On Unification of the Strong Convergence Theorems for a Finite Family of Total Asymptotically Nonexpansive Mappings in Banach Spaces

Definition 1.1. A mapping T : K → K is said to be

• (i)

  nonexpansive if ∥Tx − Ty∥   ≤   ∥x − y∥ for all x, y ∈ K,

• (ii)

  asymptotically nonexpansive if there exists a sequence {λ_n}⊂[1, ∞) with lim _n→∞ λ_n = 1 such that ∥Tⁿx − Tⁿy∥≤λ_n∥x − y∥ for all x, y ∈ K and n ∈ ℕ,

• (iii)

  asymptotically nonexpansive in the intermediate sense, if it is continuous and the following inequality holds:

Remark 1.2. Observe that if we define

then σ_n → 0 as n → ∞, and (1.1) reduces to

Definition 1.3. Let K be a nonempty closed subset of a real normed linear space X.T : K → K is called a total asymptotically nonexpansive mapping if there exist nonnegative real sequence {μ_n} and {λ_n} with μ_n, λ_n → 0 as n → ∞ and strictly increasing continuous function ϕ : ℝ⁺ → ℝ⁺ with ϕ(0) = 0 such that for all x, y ∈ K,

Remark 1.4. If ϕ(ξ) = ξ, then (1.4) reduces to

In addition, if λ_n = 0 for all n ≥ 1, then total asymptotical nonexpansive mappings coincide with asymptotically nonexpansive mappings. If μ_n = 0 and λ_n = 0 for all n ≥ 1, we obtain from (1.5) the class of mappings that includes the class of nonexpansive mappings. If μ_n = 0 and λ_n = σ_n = max {0, a_n}, where a_n : = sup _x,y∈K  (∥Tⁿx − Tⁿy∥−∥x − y∥) for all n ≥ 1, then (1.5) reduces to (1.3) which has been studied as mappings asymptotically nonexpansive in the intermediate sense.

Definition 1.5. Let T : K → K, I : K → K be two mappings of a nonempty subset K of a real normed linear space X, then T is said to be

• (i)

  I-nonexpansive if ∥Tx − Ty∥≤∥Ix − Iy∥ for all x, y ∈ K,

• (ii)

  asymptotically I-nonexpansive, if there exists a sequence {λ_n}⊂[1, ∞) with lim _n→∞  λ_n = 1 such that ∥Tⁿx − Tⁿy∥≤λ_n∥Iⁿx − Iⁿy∥ for all x, y ∈ K and n ≥ 1.

Definition 1.6. Let T : K → K, I : K → K be two mappings of a nonempty subset K of a real normed linear space X, then T is said to be a total asymptotically I-nonexpansive mapping if there exist nonnegative real sequences {μ_n} and {λ_n} with μ_n, λ_n → 0 as n → ∞ and the strictly increasing continuous function ϕ : ℝ⁺ → ℝ⁺ with ϕ(0) = 0 such that for all x, y ∈ K,

Example 1.7. Let us consider the space ℓ₁, and let B₁ = {x ∈ ℓ₁ : ∥x∥₁ ≤ 1}. Define a nonlinear operator T : ℓ₁ → ℓ₁ by

Let ∥x∥₁ ≤ 1, then from one gets T(B₁) ⊂ B₁.

One can find that

Hence, From x, y ∈ B₁, we have So, it follows from (1.10) and (1.11) that

Now consider a new Banach space ℝ × ℓ₁ with a norm ∥𝕏∥ = |x | +∥x∥₁, where 𝕏 = (x, x) and define a new mapping S : ℝ × ℓ₁ → ℝ × ℓ₁ by

Let K = [0,1] × B₁, then it is clear that S(K) ⊂ K. One can see that $$. Therefore, using (1.14), we obtain We let $$ and μ_k = α^k. It is clear that ϕ(0) = 0 and ϕ is strictly increasing, and moreover, (1.14) implies that is S is a totally asymptotically I-nonexpansive mapping. Here, I is the identity mapping of ℝ × ℓ₁.

Now we are going to show that S is not asymptotically nonexpansive. Namely, we will establish that for any sequence of positive numbers {λ_n} with λ_n → 0 and any k ∈ ℕ, one can find 𝕏₀, 𝕐₀ such that

In fact, choose 𝕏₀, 𝕐₀ as follows: where

From (1.10), one finds that

The last equalities with (1.18) imply that This yields the required assertion. Note that S has infinitely many fixed points in K, that is, Fix (S) = {(x, 0) : x ∈ [0,1]}.

