by the assumption that T is total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and μ₁ = 0, we have

Furthermore, we have

By Lemma 1.2(c), p = u. Hence, p ∈ Tp. This implies that p ∈ F(T), i.e., F(T) is closed.

Next, we prove that F(T) is convex. For any x, y ∈ F(T), t ∈ (0, 1), putting q = tx + (1 - t)y, we prove that q ∈ F(T). Indeed, let {u_n} be a sequence generated by

In view of the definition of ϕ(x, y), for all u_n ∈ Tu_n-1 ⊂ Tⁿq, we have

Since

Substituting (2.2) into (2.1) and simplifying it we have

By Lemma 2.1, we have u_n → q (as n → ∞). This implies that u_n+1 → q (as n → ∞). Since T is closed, we have q ∈ Tq, i.e., q ∈ F(T).

This completes the proof of Lemma2.2.

Lemma 2.3. [8] Let X be a uniformly convex Banach space, r > 0 be a positive number and B_r(0) be a closed ball of X. Then, there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that

for all x, y ∈ B_r(0) and all α, β ∈ [0, 1] with α + β = 1.

In this section, we shall use the hybrid iterative algorithm to study the iterative solutions of nonlinear operator equations with a closed and uniformly total quasi-ϕ-asymptotically nonexpansive multi-valued mapping in Banach space.

Theorem 3.1. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed and convex subset of X. Let T : C → N(C) be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences {ν_n},{μ_n} and a strictly increasing continuous function ζ : ℛ⁺ → ℛ⁺ such that μ₁ = 0, ν_n → 0, μ_n → 0 (as n → ∞) and ζ(0) = 0. Let x₀ ∈ C, C₀ = C and {x_n} be a sequence generated by

where w_n ∈ Tⁿx_n, ∀n ≥ 1, ξ_n = ν_n sup_p∈F(T) ζ(ϕ(p, x_n)) + μ_n, is the generalized projection of X onto C_n+1, {α_n} and {β_n} are sequences in [0,1] satisfies the following conditions:

(a) lim inf_n→∞ β_n(1 - β_n) > 0;

(b) 0 ≤ α_n ≤ α < 1 for some α ∈ (0, 1).

If F(T) is a nonempty and bounded subset of C, then the sequence {x_n} converges strongly to Π_F(T)x₀.

Proof. We divide the proof of Theorem 3.1 into six steps.

(I) C_n is closed and convex for each n ≥ 0.

In fact, by the assumption, C₀ = C is closed and convex. Suppose that C_n is closed and convex for some n ≥ 1. Since the condition ϕ(ν, y_n) ≤ ϕ(ν, x_n) + ξ_n is equivalent to

hence the set

is closed and convex. Therefore C_n is closed and convex for each n ≥ 0.

(II) {x_n} is bounded and {ϕ(x_n, x₀)} is a convergent sequence.

Indeed, it follows from (3.1) and Lemma 1.2(a) that for all n ≥ 0, u ∈ F(T)

This implies that {ϕ(x_n, x₀)} is bounded. By virtue of (1.2), we know that {x_n} is bounded.

In view of structure of {C_n}, we have C_n+1 ⊂ C_n, and . This implies that x_n+1 ∈ C_n and

Therefore {ϕ(x_n, x₀)} is a convergent sequence.

(III) F(T) ⊂ C_n for all n ≥ 0.

It is obvious that F(T) ⊂ C₀ = C. Suppose that F(T) ⊂ C_n for some n ≥ 1. Since X is uniformly smooth, X* is uniformly convex. For any given u ∈ F(T) ⊂ C_n and n ≥ 1 we have

Furthermore, it follows from Lemma 2.3 that for any u ∈ F(T), w_n ∈ Tⁿx_n we have

Substituting (3.3) into (3.2) and simplifying it, we have

i.e., u ∈ C_n+1 and so F(T) ⊂ C_n+1 for all n ≥ 0.

By the way, in view of the assumption on {ν_n}, {μ_n} we have

(IV) {x_n} converges strongly to some point p* ∈ C.

