Skip to content

Advertisement

  • Research
  • Open Access

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

Fixed Point Theory and Applications20112011:77

https://doi.org/10.1186/1687-1812-2011-77

  • Received: 17 July 2011
  • Accepted: 7 November 2011
  • Published:

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.

Keywords

  • multiple-set split feasibility problem
  • split feasibility problem
  • demi-closeness
  • Opial condition
  • total asymptotically strict pseudocontraction

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 : GG a mapping.

T is said to be a contraction if there exists a constant α (0,1) such that
T x - T y α x - y , x , y 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 x - y - ψ x - y , x , y G .
(1.2)

where ψ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that ψ is positive on (0, ∞), ψ(0) = 0, and limt→∞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 x - y , x , y 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 κ n x - y , n 1 , x , y 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 x - y + μ n ϕ x - y + ξ n , n 1 , x , y 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
T x - T y 2 x - y 2 + κ I - T x - I - T y 2 , x , y 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 κ n x - y 2 + κ I - T n x - I - T n y 2 , n 1 , x , y 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 : GG 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 x - y 2 + κ x - y - T n x - T n y 2 + μ n ϕ x - y + ξ n , n 1 , x , y 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 l 2 and let T : CC be a mapping defined by
T : x 1 , x 2 , . . . , 0 , x 1 2 , a 2 x 2 , a 3 x 3 , . . . ,

where {a i } is a sequence in (0, 1) such that i = 2 a i = 1 2 .

It is proven in Goebal and Kirk [3] that
  1. (i)

    T x - T y 2 x - y , x , y C ;

     
  2. (ii)

    T n x - T n y 2 j = 2 n a j x - y , x , y C , n 2 . .

     
Denote by κ 1 1 2 = 2 , κ n 1 2 = 2 j = 2 n a j , n 2 , then
lim n k n = lim n 2 j = 2 n a j 2 = 1 .
Letting μ n = κ n - 1 , n 1 , ϕ ( t ) = t 2 , t 0 , κ = 0 and {ξ n } be a nonnegative real sequence with ξ n → 0, then x , y C , n 1 , we have
T n x - T n y 2 x - y 2 + μ n ϕ x - y + κ x - y - T n x - T n y 2 + ξ 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 : GG be a (κ, {μ n }, {ξ n }, ϕ)-total asymptotically strict pseudocontractive mapping. If F ( T ) , then for each q F(T) and for each x G, the following inequalities hold and are equivalent:
x - q , T n x - q κ + 1 2 k x - q 2 + κ - 1 2 k T n x - q 2 + μ n 2 κ ϕ x - q + ξ n 2 κ ;
(1.9)
x - T n x , x - q 1 - κ 2 T n x - x 2 - μ n 2 ϕ x - q - ξ n 2 ;
(1.10)
x - T n x , q - T n x κ + 1 2 T n x - x 2 + μ n 2 ϕ x - q + ξ 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 [1315].

The SFP in an infinite-dimensional Hilbert space can be found in [12, 14, 1618].

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:
f i n d x * C s u c h t h a t A x * Q ,
(1.12)
where A : H1H2 is a bounded linear operator, S i : H1H1 and T i : H2H2, i = 1, 2, ..., N are mappings, C : i = 1 N F ( S i ) and Q : i = 1 N F ( T i ) . In the sequel, we use Γ to denote the set of solutions of (MSSFP)--(1.12), i.e.,
Γ = { x C , A x Q } .
(1.13)

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

Let E be a Banach space. A mapping T : EE 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
liminf n x n - x * < liminf n x n - y , y E with y 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. (1)
    A mapping T : HH is said to be uniformly L-Lipschitzian, if there exists a constant L > 0, such that
    T n x - T n y L x - y , x , y H and n 1 .
     
  2. (2)

    A mapping T : HH is said to be semi-compact, if for any bounded sequence {x n } H with limn→∞||x n - Tx n || = 0, then there exists a subsequence x n i x n 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
limsup n x n - y 2 = limsup n x n - z 2 + z - y 2 y H .

