Multiple-set split feasibility problems for total asymptotically strict pseudocontractions mappings

Abstract

The purpose of this article is to propose and investigate an algorithm for solving the multiple-set split feasibility problems for total asymptotically strict pseu-docontractions mappings in infinite-dimensional Hilbert spaces. The results presented in this article improve and extend some recent results of A. Moudafi, H. K. Xu, Y. Censor, A. Segal, T. Elfving, N. Kopf, T. Bortfeld, X. A. Motova, Q. Yang, A. Gibali, S. Reich and others.

2000 AMS Subject Classification: 47J05; 47H09; 49J25.

1. Introduction and preliminaries

Throughout this article, we always assume that H₁, H₂ are real Hilbert spaces, "→", "⇀" are denoted by strong and weak convergence, respectively, and F(T) is the fixed point set of a mapping T.

Let G be a nonempty closed convex subset of H₁ and T : G → G a mapping.

T is said to be a contraction if there exists a constant α ∈ (0,1) such that

$$∥T_x-T_y∥\le \alpha ∥x-y∥,\forall x,y\in G.$$

(1.1)

Banach contraction principle guarantees that every contractive mapping defined on complete metric spaces has a unique fixed point.

T is said to be a weak contraction if

$$∥T_x-T_y∥\le ∥x-y∥-\psi \left(∥x-y∥\right),\forall x,y\in G.$$

(1.2)

where ψ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that ψ is positive on (0, ∞), ψ(0) = 0, and lim_t→∞ψ(t) = ∞. We remark that the class of weak contractions was introduced by Alber and Guerre-Delabriere [1]. In 2001, Rhoades [2] showed that every weak contraction defined on complete metric spaces has a unique fixed point.

T is said to be nonexpansive if

$$∥T_x-T_y∥\le ∥x-y∥,\forall x,y\in G.$$

(1.3)

T is said to be asymptotically nonexpansive if there exists a sequence {k _n } ⊂ [1, ∞) with k _n → 1 as n → ∞ such that

$$∥T^{n}x-T^{n}y∥\le {\kappa }_n∥x-y∥,\forall n\ge 1,x,y\in G.$$

(1.4)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [3] as a generalization of the class of nonexpansive mappings. They proved that if G is a nonempty closed convex bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping on G, then T has a fixed point.

T is said to be total asymptotically nonexpansive if

$$∥T^{n}x-T^{n}y∥\le ∥x-y∥+{\mu }_n\varphi \left(∥x-y∥\right)+{\xi }_n,\forall n\ge 1,x,y\in G.$$

(1.5)

where ϕ : [0, ∞) → [0, ∞) is a continuous and strictly increasing function with ϕ(0) = 0, and {μ _n } and {ξ _n } are nonnegative real sequences such that μ _n → 0 and ξ _n → 0 as n → ∞. The class of mapping was introduced by Alber et al. [4]. From the definition, we see that the class of total asymptotically nonexpansive mappings includes the class of asymptotically nonexpansive mappings as special cases, see [5, 6] for more details.

T is said to be strictly pseudocontractive if there exists a constant κ ∈ [0, 1) such that

$${∥Tx-Ty∥}^2\le {∥x-y∥}^2+\kappa {∥\left(I-T\right)x-\left(I-T\right)y∥}^2,\forall x,y\in G.$$

(1.6)

The class of strict pseudocontractions was introduced by Browder and Petryshyn [7] in a real Hilbert space. In 2007, Marino and Xu [8] obtained a weak convergence theorem for the class of strictly pseudocontractive mappings, see [8] for more details.

T is said to be an asymptotically strict pseudocontraction if there exist a constant κ ∈ [0, 1) and a sequence {k _n } ⊂ [1, ∞) with k _n → 1 as n → ∞ such that

$${∥T^{n}x-T^{n}y∥}^2\le {\kappa }_n{∥x-y∥}^2+\kappa {∥\left(I-T^n\right)x-\left(I-T^n\right)y∥}^2,{\forall }_n\ge 1,x,y\in G.$$

(1.7)

The class of asymptotically strict pseudocontractions was introduced by Qihou [9] in 1996. Kim and Xu [10] proved that the class of asymptotically strict pseudocontractions is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [10] for more details.

In this article, we introduce the following mapping.

Definition 1.1 Let H be a real Hilbert space, and G be a nonempty closed convex subset of H. A mapping T : G → G is said to be (κ, {μ _n }, {ξ _n }, ϕ)-total asymptotically strict pseudocontractive, if there exists a constant κ ∈ [0, 1) and sequences {μ _n } ⊂ [0, ∞), {ξ _n } ⊂ [0, ∞) with μ _n → 0 and ξ _n → 0 as n → ∞, and a continuous and strictly increasing function ϕ : [0, ∞) → [0, ∞) with ϕ(0) = 0 such that

$${∥T^{n}x-T^{n}y∥}^2\le {∥x-y∥}^2+\kappa {∥x-y-\left(T^{n}x-T^{n}y\right)∥}^2+{\mu }_n\varphi \left(∥x-y∥\right)+{\xi }_n,{\forall }_n\ge 1,x,y\in G.$$

(1.8)

Now, we give an example of total asymptotically strict pseudocontractive mapping.