Example 1.8. Let us consider the Banach space ℝ × ℓ₁ defined as before, and let f be a mapping of a segment C ⊂ ℝ to itself, that is, f : C → C with f(0) = 0 and

where c_n → 0. Note that such kind of functions do exist. One can take (see for more details [8]) C = [−1/π, 1/π] and

Define a new mapping S_f : C × B₁ → C × B₁ by

here T is defined as above (see (1.7)). Using the same argument as the above Example 1.7, we can establish that Moreover, such a mapping is not asymptotically nonexpansive. Note that the mapping $$ with the function f_κ has a unique fixed point in C × B₁.

Remark 1.9. To the best our knowledge, we should stress that the constructed examples are currently only unique examples of totaly asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Before, no such examples were known in the literature.

Lemma 2.1 (see [16].)Let {a_n}, {b_n}, and {c_n} be three sequences of nonnegative real numbers with $$, $$. If the following condition is satisfied:

• (i)

  a_n+1 ≤ (1 + b_n)a_n + c_n,   n ≥ 1,

then the limit lim _n→∞ a_n exists.

Lemma 2.2 (see [34].)Let X be a uniformly convex Banach space and t ∈ (0,1). Suppose that {x_n},{y_n} are two sequences in X such that

hold some d ≥ 0, then lim _n→∞  ∥x_n − y_n∥ = 0.

Lemma 2.3 (see [14].)Let X be a uniformly convex Banach space, and let b, c be two constants with 0 < b < c < 1. Suppose that {t_n} is a sequence in [b, c] and {x_n},{y_n} are two sequences in X such that

hold some d ≥ 0, then lim _n→∞ ∥x_n − y_n∥ = 0.

Lemma 3.1. Let X be a uniformly convex Banach space and $$ any constants with $$. Suppose that $$ are sequences in X such that

hold some d ≥ 0, then lim _n→∞ ∥z_in∥ = d and lim _n→∞ ∥z_in − z_jn∥ = 0 for any $$.

Proof. Let us first prove lim _n→∞ ∥z_in∥ = d for any $$. Indeed, it follows from (3.1) that

We then get that $$, which means lim _n→∞ ∥z_in∥ = d.

Now we prove the statement lim _n→∞ ∥z_in − z_jn∥ = 0 by means of mathematical induction with respect to m. For m = 2, the statement immediately follows from Lemma 2.2. Assume that the statement is true, for m = k − 1. Let us prove for m = k. To do this, denote

Since $$, we get $$. On the other hand, one has We then obtain $$ which means lim _n→∞ ∥t_n∥ = d. In this case, according to the assumption of induction with the sequence t_n, we can conclude that lim _n→∞ ∥z_in − z_jn∥ = 0, if 1 ≤ i, j ≤ k − 1.

Since lim _n→∞ ∥(1 − α_k)t_n + α_kz_kn∥ = d due to Lemma 2.2, one gets

If 1 ≤ j ≤ k − 1, then the following inequality implies that lim _n→∞ ∥z_jn − z_kn∥ = 0. This completes the proof.

Lemma 3.2. Let X be a uniformly convex Banach space, and let α_*,   α^* be two constants with 0 < α_* < α^* < 1. Suppose that $$, $$ are any sequences with $$ for all n ∈ ℕ. Suppose that $$, $$ are sequences in X such that

hold for some d ≥ 0, then lim _n→∞ ∥z_in∥ = d and lim _n→∞ ∥z_in − z_jn∥ = 0 for any $$.

Proof. Analogously as in the proof of Lemma 3.1, it is easy to show that lim _n→∞ ∥z_in∥ = d. Therefore, let us prove the statement lim _n→∞ ∥z_in − z_jn∥ = 0 for any $$. Suppose to the contrary, that there exist two numbers i₀, j₀ such that

then there exists a subsequence $$ of $$ such that $$.

Let us consider the subsequences $$ of $$, here $$. Since $$, there exists a subsequence $$ of $$ such that $$ for all $$. Since $$, for all n ∈ ℕ, one gets $$, and α_i ∈ [α_*, α^*], for all $$. We know that

It then follows that $$. On the other hand, we have Therefore, $$. Consequently, Lemma 3.1 implies that $$. However, it contradicts to This completes the proof.

