Approximating Common Fixed Points for a Finite Family of Asymptotically Nonexpansive Mappings Using Iteration Process with Errors Terms

Lemma 1. Let {s_n}, {t_n}, and {σ_n} be sequences of nonnegative real numbers satisfying the following conditions: for all n ≥ 1, s_n+1 ≤ (1 + σ_n)s_n + t_n, where $$ and $$. Then

• (i)

  lim _n→∞ s_n exists;

• (ii)

  in particular, if {s_n} has a subsequence $$ converging to 0, then lim _n→∞ s_n = 0.

Lemma 2. Let p > 1 and C > 0 be two fixed numbers. Then a Banach space X is uniformly convex if and only if there exists a continuous, strictly increasing, convex function g : [0, ∞)→[0, ∞) with g(0) = 0 such that

for all x, y ∈ B_C∶ = {x ∈ X : ∥x∥ ≤ C}, and λ ∈ [0,1], where w_p(λ) = λ(1 − λ) ^p + λ^p(1 − λ).

Lemma 3. Let X be a uniformly convex Banach space and B_C∶ = {x ∈ X : ∥x∥ ≤ C}, C > 0. Then there exists a continuous, strictly increasing, convex function g : [0, ∞)→[0, ∞) with g(0) = 0 such that

for all x, y, z ∈ B_C and λ, μ, ν ∈ [0,1] with λ + μ + ν = 1.

Lemma 4. Let X be a uniformly convex Banach space, K a nonempty closed convex subset of X, and T : K → K an asymptotically nonexpansive mapping. Then I − T (I is identity mapping) is demiclosed at zero; that is, if x_n → x^* weakly and x_n − Tx_n → 0 strongly, then x^* ∈ F(T), where F(T) is the set of fixed points of T.

Definition 5. A family {T_i : i ∈ {1, …, m}} of asymptotically nonexpansive mappings on K with $$ is said to satisfy condition (A) on K if there exists a nondecreasing function f : [0, ∞)→[0, ∞) with f(0) = 0, f(r) > 0, for all r ∈ (0, ∞) such that max _1≤i≤m ∥x − T_ix∥ ≥ f(d(x, F)) for all x ∈ K.

Lemma 6. Let X be a real Banach space and K a nonempty closed convex subset of X. Let T_i : K → K  (i = 1, 2, …, m) be m asymptotically nonexpansive mappings with sequence {k_n}⊂[1, ∞), $$, and $$. Suppose that $$, $$, i = 1,2, …, m are appropriate sequences in [0,1] and $$, i = 1,2, …, m are bounded sequences in K such that $$ for i = 1,2, …, m. Let {x_n} be given by (5). Then {x_n} is bounded and lim _n→∞ ∥x_n − p∥ exists for p ∈ ℱ.

Proof. For any given p ∈ ℱ, since $$, i = 1,2, …, m are bounded sequences in K, let

For each n ≥ 1, using (5), we have Then we have which leads to where Since t^m − 1 ≤ mt^m−1(t − 1) for all t ≥ 1, the only assumption $$ is enough for the boundedness for {k_n}, then k_n ⊂ [1, D], for all n ≥ 1, and for some D. Hence $$ holds for all n ≥ 1. Therefore $$ and also $$. Equation (13) and Lemma 1 guarantee that the sequence {x_n} is bounded and lim _n→∞ ∥x_n − p∥ exists.

Theorem 7. Let X be a real uniformly convex Banach space and K a nonempty closed convex subset of X. Let T_i : K → K  (i = 1, 2, …, m) be m asymptotically nonexpansive mappings with sequence {k_n}⊂[1, ∞), $$ and $$. Suppose that $$, $$, i = 1,2, …, m are appropriate sequences in [0,1] and $$, i = 1,2, …, m are bounded sequences in K such that $$ for i = 1,  2, …, m. Let {x_n} be given by (5). Suppose that

for i = 1, …, m. Then

Proof. Let p ∈ ℱ. Then by Lemma 6, lim _n→∞ ∥x_n − p∥ exists. Since $$, i = 1,2, …, m are bounded sequences in K, let M = sup _{n≥1,  i=1,2,…,m} ∥u_in − p∥; moreover, it follows that {y_n+m−i − p} is also bounded for each i ∈ {2,3, …, m}, and hence {(u_(m−i+1)n − y_n+i−1)} is also bounded for i ∈ {1,2, …, m}. By using (5), we obtain

Note that 0 ≤ θ² − 1 ≤ 2θ(θ − 1) for all θ ≥ 1, the assumption $$ implies that $$. Since {k_n} is bounded, there exists D > 0 such that k_n ∈ [1, D], n ≥ 1. Then $$ holds for all n ≥ 1. Therefore, the assumption $$ implies that $$. Then It follows from (18) and (19) that We first obtain that Now if 0 < lim  inf _n→∞ a_mn and 0 < lim  inf _n→∞ a_mn < limsup _n→∞ (a_mn + b_mn) < 1, there exist a positive integer n₀ and η, η^′ ∈ (0,1) such that 0 < η < a_mn,  a_mn + b_mn < η^′ < 1 for all n ≥ n₀. This implies by (21) that It follows from (22) that for ℓ ≥ n₀, Then $$, and therefore $$, and by property of g, we have $$. By a similar method, together with (20) and by property of g, we have for 2 ≤ i < m. Thus, we conclude that for 2 ≤ i ≤ m. From (5) and for i = 1,2, …, m This together with (25) implies that for each i = 1,2, …, m − 2 It follows from (5) that Equations (24) and (28) imply that It follows from (5) that Thus, (24) and (30) guarantee that Continuing in this fashion, for each i = 2, …, m we get, Taking the limit on both sides inequality from (33), we have Since T_m is an asymptotically nonexpansive mapping with k_n, we have Taking the limit on both sides inequality (35), and by using (24), we get Since T_m−1 is an asymptotically nonexpansive mapping with k_n, we have Also, taking the limit on both sides inequality (37), and by using (24), we get In a similar way, one can prove that for each i = 2, …, m − 1 Next, we consider It follows from (34), (36), and the above inequality (40) that It follows from (34), (38) and (42) that Continuing similar process, for each i = 0, …, m − 1 we get The proof is completed.

