A study of asymptotically non-expansive mapping iteration and weakly convergent approximation methods based on Banach spaces

The theory of stationary points is an important part of the rapidly developing theory of nonlinear generalized functional analysis, and it is also one of the key problems that people have been paying attention to. It is very closely related to many fields of modern mathematics, especially in the various branches of applied mathematics. Almost all have a wide range of applications [1–4]. It plays a very important role in establishing all kinds of equations, such as all kinds of linear or nonlinear, differential equations, integral equations, and all kinds of operator equations, as well as the existence and uniqueness of solutions to variational inequalities and variational inclusion problems [5–8].

The iterative process of immovable points of asymptotically non-expansive mappings in Banach spaces is an important research topic in general function analysis. Banach gave the first immovable point theorem, Mann introduced the Mann iteration method to study the approximation problem of immovable points of non-expansive mappings [9–11], and Ishikawa introduced Ishikawa iteration sequences for the proposed non-expansive tight mappings in the real Hilbert space and proved the corresponding convergence theorem. The convergence of iterated sequences of nonlinear operators is a very active topic that has been widely studied by many scholars at home and abroad [12–15]. The research focuses more on the three aspects of space framework, nonlinear operators and convergence of iterative sequences, which leads to many important results. In recent years, the theory of iterative approximation of nonlinear mappings with immovable points has been rapidly developed [16–19].

In this paper, a class of iterative algorithms for mapping immovable point problems with asymptotically non-expansive expansion is studied from the perspective of immovable point problems. Hybrid iterative sequences about two families of asymptotically non-expansive mappings and hybrid iterative sequences with mean error terms are constructed, and some weak convergence theorems for the convergence of the sequences to a common immovable point are proved under appropriate conditions. Finally, the practical value and the efficiency of the weakly convergent approximation of the hybrid iterated sequence with mean error term are confirmed by its application to solve the zero and equilibrium problems and two arithmetic examples.

2

Weak convergence approximation theorem for asymptotically non-expansive mapping iterations

2.1

Preparatory knowledge

In this paper let E be a real Banach space and K be a non-empty closed convex subset of E. E* be a dual space of E, and J : E → $2^{E^{'}}$ be a regular dual map defined by the following equation: 1 $J (x) = {f \in E^{*} : 〈 x, f 〉 = | | x | |^{2} = | | f | |^{2}}, \forall x \in E .$

If E* is strictly convex, then J is single-valued, denoted j(x) ∈ J(x), and when J is single-valued also denoted j

Definition 1

1)

2)

3)

Call T: K → K an asymptotically non-expansive map if there exists a sequence {k_n} ⊂ [1, ∞), $\lim_{n} \to \infty k_{n} = 1$ such that: 4 $| | T^{n} x - T^{n} y | | \leq k_{n} | | x - y | |, \forall n \geq 0, \forall x, y \in K .$

It is easy to know that a compression mapping is a non-expansive mapping, a non-expansive mapping is an asymptotically non-expansive mapping, and the converse does not hold. In this paper denote by Π_K the set of all compression mappings on K, i.e., Π_K = f : K → K is a compression mapping, and by F(T) the set of immovable points of T, i.e., F(T) = {x ∈ K : Tx = x}.

Definition 2

Let K be a nonempty closed convex subset of a Banach space E, and say that the cluster of operators from K to itself ℑ = {T(t), t ≥0} is a semigroup of nonlinear operators on K, if 1)

∀x ∈ K, T(0)x = x.

2)

T(s + t)x = T(s)T(t) x, ∀s, t > 0, ∀x ∈ K.

3)

For every x ∈ K, the mapping T(·)x is a continuous mapping from R₊ to K.

Definition 3

Let ℑ = {T(t), t ≥ 0} be a semigroup of nonlinear operators on K.

1)

Call ℑ = {T(t), t ≥ 0} non-expansive if: 5 $| | T (t) x - T (t) y | | \leq | | x - y | |, \forall x, y \in K, t \geq 0;$

2)

Call ℑ = {T(t), t ≥ 0} asymptotically non-expansive if there exist functions k :[0, ∞) → [0, ∞), $\lim_{t \to \infty} k (t) \leq 1$

3)

Call ℑ = {T(t), t ≥ 0} asymptotically non-expansive if ∀x ∈ K, there exist functions r (t, x) : [0, ∞) → [0, ∞), $\lim_{t \to \infty} r (t, x) = 0$ such that: 7 $| | T (t) x - T (t) y | | \leq | | x - y | | + r (t, x), \forall x, y \in K .$

It is easy to see that a non-expansive semigroup is an asymptotically non-expansive semigroup, an asymptotically non-expansive semigroup is an asymptotically non-expansive semigroup, and the converse does not hold. Denote by F(ℑ) the set of common immovable points of ℑ, i.e. $F (ℑ) = {x \in K : T (t) x = x, t \geq 0} = \underset{t \geq 0}{\cap} F i x (T (t))$ .

Definition 4

1)

Call E a consistent Gateaux differentiable manifold if ∀y ∈ B₁, the limit $\lim_{t \to 0} \frac{| | x + t y | | - | | x | |}{t}$ exists consistently with respect to x ∈ B₁, where B₁ = {x ∈ E :|| x ||= 1} denotes the unit sphere of the Banach space E.

2)

Let K be a bounded nonempty closed convex subset of the Banach space E, and for each x ∈ K, denote r (x, K) = sup{|| x – y ||: y ∈ K}, call r (K) = inf{r (x, K): x ∈ K} the Chebyshev radius of K, and call $N (E) = \inf {\frac{d (K)}{r (K)} : K is Bounded closed convex subset of E, and d (K) > 0}$ the coefficients of the regular structure of E, where d(K) = sup{|| x – y ||: x, y ∈ K} if N(E) >1, then E is said to have a consistent regular structure.

Banach spaces with consistent regular structure are self-inverse; consistently convex and consistently smooth Banach spaces have consistent regular structure.

Let LIM be the Banach limit, i.e., LIM ∈ (l^∞)*,|| LIM ||= 1, and ∀{a_n} ∈ l^∞ have both: 8 $\underset{n}{L I M} a_{n} = L I M a_{n + 1}, \underset{n \to \infty}{\lim \inf} a_{n} \leq \underset{n}{L I M} a_{n} \leq \underset{n \to \infty}{\lim \sup} a_{n} .$

Applying the viscosity technique to implicit midpoint rules for non-expansive mappings, the following semi-implicit algorithm is investigated to build iterative sequences: 9 $x_{n + 1} = α_{n} f (x_{n}) + (1 - α_{n}) T (\frac{x_{n} + x_{n + 1}}{2}), n \geq 0.$

Where {α_n} ⊂ [0,1], f are the compression maps in H. It is shown that the sequence {x_n} obtained from the above equation converges strongly to x* ∈ Fix(T), which is also a solution of the variational inequality.