Let C be a unit ball in a real Hilbert space l²and let T : C → C be a mapping defined by

$$T:\left(x_1,x_2,...,\right)\to \left(0,x_1^2,a_2{x}_2,a_3{x}_3,...\right),$$

where {a _i } is a sequence in (0, 1) such that ${\prod }_{i=2}^{\infty }{a}_i=\frac{1}{2}$.

It is proven in Goebal and Kirk [3] that

• (i) $∥Tx-Ty∥\le 2∥x-y∥,\forall x,y\in C$;

• (ii) $∥T^{n}x-T^{n}y∥\le 2{\prod }_{j=2}^n{a}_j∥x-y∥,\forall x,y\in C,\forall n\ge 2.$.

Denote by ${\kappa }_1^{\frac{1}{2}}=2,{\kappa }_n^{\frac{1}{2}}=2{\prod }_{j=2}^n{a}_j,n\ge 2$, then

$$\underset{n\to \infty }{lim}{k}_n=\underset{n\to \infty }{lim}{\left(2{\prod }_{j=2}^n{a}_j\right)}^2=1.$$

Letting ${\mu }_n=\left({\kappa }_n-1\right),\forall n\ge 1,\varphi \left(t\right)=t^2,\forall t\ge 0,\kappa =0$ and {ξ _n } be a nonnegative real sequence with ξ _n → 0, then $\forall x,y\in C,n\ge 1$, we have

$${∥T^{n}x-T^{n}y∥}^2\le {∥x-y∥}^2+{\mu }_n\varphi \left(∥x-y∥\right)\left.+\kappa \right|{∥x-y-\left(T^{n}x-T^{n}y\right)∥}^2+{\xi }_n.$$

Remark 1.2 If ϕ(λ) = λ² and ξ _n = 0, then total asymptotically strict pseudocontractive mapping is asymptotically strict pseudocontraction mapping.

It is easy to see the following proposition holds.

Proposition 1.3 Let T : G → G be a (κ, {μ _n }, {ξ _n }, ϕ)-total asymptotically strict pseudocontractive mapping. If $F\left(T\right)\ne \varnothing$, then for each q ∈ F(T) and for each x ∈ G, the following inequalities hold and are equivalent:

$$⟨x-q,T^{n}x-q⟩\le \frac{\kappa +1}{2k}{∥x-q∥}^2+\frac{\kappa -1}{2k}{∥T^{n}x-q∥}^2+\frac{{\mu }_n}{2\kappa }\varphi \left(∥x-q∥\right)+\frac{{\xi }_n}{2\kappa };$$

(1.9)

$$⟨x-T^{n}x,x-q⟩\ge \frac{1-\kappa }{2}{∥T^{n}x-x∥}^2-\frac{{\mu }_n}{2}\varphi \left(∥x-q∥\right)-\frac{{\xi }_n}{2};$$

(1.10)

$$⟨x-T^{n}x,q-T^{n}x⟩\le \frac{\kappa +1}{2}{∥T^{n}x-x∥}^2+\frac{{\mu }_n}{2}\varphi \left(∥x-q∥\right)+\frac{{\xi }_n}{2}.$$

(1.11)

The split feasibility problem (SFP) in finite-dimensional spaces was first introduced by Censor and Elfving [11] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [12]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [13–15].

The SFP in an infinite-dimensional Hilbert space can be found in [12, 14, 16–18].

The purpose of this article is to introduce and study the following multiple-set SFP(MSSFP) for total asymptotically strict pseudocontraction in the framework of infinite-dimensional Hilbert spaces:

$$find{x}^{*}\in CsuchthatA{x}^{*}\in Q,$$

(1.12)

where A : H₁ → H₂ is a bounded linear operator, S _i : H₁ → H₁ and T _i : H₂ → H₂, i = 1, 2, ..., N are mappings, $C:{\bigcap }_{i=1}^{N}F\left(S_i\right)$ and $Q:{\bigcap }_{i=1}^{N}F\left(T_i\right)$. In the sequel, we use Γ to denote the set of solutions of (MSSFP)--(1.12), i.e.,

$$\Gamma =\left\{x\in C,Ax\in Q\right\}.$$

(1.13)

To prove our main results, we first recall some definitions, notations, and conclusions.

Let E be a Banach space. A mapping T : E → E is said to be demi-closed at origin, if for any sequence {x _n } ⊂ E with x _n ⇀ x* and ||(I - T)x _n || → 0, then x* = Tx*.

