Convergence Theorems for Common Fixed Points of a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces

Lemma 1 (see [4].)If E is a strictly convex and smooth Banach space, then for x, y ∈ E, ϕ(x, y) = 0 if and only if x = y.

Lemma 2 (see [23].)Let E be a uniformly convex and smooth Banach space and let {y_n},  {z_n} be two sequences of E. If ϕ(y_n, z_n) → 0 and either {y_n} or {z_n} is bounded, then y_n − z_n → 0.

Lemma 3 (see [7].)Let C be a closed convex subset of a smooth Banach space E and x ∈ E. Then, x₀ = Π_Cx if and only if

Lemma 4 (see [7].)Let E be a reflexive, strictly convex, and smooth Banach space and let C be a closed convex subset of E and x∈E. Then, ϕ(y, Π_Cx) + ϕ(Π_Cx, x) ≤ ϕ(y, x) for all y ∈ C.

Lemma 5 (see [24].)Let E be a uniformly convex Banach space and B_r(0) = {x ∈ E : ∥x∥≤r} a closed ball of E. Then, there exists a continuous strictly increasing convex function g : [0, ∞)→[0, ∞) with g(0) = 0 such that

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

Lemma 6 (see [19].)Let E be a uniformly convex and uniformly smooth Banach space and let C be a closed convex subset of E. Then, for points w, x, y, z ∈ E and a real number a ∈ R, the set K : = {v ∈ C : ϕ(v, y) ≤ ϕ(v, x)+〈v, Jz − Jw〉+a} is closed and convex.

Proposition 7. Let E be a uniformly convex Banach space and B_r(0) = {x ∈ E : ∥x∥≤r} a closed ball of E. Then, there exists a continuous strictly increasing convex function g : [0, ∞)→[0, ∞) with g(0) = 0 such that

for all n ≥3,  x_i ∈ B_r(0) and λ_i ∈ [0,1] with $$.

Proof. If λ₃ + λ₄ ⋯   + λ_n ≠ 0, using Lemma 5 and the convexity of ∥·∥², we have

If λ₃ + λ₄ ⋯ +λ_n = 0, the last inequality above also holds obviously. By the same argument in the proof above, we obtain for all i, j ∈ {1,2, …, n}. Then, So,

Theorem 8. Let E be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E and T₁,  T₂,   … ,  T_N :  C → C relatively nonexpansive mappings such that $$. The sequence {x_n} is given by (9) with the following restrictions:

• (a)

  α_n + β_n + γ_n = 1,  0 ≤ α_n < 1,  0 ≤ β_n < 1,  0 < γ_n ≤ 1 for all n ≥ 0;

• (b)

  lim _n→∞α_n = 0 and lim sup _n→∞β_n < 1;

• (c)

  $$ with $$, for all n ≥ 0;

• (d)

  $$ and $$; or

•  

  d^′ $$.

Then, the sequence {x_n} converges strongly to Π_Fx, where Π_F is the generalized projection from C onto F.

Proof. We split the proof into seven steps.

Step  1. Show that P_F is well defined for every x ∈ C.

It is easy to know that F(T_i), i = 1,2, …, N are closed convex sets and so is F. What is more, F is nonempty by our assumption. Therefore, P_F is well defined for every x ∈ C.

Step  2. Show that H_n and W_n are closed and convex for all n ≥ 0.

From the definition of W_n, it is obvious W_n is closed and convex for each n ≥ 0. By Lemma 6, we also know that H_n is closed and convex for each n ≥ 0.

Step  3. Show that F ⊂ H_n⋂ W_n for all n ≥ 0.

Let u ∈ F and let n ≥ 0. Then, by the convexity of ∥·∥², we have

and then, Thus, we have u ∈ H_n. Therefore, we obtain F ⊂ H_n for all n ≥ 0.

Next, we prove F ⊂ W_n for all n ≥ 0. We prove this by induction. For n = 0, we have F ⊂ C = W₀. Assume that F ⊂ W_n. Since x_n+1 is the projection of x onto H_n⋂ W_n, by Lemma 3, we have

for any z ∈ H_n⋂ W_n. As F ⊂ H_n⋂ W_n by the induction assumption,F ⊂ W_n holds, in particular, for all u ∈ F. This together with the definition of W_n+1 implies that F ⊂ W_n+1. Hence, F ⊂ H_n⋂ W_n for all n ≥ 0.

Step  4. Show that ∥x_n+1 − x_n∥→0 as n → ∞.