On this basis, the generalised viscous implicit norm is studied and the following iteration format is given: 10 $x_{n + 1} = α_{n} f (x_{n}) + (1 - α_{n}) T (t_{n} x_{n} + (1 - t_{n}) x_{n + 1}), n \geq 0.$

Where {α_n} ⊂ [0,1],{t_n} ⊂ [0,1]. They proved that the iterated column {x_n} converges strongly to the immovable point x^* of T and solved the variational inequality.

Definition 5

Let C be a nonempty convex subset of E, let S₁, S₂ : C → C be two asymptotically non-expansive self-maps, and T₁, T₂ : C → E be three asymptotically non-expansive nonself-maps. The iteration sequence {x_n} ⊂ C is defined as follows: 11 ${\begin{array}{l} x_{1} \in C \\ x_{n + 1} = P ((1 - α_{n}) S_{1}^{n} x_{n} + α_{n} T_{1} {(P T_{1})}^{n - 1} y_{n} + u_{n}^{(1)}) \\ y_{n} = P ((1 - β_{n}) S_{2}^{n} x_{n} + β_{n} T_{2} {(P T_{2})}^{n - 1} z_{n} + u_{n}^{(2)}) \end{array}$

Where {α_n}, {β_n} is the two series in (0, 1) and ${u_{n}^{(i)}} (i = 1, 2, 3)$ is the bounded sequence in C. If ∀n ≥ 1, in (11), has $u_{n}^{(l)} = u_{n}^{(2)} = u_{n}^{(3)} = 0$ , then the iterated series (12) reduces to the following iterated series: 12 ${\begin{array}{l} x_{1} \in C \\ x_{n + 1} = P ((1 - α_{n}) S_{1}^{n} x_{n} + α_{n} T_{1} {(P T_{1})}^{n - 1} y_{n}) \\ y_{n} = P ((1 - β_{n}) S_{2}^{n} x_{n} + β_{n} T_{2} {(P T_{2})}^{n - 1} z_{n}) \end{array}$

Where {α_n},{β_n} is a series of two numbers in (0, 1).

2.2

Weak convergence of hybrid iterative sequences

2.2.1

Some citations

In order to prove the main result of this paper, it is necessary to introduce the following lemma:

Lemma 1

Let C be a non-empty closed convex subset of a uniformly convex Banach space E and T : C → C be an asymptotically non-expansive map. Then I – T is semiclosed at zero. That is, any sequence: ${x_{n}} \subset C, x_{n} \overset{}{\to} q \in C and sequence {(I - T) x_{n}} converges strongly to 0, then (I - T) q = 0.$

Lemma 2

Let E be a uniformly convex Banach space and C a nonempty closed convex subset of E, then there exists a continuous strictly increasing convex function Φ: [0, ∞) → [0, ∞), Φ(0) = 0 such that for all L-Lipschitz mappings T : C → C there is: 13 $\begin{matrix} \begin{matrix} ‖ t T x + (1 - t) T y - T (t x + (1 - t) y) ‖ \leq L Φ^{- 1} (‖ x - y ‖ - \frac{1}{L} ‖ T x - T y ‖), \\ \forall x, y \in C, t \in [0, 1] \end{matrix} \end{matrix}$

Lemma 3

Let E be a consistently convex Banach space whose dual space E* has the Kadec-Klee property with bounded sequences {x_n} ⊂ E, p, q ∈ w_w(x_n) (where w_w(x_n) denotes the set of weak limits of all subsequences of {x_n}). If ∀t ∈ [0,1], $\lim_{n \to \infty} | | t x_{n} + (1 - t) p - q | |$ exists, then p = q.

Lemma 4

Let C be a nonempty closed convex subset of a uniformly convex Banach space E and T : C → E be an asymptotically non-expansive nonself-map. Then I – T is semiclosed at zero. That is, any sequence {x_n} ⊂ C, $x_{n} \overset{weak}{\to} q \in C$ and the sequence {(I – T)x_n} converges strongly to 0. Then (I – T)q = 0.

Lemma 5

Let E be a real consistent convex Banach space and K be a non-empty closed convex subset of E. Let S₁, S₂; K → K be two asymptotically non-expansive self-maps and have real series ${k_{n}^{(1)}}$ , ${k_{n}^{(2)}} \subset [1, \infty)$ , respectively, and T₁,T₂;K → E be two asymptotically non-expansive nonself-maps and have real series ${l_{n}^{(1)}}$ , ${l_{n}^{(2)}} \subset [1, \infty)$ , respectively, such that $\sum_{n} = 1^{\infty} (k_{n}^{(i)} - 1) < \infty$ , $\sum_{n = 1}^{\infty} (l_{n}^{(i)} - 1) < \infty (i = 1, 2)$ and $F = \cap_{i = 1}^{2} (F (S_{i}) \cap F (T_{i})) \neq ϕ$ . Let {x_n} be the sequence defined by (11), where {α_n}, {β_n} is a sequence of three numbers in (0, 1) and $\sum_{n} = 1^{\infty} | | u_{n}^{(i)} | | < \infty (i = 1, 2)$ then: 1)

For any q ∈ F, $\lim_{n} \to \infty | | x_{n} - q | |$ exists.

2)

$\lim_{n \to \infty} d (x_{n}, F)$ exists.

Lemma 6

Let E be a real consistent convex Banach space and K be a nonempty closed convex subset of E. Let S₁, S₂ ; K → K be two asymptotically non-expansive self-maps and have respectively real columns ${k_{n}^{(0)}}$ , ${k_{n}^{(2)}} \subset [1, \infty)$ , and T₁,T₂ : K → E be two asymptotically non-expansive nonself-maps and have respectively real columns ${l_{n}^{(1)}}$ , ${l_{n}^{(2)}} \subset [1, \infty)$ such that $\sum_{n} = 1^{\infty} (k_{n}^{(i)} - 1) < \infty$ and $\sum_{n} = 1^{\infty} (l_{n}^{(i)} - 1) < \infty (i = 1, 2)$ , and $F = \cap_{i = 1}^{2} (F (S_{i}) \cap F (T_{i})) \neq ϕ$ . Let {x_n} be the sequence defined by (11) satisfying the following conditions: 1)

There exists ε ∈ (0,1) such that {α_n}, {β_n} and {γ_n} are three series in [ε, 1 – ε] and: $\sum_{n = 1}^{\infty} | | u_{n}^{(i)} | | < \infty (i = 1, 2, 3)$ .