In fact, since {x_n} is bounded and X is reflexive, there exists a subsequence such that (some point in C). Since C_n is closed and convex and C_n+1 ⊂ C_n, this implies that C_n is weakly closed and p* ∈ C_n for each n ≥ 0. In view of , we have

Since the norm || · || is weakly lower semi-continuous, we have

and so

This implies that , and so . Since , by virtue of Kadec-Klee property of X, we obtain that

Since {ϕ(x_n, x₀)} is convergent, this together with , which shows that lim_n→∞ ϕ(x_n, x₀) = ϕ(p*, x₀). If there exists some sequence such that , then from Lemma 1.2(a) we have that

This implies that p* = q and

(V) Now we prove that p* ∈ F(T).

In fact, since x_n+1 ∈ C_n+1 ⊂ C_n, it follows from (3.1) and (3.5) that

Since x_n → p*, by the virtue of Lemma 2.1

From (3.2) and (3.3), for any u ∈ F(T) and w_n ∈ Tⁿx_n, we have

i.e.,

By conditions (a) and (b) it shows that lim_n→∞ g(||Jx_n - Jw_n||) = 0. In view of property of g, we have

Since Jx_n → Jp*, this implies that Jw_n → Jp*. From Remark 1.1 (ii) it yields

Again since

this together with (3.7) and the Kadec-Klee property of X shows that

Let {s_n} be a sequence generated by

By the assumption that T is uniformly L-Lipschitz continuous, hence for any w_n ∈ Tⁿx_n and s_n+1 ∈ Tw_n ⊂ Tⁿ⁺¹x_n we have

This together with (3.5) and (3.8) shows that lim_n→∞ ||s_n+1 - w_n|| = 0 and lim_n→∞ s_n+1 = p*. In view of the closeness of T, it yields that p* ∈ Tp*, i.e.,

(VI) we prove that x_n → p* = Π_F(T)x₀.

Let t = Π_F(T)x₀. Since t ∈ F(T) ⊂ C_n and , we have

This implies that

In view of the definition of Π_F(T)x₀, from (3.10) we have p* = t. Therefore, x_n → p* = Π_F(T)x₀.

This completes the proof of Theorem 3.1.

From Remark 1.8, the following theorems can be obtained from Theorem 3.1 immediately.

Theorem 3.2. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed and convex subset of X. Let T : C → N(C) be a closed and quasi-ϕ-asymptotically nonexpansive multi-valued mapping with a real sequences {k_n} ⊂ [1, ∞) and k_n → 1(n → (∞). Let x₀ ∈ C, C₀ = C and {x_n} be a sequence generated by

where ξ_n = (k_n - 1) sup_p∈F(T)(ϕ(p, x_n)), is the generalized projection of X onto C_n+1, {α_n} and {β_n} are sequences in [0,1] satisfies the following conditions:

(a) lim inf_n→∞ β_n(1 - β_n) > 0;

(b) 0 ≤ α_n ≤ α < 1 for some α ∈ (0, 1).

If F(T) is a nonempty and bounded subset of C, then the sequence {x_n} converges strongly to Π_F(T)x₀.

Theorem 3.3. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed and convex subset of X. Let T : C → N(C) be a closed and quasi-ϕ nonexpansive multi-valued mapping. Let x₀ ∈ C, C₀ = C and {x_n} be a sequence generated by

where is the generalized projection of X onto C_n+1, {α_n} and {β_n} are sequences in [0,1] satisfies the following conditions:

(a) lim inf_n→∞ β_n(1 - β_n) > 0;

(b) 0 ≤ α_n ≤ α < 1 for some α ∈ (0, 1).

If F(T) is a nonempty and bounded subset of C, then the sequence {x_n} converges strongly to Π_F(T)x₀.

Theorem 3.4. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed and convex subset of X. Let T : C → N(C) be a closed and relatively nonexpansive multi-valued mapping. Let x₀ ∈ C, C₀ = C and {x_n} be a sequence generated by

where is the generalized projection of X onto C_n+1, {α_n} and {β_n} are sequences in [0,1] satisfies the following conditions:

(a) lim inf_n→∞ β_n(1 - β_n) > 0;

(b) 0 ≤ α_n ≤ α < 1 for some α ∈ (0, 1).

If F(T) is a nonempty and bounded subset of C, then the sequence {x_n} converges strongly to Π_F(T)x₀.

The authors declare that they have no competing interests.

All the authors contributed equally to the writing of the present article. And they also read and approved the final manuscript.

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