Proposition 3.3. Let X be a real Banach space, and let K be a nonempty closed-convex subset of X. Let $$: K → K be a finite family of total asymptotically I_i-nonexpansive mappings with sequences $$, $$, where $$, and let $$: K → K be a finite family of total asymptotically nonexpansive mappings with sequences $$, $$, where $$. Suppose that there exist $$, $$ such that $$, for all ξ_i ≥ M_i and $$ for all ζ_i ≥ N_i, where $$, then the following holds for any x, y ∈ K and for any $$:

Proof. Since ϕ_i, φ_i: ℝ⁺ → ℝ⁺ are the strictly increasing continuous functions, where $$, it follows that ϕ_i(ξ_i) ≤ ϕ_i(M_i) and φ_i(ζ_i) ≤ ϕ_i(N_i) whenever ξ_i ≤ M_i and ζ_i ≤ N_i, where $$. By the hypothesis of Proposition 3.3, for all ξ_i, ζ_i ≥ 0 and $$, we then get

Since $$, $$ are total asymptotically I_i-nonexpansive and total asymptotically nonexpansive mappings, respectively, from (3.14) and (3.15), one gets

Lemma 3.4. Let X be a uniformly convex real Banach space, and let K be a nonempty closed-convex subset of X. Let $$: K → K be a finite family of total asymptotically I_i-nonexpansive mappings with sequences $$, $$, where $$, and let $$: K → K be a finite family of total asymptotically nonexpansive mappings with sequences $$, $$, where $$, such that $$. Suppose that $$, $$, $$, $$ for all $$, and there exist $$, $$ such that $$, for all ξ_i ≥ M_i and $$ for all ζ_i ≥ N_i, where $$. If {x_n} is the explicit iterative sequence defined by (1.27), then for each p ∈ F, the limit lim _n→∞ ∥x_n − p∥ exists.

Proof. Since F ≠ ∅, for any given p ∈ F, it follows from (1.27) and (3.13) that

Again from (1.27) and (3.12), we derive that

Then from (3.17) and (3.18), one finds

Here Denoting a_n = ∥x_n − p∥ in (3.19), one gets Since $$ and $$, it follows from Lemma 2.1 the existence of the limit lim _n→∞ a_n. This means the limit exists, where d ≥ 0 is a constant. This completes the proof.

Theorem 3.5. Let X be a uniformly convex real Banach space, and let K be a nonempty closed-convex subset of X. Let $$ be a finite family of total asymptotically I_i-nonexpansive continuous mappings with sequences $$, where $$, and let $$ be a finite family of total asymptotically nonexpansive continuous mappings with sequences $$, $$, where $$, such that $$. Suppose that $$, $$, $$, $$ for all $$, and there exist $$ such that $$, for all ξ_i ≥ M_i and $$ for all ζ_i ≥ N_i, where $$, then the explicit iterative sequence {x_n} defined by (1.27) converges strongly to a common fixed point in F if and only if

Proof. The necessity of condition (3.23) is obvious. Let us prove the sufficiency part of the theorem.

Since $$ are continuous mappings, the sets F(T_i) and F(I_i) are closed. Hence, $$ is a nonempty closed set.

For any given p ∈ F, we have (see (3.19))

Hence, one finds From (3.25) due to Lemma 2.1, we obtain the existence of the limit lim _n→∞ d(x_n, F). By condition (3.23), one gets

Let us prove that the sequence {x_n} converges strongly to a common fixed point in F. We first show that {x_n} is Cauchy sequence in X. In fact, due to 1 + t ≤ exp (t) for all t > 0, and from (3.24), we obtain

Thus, for any positive integers m, n, from (3.27) with $$,$$, we find Therefore, we get for all p ∈ F, where $$. Taking infimum over p ∈ F in (3.29) gives

Since lim _n→∞ d(x_n, F) = 0 and $$, given ɛ > 0, there exists an integer N₀ > 0 such that for all n > N₀, we have d(x_n, F) < (ɛ/2W) and $$. Consequently, for all integers n ≥ N₀ and m ≥ 1 and from (3.30), we derive

which means that {x_n} is Cauchy sequence in X, and since X is complete, there exists x^* ∈ X such that the sequence {x_n} converges strongly to x^*.

Now we show that x^* is a common fixed point in F. Suppose for contradiction that x^* ∉ F. Since F is closed subset of X, we have that d(x^*, F) > 0. However, for all p ∈ F, we have

This implies that so that as n → ∞ we obtain d(x^*, F) = 0 which contradicts d(x^*, F) > 0. Hence, x^* is a common fixed point in F. This proves the required assertion.

