Throughout this article we assume that E is a real Banach space with norm ||·||, E* is the dual space of E; 〈·, ·〉 is the duality pairing between E and E*; C is a nonempty closed convex subset of E; ℕ denotes the natural number set; ℜ⁺ is the set of nonnegative real numbers; The mapping J : E → 2 E * defined by

J ( x ) = f * ∈ E * : x , f * = x 2 ; f * = x , x ∈ E

is called the normalized duality mapping. We denote a single valued normalized duality mapping by j.

Let T: C → C be a nonlinear mapping; F(T) denotes the set of fixed points of mapping T, i.e., F(T) := {x ∈ C, x = Tx}. We use "→" to stand for strong convergence and "⇀" for weak convergence. For a given sequence {x_n} ⊂ C, let ω_w(x_n) denote the weak ω-limit set.

Recall that T is said to be pseudocontractive if for all x, y ∈ C, there exists j(x - y) ∈ J(x - y) such that

T x - T y , j ( x - y ) ≤ x - y 2 ;

T is said to be strongly pseudocontr active if there exists a constant α ∈ (0,1), such that for any x, y ∈ C, there exists j(x - y) ∈ J(x - y)

T x - T y , j ( x - y ) ≤ α x - y 2 .

In recent years, many authors have focused on the studies about the existence and convergence of fixed points for the class of pseudocontractions. Especially in 1974, Deimling 1 proved the following existence theorem of fixed point for a continuous and strong pseudocontraction in a nonempty closed convex subset of Banach spaces.

Theorem D. Let E be a Banach space, C be a nonempty closed convex subset of E and T: C → C be a continuous and strong pseudocontraction. Then T has a unique fixed point in C.

Recently, the problems of convergence of an implicit iterative algorithm to a common fixed point for a family of nonexpansive mappings or pseudocontractive mappings have been considered by several authors, see 2345. In 2001, Xu and Ori 2 firstly introduced an implicit iterative x_n = α_nx_n-1 + (1 - α_n)T_nx_n, n ∈ ℕ, x₀ ∈ C for a finite family of nonexpansive mappings T i i = 1 N and proved some weak convergence theorems to a common fixed point for a finite family of nonexpansive mappings in a Hilbert space. In 2004, Osilike 3 improved the results of Xu and Ori 2 from nonexpansive mappings to strict pseudocontractions in the framework of Hilbert spaces. In 2006, Chen et al. 4 extended the results of Osilike 3 to more general Banach spaces.

On the other hand, the convergence problems of semi-groups have been considered by many authors recently. Suzuki 6 considered the strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Xu 7 gave strong convergence theorem for contraction semigroups in Banach spaces. Chang et al. 8 proved the strong convergence theorem for nonexpansive semi-groups in Banach space. He also studied the weak convergence problems of the implicit iteration process for Lipschitzian pseudocontractive semi-groups in the general Banach spaces 9. The pseudocontractive semi-groups is defined as follows.

Definition 1.1 (1) One-parameter family T: = {T(t): t ≥ 0} of mappings from C into itself is said to be a pseudo-contraction semigroup on C, if the following conditions are satisfied:

(a). T(0)x = x for each x ∈ C;

(b). T(t + s)x = T(s)T(t) for any t, s ∈ ℜ⁺ and x ∈ C;

(c). For any x ∈ C, the mapping t → T(t)x is continuous;

(d). For all x, y ∈ C, there exists j(x - y) ∈ J(x - y) such that

T ( t ) x - T ( t ) y , j ( x - y ) ≤ x - y 2 , f o r a n y t > 0 .

(2) A pseudo-contraction semigroup of mappings from C into itself is said to be a Lipschitzian if the condition (a)-(d) and following condition (f) are satisfied.

(f) there exists a bounded measurable function L: [0, ∞) → [0, ∞) such that for any x, y ∈ C,

T ( t ) x - T ( t ) y ≤ L ( t ) x - y

for any t > 0. In the sequel, we denote it by

L = sup t ≥ 0 L ( t ) < ∞

Cho et al. 10 considered viscosity approximations with continuous strong pseudocontractions for a pseudocontraction semigroup and prove the following theorem.