2)

For ∀x, y ∈ K and i = 1,2,3, there is ||x – T_iy ||≤|| S_ix – T_iy||, then for i = 1,2,3, there is $\lim_{n} \to \infty | | x_{n} - S_{i} x_{n} | | = \lim_{n \to \infty} | | x_{n} - T_{i} x_{n} | | = 0$ .

2.2.2

Findings

This chapter studies the strong and weak convergence of the iterated sequence {x_n} defined by equations (11) and (12) in a real consistent convex Banach space E. We denote by $F = \cap_{i = 1}^{3} (F (S_{i}) \cap F (T_{i}))$ the set of common immovable points of S₁, S₂, S₃, T₁, T₂, T₃ and by $d (z, A) = \inf_{ve} | | z - x | |$ the distance of a point z in E from a set A.

Theorem 1

Let E be a real consistent convex Banach space and satisfy the Opial condition, and K be a nonempty closed convex subset of E. Let S₁, S₂, S₃: K → K be three asymptotically non-expansive self-maps with real columns ${k_{n}^{(1)}}$ , ${k_{n}^{(2)}}$ , ${k_{n}^{(3)}} \subset [1, \infty)$ and T₁,T₂,T₃; K → E be three asymptotically non-expansive nonself-maps with real columns ${l_{n}^{(1)}}$ , ${l_{n}^{(2)}}$ , ${l_{n}^{(3)}} \subset [1, \infty)$ such that $\sum_{n} = 1^{\infty} (k_{n}^{(i)} - 1) < \infty$ and $\sum_{n} = 1^{\infty} (l_{n}^{(i)} - 1) < \infty (i = 1, 2, 3)$ , and $F = \cap_{i = 1}^{3} (F (S_{i}) \cap F (T_{i})) \neq ϕ$ . Let {x_n} be the sequence defined by (11) satisfying the following conditions: 1)

There exists ε ∈ (0,1) such that {α_n}, {β_n} and {γ_n} are the three series in [ε, 1 – ε] and $\sum_{n = 1}^{\infty} | | u_{n}^{(i)} | | < \infty (\begin{matrix} i = 1, 2, 3 \end{matrix})$ .

2)

∀x, y ∈ K and i = 1,2,3 with ||x – T_iy||≤||S_ix – T_iy||.

Then the sequence {x_n} converges weakly to a common immovable point of S₁, S₂, S₃, T₁, T₂ and T₃.

Proof By Lemma 5 we know that {x_n} is a bounded sequence, and since E is a consistently convex Banach space, then there exists a subsequence {x_nk}_k≥1 of {x_n}_n≥1 such that {x_nk}_k≥1 converges weakly to some point p ∈ K. By Lemma 6, for i = 1,2,3, there is $\lim_{k} \to \infty | | x_{n_{k}} - S_{i} x_{n_{k}} | | = \lim_{k \to \infty} | | x_{n_{k}} - T_{i} x_{n_{k}} | | = 0$ . Thus we have p ∈ F by Lemma 1 and Lemmas 2–4.

In the following we prove that {x_n} weakly converges to p. Suppose that there exists a subcolumn {x_mj}_j ≥ 1 of {x_n}_n ≥ 1 such that {x_mj}_j≥1 weakly converges to q ∈ K and p ≠ q. By Lemma 1, we have the following two limits exist: 14 $\lim_{n \to \infty} | | x_{n} - p | | = a, \lim_{n \to \infty} | | x_{n} - q | | = c$

Thus, by the Opial condition, we have: 15 $\begin{array}{l} a = \lim_{k \to \infty} \sup | | x_{n_{k}} - p | | < \lim_{k \to \infty} \sup | | x_{n_{k}} - q | | \\ = c = \lim_{j \to \infty} \sup | | x_{m_{j}} - q | | < \lim_{j \to \infty} \sup | | x_{m_{j}} - p | | = a \end{array}$

Clearly a contradiction. So p = q. This proves that {x_n} weakly converges to p. Proof.

Lemma 7

Let E be a real consistent convex Banach space and K be a nonempty closed convex subset of E. Let S₁, S₂, S₃; K → K be three asymptotically non-expansive self-maps and have real series ${k_{n}^{(0)}}$ , ${k_{n}^{(2)}}$ , ${k_{n}^{(3)}} \subset [1, \infty)$ , T₁,T₂,T₃; K → E be three asymptotically non-expansive nonself-maps and have real series ${l_{n}^{(1)}}$ , ${l_{n}^{(2)}}$ , ${l_{n}^{(3)}} \subset [1, \infty)$ , such that $\sum_{n} = 1^{\infty} (k_{n}^{(i)} - 1) < \infty$ , $\sum_{n = 1}^{\infty} (l_{n}^{(i)} - 1) < \infty (i = 1, 2, 3)$ and $F = \cap_{i = 1}^{3} (F (S_{i}) \cap F (T_{i})) \neq ϕ$ . Let {x_n} be the sequence defined by (12), where {α_n}, {β_n} and {γ_n} are the three series in (0, 1), then the limit is for all p, q ∈ F and everything t ∈ [0,1]: 16 $\lim_{n \to \infty} | | t x_{n} + (1 - t) p - q | |$

Existence.

Prove that ∀t ∀ [0,1], such that a_n (t) =||tx_n + (1 – t) p – q|| then $\lim_{n} \to \infty a_{n} (0) = | | p - q | |$ , and $\lim_{n} \to \infty a_{n} (1) = \lim_{n \to \infty} | | x_{n} - q | |$ exists by Lemma 5. The following proof also holds for t ∈ (0,1) Lemma 7.

Define the mapping H_n : K → K as follows:

∀x ∈ K, Lemma: 17 $\begin{array}{l} H_{n} x = P [(1 - α_{n}) S_{1}^{n} x + α_{n} T_{1} {(P T_{1})}^{n - 1} \\ P ((1 - β_{n}) S_{2}^{n} x + β_{n} T_{2} {(P T_{2})}^{n - 1} P ((1 - γ_{n}) S_{3}^{n} x + γ_{n} T_{3} {(P T_{3})}^{n - 1} x))] \end{array}$

ECS for all x, y ∈ K: 18 $| | H_{n} x - H_{n} y | | \leq h_{n}^{3} | | x - y | |$

One of the $h_{n} = * m a x {k_{n}^{(1)}, k_{n}^{(2)}, k_{n}^{(3)}, l_{n}^{(1)}, l_{n}^{(2)}, l_{n}^{(3)}}$ Order: 19 $S_{n, m} = H_{n + m - 1} H_{n + m - 2} ... H_{n}, m \geq 1$ 20 $b_{n, m} = | | S_{n, m} (t x_{n} + (1 - t) p) - (t S_{n, m} x_{n} + (1 - t) S_{n, m} p) | |$

It is not difficult to prove that for all x, y ∈ K, ||S_n,mx – S_n,my||≤ L_n||x – y||.