A Banach space E is said to have the Opial property, if for any sequence {x _n } with x _n ⇀ x*, then

$$\underset{n\to \infty }{liminf}∥x_n-x^{*}∥<\underset{n\to \infty }{liminf}∥x_n-y∥,\forall y\in E\mathsf{\text{with}}y\ne x^{*}.$$

Remark 1.4 It is well known that each Hilbert space possesses the Opial property.

Definition 1.5 Let H bea real Hilbert space.

• (1) A mapping T : H → H is said to be uniformly L-Lipschitzian, if there exists a constant L > 0, such that

  $$∥T^{n}x-T^{n}y∥\le L∥x-y∥,\forall x,y\in H\mathsf{\text{and}}n\ge 1.$$

• (2) A mapping T : H → H is said to be semi-compact, if for any bounded sequence {x _n } ⊂ H with lim_n→∞||x _n - Tx _n || = 0, then there exists a subsequence $\left\{x_{n_i}\right\}\subset \left\{x_n\right\}$ such that $x_{n_i}$ converges strongly to some point x* ∈ H.

Lemma 1.6 [10] Let H be a real Hilbert space. If {x _n } is a sequence in H weakly convergent to z, then

$$\underset{n\to \infty }{limsup}{∥x_n-y∥}^2=\underset{n\to \infty }{limsup}{∥x_n-z∥}^2+{∥z-y∥}^2\forall y\in H.$$

Proposition 1.7 Assume that G is a closed convex subset of a real Hilbert space H and let T : G → G be a (κ, {μ _n }, {ξ _n }, ϕ)-total asymptotically strict pseudocon-traction mapping and uniformly L-Lipschitzian. Then the demiclosedness principle holds for I - T in the sense that if {x _n } is a sequence in G such that x _n ⇀ x*, and lim sup_m→∞lim sup_n→∞||x _n - T^mx _n || = 0 then (I - T)x* = 0. In particular, x _n ⇀ x*, and (I - T)x _n → 0 ⇒ (I - T)x* = 0, i.e., T is demiclosed at origin.

Proof Since {x _n } is bounded, we can define a function f on H by

$$f\left(x\right)=\underset{n\to \infty }{limsup}{∥x_n-x∥}^2,\forall x\in H.$$

By Lemma 1.6, the weak convergence x _n ⇀ x* implies that

$$f\left(x\right)=f\left(x^{*}\right)+{∥x-x^{*}∥}^2,\forall x\in H.$$

In particular, for each m ≥ 1,

$$f\left(T^m{x}^{*}\right)=f\left(x^{*}\right)+{∥T^m{x}^{*}-x^{*}∥}^2.$$

(1.14)

On the other hand, since T is a (κ, {μ _n }, {ξ _n })-total asymptotically strict pseudo-contraction mapping, by (1.8), we get

$$\begin{array}{ll}\hfill f\left(T^m{x}^{*}\right)& =\underset{n\to \infty }{limsup}{∥x_n-T^m{x}^{*}∥}^2\\ =\underset{n\to \infty }{limsup}{∥x_n-T^m{x}_n+T^m{x}_n-T^m{x}^{*}∥}^2\\ =\underset{n\to \infty }{limsup}\left({∥x_n-T^m{x}_n∥}^2+2⟨x_n-T^m{x}_n,T^m{x}_n-T^m{x}^{*}⟩+{∥T^m{x}_n-T^m{x}^{*}∥}^2\right)\\ \le \underset{n\to \infty }{limsup}∥x_n-T^m{x}_n∥\left(∥x_n-T^m{x}_n∥+2L∥x_n-x^{*}∥\right)\\ +\underset{n\to \infty }{limsup}\left({∥x_n-x^{*}∥}^2+k{∥x_n-T^m{x}_n-\left(x^{*}-T^m{x}^{*}\right)∥}^2+{\mu }_m\varphi \left(∥x_n-x^{*}∥\right)+{\xi }_m\right)\end{array}$$

Taking lim sup_m→∞on both sides and observing the facts that lim_m→∞μ _m = 0, lim_m→∞ξ _m = 0 and lim sup_m→∞lim sup_n→∞||x _n - T^mx _n || = 0, we derive that

$$\underset{m\to \infty }{limsup}f\left(T^m{x}^{*}\right)\le \underset{n\to \infty }{limsup}{∥x_n-x^{*}∥}^2+k\underset{m\to \infty }{limsup}{∥x^{*}-T^m{x}^{*}∥}^2$$

(1.15)

Since lim sup_m→∞f(T^mx*) = f(x*)+lim sup_m→∞||T^mx* - x*||², and f(x*) = lim sup_n→∞ ||x _n - x*||², it follows from (1.15) that lim sup_m→∞||x* - T^mx*||² = 0. That is, T^mx* → x*; hence Tx* = x*.