Theorem 8. Let X be a real uniformly convex Banach space and K a nonempty closed convex subset of X. Let T_i : K → K  (i = 1, 2, …, m) be m asymptotically nonexpansive mappings with sequence {k_n}⊂[1, ∞), $$ and $$. Suppose that $$, $$, i = 1,2, …, m are appropriate sequences in [0,1] and $$, i = 1,2, …, m are bounded sequences in K such that $$ for i = 1,2, …, m. Suppose that

for i = 1, …, m. If one of {T_i} is either completely continuous or semicompact, for some i ∈ {1,2, …, m}, then the sequence {x_n} generated by (5) converges strongly to an element of ℱ.

Proof. Assume that there exists ℓ ∈ {1,2, …, m} such that T_ℓ is semi-compact. Since {x_n} is bounded and by Theorem 7, ∥x_n − T_ℓx_n∥ → 0 as n → ∞, there exists a subsequence $$ of {x_n} such that $$ converges strongly to p ∈ K. Since $$, it follows from Lemma 4 that T_ℓp = p. Also, from Theorem 7  $$, i = 1,2, …, m. Therefore, from Lemma 4 we obtain that $$. So {x_n} converges strongly to p.

If one of T_i’s is completely continuous, say T_ℓ, since {x_n} is bounded, there exists a subsequence $$ of {x_n} such that $$ converges strongly to p ∈ K. By Theorem 7, $$. It follows from continuity of ∥·∥ that

Using $$ as j → ∞, $$, i = 1,2, …, m and Lemma 4, we obtain that $$. Also using $$ as j → ∞ and Lemma 6, we obtain that lim _n→∞ ∥x_n − p∥ = 0. This completes the proof.

Theorem 9. Let X be a real uniformly convex Banach space and K a nonempty closed convex subset of X. Let T_i : K → K  (i = 1, 2, …, m) be m asymptotically nonexpansive mappings with sequence {k_n}⊂[1, ∞), $$ and satisfying the condition (A). Suppose that $$, $$, i = 1,2, …, m are appropriate sequences in [0,1] and $$, i = 1,2, …, m are bounded sequences in K such that $$ for i = 1,2, …, m. Suppose that $$ and 0 < lim  inf _n→∞ a_in < lim  sup _n→∞ (a_in + b_in) < 1 for i = 1, …, m. Then the sequence {x_n} generated by (5) converges strongly to an element of ℱ.

Proof. Since lim _n→∞ ∥x_n − p∥ exists for all p ∈ ℱ by Lemma 6, then, for any p ∈ ℱ such that

we have that lim _n→∞ ∥x_n − p∥ exists. It follows from (47) that lim _n→∞ d(x_n, ℱ) exists. From condition (A) where $$ is max _1≤i≤m ∥x_n − T_ix_n∥. From Theorem 7  $$. It then follows (48) that lim _n→∞ f(d(x_n, ℱ)) = 0. By property of f, lim _n→∞ d(x_n, ℱ) = 0. It also follows from (47) that lim _n→∞ ∥x_n − p∥ = 0. Therefore lim _n→∞ x_n = p ∈ ℱ.

Theorem 10. Let X be a uniformly convex Banach space satisfying Opial’s condition, and let K be a nonempty closed convex subset of X. Let T_i : K → K  (i = 1, 2, …, m) be m asymptotically nonexpansive mappings with sequence {k_n}, and let the sequences $$, $$, and $$, i = 1,2, …, m be the same as in Theorem 7. Then the sequence {x_n} defined by (5) converges weakly to a common fixed point of {T_i : i = 1, …, m}.

Proof. It follows from Lemma 6 that lim _n→∞ ∥x_n − p∥ exists. Therefore, {x_n − p} is a bounded sequence in X. Then by the reflexivity of X and the boundedness of {x_n}, there exists a subsequence $$ of {x_n} such that $$ weakly. By Theorem 7, lim _n→∞ ∥x_n − T_ix_n∥ = 0, and I − T_i is demiclosed at 0 for i = 1,2, …, m. So we obtain T_ip = p for i = 1,2, …, m. Finally, we prove that {x_n} converges to p. Suppose p, q ∈ w({x_n}), where w({x_n}) denotes the weak limit set of {x_n}. Let $$ and $$ be two subsequences of {x_n} which converge weakly to p and q, respectively. Opial’s condition ensures that ω(x_n) is a singleton set. It follows that p = q. Thus {x_n} converges weakly to an element of ℱ. This completes the proof.