Here $L_{n} = \prod_{j = n}^{n + m - 1} h_{j}^{3}$ , and S_n,mx_n = x_n+m, and for all p ∈ F there is S_n,m p = p, i.e.: 21 $\begin{array}{l} a_{m + n} (t) & = ‖ t x_{n + m} + (1 - t) p - q ‖ \leq b_{n, m} + ‖ S_{n + m} (t x_{n} + (1 - t) p) - q ‖ \\ \leq b_{n, m} + L_{n} a_{n} (t) \leq b_{n, m} + (\prod_{j = n}^{\infty} h_{j}^{3}) a_{n} (t) \end{array}$

It follows from Lemma 2: 22 $\begin{array}{l} b_{n, m} \leq L_{n} ϕ^{- 1} (‖ x_{n} - p ‖ - L_{n}^{- 1} ‖ x_{n + m} - p ‖) \\ \leq (\prod_{j = n}^{\infty} h_{j}^{3}) ϕ^{- 1} (‖ x_{n} - p ‖ - {(\prod_{j = n}^{\infty} h_{j}^{3})}^{- 1} ‖ x_{n + m} - p ‖) \end{array}$

So for fixing n, there is: 23 $\lim_{m \to \infty} \sup a_{m} (t) \leq (\prod_{j = n}^{\infty} h_{j}^{3}) a_{n} (t) + (\prod_{j = n}^{\infty} h_{j}^{3}) ϕ^{- 1} (‖ x_{n} - p ‖ - {(\prod_{j = n}^{\infty} h_{j}^{3})}^{- 1} \lim_{m \to \infty} ‖ x_{m} - p ‖)$

By lim $\lim_{n \to \infty} \prod_{n \to \infty} h_{j}^{3} = 1$ and Lemma 5, and then let n → ∞ take the lower limit there: 24 $\lim_{n \to \infty} \sup a_{n} (t) \leq * l i m i n f_{n \to \infty} a_{n} (t) + ϕ^{- 1} (0) = * l i m_{n \to \infty} \inf a_{n} (t)$

So for all t ∈ (0,1), $\lim_{n \to \infty} | | t x_{n} + (1 - t) p - q | |$ exist. Proof.

Lemma 8

Let E be a real consistent convex Banach space with Frechet differentiable parametrization, and K be a nonempty closed convex subset of E. Let S₁, S₂, S₃: K → K be three asymptotically non-expansive self-maps with real series ${k_{n}^{(1)}}$ , ${k_{n}^{(2)}}$ , ${k_{n}^{(3)}} \subset [1, \infty)$ and T₁, T₂, T₃ : K → E be three asymptotically non-expansive nonself-maps with real series ${l_{n}^{(1)}}$ , ${l_{n}^{(2)}}$ , ${l_{n}^{(3)}} \subset [1, \infty)$ such that:

$\sum_{n} = 1^{\infty} (k_{n}^{(i)} - 1) < \infty$ , $\sum_{n} = 1^{\infty} (l_{n}^{(i)} - 1) < \infty (i = 1, 2, 3)$ , and $F = \cap_{i = 1}^{3} (F (S_{i}) \cap F (T_{i})) \neq ϕ$ . Set {x_n} is the sequence defined by (12), where {α_n}, {β_n} and {γ_n} are three sequences in (0, 1), then for all P, q ∈ F: 25 $\lim_{n \to \infty} 〈 x_{n}, j (p - q) 〉$ exists. Further, if W_w({x_n}) denotes the set of weak limits of all subsequences of {x_n}, then for all p,q ∈ F and x*,y* ∈ W_w({x_n}), 〈x* – y*, ∈ W_w(p – q)〉 = 0.

The proof is the same as above.

Theorem 2

Let E be a real consistent convex Banach space with Frechet differentiable parametrization, and K be a nonempty closed convex subset of E. Let S₁, S₂, S₃; K → K be three asymptotically non-expansive self-maps and have real columns ${k_{n}^{(1)}}$ , ${k_{n}^{(2)}}$ , ${k_{n}^{(3)}} \subset [1, \infty)$ , T₁, T₂, T₃; K → E be three asymptotically non-expansive nonself-maps and have real columns ${l_{n}^{(1)}}$ , ${l_{n}^{(2)}}$ , ${l_{n}^{(3)}} \subset [1, \infty)$ such that $\sum_{n} = 1^{\infty} (k_{n}^{(i)} - 1) < \infty$ and $\sum_{n} = 1^{\infty} (l_{n}^{(i)} - 1) < \infty (i = 1, 2, 3)$ , and $F = \cap_{i = 1}^{3} (F (S_{i}) \cap F (T_{i})) \neq ϕ$ Let {x_n} be the sequence defined by (12) satisfying the following conditions: 1)

There exists ε ∈ (0,1) such that {α_n}, {β_n} and {γ_n} are the three number series in [ε, 1 – ε].

2)

∀x, y ∈ K and i = 1,2,3 such that ||x – T_iy||≤|| S_ix – T_iy|| exists.

Then {x_n} converges weakly to a common immovable point of S₁, S₂, S₃, T₁, T₂ and T₃.

Proof Imitating Theorem 1, we can show that there exists a subcolumn {x_nk}_k≥1 of {x_n}_n≥1 which converges weakly to p ∈ F Suppose that there exists a subcolumn {x_mj}_j≥1 of {x_n}_n ≥ 1 such that {x_mj}_j ≥ 1 converges weakly to q ∈ K. Then, using the same method as given above, we can also prove q ∈ F. so p, q ∈ F ∩ W_w({x_n}). by Lemma 8: 26 $| | p - q | |^{2} = 〈 p - q, j (p - q) 〉 = 0$

Thus, p = q, which proves that {x_n} weakly converges to p provably.

Theorem 3

Let E be a real consistent convex Banach space and the dual space E* of E have the Kadec-Klee property, and K be a nonempty closed convex subset of E. Let S₁, S₂, S₃: K → K be three asymptotically non-expansive self-maps with real series ${k_{n}^{(1)}}$ , ${k_{n}^{(2)}}$ , ${k_{n}^{(3)}} \subset [1, \infty)$ , T₁, T₂, T₃ : K → E be three asymptotically non-expansive nonself-maps with real series ${l_{n}^{(1)}}$ , ${l_{n}^{(2)}}$ , ${l_{n}^{(3)}} \subset [1, \infty)$ such that $\sum_{n} = 1^{\infty} (k_{n}^{(i)} - 1) < \infty$ and $\sum_{n = 1}^{\infty} (l_{n}^{(i)} - 1) < \infty (i = 1, 2, 3)$ , and $F = \cap_{i = 1}^{3} (F (S_{i}) \cap F (T_{i})) \neq ϕ$ . Let {x_n} be the sequence defined by (12) satisfying the following conditions: 1)

There exists ε ∈ (0,1) such that {α_n}, {β_n} and {γ_n} are the three number series in [ε, 1 – ε].

