Strong Convergence Theorems for a Common Fixed Point of a Family of Asymptotically k-Strict Pseudocontractive Mappings

Theorem Q (see [5].)Let K be a closed convex and bounded subset of a Hilbert space H. Let T : K → K be completely continuous asymptotically k-strict pseudocontractive mapping for some 0 ≤ k < 1 with sequence {l_n}⊂[0, ∞) such that ∑ (l_n − 1) < ∞ and F(T) ≠ ∅. Let {x_n} be a sequence generated by the modified Mann′s iteration method:

where {α_n} is a real sequence satisfying ϵ ≤ α_n ≤ 1 − k − ϵ for all n ≥ 1 and some ϵ > 0. Then, {x_n} converges strongly to a fixed point of  T.

Theorem KX (see [12].)Let K be a closed and convex subset of a Hilbert space H. Let T : K → K be an asymptotically k-strict pseudocontractive mapping for some 0 ≤ k < 1 with sequence {l_n}⊂[0, ∞) such that ∑ (l_n − 1) < ∞ and F(T) ≠ ∅. Let {x_n} be a sequence generated by the modified Mann′s iteration method:

where {α_n} is a real sequence satisfying k + λ ≤ α_n ≤ 1 − λ, for all n ≥ 1, and λ ∈ (0,1). Then, {x_n} converges weakly to a fixed point of  T.

Example 1. Let $$ and $$. Define $$ by $$, where $$ is a real sequence satisfying 0 < a_k < 1, k ≥ 2, and $$. Then it is shown in [13] that T is asymptotically k-strict pseudocontractive mapping.

Lemma 2 (see [19].)Let C be a nonempty close convex subset of a real Banach space E which has the Fréchet differentiable norm. For x ∈ E, let ρ be defined for 0 < t < ∞ by

Then, lim_t→0ρ(t) = 0 and

Lemma 3. Let E be a real Banach space. Then the following inequality holds:

Lemma 4 (see [20].)Let E be a uniformly convex Banach space and B_R(0) a closed ball of E. Then, there exists a continuous strictly increasing convex function g : [0, ∞)→[0, ∞) with g(0) = 0 such that

for each α_i ∈ (0,1) and for x_i ∈ B_R(0): = {x ∈ E:| | x|| ≤ R}, i = 0,1, 2, …, k with $$.

Lemma 5 (see [21].)Let {a_n} be a sequence of nonnegative real numbers satisfying the following relation:

where {α_n}⊂(0,1) and {δ_n} ⊂ R satisfying the following conditions: $$, and limsup_n→∞δ_n ≤ 0. Then, lim_n→∞a_n = 0.

Lemma 6 (see [17].)Let C be a nonempty closed convex subset of a real uniformly convex Banach space E which has the Fréchet differentiable norm. Let T : C → C be an asymptotically k-strict pseudocontractive mapping with fixed point of T, F(T): = {x ∈ C : Tx = x} ≠ ∅. Then (I − T) is demiclosed at zero, that is, if x_n⇀x and Tx_n − x_n → 0, as n → ∞, then x = T(x).

Lemma 7 (see [22].)Let {a_n} be sequences of real numbers such that there exists a subsequence {n_i} of {n} such that $$ for all i ∈ N. Then there exists a nondecreasing sequence {m_k} ⊂ N such that m_k → ∞ and the following properties are satisfied by all (sufficiently large) numbers k ∈ N:

In fact, m_k = max{j ≤ k : a_j < a_j+1}.

Theorem 8. Let C be a nonempty, closed, and convex subset of a real uniformly convex Banach space E which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from E into E^*. Let T_i : C → C be asymptotically k_i-strict pseudocontractive mappings for 0 ≤ k_i < 1 with sequences {l_n,i}⊂[1, ∞), for i = 1,2, …, N. Assume that $$ is nonempty. Let {x_n} be a sequence defined by x₁ = u ∈ C and

where $$, such that θ_n,1 + θ_n,2 + ⋯+θ_n,N = 1, for each n ≥ 1, {α_n}, {θ_n,i}⊂(0, c)⊂(0,1), satisfying liminf_nθ_n,i > 0, lim_n→∞α_n = 0, ∑ α_n = ∞, lim_n→∞((l_n,i − 1)/α_n) = 0, for i = 1,2, …, N and {β_n}⊂[a, b]⊂(0, k) (a, b, and c constants), for k = min_1≤i≤N{k_i}, Then the sequence {x_n} generated by (14) converges strongly to a common fixed point of {T_i : i = 1,2, …, N}.