Theorem Cho. Let E be a real uniformly convex Banach space with a uniformly Gâteaux differentiable norm, and C be a nonempty closed convex subset of E. Let T(t): t ≥ 0 be a strongly continuous L-Lipschitz semigroup of pseudocontractions on C such that Ω ≠ 0̸ , where Ω is the set of common fixed points of semi-group T(t). Let f: C → C be a fixed bounded, continuous and strong pseudocontraction with the coefficient α in (0,1), let α_n and t_n be sequences of real numbers satisfying α_n ∈ (0, 1), t_n > 0, and lim n → ∞ t n = lim n → ∞ α n t n = 0 ; Let {x_n} be a sequence generated in the following manner:

x n = ( 1 - α n ) f ( x n ) + α n T ( t n ) x n , ∀ n ≥ 1 .

Assume that LIM||T(t)x_n - T(t)x*|| ≤ ||x_n - x*||, ∀x* ∈ K, t ≥ 0, where K := {x* ∈ C: Φ(x*) = min_x∈C Φ(x)} with Φ(x) = LIM||x_n - x||², ∀x ∈ C. Then x_n converges strongly to x* ∈ Ω which solves the following variational inequality: 〈(I - f)x*, j(x* - x)〉 ≤ 0, ∀x ∈ Ω.

Qin and Cho 11 established the theorems of weak convergence of an implicit iterative algorithm with errors for strongly continuous semigroups of Lipschitz pseudocontractions in the framework of real Banach spaces.

Theorem Q. Let E be a reflexive Banach space which satisfies Opial's condition and K a nonempty closed convex subset of E. Let T : = { T ( t ) : t ≥ 0 } be a strongly continuous semigroup of Lipschitz pseudocontractions from K into itself with F : = ⋂ t ≥ 0 F ( T ( t ) ) ≠ 0̸ ; Assume that sup_t≥0{L(t)} < ∞, where L(t) is the Lipschitz constant of the mapping T(t). Let {x_n} be a sequence generated by the following iterative process:

x 0 ∈ K ; x n = α n x n - 1 + β n T ( t n ) x n + γ n u n ; ∀ n ≥ 1 ;

where {α_n}, {β_n}, {γ_n} are sequences in (0,1), {t_n} is a sequence in (0, ∞) and {u_n} is a bounded sequence in K. Assume that the following conditions are satisfied:

(a) α_n + β_n + γ_n = 1;

(b) lim n → ∞ t n = lim n → ∞ α n + γ n t n = 0 .

Then the sequence {x_n} generated in (7) converges weakly to a common fixed point of the semigroup T := {T(t) : t ≥ 0} ;

Agarwal et al. 12 studied strongly continuous semigroups of Lipschitz pseudocontractions and proved the strong convergence theorems of fixed points in an arbitrary Banach space based on an implicit iterative algorithm.

Theorem A. Let E be an arbitrary Banach space and K a nonempty closed convex subset of E. Let T := {T(t) : t ≥ 0} be a strongly continuous semigroup of Lipschitz pseudocontractions from K into itself with F := ⋂ t≥0F(T(t))≠0̸ . Assume that sup_t≥0{L(t)} < ∞, where L(t) is the Lipschitz constant of the mapping T(t). Let {x_n} be a sequence in

x 0 ∈ K ; x n = α n x n - 1 + β n T ( t n ) x n + γ n u n ; ∀ n ≥ 1 ,

where {α_n}, {β_n}, {γ_n} are sequences in (0,1) such that α_n + β_n + γ_n = 1, {t_n} is a sequence in (0, ∞) and {u_n} is a bounded sequence in K. Assume that limn→∞tn = limn→∞αn + γn tn = 0 , lim n → ∞ γ n α n + γ n < ∞ and there is a nondecreasing function f: (0, ∞) → (0, ∞) with f(0) = 0 and f(t) > 0 for all t ∈ (0, ∞) such that, for all x ∈ C, sup { x - T ( t ) x : t ≥ 0 } ≥ f ( dist ( x , F ) ) . Then the sequence {x_n} converges strongly to a common fixed point of the semigroup T := {T(t) : t ≥ 0} .