2)

∀x, y ∈ K and i = 1,2,3 such that ||x – T_iy||≤||S_ix – T_iy|| exists.

Then {x_n} converges weakly to a common immovable point of S₁, S₂, S₃, T₁, T₂ and T₃.

Proof Using the method given in the proof of Theorem 1, we can show that there exists a subcolumn {x_{n_k}}_k≥1 of {x_n}_n≥1 which converges weakly to p ∈ F Suppose that there exists a subcolumn {x_{m_j}}_j≥1 of {x_n}_n ≥ 1 such that {x_{m_j}}_j≥1 converges weakly to q ∈ K Then, using the same method given above, we can also prove q ∈ F. By Lemma 7, for all t ∈ [0,1]: 27 $\lim_{n \to \infty} | | t x_{n} + (1 - t) p - q | |$ exists, and noting again that p, q ∈ W_w ({x_n}), by Lemma 3, we have p = q, which proves that {x_n} weakly converges to p provably.

2.3

Weak convergence of hybrid iterative sequences with mean error terms

Weak convergence of hybrid iterated sequences has been studied in the previous section, and in this section we study hybrid iterated sequences with an average error term.

Define the iterative sequence {x_n} as follows: 28 ${\begin{matrix} x_{1} \in K \\ x_{n + 1} = P (a_{n} S_{1}^{n} x_{n} + b_{n} T_{1} {(P T_{1})}^{n - 1} y_{n} + c_{n} u_{n}) \\ y_{n} = P (a_{n}^{'} S_{2}^{n} x_{n} + b_{n}^{'} T_{2} {(P T_{2})}^{n - 1} x_{n} + c_{n}^{'} u_{n}^{'}), n \geq 1 \end{matrix}$

Where $0 \leq a_{n}, a_{n}^{'}, b_{n}, b_{n}^{'}, c_{n}, c_{n}^{'} < 1$ and satisfies $a_{n} + b_{n} + c_{n} = 1 = a_{n}^{'} + b_{n}^{'} + c_{n}^{'}, n \geq 1$ and ${u_{n}^{'}}$ are bounded columns in E.

Let E be a real Banach space, E* be the dual space of E, and J: E → 2^E* be a regular dual map defined as follows: 29 $J (x) = (f \in E^{*} : 〈 x, f 〉 = | | x | | | | f | |, | | | f | = | | x | |), \forall x \in E,$ where 〈·,·〉 is a pairing between E and E*. The single-valued regular pairwise mapping is denoted j.

A Banach space E is said to have Frechet differentiable parametrization if for everything x ∈ U = {x ∈ E :||x||= 1}: 30 $\lim_{t \to 0} \frac{| | x + t y | | - | | x | |}{t}$ both exist and hold consistently for y ∈ U.

The Banach space E is said to have the Kadec-Klee property if for every sequence {x_n} ⊂ E, $x_{n} \overset{59}{\to} x$ and ||x_n||→||x|| in E, when x_n → x. (Here $\overset{weak}{\to}$ and → denote weak and strong convergence, respectively.)

The Banach space E is said to satisfy Opial’s condition if for any sequence {x_n} ⊂ E in E, when $x_{n} \overset{weak}{\to} x$ : 31 $\lim_{n \to \infty} \sup | | x_{n} - x | | < \lim_{n \to \infty} \sup | | x_{n} - y | |, \forall y \in E and y \neq x .$

2.3.1

Some citations

Lemma 9

Let E be a uniformly convex Banach space and K a convex subset of E. Then there exists a strictly increasing continuous convex function γ :[0, ∞) → [0, ∞), γ(0) = 0 such that for every mapping S : K → K with Lipschitz constant L > 0: 32 $\begin{matrix} ‖ α S x + (1 - α) S y - S [α S x + (1 - α) S y] ‖ \leq L γ^{- 1} (‖ x - y ‖ - \frac{1}{L} ‖ S x - S y ‖) \\ \forall x, y \in K and 0 < α < 1. \end{matrix}$

Lemma 10

Let E be a uniformly convex Banach space and the dual space E* of E have the Kadec-Klee property. If {x_n} is a bounded sequence and f₁, f₂ ∈ W_ω({x_n}) (where W_ω({x_n}) denotes the set of weak convergence points of all subsequences of the sequence {x_n}) such that ∀ α ∈ [0,1]: 33 $\lim_{n \to \infty} | | α x_{n} + (1 - α) f_{1} - f_{2} | |$ both exist, then f₁ = f₂.

Lemma 11

Let E be a consistent convex Banach space, K be a nonempty closed convex subset of E, and T : K → E be an asymptotically non-expansive nonself-map with real series {k_n} ⊂ [1, ∞) and k_n → 1, then I – T is semiclosed at zero. That is, if $x_{n} \overset{weak}{\to} x$ and x_n – Tx_n → 0, then x ∈ F(T) (F(T) is an immovable set of points of T ).

2.3.2

Findings

Lemma 12

Under the conditions of Lemma 3, then: 34 $\forall q_{1}, q_{2} \in F = F (S_{1}) \cap F (S_{2}) \cap F (T_{1}) \cap F (T_{2}) and t \in [0, 1]$ 35 $\lim_{n \to \infty} | | t x_{n} + (1 - t) q_{1} - q_{2} | |$ exists, where {x_n} is defined by (28).

Prove that if we set a_n(t) =||x_n + (1 – t)q₁ – q₂|| then $\lim_{n} \to \infty a_{n} (0) = | | q_{1} - q_{2} | |$ , and $\lim_{n \to \infty} a_{n} (1) = | | x_{n} - q_{2} | |$ exists by Lemma 3. Therefore, in the following we only need to show that Lemma 12 holds for all t ∈ (0,1).