Lemma 1.8 [19] Let {a _n }, {b _n } and {δ _n } be sequences of nonnegative real numbers satisfying

$$a_{n+1}\le \left(1+{\delta }_n\right)a_n+b_n,\forall n\ge 1.$$

If ${\sum }_{i=1}^{\infty }{\delta }_n<\infty$ and ${\sum }_{i=1}^{\infty }{b}_n<\infty$, then the limit lim_n→∞a _n exists.

2. Multiple-set split feasibility problem

For solving the multiple-set split feasibility problem (1.12), let us assume that the following conditions are satisfied:

• 1. H₁ and H₂ are two real Hilbert spaces, A : H₁ → H₂ is a bounded linear operator;

• 2. Let G, $\tilde{G}$ be a nonempty closed convex subset of H₁ and H₂ respectively, S _i : G → G, i = 1, 2,...,N, is a uniformly L _i -Lipschitzian and (β _i , {μ _i,n }, {ξ _i,n }, ϕ _i )-total asymptotically strictly pseudocontractive mapping and $T_i:\tilde{G}\to \tilde{G},i=1,2,...,N$, is a uniformly ${\tilde{L}}_i$-Lipschitzian and $\left(k_i\left\{{\tilde{\mu }}_{i,n}\right\},\left\{{\tilde{\xi }}_{i,n}\right\},{\tilde{\varphi }}_i\right)$-total asymptotically strictly pseudocontractive mapping which satisfy the following conditions:

• (i) $C:{\bigcap }_{i=1}^{N}F\left(S_i\right)\ne \varnothing ,Q:={\bigcap }_{i=1}^{N}F\left(T_i\right)\ne \varnothing$;

• (ii) $\beta =\underset{1\le i\le N}{max}{\beta }_i<1,\kappa =\underset{1\le i\le N}{max}{\kappa }_i<1;$;

• (iii) $L:=\underset{1\le i\le N}{max}{L}_i<\infty ,\tilde{L}:=\underset{1\le i\le N}{max}{\tilde{L}}_i<\infty$;

• (iv) ${\mu }_n=\underset{1\le i\le N}{max}\left\{{\mu }_{i,n},{\tilde{\mu }}_{i,n}\right\},{\xi }_n=\underset{1\le i\le N}{max}\left\{{\xi }_{i,n},{\tilde{\xi }}_{i,n}\right\}$ and ${\sum }_{i=1}^{\infty }{\mu }_n<\infty ,{\sum }_{i=1}^{\infty }{\xi }_n<\infty .$.

• (v) $\varphi =\underset{1\le i\le N}{max}\left\{{\varphi }_i,\left\{{\tilde{\varphi }}_i\right.\right\}$

We are now in a position to give the following result:

Theorem 2.1 Let $H_1,H_2,G,\tilde{G},A,\left\{S_i\right\},\left\{T_i\right\},C,Q,\beta ,\kappa ,L,\tilde{L},\left\{{\mu }_n\right\},\left\{{\xi }_n\right\}$ and ϕ be the same as above. In addition, there exist positive constants M and M* such that ϕ(λ) ≤ M*λ² for all λ ≥ M. Let {x _n } be the sequence generated by:

$$\left\{\begin{array}{c}\hfill x_1\in G\mathsf{\text{chosen}}\mathsf{\text{arbitrarily}}\hfill \\ \hfill x_{n+1}=\left(1-{\alpha }_n\right)u_n+{\alpha }_n{S}_n^n\left(u_n\right),\hfill \\ \hfill u_n=x_n+\gamma A^{*}\left(T_n^n-I\right)A{x}_n,\forall n\ge 1,\hfill \end{array}\right.$$

(2.1)

where $S_n^n=S_{n\left(modN\right)}^n,T_n^n=T_{n\left(modN\right)}^n\forall n\ge 1,\left\{{\alpha }_n\right\}$ is a sequence in [0, 1] and γ > 0 is a constant satisfying the following conditions:

• (vi) ${\alpha }_n\in \left(\delta ,1-\beta \right),\forall n\ge 1$ and $\gamma \in \left(0,\frac{1-\kappa }{{∥A∥}^2}\right)$, where δ ∈ (0, 1 - β) is a positive constant.

• (I) If $\Gamma \ne \varnothing$ (where Γ is the set of solutions to (MSSFP)--(1.12)), then {x _n } converges weakly to a point x* ∈ Γ.

• (II) In addition, if there exists a positive integer j such that S _j is semi-compact, then {x _n } and {u _n } both converge strongly to x* ∈ Γ.