Proof. Fix x^* ∈ F. Let y_n = (1 − β_n)x_n + β_nS_nx_n and l_n : = max{l_n,i : i = 1,2, …, N}. Then, using Lemma 2 and (3) we have that

On the other hand using Lemma 4 we get that

Now substituting (16) into (15) we obtain that

since (k − β_n) ≥ 0 for each n ≥ 1. Then now, from (14) and (18) we get that where N₀ is a positive integer such that 2(1  −  α_n)β_n(l_n  −  1)/α_n < ϵ, for all n ≥ N₀, for some ϵ > 0. Therefore, by induction, which implies that {x_n} and hence {y_n} are bounded.

Furthermore, from (14), Lemma 3, and (17) we get that

for some M > 0.

Now, the rest of the proof is divided into two parts.

Case 1. Suppose that there exists N₁ ≥ 0 such that {| | x_n − x^*||} is decreasing for all n ≥ N₁. Then we have that {| | x_n − x^*||)} is convergent. Then from (21) and the assumptions on {β_n}, {α_n}, and {l_n} we have that $$, as n → ∞, which implies that

for i = 1,2, …, N. Then from (14) we obtain that Again, from (23) we get that as n → ∞. Thus, (24) and (25) imply that Therefore, since each T_i, for i = 1,2, …, N, is uniformly L-Lipschitzian and we have from (23), (26), and uniform continuity of T_i that for each i = 1,2, …, N. Furthermore, the fact that {x_n} is bounded and E is reflexive implies that we can choose a subsequence $$ of {x_n+1} such that $$ and Now, from (26) we get that $$ and from Lemma 6 we have that z ∈ F(T_i), for each i = 1,2, …, N. Hence, $$. Therefore, putting x^* = z in (29) and using the fact that J is weakly sequentially continuous we immediately obtain that $$. Again, putting x^* = z in inequality (22), we get that and, hence, it follows from (30) and Lemma 5 that | | x_n − z|| → 0, as n → ∞. Consequently, x_n → z.

Case 2. Suppose that there exists a subsequence {n_i} of {n} such that

for all i ∈ N. Then, by Lemma 7, there exists a nondecreasing sequence {m_j} ⊂ N such that m_j → ∞, $$ and $$ for all j ∈ N. Then from (21) and following the method of Case 1, we get that for each i = 1,2, …, N. Thus, again following the method of Case 1, we obtain that $$ and $$, as j → ∞, for each i = 1,2, …, N and there exists $$ such that Then now, putting x^* = z^* in (22) we have that Since $$, (34) implies that Moreover, since $$, inequality (35) gives that Then, from (33) and the fact that $$, we obtain that $$, as j → ∞. This together with (34) gives that $$, as j → ∞. But $$, for all j ∈ N; thus we obtain that x_j → z^*. Therefore, from the above two cases, we can conclude that {x_n} converges strongly to an element of F and the proof is complete.

Corollary 9. Let C be a nonempty, closed, and convex subset of a real uniformly convex Banach space E which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from E into E^*. Let T : C → C be an asymptotically k-strict pseudocontractive mapping for 0 ≤ k < 1 with sequences {l_n}⊂[1, ∞). Assume that F(T) is nonempty. Let {x_n} be a sequence defined by x₁ = u ∈ C and

where {α_n}⊂(0, c)⊂(0,1), satisfying lim_n→∞α_n = 0, ∑ α_n = ∞, lim_n→∞((l_n − 1)/α_n) = 0, and {β_n}⊂[a, b]⊂(0, k) (a, b, and c constants). Then the sequence {x_n} generated by (37) converges strongly to a fixed point of T.

Proof. Putting T = T₁ = T₂ = ⋯ = T_N in (14), we get that S_n = Tⁿ and the scheme reduces to scheme (37) and following the method of proof of Theorem 8 we get that (see (21) and (22))

for some M^′ > 0. Now, considering cases, as in the proof of Theorem 8, we obtain the required result.

