Three-Step Fixed Point Iteration for Generalized Multivalued Mapping in Banach Spaces

Definition 1.1. A multivalued mapping T : K → CB(K) is said to satisfy Condition (A) if there is a nondecreasing function f : [0, ∞)→[0, ∞) with f(0) = 0, f(x) > 0 for x ∈ (0, ∞) such that

where F(T) ≠ ∅  is the fixed point set of the multivalued mapping T. From now on, F(T) stands for the fixed point set of the multivalued mapping  T.

Lemma 2.1 (see [7], [11].)Let {x_n}, {y_n}, and {z_n} be sequence in uniformly convex Banach space X. Suppose that {α_n}, {β_n}, and {γ_n} are sequence in [0,1] with α_n + β_n + γ_n = 1,   limsup _n∥x_n∥ ≤ d,   limsup _n∥y_n∥≤d,   limsup _n∥z_n∥≤d, and lim _n∥α_nx_n + β_ny_n + γ_nz_n∥ = d. If liminf _nα_n > 0 and  liminf _nβ_n > 0, then lim _n∥x_n − y_n∥ = 0.

Lemma 2.2 (see [7], [11].)Let X be a uniformly convex Banach space and B_r : = {x ∈ X : ∥x∥≤r}, r > 0. Then there exists a continuous strictly increasing convex function g : [0, ∞)→[0, ∞) with g(0) = 0 such that

for all x, y, z, ω ∈ B_r and λ, μ, ξ, ϑ ∈ [0,1] with λ + μ + ξ + ϑ = 1.

Lemma 3.1. Let X be a real Banach space and K be a nonempty convex subset of X, T : K → P(K) be a generalized multivalued nonexpansive mapping with F(T) ≠ ∅ such that P_T is nonexpansive. Let {x_n} be a sequence in K defined by (1.8), then one has the following conclusion:

Proof. Let p ∈ F(T), then p ∈ P_T(p) = {p}. Since T is quasi-nonexpansive, thus we obtain

similarly ∥y_n − p∥≤∥x_n − p∥, then we have Then {∥x_n − p∥} is a decreasing sequence and hence lim _n∥x_n − p∥ exists for any p ∈ F(T).

Lemma 3.2. Let X be a uniformly convex Banach space and K be a nonempty convex subset of X, T : K → P(K) be a generalized multivalued nonexpansive mapping with F(T) ≠ ∅ such that P_T is nonexpansive. Let {x_n} be a sequence in K defined by (1.8), if the coefficient satisfy one of the following control conditions:

• (i)

  liminf _nα_n > 0 and one of the following holds:

  • (a)

    limsup _n(α_n + β_n + γ_n) < 1 and limsup _n(b_n + c_n) < 1,

  • (b)

    0 < liminf _nβ_n ≤ limsup _n(α_n + β_n + γ_n) < 1 and limsup _nc_n < 1,

  • (c)

    0 < liminf _nb_n ≤ limsup _n(b_n + c_n) < 1 and limsup _na_n < 1,

  • (d)

    0 < liminf _nc_n ≤ limsup _n(b_n + c_n) < 1;

• (ii)

  0 < liminf _nβ_n ≤ limsup _n(α_n + β_n + γ_n) < 1 and limsup _na_n < 1;

• (iii)

  0 < liminf _nγ_n ≤ limsup _n(α_n + β_n + γ_n) < 1;

• (iv)

  0 < liminf _n(α_nb_n + β_n) and 0 < liminf _na_n ≤ limsup _na_n < 1;  

then we have lim _nd(x_n, Tx_n) = 0.

Proof. By Lemma 3.1, we know that lim _n∥x_n − p∥ exists for any p ∈ F(T), then it follows that {s_n − p}, {t_n − p}, and {r_n − p} are all bounded. We may assume that these sequences belong to B_r where r > 0. Note that p ∈ P_T(p) = {p} for any fixed point p ∈ F(T) and T is quasi-nonexpansive. By Lemma 2.2, we get

and therefore we have Then Since lim _n∥x_n − p∥ exists for any p ∈ F(T), it follows from (3.6) that lim _n(1 − α_n − β_n − γ_n)α_ng(∥x_n − r_n∥) = 0. From g is continuous strictly increasing with g(0) = 0 and 0 < liminf _nα_n ≤ limsup _n(α_n + β_n + γ_n) < 1, then Using a similarly method together with inequalities (3.7) and 0 < liminf _nβ_n ≤ limsup _n(α_n + β_n + γ_n) < 1, then Similarly, from (3.8) and 0 < liminf _nγ_n ≤ limsup _n(α_n + β_n + γ_n) < 1, we have lim _n∥x_n − s_n∥ = 0, since s_n ∈ Tx_n, then 0 ≤ lim _nd(x_n, Tx_n) ≤ lim _n∥x_n − s_n∥ = 0, thus we get (iii). In the sequence we prove (i) (a). From iterative scheme (1.8), we have To show that lim _n∥x_n − s_n∥ = 0, it suffices to show that there exist a subsequence {n_j} of {n} such that  $$. If  $$, it follows from (3.9) that Since lim _n∥x_n − p∥ exists for any p ∈ F(T), we have From g is continuous strictly increasing with $$ and $$, we have This together with (3.10), (3.12), (3.15) gives Since $$, we have $$. On the other hand, if $$, then we may extract a subsequence $$ of $$ so that $$. This together with (i) (a) and (3.10), (3.12) gives By Double Extract Subsequence Principle, we obtain the result.