Proposition 3.6. Let X be a uniformly convex real Banach space, and let K be a nonempty closed-convex subset of X. Let $$ be a finite family of total asymptotically I_i-nonexpansive continuous mappings with sequences $$, where $$, and let $$: K → K be a finite family of total asymptotically nonexpansive continuous mappings with sequences $$, $$, where $$, such that $$. Suppose that $$, $$,$$, $$ for all $$, and $$ are sequences with $$ and $$, for all $$, here 0 < α_* < α^* < 1, 0 < β_* < β^* < 1, and there exist $$, $$ such that $$, for all ξ_i ≥ M_i and $$ for all ζ_i ≥ N_i, where $$. then the explicit iterative sequence {x_n} defined by (1.27) satisfies the following:

for all $$.

Proof. According to Lemma 3.4 for any p ∈ F, we have lim _n→∞ ∥x_n − p∥ = d. It follows from (1.27) that

as n → ∞. By means of $$, $$, for all $$, from (3.18), one yields that and from (3.13), (3.37), we have for all $$. Now using with (3.38) and applying Lemma 3.2 to (3.36), one finds for all $$. Now from (1.27) and (3.40), we infer that On the other hand, from (3.13), we have which implies The last inequality with (3.22), (3.40) yields Combining (3.44) with (3.37), we get Again from (1.27), we can see that From (3.12) and (3.22), one finds for all $$. Now applying Lemma 3.2 to (3.46), we obtain for all $$. We then have Consider for all $$. Then from (3.40) and (3.49), we get for all $$.

Theorem 3.7. Let X be a uniformly convex real Banach space, and let K be a nonempty closed-convex subset of X. Let $$ be a finite family of total asymptotically I_i-nonexpansive continuous mappings with sequences $$, where $$, and let $$ be a finite family of total asymptotically nonexpansive continuous mappings with sequences $$, $$, where $$, such that $$. Suppose that $$, $$, $$, $$ for all $$, and $$ are sequences with $$ and $$, for all $$, here 0 < α_* < α^* < 1, 0 < β_* < β^* < 1, and there exist $$, $$ such that $$, for all ξ_i ≥ M_i and $$ for all ζ_i ≥ N_i, where $$. If at least one mapping of the mappings $$ and $$ is compact, then the explicitly iterative sequence {x_n} defined by (1.27) converges strongly to a common fixed point of $$ and $$.

Proof. Without any loss of generality, we may assume that T₁ is compact. This means that there exists a subsequence $$ of $$ such that $$ converges strongly to x^* ∈ K. Then from (3.34), we have that $$ converges strongly to x^*. Also from (3.34), we obtain that $$ converges strongly to x^*, for all $$. Since $$ are continuous mappings, so $$ converges strongly to T_ix^*, for all $$. On the other hand, from (3.35) and continuousness of $$, we obtain that $$ converges strongly to x^*, and $$ converges strongly to I_ix^*, for all $$. Due to (3.41), $$ converges to 0, as k → ∞. Then, $$ converges strongly to x^* and moreover, (3.13) and (3.12) imply that $$ and $$ converge to 0, as k → ∞, for all $$. From (3.34), (3.35), it yields that $$ and $$ converge to 0 as k → ∞, for all $$. Observe that

for all $$. Taking limit as k → ∞, we have that x^* = T_ix^* and x^* = I_ix^*, for all $$, which means x^* ∈ F. However, by Lemma 3.4, the limit lim _n→∞ ∥x_n − x^*∥ exists, then which means {x_n} converges strongly to x^* ∈ F. This completes the proof.

Remark 3.8. If one has that all I_i are identity mappings, then the obtained results recover and correctly prove the main result of [24].

Remark 3.9. Suppose that we are given two families $$ and $$ of total asymptotically nonexpansive continuous mappings such that $$. Define the following explicit iterative process:

such that $$ and $$.

Remark 3.10. Let $$: K → K be a finite family of total asymptotically nonexpansive continuous mappings with sequences $$, $$, where $$. It is clear for each operator T_i that one has

and this means that T_i is total asymptotically T_i-nonexpansive mappings with sequence $$ and the function ϕ(λ) = λ. Hence, our iteration scheme can be written as follows: where $$ in (0,1), ($$ with $$, $$.