The purpose of this article is to prove the strong and weak convergence of implicit iterative process

x n = ( 1 - α n ) x n - 1 + α n T ( t n ) x n , n ∈ ℕ , x 0 ∈ C

for a pseudocontraction semigroup T: = {T(t): t ≥ 0} in the framework of Banach spaces, which improves and extends the corresponding results of many author's. We need the following Lemma.

Lemma 1.1 9 Let E be a real reflexive Banach space with Opial condition. Let C be a nonempty closed convex subset of E and T: C → C be a continuous pseudocontractive mapping. Then I - T is demiclosed at zero, i.e., for any sequence {x_n} ⊂ E, if x_n ⇀ y and ||(I - T)x_n|| → 0, then (I - T)y = 0.

Theorem 2.1 Let E be a real Banach space and C be a nonempty compact convex subset of E. Let T: = {T(t): t ≥ 0}: C → C be a Lipschitian and pseudocontraction semigroup defined by Definition 1.1 with a bounded measurable function L: [0, ∞) → [0, ∞). Suppose F ( T ) : = ⋂ t ≥ 0 F ( T ( t ) ) ≠ 0̸ . Let α_n and t_n be sequences of real numbers satisfying t_n > 0, α_n ∈ [a, 1) ⊂ (0, 1) and lim_n→∞ α_n = 1. Then the sequence {x_n} defined by (9) converges strongly to a common fixed point x* ∈ F(T) in C.

Proof. We divide the proof into five steps.

(I). The sequence {x_n} defined by x_n = (1 - α_n)x_n-1 + α_nT(t_n)x_n, n ∈ ℕ, x₀ ∈ C is well defined.

In fact for all n ∈ ℕ, we define a mapping S_n as follows:

S n x = ( 1 - α n ) x n - 1 + α n T ( t n ) x , n ∈ , ∀ x ∈ C .

Then we have

S n x - S n y , j ( x - y ) = α n T ( t n ) x - T ( t n ) y , j ( x - y ) ≤ α n x - y 2 .

So S_n is strongly pseudo-contraction, thus from Theorem D, there exists a point x_n such that x_n = (1 - α_n)x_n-1 + α_nT(t_n)x_n, that is the sequence {x_n} defined by x_n = (1 - α_n)x_n-1 + α_nT(t_n)x_n, n ∈ ℕ, x₀ ∈ C is well defined.

(II). Since the common fixed-point set F(T) is nonempty let p ∈ F(T). For each p ∈ F(T), we prove that lim_n→∞ ||x_n - p|| exists.

In fact

x n - p 2 = x n - p , j ( x n - p ) = ( 1 - α n ) ( x n - 1 - p ) + α n ( T ( t n ) x n - p ) , j ( x - p ) ≤ ( 1 - α n ) x n - 1 - p x n - p + α n x n - p 2 .

So we get ||x_n - p|| ≤ (1 - α_n)||x_n-1 - p|| + α_n||x_n - p||, that is

x n - p ≤ x n - 1 - p .

This implies that the limit lim_n→∞ ||x_n - p|| exists.

(III). We prove lim_n→∞ ||T(t_n)x_n - x_n|| = 0.

The sequence {||x_n - p||_n∈ℕ} is bounded since lim_n→∞ ||x_n - p|| exists, so the sequence {x_n} is bounded. Since

T ( t n ) x n = x n - ( 1 - α n ) x n - 1 α n ≤ x n α n + ( 1 - α n ) x n - 1 α n ≤ x n a + ( 1 - α n ) x n - 1 a ,

This shows that {T(t_n)x_n} is bounded. In view of

x n - T ( t n ) x n = ( 1 - α n ) ( x n - 1 - T ( t n ) x n ) = 1 - α n ⋅ x n - 1 - T ( t n ) x n

and condition lim_n→∞ α_n = 1, we have

lim n → ∞ T ( t n ) x n - x n = 0 .