Proposition 1.7 Assume that G is a closed convex subset of a real Hilbert space H and let T : GG 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 supm→∞lim supn→∞||x n - T m x 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 ( x ) = limsup n x n - x 2 , x H .
By Lemma 1.6, the weak convergence x n x* implies that
f ( x ) = f ( x * ) + x - x * 2 , x H .
In particular, for each m ≥ 1,
f ( T m x * ) = f ( x * ) + 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
f T m x * = limsup n x n - T m x * 2 = limsup n x n - T m x n + T m x n - T m x * 2 = limsup n 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 limsup n x n - T m x n x n - T m x n + 2 L x n - x * + limsup n x n - x * 2 + k x n - T m x n - x * - T m x * 2 + μ m ϕ x n - x * + ξ m
Taking lim supm→∞on both sides and observing the facts that limm→∞μ m = 0, limm→∞ξ m = 0 and lim supm→∞lim supn→∞||x n - T m x n || = 0, we derive that
limsup m f T m x * limsup n x n - x * 2 + k limsup m x * - T m x * 2
(1.15)

Since lim supm→∞f(T m x*) = f(x*)+lim supm→∞||T m x* - x*||2, and f(x*) = lim supn→∞ ||x n - x*||2, it follows from (1.15) that lim supm→∞||x* - T m x*||2 = 0. That is, T m x* → 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 ( 1 + δ n ) a n + b n , n 1 .

If i = 1 δ n < and i = 1 b n < , then the limit limn→∞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. 1.

    H1 and H2 are two real Hilbert spaces, A : H1H2 is a bounded linear operator;

     
  2. 2.

    Let G, G ̃ be a nonempty closed convex subset of H1 and H2 respectively, S i : GG, 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 : G ̃ G ̃ , i = 1 , 2 , . . . , N , is a uniformly L ̃ i -Lipschitzian and k i μ ̃ i , n , ξ ̃ i , n , ϕ ̃ i -total asymptotically strictly pseudocontractive mapping which satisfy the following conditions:

     
  3. (i)

    C : i = 1 N F ( S i ) , Q : = i = 1 N F ( T i ) ;

     
  4. (ii)

    β = max 1 i N β i < 1 , κ = max 1 i N κ i < 1 ; ;

     
  5. (iii)

    L : = max 1 i N L i < , L ̃ : = max 1 i N L ̃ i < ;

     
  6. (iv)

    μ n = max 1 i N μ i , n , μ ̃ i , n , ξ n = max 1 i N ξ i , n , ξ ̃ i , n and i = 1 μ n < , i = 1 ξ n < . .

     
  7. (v)

    ϕ = max 1 i N ϕ i , ϕ ̃ i

     

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

Theorem 2.1 Let H 1 , H 2 , G , G ̃ , A , { S i } , { T i } , C , Q , β , κ , L , L ̃ , { μ n } , { ξ n } and ϕ be the same as above. In addition, there exist positive constants M and M* such that ϕ(λ) ≤ M2 for all λ ≥ M. Let {x n } be the sequence generated by:
x 1 G chosen arbitrarily x n + 1 = ( 1 - α n ) u n + α n S n n ( u n ) , u n = x n + γ A * ( T n n - I ) A x n , n 1 ,
(2.1)
where S n n = S n ( mod N ) n , T n n = T n ( mod N ) n n 1 , { α n } is a sequence in [0, 1] and γ > 0 is a constant satisfying the following conditions:
  1. (vi)

    α n ( δ , 1 - β ) , n 1 and γ 0 , 1 - κ A 2 , where δ (0, 1 - β) is a positive constant.

     
  2. (I)

    If Γ (where Γ is the set of solutions to (MSSFP)--(1.12)), then {x n } converges weakly to a point x* Γ.