Define the mapping H_n : K → K as follows: 36 $H_{n} x = P [a_{n} S_{1}^{n} x + b_{n} T_{1} {(P T_{1})}^{n - 1} P (a_{n}^{'} S_{2}^{n} x + b_{n} T_{2} (P T_{2}) x + c_{n}^{'} u_{n}^{'}) + c_{n} u_{n}], \forall x \in K .$

Easy Certificates ∀x, y ∈ K, there: 37 $| | H_{n} x - H_{n} y | | \leq h_{n}^{2} | | x - y | |$

Where $h_{n} = \max {k_{n}, k_{n}^{'}, l_{n}, l_{n}^{'}}$ , note h_n = 1 + v_n, is $\lim_{n \to \infty} \prod_{j = n}^{\infty} h_{j}^{2} = 1$ from 1 $1 \leq \prod_{j} = n^{\infty} h_{j}^{2} \leq e^{2 \sum_{j = m}^{\infty} ν_{j}}$ and $\sum_{n} = 1^{\infty} ν_{n} < \infty$ Set: 38 $S_{n, m} = H_{n + m - 1} H_{n + m - 2} \dots H_{n}, \forall m \geq 1$

By (37) and (38), ∀x, y ∈ K can be obtained: 39 $| | S_{n, m} x - S_{n, m} y | | \leq \prod_{j = n}^{n + m - 1} h_{j}^{2} | | x - y | |$ and S_n,m,x_n = x_n+m Below we show that ∀q ∈ F, m ≥ 1, $| | S_{n, m} q - q | | \overset{accord}{\to} 0, (n \to \infty)$ .

In fact, for any q ∈ F, it follows that: 40 $\begin{matrix} H_{n} q - q \leq a_{n} S_{1}^{n} q - q + b_{n} T_{1} {(P T_{1})}^{n - 1} P (a_{n}^{'} S_{2}^{n} q + b_{n}^{'} T_{2} (P T_{2}) q + c_{n}^{'} u_{n}^{'}) \\ - q + c_{n} u_{n} - q \leq M (c_{n}^{'} + c_{n}) \end{matrix}$ where $M = \max {\sup_{n \geq 1} h_{n} ‖ u_{n} - q ‖, \sup_{n \geq 1} h_{n} ‖ u_{n}^{'} - q ‖}$ is thus available: 41 $\begin{array}{l} ‖ S_{n, m} q - q ‖ \leq ‖ H_{n + m - 1} \dots H_{n} q - H_{n + m - 1} \dots H_{n + 1} q ‖ + \dots + ‖ H_{n + m - 1} q - q ‖ \\ \leq (\prod_{j = n + 1}^{n + m - 1} h_{j}^{2}) ‖ H_{n} q - q ‖ + (\prod_{j = n + 2}^{n + m - 1} h_{j}^{2}) ‖ H_{n + 1} q - q ‖ + \dots + ‖ H_{n + m - 1} q - q ‖ \\ \leq M \prod_{j = n + 1}^{\infty} h_{j}^{2} \sum_{j = n}^{\infty} (c_{j}^{'} + c_{j}) \end{array}$

Set $δ_{n} = M \prod_{j = n + 1}^{\infty} h_{j}^{2} \sum_{j = n}^{\infty} (c_{j}^{'} + c_{j})$ and by $\sum_{n} = 1^{\infty} (c_{n}^{'} + c_{n}) < \infty$ and $\lim_{n} \to \infty Π_{j = n}^{\infty} h_{j}^{2} = 1$ we know that when n →∞, δ_n → 0. Set: 42 $b_{n, m} = | | t S_{n, m} x_{n} + (1 - t) S_{n, m} q_{1} - S_{n, m} (t x_{n} + (1 - t) q_{1}) | |$

Noted: 43 $\begin{array}{l} a_{n + m} (t) \leq | | S_{n, m} (t x_{n} + (1 - t) q_{1}) - q_{2} | | + b_{n, m} \\ \leq | | S_{n, m} (t x_{n} + (1 - t) q_{1}) - S_{n, m} q_{2} | | + | | S_{n, m} q_{2} - q_{2} | | + b_{n, m} \\ \leq (\prod_{j = n}^{\infty} h_{j}^{2}) a_{n} (t) + | | S_{n, m} q_{2} - q_{2} | | + b_{n, m} \end{array}$

It follows from (42) and Lemma 9: 44 $\begin{array}{l} b_{n, m} \leq (\prod_{j = n + 1}^{n + m - 1} h_{j}^{2}) γ^{- 1} (| | x_{n} - q_{1} | | - {(\prod_{j = n + 1}^{n + m - 1} h_{j}^{2})}^{- 1} | | S_{n, m} x_{n} - S_{n, m} q_{1} | | \\ \leq (\prod_{j = n}^{\infty} h_{j}^{2}) γ^{- 1} (| | x_{n} - q_{1} | | - {(\prod_{j = n}^{\infty} h_{j}^{2})}^{- 1} (| | x_{n + m} - q_{1} | | + | | S_{n, m} q_{1} - q_{1} | |) \end{array}$

By Lemmas 3 and $\lim_{n} \to \infty \prod_{j = n}^{\infty} h_{j}^{2} = 1$ , it follows that $\lim_{n} \to \infty b_{n, m} = 0$ holds consistently for all m.

Thus in equation (44), fixing n when m → ∞ yields: 45 $\begin{matrix} \lim_{m \to \infty} \sup a_{m} (t) \leq (\prod_{j = n}^{\infty} h_{j}^{2}) γ^{- 1} (| | x_{n} - q_{1} | | \\ - {(\prod_{j = n}^{\infty} h_{j}^{2})}^{- 1} (\lim_{m \to \infty} | | x_{m} - q_{1} | | - δ_{n})) + (\prod_{j = n}^{\infty} h_{j}^{2}) a_{n} (t) + δ_{n} \end{matrix}$

Then let n → ∞, can be obtained: 46 $\lim_{n \to \infty} \sup a_{n} (t) \leq γ^{- 1} (0) + \underset{n \to \infty}{\lim \inf} a_{n} (t) = \underset{n \to \infty}{\lim \inf} a_{n} (t)$

That is, ∀t ∈ (0,1), $\lim_{n \to \infty} | | t x_{n} + (1 - t) q_{1} - q_{2} | |$ exists. Proof of Bi.

Lemma 13

Under the conditions of Lemma 3, if E has a Fre’chet differentiable paradigm, then for ∀q₁, q₂ ∈ F = F(S₁) ∩ F(S₂) ∩ F(T₁) ∩ F(S₂), the limit: 47 $\lim_{n \to \infty} 〈 x_{n}, j (q_{1} - q_{2}) 〉$ exists, where {x_n} is defined by (28). Further if W_ω ({x_n} denotes the set of weak convergence points of all subsequences of the sequence {x_n}, then ∀q₁, q₂ ∈ F and x*, y* ∈ W_ω ({x_n}) have 〈x* – y*, j(q₁ – q₂)〉 = 0, and it follows that Lemma 13 holds.

Theorem 4

Under the conditions of Lemma 3, if E has a Fre’ chet differentiable paradigm, then {x_n} is defined by (28) to converge weakly to a common immovable point of S₁, S₂, T₁ and T₂.

