Strong convergence theorems by Halpern-Mann iterations for multi-valued relatively nonexpansive mappings in Banach spaces with applications

for A₁,A₂ ∈ CB(D), where d(x, A₁) = inf{∥x - y∥, y ∈ A₁}. The multi-valued mapping T : D → CB(D) is called nonexpansive if H(T(x),T(y)) ≤ ∥x - y∥ for all x, y ∈ D. An element p ∈ D is called a fixed point of T : D → N(D) if p ∈ T(p). The set of fixed points of T is represented by F(T).

where 〈·,·〉 denotes the generalized duality pairing.

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

for all λ ∈ [0,1] and x,y,z ∈ E.

Let D be a nonempty subset of a smooth Banach space. A mapping T : D → E is relatively expansive [3–5], if the following properties are satisfied:

(R1) $F\left(T\right)\ne \varnothing$;

(R2) ϕ(p,Tx) ≤ ϕ(p,x) for all p ∈ F(T) and x ∈ D;

(R3) I - T is demi-closed at zero, that is, whenever a sequence {x _n } in D converges weakly to p and {x _n - Tx _n } converges strongly to 0, it follows that p ∈ F(T).

If T satisfies (R1) and (R2), then T is called quasi-ϕ-nonexpansive [6].

where D is nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, Π _D is the generalized projection of E onto D and {α _n } and {β _n } are two sequences in [0,1].

They proved strong convergence theorems in uniformly convex and uniformly smooth Banach space E.

Iterative methods for approximating fixed points of multi-valued mappings in Banach spaces have been studied by some authors, see for instance [8–15].

Let D be a nonempty closed convex subset of a smooth Banach space E. We define a relatively nonexpansive multi-valued mapping as follows.

Definition 1.2. A multi-valued mapping T : D → N(D) is called relatively nonexpansive, if the following conditions are satisfied:

(S1) $F\left(T\right)\ne \varnothing$

(S2) ϕ(p,z) ≤ ϕ(p, x), ∀x ∈ D, z ∈ T(x), p ∈ F(T);

(S3) I - T is demi-closed at zero, that is, whenever a sequence {x _n } in D which weakly to p and lim_n→∞d(x _n , T(x _n )) = 0, it follows that p ∈ F(T).

If T satisfies (S1) and (S2), then multi-valued mapping T is called quasi-ϕ-nonexpansive.

where w _n ∈ Tx _n for all n ∈ ℕ, D is a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, Π _D is the generalized projection of E onto D and {α _n }, {β _n } are sequences in [0,1]. We proved the strong convergence theorems in uniformly convex and uniformly smooth Banach space E.

In the sequel, we denote the strong convergence and weak convergence of the sequence {x _n } by x _n → x and x _n ⇀ x, respectively.

First, we recall some conclusions.

Remark 2.2. If E is a real Hilbert space H, then ϕ(x, y) = ∥x - y∥² and Π _C is the metric projection P _C of H onto C.

In what follows, we need the following lemmas for proof of our main results.

Lemma 2.4 [17]. Let E be a uniformly convex and smooth Banach space and let {x _n } and {y _n } be two sequences of E such that {x _n } or {y _n } is bounded. If lim_n→∞ϕ(x _n , y _n ) = 0. Then lim_n→∞∥x _n -y _n ∥ = 0.

for all x ∈ E and x* ∈ E*. Obviously, V(x, x*) = ϕ(x, J^-1(x*)) for all x ∈ E and x* ∈ E*. We know the following lemma.

for all x ∈ E and x*,y* ∈ E*.

Then lim_n→∞α _n = 0.

In fact, m _k = max{j ≤ k : α _j < α_j+1}.

Lemma 3.1 Let E be a strictly convex and smooth Banach space, and D a nonempty closed subset of E. Suppose T : D → N(D) is a quasi-ϕ-nonexpansive multi-valued mapping. Then F(T) is closed and convex.

By Lemma 2.1(c), we obtain p = w. Hence, T(p) = {p}. So, we have p ∈ F(T). Therefore, F(T) is convex.

where w _n ∈ Tx _n for all n ∈ ℕ, {α _n }, {β _n } are sequences in [0,1]. Then {x _n } is bounded.

This implies that {x _n } is bounded.

Theorem 3.4 Let D be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and T : D → N(D) be a relatively nonexpansive multivalued mapping. Let {α _n } and {β _n } be sequences in (0,1) satisfying

(C1) lim_n→∞, α _n = 0;

(C2) ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$;

(C3) lim inf_n→∞β _n (1- β _n ) > 0.

Then {x _n } defined by (3.1) converges strongly to Π_F(T)u, where Π_F(T)is the generalized projection from E onto F(T).

The rest of proof will be divided into two parts:

Hence the conclusion follows by Lemma 2.5.

But $\varphi \left(u^{*},x_k\right)\le \varphi \left(u^{*},x_{m_k+1}\right)$ for all k ∈ ℕ. We conclude that x _k → u*.

This implies that lim_n→∞x _n = u* and the proof is finished.

Letting β _n = β gives the following result.

where w _n ∈ Tx _n for all n ∈ ℕ, {α _n } is a sequence in [0,1] satisfying condition (C1) and (C2), and β ∈ (0,1). Then {x _n } converges strongly to ∏_F(T)u.

Let E be a smooth, strictly convex, and reflexive Banach space. An operator A : E → 2^E*is said to be monotone, if 〈x - y, x* - y*〉 ≥ 0 whenever x, y ∈ E, x* ∈ Ax, y* ∈ Ay. We denote the zero point set {x ∈ E : 0 ∈ Ax} of A by A^-10. A monotone operator A is said to be maximal, if its graph G(A) := {(x, y) : y ∈ Ax} is not properly contained in the graph of any other monotone operator. If A is maximal monotone, then A^-10 is closed and convex. Let A be a maximal monotone operator, then for each r > 0 and x ∈ E, there exists a unique x _r ∈ D(A) such that J(x) ∈ J(x _r ) + rA(x _r ) (see, for example, [19]). We define the resolvent of A by J _r x = x _r . In other words J _r = (J + rA)-¹ J, ∀r > 0. We know that J _r is a single-valued relatively nonexpansive mapping and A^-10 = F(J _r ),∀r > 0, where F(J _r ) is the set of fixed points of J _r .

We have the following

then {x _n } converges strongly to Π _A -1₀x₁.

Proof. In Theorem 3.4 taking D = E, T = J _r , r > 0, then T : E → E is a single-valued relatively nonexpansive mapping and A^-10 = F(T) = F(J _r ),∀r > 0 is a nonempty closed convex subset of E. Therefore all the conditions in Theorem 3.4 are satisfied. The conclusion of Theorem 4.1 can be obtained from Theorem 3.4 immediately.