In view of (19) and Lemma 4, we have $$, which means that, for any z ∈ W_n,

Since x_n+1 ∈ W_n and u ∈ F ⊂ W_n, we obtain for all n ≥ 0. Consequently, lim _n→∞ϕ(x_n, x) exists and {x_n} is bounded. By using Lemma 4, we have as n → ∞. By using Lemma 2, we obtain ∥x_n+1 − x_n∥→0 as n → ∞.

Step  5. Show that ∥x_n − z_n∥→0 as n → ∞.

From $$, we have

as n → ∞. By Lemma 2, we also have ∥x_n+1 − y_n∥→0, and then, as n → ∞. We observe that So, Since lim _n→∞α_n = 0, limsup _n→∞β_n < 1, ϕ(y_n, x_n) → 0, and ∥z_n − y_n∥∥Jy_n − Jx_n∥→0 as n → ∞, we have as n → ∞. Using Lemma 2, we obtain ∥x_n − z_n∥→0 as n → ∞.

Step  6. Show that ∥x_n − T_ix_n∥→0,  i = 1,2, …, N.

Since {x_n} is bounded and ϕ(p, T_ix_n) ≤ ϕ(p, x_n), where p ∈ F, i = 1,2, …, N, we also obtain that {Jx_n},  {JT₁x_n}, …, {JT_Nx_n} are bounded, and hence, there exists r > 0 such that {Jx_n},  {JT₁x_n}, …, {JT_Nx_n}⊂B_r(0). Therefore, Proposition 7 can be applied and we observe that

where g : [0, ∞)→[0, ∞) is a continuous strictly increasing convex function with g(0) = 0. And as n → ∞. From the properties of the mapping g, we have for all i, j ∈ {1,2, …, N}. From the condition (d^′), we have ∥x_n − T_ix_n∥→0 immediately, as n → ∞, i = 1,2, …, N; from the condition (d), we can also have ∥x_n − T_ix_n∥→0, as n → ∞,   i = 1,2, …, N. In fact, since $$, it follows that for all i, j ∈ {1,2, …, N}. Next, we note by the convexity of ∥·∥² and (9) that as n → ∞. By Lemma 2, we have lim _n→∞∥T_ix_n − z_n∥ = 0 and as n → ∞ for all i ∈ {1,2, …, N}.

Step  7. Show that x_n → Π_Fx, as n → ∞.

From the result of Step 6, we know that if $$ is a subsequence of {x_n} such that $$, then $$. Because E is a uniformly convex and uniformly smooth Banach space and {x_n} is bounded, so we can assume $$ is a subsequence of {x_n} such that $$ and ω = Π_Fx. For any n ≥ 1, from $$ and ω ∈ F ⊂ H_n⋂ W_n, we have

On the other hand, from weakly lower semicontinuity of the norm, we have From the definition of Π_Fx, we obtain $$, and hence, $$. So, we have $$. Using the Kadec-klee property of E, we obtain that $$ converges strongly to Π_Fx. Since $$ is an arbitrary weakly convergent sequence of {x_n}, we can conclude that {x_n} converges strongly to Π_Fx.

Corollary 9. Let C be a nonempty closed convex subset of a Hilbert space H and T₁,  T₂,   … ,  T_N :  C → C relatively nonexpansive mappings such that $$. The sequence {x_n} is given by (9) with the following restrictions:

• (a)

  α_n + β_n + γ_n = 1,  0 ≤ α_n < 1,  0 ≤ β_n < 1,  0 ≤ γ_n ≤ 1 for all n ≥ 0;

• (b)

  lim _n→∞α_n = 0 and lim sup _n→∞β_n < 1;

• (c)

  $$ with $$, i = 0,1, 2, …, N, for all n ≥ 0;

• (d)

  $$ and $$; or

•  

  d^′ $$.

Then, the sequence {x_n} converges strongly to P_Fx, where P_F is the metric projection from C onto F.

Proof. It is true because the generalized projection Π_F is just the metric projection P_F in Hilbert spaces.

Remark 10. The results of Nakajo and Takahashi [18] and Song et al. [11] are the special cases of our results in Corollary 9. And in our results of Theorem 8, if $$ and α_n = 0 for all n ≥ 0, then, we obtain Theorem 4.1 of Matsushita and Takahashi [10]; if T₁ = T₂ = ⋯ = T_N−1 and α_n = 0 for all n ≥ 0, then, we obtain Theorem 3.1 of Plubtieng and Ungchittrakool [19]; if T₁ = T₂ = ⋯ = T_N−1 and β_n = 0 for all n ≥ 0, then, we obtain Theorem 3.2 of Plubtieng and Ungchittrakool [19]. So, our results improve and extend the corresponding results by many others.