     
  3. (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
lim n x n - p and lim n 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) ≤ M2, if λ ≥ M. In either case, we can obtain that
ϕ ( λ ) ϕ ( M ) + M * λ 2 , λ 0 .
(2.3)
Since p Γ, then p C : = i = 1 N F ( S i ) and A p Q : = i = 1 N F ( T i ) . From (2.1) and (1.10) we have
x n + 1 - p 2 = u n - p - α n u n - S n n u n 2 = u n - p 2 - 2 α n u n - p , u n - S n n u n + α n 2 u n - S n n u n 2 u n - p 2 - α n ( 1 - β ) u n - S n n u n 2 + α n μ n ϕ u n - p + α n ξ n + α n 2 u n - S n n u n 2 ( b y ( 1 . 1 0 ) ) u n - p 2 - α n 1 - β - α n u n - S n n u n 2 + α n μ n ϕ ( M ) + M * u n - p 2 + α n ξ n = 1 + α n μ n M * u n - p 2 - α n 1 - β - α n u n - S n n u n 2 + α n μ n ϕ ( M ) + α n ξ n
(2.4)
On the other hand, since
u n - p 2 = x n - p + γ A * T n n - I A x n 2 = x n - p 2 + γ 2 A * T n n - I A x n 2 + 2 γ x n - p , A * T n n - I A x n ,
(2.5)
and
A * T n n - I A x n 2 = A * T n n - I A x n , A * T n n - I A x n = A A * T n n - I A x n , T n n - I A x n A 2 T n n A x n - A x n 2 ,
(2.6)
It follows from (1.11) we have
x n - p , A * T n n - I A x n = A x n - A p , T n n - I A x n = A x n - A p + T n n - I A x n - T n n - I A x n , T n n - I A x n = T n n A x n - A p , T n n A x n - A x n - T n n - I A x n 2 . 1 + κ 2 T n n - I A x n 2 + μ n 2 ϕ A x n - A p + ξ n 2 - T n n - I A x n 2 . κ - 1 2 T n n - I A x n 2 + μ n 2 ϕ ( M ) + M * A x n - A p 2 + ξ n 2 . κ - 1 2 T n n - I A x n 2 + μ n 2 M * A x n - A p 2 + μ n 2 ϕ ( M ) + ξ n 2 .
(2.7)
Substituting (2.6) and (2.7) into (2.5) and simplifying it, we have
u n - p 2 x n - p 2 + γ 2 A 2 T n n A x n - A x n 2 + γ κ - 1 T n n - I A x n 2 + γ μ n M * A x n - A p 2 + γ μ n ϕ ( M ) + γ ξ n = x n - p 2 - γ 1 - κ - γ A 2 T n n A x n - A x n 2 + γ μ n M * A x n - A p 2 + γ μ n ϕ ( M ) + γ ξ n 1 + γ μ n M * A 2 x n - p 2 - γ 1 - κ - γ A 2 T n n A x n - A x n 2 + γ μ n ϕ ( M ) + γ ξ n
(2.8)
Substituting (2.8) into (2.4) and after simplifying we have
x n + 1 - p 2 1 + α n μ n M * 1 + γ μ n M * A 2 x n - p 2 - γ 1 - κ - γ A 2 T n n A x n - A x n 2 + γ μ n ϕ ( M ) + γ ξ n - α n 1 - β - α n u n - S n n u n 2 + α n μ n ϕ ( M ) + α n ξ n 1 + δ n x n - p 2 - γ 1 - κ - γ A 2 T n n A x n - A x n 2 - α n 1 - β - α n u n - S n n u n 2 + b n
(2.9)
where
δ n = α n μ n M * + γ μ n M * A 2 + γ A 2 α n μ n 2 ( M * ) 2 b n = 1 + α n μ n M * γ + α n μ n ϕ ( M ) + 1 + α n μ n M * γ + α n ξ n
By condition (vi) we have
x n + 1 - p 2 1 + δ n x n - p 2 + b n
By condition (iv), n = 1 δ n < and n = 1 b n < . Hence, from Lemma 1.8 we know that the following limit exists
lim n x n - p .
(2.10)
Consequently, from (2.9) and (2.10) we have that
γ 1 - κ - γ A 2 T n n - I A x n 2 + α n 1 - β - α n u n - S n n u n 2 x n - p 2 - x n + 1 - p 2 + δ n x n - p 2 + b n 0 a s n .
This together with the condition (vi) implies that
lim n u n - S n n u n = 0 ;
(2.11)
and
lim n T n n - I 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
lim n x n + 1 - x n = 0 and lim n u n + 1 - u n = 0 .
(2.13)
In fact, it follows from (2.1) that
x n + 1 - x n = 1 - α n u n + α n S n n ( u n ) - x n = 1 - α n x n + γ A * T n n - I A x n + α n S n n ( u n ) - x n = 1 - α n γ A * T n n - I A x n + α n S n n ( u n ) - x n = 1 - α n γ A * T n n - I A x n + α n S n n ( u n ) - u n + α n u n - x n = 1 - α n γ A * T n n - I A x n + α n S n n ( u n ) - u n + α n γ A * T n n - I A x n = γ A * T n n - I A x n + α n S n n ( u n ) - u n .
In view of (2.11) and (2.12) we have that
lim n x n + 1 - x n = 0 .
(2.14)
Similarly, it follows from (2.1), (2.12), and (2.14) that
u n + 1 - u n = x n + 1 + γ A * T n + 1 n + 1 - I A x n + 1 - x n + γ A * T n n - I A x n x n + 1 - x n + γ A * T n + 1 n + 1 - I A x n + 1 + γ A * T n n - I A x n 0 as n .
(2.15)