The proof of conclusion (I)

(1) First we prove that for each p ∈ Γ, the following limits exist

$$\underset{n\to \infty }{lim}∥x_n-p∥\mathsf{\text{and}}\underset{n\to \infty }{lim}∥u_n-p∥.$$

(2.2)

In fact, since ϕ is an increasing function, it results that ϕ(λ) ≤ ϕ(M), if λ ≤ M and ϕ(λ) ≤ M*λ², if λ ≥ M. In either case, we can obtain that

$$\varphi \left(\lambda \right)\le \varphi \left(M\right)+M^{*}{\lambda }^2,\forall \lambda \ge 0.$$

(2.3)

Since p ∈ Γ, then $p\in C:={\bigcap }_{i=1}^{N}F\left(S_i\right)$ and $Ap\in Q:={\bigcap }_{i=1}^{N}F\left(T_i\right)$. From (2.1) and (1.10) we have

$$\begin{array}{c}{∥x_{n+1}-p∥}^2={∥u_n-p-{\alpha }_n\left(u_n-S_n^n{u}_n\right)∥}^2\\ ={∥u_n-p∥}^2-2{\alpha }_n⟨u_n-p,u_n-S_n^n{u}_n⟩+{\alpha }_n^2{∥u_n-S_n^n{u}_n∥}^2\\ \le {∥u_n-p∥}^2-{\alpha }_n\left(1-\beta \right){∥u_n-S_n^n{u}_n∥}^2\\ +{\alpha }_n{\mu }_n\varphi \left(∥u_n-p∥\right)+{\alpha }_n{\xi }_n+{\alpha }_n^2{∥u_n-S_n^n{u}_n∥}^2\left(by\left(1.10\right)\right)\\ \le {∥u_n-p∥}^2-{\alpha }_n\left(1-\beta -{\alpha }_n\right){∥u_n-S_n^n{u}_n∥}^2\\ +{\alpha }_n{\mu }_n\left(\varphi \left(M\right)+M^{*}{\left(∥u_n-p∥\right)}^2\right)+{\alpha }_n{\xi }_n\\ =\left(1+{\alpha }_n{\mu }_n{M}^{*}\right){∥u_n-p∥}^2-{\alpha }_n\left(1-\beta -{\alpha }_n\right){∥u_n-S_n^n{u}_n∥}^2\\ +{\alpha }_n{\mu }_n\varphi \left(M\right)+{\alpha }_n{\xi }_n\end{array}$$

(2.4)

On the other hand, since

$$\begin{array}{ll}\hfill {∥u_n-p∥}^2& ={∥x_n-p+\gamma A^{*}\left(T_n^n-I\right)A{x}_n∥}^2\\ ={∥x_n-p∥}^2+{\gamma }^2{∥A^{*}\left(T_n^n-I\right)A{x}_n∥}^2+2\gamma ⟨x_n-p,A^{*}\left(T_n^n-I\right)A{x}_n⟩,\end{array}$$

(2.5)

and

$$\begin{array}{ll}\hfill {∥A^{*}\left(T_n^n-I\right)A{x}_n∥}^2& =⟨A^{*}\left(T_n^n-I\right)A{x}_n,A^{*}\left(T_n^n-I\right)A{x}_n⟩\\ =⟨A{A}^{*}\left(T_n^n-I\right)A{x}_n,\left(T_n^n-I\right)A{x}_n⟩\\ \le {∥A∥}^2{∥T_n^{n}A{x}_n-A{x}_n∥}^2,\end{array}$$

(2.6)

It follows from (1.11) we have

$$\begin{array}{c}⟨x_n-p,A^{*}\left(T_n^n-I\right)A{x}_n⟩=⟨A{x}_n-Ap,\left(T_n^n-I\right)A{x}_n⟩\\ =⟨\left(A{x}_n-Ap\right)+\left(T_n^n-I\right)A{x}_n-\left(T_n^n-I\right)A{x}_n,\left(T_n^n-I\right)A{x}_n⟩\\ =⟨T_n^{n}A{x}_n-Ap,T_n^{n}A{x}_n-A{x}_n⟩-{∥\left(T_n^n-I\right)A{x}_n∥}^2.\\ \le \frac{1+\kappa }{2}{∥\left(T_n^n-I\right)A{x}_n∥}^2+\frac{{\mu }_n}{2}\varphi \left(∥A{x}_n-Ap∥\right)+\frac{\xi n}{2}-{∥\left(T_n^n-I\right)A{x}_n∥}^2.\\ \le \frac{\kappa -1}{2}{∥\left(T_n^n-I\right)A{x}_n∥}^2+\frac{{\mu }_n}{2}\left(\varphi \left(M\right)+M^{*}{∥A{x}_n-Ap∥}^2\right)+\frac{{\xi }_n}{2}.\\ \le \frac{\kappa -1}{2}{∥\left(T_n^n-I\right)A{x}_n∥}^2+\frac{{\mu }_n}{2}{M}^{*}{∥A{x}_n-Ap∥}^2+\frac{{\mu }_n}{2}\varphi \left(M\right)+\frac{{\xi }_n}{2}.\end{array}$$

(2.7)