Corollary 10. Let K be a nonempty, closed, and convex subset of l_p, 1 < p < ∞. Let T_i : C → C be asymptotically k_i-strict pseudocontractive mappings for 0 ≤ k_i < 1 with sequences {l_n,i}⊂[1, ∞), for i = 1,2, …, N. Assume that $$ is nonempty. Let {x_n} be a sequence defined by x₁ = u ∈ C and

where $$, such that θ_n,1 + θ_n,2 + ⋯+θ_n,N = 1, for each n ≥ 1, {α_n}, {θ_n,i}⊂(0, c)⊂(0,1), satisfying liminf_n→∞θ_n,i > 0, lim_n→∞α_n = 0, ∑ α_n = ∞, lim_n→∞((l_n,i − 1)/α_n) = 0 and {β_n}⊂[a, b]⊂(0, k) (for a, b, and c constants), for k = min_1≤i≤N{k_i}. Then the sequence {x_n} generated by (39) converges strongly to a common fixed point of {T_i : i = 1,2, …, N}.

Proof. We note that l_p, 1 < p < ∞, spaces are uniformly convex which have Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from E into E^* (see, e.g., [18]). Thus, the result follows from Theorem 8.

Corollary 11. Let K be a nonempty, closed, and convex subset of l_p, 1 < p < ∞. Let T : C → C be an asymptotically k-strict pseudocontractive mapping for some 0 ≤ k < 1 with sequences {l_n}⊂[1, ∞). Assume that F(T) is nonempty. Let {x_n} be a sequence defined by x₁ = u ∈ C and

where {α_n}⊂(0, c)⊂(0,1), and {β_n}⊂[a, b]⊂(0, k) (for a, b, and c constants) satisfying lim_n→∞α_n = 0, ∑ α_n = ∞ and lim_n→∞((l_n − 1)/α_n) = 0. Then the sequence {x_n} converges strongly to a fixed point of T.

Corollary 12. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let T_i : C → C be asymptotically k_i-strict pseudocontractive mappings for 0 ≤ k_i < 1 with sequences {l_n,i}⊂[1, ∞), for i = 1,2, …, N. Assume that $$ is nonempty. Let {x_n} be a sequence defined by x₁ = u ∈ C and

where $$, such that θ_n,1 + θ_n,2 + ⋯+θ_n,N = 1, for each n ≥ 1, {α_n}, {θ_n,i}⊂(0, c)⊂(0,1), satisfying liminf_nθ_n,i > 0, lim_n→∞α_n = 0, ∑ α_n = ∞, lim_n→∞((l_n,i − 1)/α_n) = 0 and {β_n}⊂[a, b]⊂(0, k) (for a, b, and c constants), for k = min_1≤i≤N{k_i}. Then the sequence {x_n} generated by (41) converges strongly to a common fixed point of {T_i : i = 1,2, …, N}.

Corollary 13. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let T : C → C be an asymptotically k-strict pseudocontractive mapping for some 0 ≤ k < 1 with sequences {l_n}⊂[1, ∞). Assume that F(T) is nonempty. Let {x_n} be a sequence defined by x₁ = u ∈ C and

where {α_n}⊂(0, c)⊂(0,1), and {β_n}⊂[a, b]⊂(0, k) (for a, b, and c constants) satisfying lim_n→∞α_n = 0, ∑ α_n = ∞ and lim_n→∞((l_n − 1)/α_n) = 0. Then the sequence {x_n} converges strongly to a fixed point of T.

Remark 14. We note that Corollary 9 generalizes several recent results of this nature. Particularly, it extends Theorem KX of [12], Theorem 2 of Liu [5], and corresponding theorem of Schu [7] in the sense that our convergence is strong in more general Banach spaces possessing weakly sequentially continuous duality mappings without the requirement that T be completely continuous.

Remark 15. Corollary 9 is an improvement of Theorem 3.2 of Osilike et al. [13] and Theorems 3.1 and 3.2 of Zhang and Xie [17] in the sense that our convergence is strong without the requirement that liminf_n→∞d(x_n, F(T)) = 0, provided that E possesses weakly sequentially continuous duality mappings.