Proof Since E is a consistently convex Banach space, by Lemma 3 sequence {x_n} is bounded, so there exists a subsequence {x_{n_k}} ⊂ {x_n} weakly convergent to some q ∈ K by Lemma 4, for i = 1,2 there: 48 $\lim_{k \to \infty} | | x_{n_{k}} - S_{i} x | | = \lim_{k \to \infty} | | x_{n_{k}} - T_{i} x_{n_{k}} | | = 0$

By Lemma 11, q ∈ F = F(S₁) ∩ F(S₂) ∩ F(T₁) ∩ F(T₂).

Now, we show that the sequence {x_n} converges weakly to q. Suppose that there exists another sub-series {x_{m_j}} ⊂ {x_n} that converges weakly to q₁ ∈ K. Imitating the above we know that q ∈ F, and hence q, q₁ ∈ F ∩ W_ω ({x_n}) follows by Lemma 13: 49 $| | q - q_{1} | |^{2} = 〈 q - q_{1}, j (q - q_{1}) 〉 = 0$

Thus, q₁ = q, the sequence {x_n} of proofs converges weakly to q. proofs graduate.

Theorem 5

Under the conditions of Lemma 4, if the dual space E* of E has the Kadec-Klee property, then {x_n} defined by (28) converges weakly to a common immovable point of S₁, S₂, T₁ and T₂.

Proof The proof of Theorem 1 shows that there exists a subsequence {x_{n_k}} ⊂ {x_n} weakly convergent to some q ∈ F = F(S₁) ∩ F(S₂) ∩ F(T₁) ∩ F(T₂).

Theorem 6

Under the conditions of Lemma 4, if E satisfies Opial’s condition, then {x_n} defined by (28) converges weakly to a common immovable point of S₁, S₂, T₁ and T₂. 50 $\lim_{k \to \infty} | | x_{n_{k}} - S_{i} x_{n_{k}} | | = \lim_{k \to \infty} | | x_{n_{k}} - T_{i} x_{n_{k}} | | = 0$

Again, by Lemma 11, I – S_i, I – T_i, (i = 1,2) are both closed at zero and a half, then q ∈ F = F (S₁) ∩ F(S₂) ∩ F(T₁) ∩ F(T₂).

Now, we prove that the sequence {x_n} weakly converges to q. Suppose that there exists another subsequence {x_{n_j}} ⊂ {x_n} such that {x_{m_j}} weakly converges to a point $\bar{q} \in K$ and $\bar{q} \neq q$ then the proof follows $\bar{q} \in F$ in imitation of q. By Lemma 3, the following two limits are known: 51 $\lim_{n \to \infty} | | x_{n} - q | | = d, \lim_{n \to \infty} | | x_{n} - \bar{q} | | = \bar{d}$ are present. Thus by Opial’s condition: 52 $\begin{array}{l} d & = \lim_{k \to \infty} \sup | | x_{n_{k}} - q | | < \lim_{k \to \infty} \sup | | x_{n_{k}} - \bar{q} | | \\ = \lim \sup | | x_{m_{j}} - \bar{q} | | < \lim \sup | | x_{m_{j}} - q | | = d \end{array}$

3

Applications and numerical simulations

3.1

Main applications

3.1.1

Application to the zero problem

Zero Problem: Find x ∈ E such that: 53 $0 \in A x$ where A ⊂ E × E is the augmentation operator. The pre-solution operator J_r = (I + rA)⁻¹ is a non-expansive map and A⁻¹(0) = F (J_r), A⁻¹(0) = {x ∈ E : 0 ∈ Ax} is the set of zeros of A and F(J_r) is the set of immovable points of J_r. Thus the main result can be applied by ordering T = J_r.

Corollary 1

E is a non-empty closed convex subset of a uniformly smooth Banach space X and has a weakly continuous dual mapping. A is the m– augmentation operator in E such that A⁻¹(0) ≠ ∅ f : E → E is a compression mapping. Then for a given x₀ ∈ E, the sequence {x_n} is generated in the following way: 54 $x_{n + 1} = α_{n} x_{n} + β_{n} f (x_{n}) + γ_{n} J_{r} (t_{n} x_{n} + (1 - t_{n}) x_{n + 1})$ where {α_n}, {β_n}, {γ_n}, {t_n} ⊂ (0,1) satisfies the following conditions: 1)

α_n + β_n + γ_n = 1, $\lim_{n \to \infty} β_{n} = 0$ , $Σ_{n = 0}^{\infty} β_{n} = \infty$

2)

$0 < * l i m_{n \to \infty} \inf α_{n} \leq \lim \sup_{n \to \infty} α_{n} < 1$ , $\lim_{n \to \infty} | α_{n + 1} - α_{n} | = 0$ , $\lim_{n \to \infty} | β_{n + 1} - β_{n} | = 0$

3)

0 < t_n ≤ t_n+1 < 1

Then {x_n} converges weakly to x* ∈ A⁻¹ (0) and is also a solution of the following variational inequality: 55 $〈 (I - f) p, j (p - y) 〉 \leq 0, \forall y \in A^{- 1} (0)$

G(x, y) ≥ 0, holds for all y ∈ B.

3.1.2

Application to equilibrium problems

B is a non-empty closed convex subset of Banach space H. Equilibrium problem: Find x ∈ B such that G : B × B → R of them is a bifunction satisfying the following conditions:

(H1) for all x ∈ B, G(x, x) = 0

(H2) for all x, y ∈ B, G(x, y) + G(y, x) ≤ 0

(H3) for every x, y, z, ∈ B, $\lim_{t} \to \infty G (t z + (1 - t) x, y) \leq G (x, y)$

(H4) is convex and weakly lower semicontinuous for all x ∈ B, G(x, y).

Assume that G satisfies condition H(1) – H(4). For r > 0 and x ∈ H, define T_r : H → B as $T_{r} = {u \in B : G (u, y) + \frac{1}{r} 〈 y - u, u - x 〉 \geq 0, \forall y \in B}$ and the bounded set of the equilibrium problem is noted as EP. The single-valued mapping T_r is strongly non-expansive and EP(G) = F(T_r), where EP(G) is a closed convex set.

3.2

Numerical simulation

In this section, two arithmetic examples are given, the first one to illustrate the convergence of the proposed algorithm and the second one will be compared with the convergence rate of several iterative algorithms.