Substituting (2.6) and (2.7) into (2.5) and simplifying it, we have

$$\begin{array}{c}{∥u_n-p∥}^2\le {∥x_n-p∥}^2+{\gamma }^2{∥A∥}^2{∥T_n^{n}A{x}_n-A{x}_n∥}^2+\gamma \left(\kappa -1\right){∥\left(T_n^n-I\right)A{x}_n∥}^2\\ +\gamma {\mu }_n{M}^{*}{∥A{x}_n-Ap∥}^2+\gamma {\mu }_n\varphi \left(M\right)+\gamma {\xi }_n\\ ={∥x_n-p∥}^2-\gamma \left(1-\kappa -\gamma {∥A∥}^2\right){∥T_n^{n}A{x}_n-A{x}_n∥}^2\\ +\gamma {\mu }_n{M}^{*}{∥A{x}_n-Ap∥}^2+\gamma {\mu }_n\varphi \left(M\right)+\gamma {\xi }_n\\ \le \left(1+\gamma {\mu }_n{M}^{*}{∥A∥}^2\right){∥x_n-p∥}^2-\gamma \left(1-\kappa -\gamma {∥A∥}^2\right){∥T_n^{n}A{x}_n-A{x}_n∥}^2\\ +\gamma {\mu }_n\varphi \left(M\right)+\gamma {\xi }_n\end{array}$$

(2.8)

Substituting (2.8) into (2.4) and after simplifying we have

$$\begin{array}{c}{∥x_{n+1}-p∥}^2\le \left(1+{\alpha }_n{\mu }_n{M}^{*}\right)\left\{\left(1+\gamma {\mu }_n{M}^{*}{∥A∥}^2\right)\right.{∥x_n-p∥}^2\\ \left.-\gamma \left(1-\kappa -\gamma {∥A∥}^2\right){∥T_n^{n}A{x}_n-A{x}_n∥}^2+\gamma {\mu }_n\varphi \left(M\right)+\gamma {\xi }_n\right\}\\ -{\alpha }_n\left(1-\beta -{\alpha }_n\right){∥u_n-S_n^n{u}_n∥}^2+{\alpha }_n{\mu }_n\varphi \left(M\right)+{\alpha }_n{\xi }_n\\ \le \left(1+{\delta }_n\right){∥x_n-p∥}^2-\gamma \left(1-\kappa -\gamma {∥A∥}^2\right){∥T_n^{n}A{x}_n-A{x}_n∥}^2\\ -{\alpha }_n\left(1-\beta -{\alpha }_n\right){∥u_n-S_n^n{u}_n∥}^2+b_n\end{array}$$

(2.9)

where

$$\begin{array}{c}{\delta }_n={\alpha }_n{\mu }_n{M}^{*}+\gamma {\mu }_n{M}^{*}{∥A∥}^2+\gamma {∥A∥}^2{\alpha }_n{\mu }_n^2{\left(M^{*}\right)}^2\\ b_n=\left(\left(1+{\alpha }_n{\mu }_n{M}^{*}\right)\gamma +{\alpha }_n\right){\mu }_n\varphi \left(M\right)+\left(\left(1+{\alpha }_n{\mu }_n{M}^{*}\right)\gamma +{\alpha }_n\right){\xi }_n\end{array}$$

By condition (vi) we have

$${∥x_{n+1}-p∥}^2\le \left(1+{\delta }_n\right){∥x_n-p∥}^2+b_n$$

By condition (iv), ${\sum }_{n=1}^{\infty }{\delta }_n<\infty$ and ${\sum }_{n=1}^{\infty }{b}_n<\infty$. Hence, from Lemma 1.8 we know that the following limit exists

$$\underset{n\to \infty }{lim}∥x_n-p∥.$$

(2.10)

Consequently, from (2.9) and (2.10) we have that

$$\begin{array}{c}\gamma \left(1-\kappa -\gamma {∥A∥}^2\right){∥\left(T_n^n-I\right)A{x}_n∥}^2+{\alpha }_n\left(1-\beta -{\alpha }_n\right){∥u_n-S_n^n{u}_n∥}^2\\ \le {∥x_n-p∥}^2-{∥x_{n+1}-p∥}^2+{\delta }_n{∥x_n-p∥}^2+b_n\to 0\left(asn\to \infty \right).\end{array}$$

This together with the condition (vi) implies that

$$\underset{n\to \infty }{lim}∥u_n-S_n^n{u}_n∥=0;$$

(2.11)

and

$$\underset{n\to \infty }{lim}∥\left(T_n^n-I\right)A{x}_n∥=0.$$

(2.12)

It follows from (2.5), (2.10) and (2.12) that the limit ||u _n - p|| exists.

The conclusion (1) is proved.