(IV). Now we prove that for all t > 0, lim_n→∞ ||T(t)x_n - x_n|| = 0.

Since pseudocontraction semigroup T: = {T(t) : t ≥ 0} is Lipschitian, for any k ∈ ℕ,

T ( ( k + 1 ) t n ) x n - T ( k t n ) x n = T ( k t n ) T ( t n ) x n - T ( k t n ) x n ≤ L ( k t n ) T ( t n ) x n - x n ≤ L T ( t n ) x n - x n .

Because lim_n→∞ ||T(t_n)x_n - x_n|| = 0, so for any k ∈ ℕ,

lim n → ∞ T ( ( k + 1 ) t n ) x n - T ( k t n ) x n = 0 .

Since

T ( t ) x n - T t t n t n x n = T t t n t n T t - t t n t n x n - T t t n t n x n ≤ L T t - t t n t n x n - x n

and T(·) is continuous, we have

lim n → ∞ T t t n t n x n - T ( t ) x n = 0 .

So from

x n - T ( t ) x n ≤ ∑ k = 0 t t n - 1 T ( ( k + 1 ) t n ) x n - T ( k t n ) x n + T t t n t n x n - T ( t ) x n ,

and lim_n→∞ ||T((k+1)t_n)x_n - T(kt_n)x_n|| = 0 as well as lim n → ∞ T t t n t n x n - T ( t ) x n = 0 , we can get

lim n → ∞ T ( t ) x n - x n = 0 .

(V). We prove {x_n} converges strongly to an element of F(T).

Since C is a compact convex subset of E, we know there exists a subsequence x n j ⊂ x n , such that x n j → x ∈ C . So we have lim j → ∞ T ( t ) x n j - x n j = 0 from lim_n→∞ ||T(t)x_n - x_n|| = 0, and

x - T ( t ) x = lim j → ∞ T ( t ) x n j - x n j = 0 .

This manifests that x ∈ F(T). Because for any p ∈ F(T), lim_n→∞ ||x_n - p|| exists, and lim n → ∞ x n - x = lim j → ∞ x n j - x = 0 , we have that {x_n} converges strongly to an element of F(T). This completes the proof of Theorem 2.1.

Theorem 2.2 Let E be a reflexive Banach space satisfying the Opial condition and C be a nonempty closed convex subset of E. Let T: = {T(t): t ≥ 0}: C → C be a Lipschitian and pseudocontraction semigroup defined by Definition 1.1 with a bounded measurable function L: [0, ∞) → [0, ∞). Suppose F(T) := ⋂ t≥0F(T(t))≠0̸ . Let α_n and t_n be sequences of real numbers satisfying t_n > 0, α_n ∈ [a, 1) ⊂ (0,1) and lim_n→∞ α_n = 1. Then the sequence {x_n} defined by x_n = (1 - α_n)x_n-1 + α_nT(t_n)x_n, x₀ ∈ C, n ∈ ℕ, converges weakly to a common fixed point x* ∈ F(T) in C.

Proof. It can be proved as in Theorem 2.1, that for each p ∈ F(T), the limit lim_n→∞ ||x_n - p|| exists and {T(t_n)x_n} is bounded, for all t > 0, lim_n→∞ ||T(t)x_n - x_n|| = 0. Since E is reflexive, C is closed and convex, {x_n} is bounded, there exist a subsequence xnj ⊂ xn such that x n j ⇀ x . For any t > 0, we have lim n j → ∞ T ( t ) x n j - x n j = 0 . By Lemma 1.1, x ∈ F(T(t)), ∀t > 0. Since the space E satisfies Opial condition, we see that ω_w(x_n) is a singleton. This completes the proof.

Remark 2.1 There is no other condition imposed on t_n in the Theorems 2.1 and 2.2 except that in the definition of pseudo-contraction semigroups. So our results improve corresponding results of many authors such as 101112, of cause extend many results in 45678.

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.