If 0 < liminf _nβ_n ≤ limsup _n(α_n + β_n + γ_n) < 1 and limsup _na_n < 1, we will prove (ii),

Since limsup _na_n < 1, then This together with (3.11), (3.18), we obtain the result.

We will prove (i) (b), let p ∈ F(T). By Lemma 3.1, we let lim _n∥x_n − p∥ = d for some d ≥ 0. From iterative scheme (1.8), we know

From Lemma 3.1, we have known that ∥z_n − p∥≤∥x_n − p∥ and ∥y_n − p∥≤∥x_n − p∥, then From (3.20) and Lemma 2.1, we have Notice that Since limsup _nc_n < 1, we have lim _n∥s_n − x_n∥ = 0, therefore 0 ≤ lim _nd(x_n, Tx_n) ≤ lim _n∥x_n − s_n∥ = 0.

We will prove (i) (c). From iterative scheme (1.8) and Lemma 3.1, we have

which implies Notice that liminf _nα_n > 0 and lim _n∥x_n − p∥ exists. Hence from (3.25) we have Therefore, from iterative scheme (1.8) we have From Lemma 2.1, we have Notice that Since limsup _na_n < 1, then 0 ≤ lim _nd(x_n, Tx_n) ≤ lim _n∥x_n − s_n∥ = 0.

By (3.27) and Lemma 2.1, we can similarly prove (i) (d).

Finally, we will prove (iv). From iterative scheme (1.8) and Lemma 3.1, we have

which implies Notice that Hence we have Thus, we have By Lemma 2.1  and 0 < liminf _na_n ≤ limsup _na_n < 1, we have  0 ≤ lim _nd(x_n, Tx_n) ≤ lim _n∥x_n − s_n∥ = 0.

Theorem 3.3. Let X be a uniformly convex Banach space and K be a nonempty convex subset of X, T : K → P(K) be a generalized multivalued nonexpansive mapping with F(T) ≠ ∅ such that P_T is nonexpansive. Let {x_n} be a sequence in K defined by (1.8), the coefficient satisfy the control conditions in Lemma 3.2 and T satisfies Condition (A) with respect to the sequence {x_n}, then {x_n} converges strongly to a fixed point of T.

Proof. By Lemma 3.2, we have lim _nd(x_n, Tx_n) = 0. Since T satisfies Condition (A) with respect to {x_n}. Then

Thus, we get lim _nd(x_n, F(T)) = 0. The remainder of the proof is the same as in [6, Theorem 2.4], we omit it.

Theorem 3.4. Let X be a uniformly convex Banach space and K be a nonempty convex subset of X, T : K → P(K) be a generalized multivalued nonexpansive mapping with F(T) ≠ ∅ such that P_T is nonexpansive. Let {x_n} be a sequence in K defined by (1.8), the coefficient satisfy the control conditions in Lemma 3.2 and T is hemicompact, then {x_n} converges strongly to a fixed point of T.

Proof. By Lemma 3.2, we have lim _nd(x_n, Tx_n) = 0. Since T is hemicompact, then there exist a subsequence $$ of {x_n} such that $$ for some q ∈ K. Thus,

Hence, q is a fixed point of T. Now on take on q in place of p, we get that lim _n→∞∥x_n − q∥ exists. It follows that x_n → q as n → ∞. This completes the proof.

Theorem 3.5. Let X, T and {x_n} be the same as in Lemma 3.2. If K be a nonempty weakly compact convex subset of a Banach space X and X satisfies Opial′s condition, then {x_n} converges weakly to a fixed point of T.

Proof. The proof of the Theorem is the same as in [6, Theorem 2.5], we omit it.

Remark 3.6. From the definition of iterative scheme (1.8), Theorems 3.3, 3.4, and 3.5 extend some results in [6, 12], and also give some new results are different from the [5]. In fact, we can present an example of a multivalued map T for which P_T is nonexpansive. A multivalued map T : D → CB(X) is *-nonexpansive [13] if for all x, y ∈ D and u_x ∈ T(x) with d(x, u_x) = inf {d(x, z) : z ∈ T(x)}, there exists u_y ∈ T(y) with d(y, u_y) = inf {d(y, w) : ω ∈ T(y)} such that

It is clear that if T is *-nonexpansive, then P_T is nonexpansive. It is known that *-nonexpansiveness is different from nonexpansiveness for multivalued maps. Let D = [0, ∞) and T be defined by Tx = [x, 2x] for x ∈ D [14]. Then P_T(x) = x for x ∈ D and thus it is nonexpansive. Note that T is *-nonexpansive but not nonexpansive (see [14]).