(2) Next we prove that

$$\underset{n\to \infty }{lim}∥x_{n+1}-x_n∥=0\mathsf{\text{and}}\underset{n\to \infty }{lim}∥u_{n+1}-u_n∥=0.$$

(2.13)

In fact, it follows from (2.1) that

$$\begin{array}{ll}\hfill ∥x_{n+1}-x_n∥& =∥\left(1-{\alpha }_n\right)u_n+{\alpha }_n{S}_n^n\left(u_n\right)-x_n∥\\ =∥\left(1-{\alpha }_n\right)\left(x_n+\gamma A^{*}\left(T_n^n-I\right)A{x}_n\right)+{\alpha }_n{S}_n^n\left(u_n\right)-x_n∥\\ =∥\left(1-{\alpha }_n\right)\gamma A^{*}\left(T_n^n-I\right)A{x}_n+{\alpha }_n\left(S_n^n\left(u_n\right)-x_n\right)∥\\ =∥\left(1-{\alpha }_n\right)\gamma A^{*}\left(T_n^n-I\right)A{x}_n+{\alpha }_n\left(S_n^n\left(u_n\right)-u_n\right)+{\alpha }_n\left(u_n-x_n\right)∥\\ =∥\left(1-{\alpha }_n\right)\gamma A^{*}\left(T_n^n-I\right)A{x}_n+{\alpha }_n\left(S_n^n\left(u_n\right)-u_n\right)+{\alpha }_n\gamma A^{*}\left(T_n^n-I\right)A{x}_n∥\\ =∥\gamma A^{*}\left(T_n^n-I\right)A{x}_n+{\alpha }_n\left(S_n^n\left(u_n\right)-u_n\right)∥.\end{array}$$

In view of (2.11) and (2.12) we have that

$$\underset{n\to \infty }{lim}∥x_{n+1}-x_n∥=0.$$

(2.14)

Similarly, it follows from (2.1), (2.12), and (2.14) that

$$\begin{array}{c}∥u_{n+1}-u_n∥=∥x_{n+1}+\gamma A^{*}\left(T_{n+1}^{n+1}-I\right)A{x}_{n+1}-\left(x_n+\gamma A^{*}\left(T_n^n-I\right)A{x}_n\right)∥\\ \le ∥x_{n+1}-x_n∥+\gamma ∥A^{*}\left(T_{n+1}^{n+1}-I\right)A{x}_{n+1}∥\\ +\gamma ∥A^{*}\left(T_n^n-I\right)A{x}_n∥\to 0\left(\mathsf{\text{as}}n\to \infty \right).\end{array}$$

(2.15)

The conclusion (2.13) is proved.

(3) Next we prove that for each j = 1, 2,..., N - 1,

$$∥u_{iN+j}-S_j{u}_{iN+j}∥\to 0\mathsf{\text{and}}∥A{x}_{iN+j}-T_{j}A{x}_{iN+j}∥\to 0\left(asi\to \infty \right),$$

(2.16)

In fact, from (2.11) we have

$${\eta }_{iN+j}:=∥u_{iN+j}-S_j^{iN+j}{u}_{iN+j}∥\to 0\left(asi\to \infty \right).$$

(2.17)

Since S _j is uniformly L _j -Lipschitzian continuous, it follows from (2.13) and (2.17) that

$$\begin{array}{c}∥u_{iN+j}-S_j{u}_{iN+j}∥=∥u_{iN+j}-S_j^{iN+j}{u}_{iN+j}∥+∥S_j^{iN+j}{u}_{iN+j}-S_j{u}_{iN+j}∥\\ \le {\eta }_{iN+j}+L_j∥S_j^{iN+j-1}{u}_{iN+j}-u_{iN+j}∥\\ \le {\eta }_{iN+j}+L_j\left\{∥S_j^{iN+j-1}{u}_{iN+j}-S_j^{iN+j-1}{u}_{iN+j-1}∥\right.\\ \left.+L_j∥S_j^{iN+j-1}{u}_{iN+j-1}-u_{iN+j}∥\right\}\\ \le {\eta }_{iN+j}+L_j^2∥u_{iN+j}-u_{iN+j-1}∥\\ +L_j∥S_j^{iN+j-1}{u}_{iN+j-1}-u_{iN+j-1}+u_{iN+j-1}-u_{iN+j}∥\\ \le {\eta }_{iN+j}+L_j\left(1+L_j\right)∥u_{iN+j}-u_{iN+j-1}∥+L_j{\eta }_{iN+j-1}\to 0\left(asi\to \infty \right)\end{array}$$

Similarly, for each j = 1, 2,..., N - 1, from (2.13) we have

$${\varsigma }_{iN+j}:=∥A{x}_{iN+j}-T_j^{iN+j}A{x}_{iN+j}∥\to 0\left(\mathsf{\text{as}}i\to \infty \right).$$

(2.18)

Since T _j is uniformly ${\tilde{L}}_j$-Lipschitzian continuous, by the same way as above, from (2.13) and (2.18), we can also prove that

$$∥A{x}_{iN+j}-T_{j}A{x}_{iN+j}∥\to 0\left(asi\to \infty \right).$$

(2.19)

(4) Finally we prove that x _n ⇀ x* and u _n ⇀ x* which is a solution of (MSSFP)--(1.12).