3.2.1

Example 1

Let the inner product 〈·,·〉 : R³ × R³ → R³ R be 〈x,y〉 = x₁y₁ + x₂y₂ + x₃y₃. Let $T^{n} x = (1 + \frac{1}{3 n}) x$ and $f (x) = \frac{1}{4} x$ where x = (x₁, x₂, x₃) ∈ R³ is taken to bean $α_{n} = \frac{1}{3} + \frac{1}{n}$ , $β_{n} = \frac{1}{n}$ , $γ_{n} = 2 (\frac{1}{3} - \frac{1}{n})$ , $t_{n} = 1 - \frac{1}{3 n}$ for all n ∈ N. then it is easy to see that $k_{n} = 1 + \frac{1}{3 n}$ , $o = \frac{1}{3}$ , $o = \frac{1}{4}$ satisfies condition (i) – (iv) Theorems 4 to 6. then it follows: 56 $x_{n + 1} = \frac{108 n^{3} - 81 n^{2} - 8 n + 24}{108 n^{3} - 24 n^{2} + 64 n + 24} x_{n}$

Using the algorithm in the theorem, starting from x₁ = (1,2,3), the numerical results of Figures 1 and 2 are obtained.

3.2.2

Example 2

Set $ω_{n}^{i} = \frac{3}{4}$ , $μ = \frac{3}{2}$ the other conditions of this calculus satisfy the rest of the assumptions in Calculation 1. Calculation obtained $T_{i}^{n} = \frac{1}{4} I + \frac{3}{4} T_{i}$ To make the numerical results more obvious, consider the case of N = 3.

Firstly, the sequence {x_n} produced by Algorithm (28) can be expressed as: 57 $\begin{array}{l} x_{n + 1} & = α_{n} γ V (x_{n}) + (I - μ α_{n} F) T_{N}^{n} T_{N - 1}^{n} \dots T_{1}^{n} (t_{n} x_{n} + (1 - t_{n}) x_{n + 1}) \\ = \frac{1}{32 n} x_{n} + (1 - \frac{3}{4 n}) (\frac{10}{96} x_{n} + \frac{10}{48} x_{n + 1}) = \frac{40 n - 18}{304 n + 60} x_{n} \end{array}$

When n = 1, T₁x = Tx = x, the sequence {x_n} generated by Eq. (12) can be expressed as: 58 $\begin{array}{l} x_{n + 1} & = P_{C} [α_{n} f (x_{n}) + (I - μ α_{n} F) T (t_{n} x_{n} + (1 - t_{n}) x_{n + 1})] \\ = \frac{1}{8 n} x_{n} + (1 - \frac{3}{4 n}) (\frac{1}{3} x_{n} + \frac{2}{3} x_{n + 1}) \\ = \frac{8 n - 3}{8 n + 12} x_{n} \end{array}$

The sequence {x_n} generated by algorithm (10) can be represented as: 59 $\begin{array}{l} x_{n + 1} & = α_{n} f (x_{n}) + (1 - α_{n}) T (t_{n} x_{n} + (1 - t_{n}) x_{n + 1}) \\ = \frac{1}{8 n} x_{n} + (1 - \frac{1}{2 n}) (\frac{1}{3} x_{n} + \frac{2}{3} x_{n + 1}) \\ = \frac{8 n - 1}{8 n + 8} x_{n} \end{array}$

The sequence {x_n} produced by algorithm (9) can be represented as: 60 $\begin{array}{l} x_{n + 1} & = α_{n} f (x_{n}) + (1 - α_{n}) T (\frac{x_{n} + x_{n + 1}}{2}) \\ = \frac{1}{8 n} x_{n} + (1 - \frac{1}{2 n}) (\frac{1}{2} x_{n} + \frac{1}{2} x_{n + 1}) \\ = \frac{4 n - 1}{4 n + 2} x_{n} \end{array}$

The numerical comparison of convergence between the algorithms is shown in Table 1 and Figure 3, from which it can be seen that the iterative algorithm (28) converges faster. From Table 1, it can be seen that Algorithm (28) not only converges faster but also converges to zero early compared to the iterations of Algorithms (9), (10) and (12), which do not converge to zero until the 100th term.

Table 1.

Numerical comparison of convergence between algorithms

n	Algorithm 9	Algorithm 10	Algorithm 12	Algorithm 28
1	100	100	100	100
2	39.9973	34.9969	19.9997	4.8332
3	27.9969	21.8754	9.2945	0.4462
4	22.0014	15.7271	5.4182	0.043
5	18.3379	12.1843	3.5687	0.0064
6	15.8334	9.9048	2.5476	5E-4
7	14.0063	8.3063	1.8985	1E-4
8	12.6092	7.14	1.4791	0
9	11.49	6.2552	1.1944	0
…	…	…	…	…
96	7.3599	3.2601	0.4099	0
97	7.0265	3.0455	0.3651	0
98	6.7271	2.8536	0.2187	0
99	6.4581	2.678	0.1097	0
100	6.2034	2.5275	0	0

4

Conclusion

The main work of this paper is based on the existing results of many scholars, which have been improved, extended, and supplemented.

This paper focuses on the convergence of asymptotically non-expansive mappings under two different iterative sequences, including hybrid iterative sequences and hybrid iterative sequences with an average error term, and some weakly convergent approximation theorems for this iterative sequence are proved in Banach spaces.

In addition, the main results are applied to the zero and equilibrium problems, and a numerical example is given for convergence analysis. Compared with the rest of the iterative sequences, the hybrid asymptotically non-expansive mapping iterative sequences with averaging error terms not only have faster convergence but also converge to zero in advance, and they can be better applied in practice.

The current study is focused on the Banach space. In the research problem combined in this paper, the convergence of iterative algorithms in two or even more spaces, such as the study of iterative algorithms, provides new ideas for subsequent research. Iterative algorithms have practical significance in real-life applications, such as image reconstruction and signal processing. The optimization of these real-life practical problems and in-depth application-oriented research also require further consideration.

Język:: Angielski

Częstotliwość wydawania:: 1 razy w roku
Dziedziny czasopisma:: Nauki biologiczne, Nauki biologiczne, inne, Matematyka, Matematyka stosowana, Matematyka ogólna, Fizyka, Fizyka, inne

Kanał RSS czasopisma

A study of asymptotically non-expansive mapping iteration and weakly convergent approximation methods based on Banach spaces

Zhirong Guo

Data publikacji: 03 lut 2025

Otrzymano: 25 wrz 2024

Przyjęty: 04 sty 2025

DOI: https://doi.org/10.2478/amns-2025-0040

Słowa kluczoweBanach spaces, Asymptotically non-expansive mappings, Consistent Gateaux differentiable paradigms, Mixed iterated sequences, Weakly convergent approximations

© 2025 Zhirong Guo, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Słowa kluczowe
Banach spaces, Asymptotically non-expansive mappings, Consistent Gateaux differentiable paradigms, Mixed iterated sequences, Weakly convergent approximations