Since {u _n } is bounded. There exists a subsequence $\left\{u_{n_i}\right\}\subset \left\{u_n\right\}$ such that $u_{n_i}\rightharpoonup x^{*}$ (some point in H₁). Hence, for any positive integer j = 1, 2,..., N, there exists a subsequence {n _i (j)} ⊂ {n _i } with n _i (j)(modN) = j such that $u_{n_i\left(j\right)}\rightharpoonup x^{*}$. Again from (2.16) we have

$$∥u_{n_i\left(j\right)}-S_j{u}_{n_i\left(j\right)}∥\to 0\left(\mathsf{\text{as}}{n}_i\left(j\right)\to \infty \right)$$

(2.20)

Since S _j is demiclosed at zero (see Proposition 1.7), it gets that x* ∈ F(S _j ). By the arbitrariness of j = 1, 2,..., N, we have $x^{*}\in C:={\bigcap }_{j=1}^{N}F\left(S_j\right)$.

Moreover, from (2.1) and (2.12) we have

$$x_{n_i}=u_{n_i}-\gamma A^{*}\left(T_{n_i}^{n_i}-I\right)A{x}_{n_i}\rightharpoonup x^{*}.$$

Since A is a linear bounded operator, it gets $A{x}_{n_i}\rightharpoonup A{x}^{*}$. For any positive integer k = 1, 2,..., N, there exists a subsequence {n _i (k)} ⊂ {n _i } with n _i (k)(modN) = k such that $A{x}_{n_i\left(k\right)}\rightharpoonup A{x}^{*}$. In view of (2.16) we have

$$∥A{x}_{n_i\left(k\right)}-T_{k}A{x}_{n_i\left(k\right)}∥\to 0\left(\mathsf{\text{as}}{n}_i\left(k\right)\to \infty \right).$$

Since T _k is demiclosed at zero, we have Ax* ∈ F(T _k ). By the arbitrariness of k = 1, 2,..., N, it yields $A{x}^{*}\in Q:={\bigcap }_{k=1}^{N}F\left(T_k\right)$. This together with x* ∈ C shows that x* ∈ Γ, i.e., x* is a solution to the (MSSFP)--(1.12).

Now we prove that x _n ⇀ x* and u _n ⇀ x*.

In fact, if there exists another subsequence $\left\{u_{n_i}\right\}\subset \left\{u_n\right\}$ such that $u_{n_i\left(j\right)}\rightharpoonup y^{*}\in \Gamma$ with y* ≠ x*. Consequently, by virtue of (2.2) and the Opial property of Hilbert space, we have

$$\begin{array}{ll}\hfill \underset{n_i\to \infty }{liminf}∥u_{n_i}-x^{*}∥& <\underset{n_i\to \infty }{liminf}∥u_{n_i}-y^{*}∥=\underset{n\to \infty }{lim}∥u_n-y^{*}∥\\ =\underset{n_i\to \infty }{liminf}∥u_{n_i}-y^{*}∥<\underset{n_j\to \infty }{lim}∥u_{n_j}-x^{*}∥\\ =\underset{n\to \infty }{liminf}∥u_n-x^{*}∥=\underset{n_i\to \infty }{lim}∥u_{n_i}-x^{*}∥.\end{array}$$

This is a contradiction. Therefore, u _n ⇀ x*. By using (2.1) and (2.12), we have

$$x_n=u_n-\lambda A^{*}\left(T_n^n-I\right)A{x}_n\rightharpoonup x^{*}.$$

The proof of conclusion (II).

Without loss of generality, we can assume that S₁ is semi-compact. It follows from (2.20) that

$$∥u_{n_i\left(1\right)}-S_1{u}_{n_i\left(1\right)}∥\to 0\left(\mathsf{\text{as}}{n}_i\left(1\right)\to \infty \right)$$

(2.21)

Therefore, there exists a subsequence of $\left\{u_{n_i\left(1\right)}\right\}$ (for the sake of convenience we still denote it by $\left\{u_{n_i\left(1\right)}\right\}$ such that $u_{n_i\left(1\right)}\to u^{*}\in H_1$ (some point in H₁). Since $u_{n_i\left(1\right)}\rightharpoonup x^{*}$. This implies that x* = u*, and so $u_{n_i\left(1\right)}\to x^{*}\in \Gamma$. By virtue of (2.2) we know that lim_n→∞||u _n - x*|| = 0 and lim_n→∞||x _n - x*|| = 0, i.e., {u _n } and {x _n } both converge strongly to x* ∈ Γ.

This completes the proof of Theorem 2.1.

Remark 2.2 Since the class of total asymptotically strict pseudocontractive mappings includes the class of asymptotically strict pseudocontractions mappings and the class of strict pseudocontractions mappings as special cases, Theorem 2.1 improves and extend the corresponding results of Censor et al. [14, 15], Yang [17], Moudafi [20], Xu [21], Censor and Segal [22], Censor et al. [23] and others.