The conclusion (2.13) is proved.

(3) Next we prove that for each j = 1, 2,..., N - 1,
u i N + j - S j u i N + j 0 and A x i N + j - T j A x i N + j 0 a s i ,
(2.16)
In fact, from (2.11) we have
η i N + j : = u i N + j - S j i N + j u i N + j 0 a s i .
(2.17)
Since S j is uniformly L j -Lipschitzian continuous, it follows from (2.13) and (2.17) that
u i N + j - S j u i N + j = u i N + j - S j i N + j u i N + j + S j i N + j u i N + j - S j u i N + j η i N + j + L j S j i N + j - 1 u i N + j - u i N + j η i N + j + L j S j i N + j - 1 u i N + j - S j i N + j - 1 u i N + j - 1 + L j S j i N + j - 1 u i N + j - 1 - u i N + j η i N + j + L j 2 u i N + j - u i N + j - 1 + L j S j i N + j - 1 u i N + j - 1 - u i N + j - 1 + u i N + j - 1 - u i N + j η i N + j + L j 1 + L j u i N + j - u i N + j - 1 + L j η i N + j - 1 0 ( a s i )
Similarly, for each j = 1, 2,..., N - 1, from (2.13) we have
ς i N + j : = A x i N + j - T j i N + j A x i N + j 0 ( as i ) .
(2.18)
Since T j is uniformly L ̃ j -Lipschitzian continuous, by the same way as above, from (2.13) and (2.18), we can also prove that
A x i N + j - T j A x i N + j 0 ( a s i ) .
(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 u n i u n such that u n i 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 ( j ) x * . Again from (2.16) we have
u n i ( j ) - S j u n i ( j ) 0 ( as n i ( j ) )
(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 * C : = j = 1 N F S j .

Moreover, from (2.1) and (2.12) we have
x n i = u n i - γ A * T n i n i - I A x n i x * .
Since A is a linear bounded operator, it gets A x n i 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 ( k ) A x * . In view of (2.16) we have
A x n i ( k ) - T k A x n i ( k ) 0 ( as n i ( k ) ) .

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 * Q : = k = 1 N F ( T k ) . 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 u n i u n such that u n i ( j ) y * Γ with y* ≠ x*. Consequently, by virtue of (2.2) and the Opial property of Hilbert space, we have
liminf n i u n i - x * < liminf n i u n i - y * = lim n u n - y * = liminf n i u n i - y * < lim n j u n j - x * = liminf n u n - x * = lim n i u n i - x * .
This is a contradiction. Therefore, u n x*. By using (2.1) and (2.12), we have
x n = u n - λ A * T n n - I A x n 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 ( 1 ) - S 1 u n i ( 1 ) 0 ( as n i ( 1 ) )
(2.21)

Therefore, there exists a subsequence of u n i ( 1 ) (for the sake of convenience we still denote it by u n i ( 1 ) such that u n i ( 1 ) u * H 1 (some point in H1). Since u n i ( 1 ) x * . This implies that x* = u*, and so u n i ( 1 ) x * Γ . By virtue of (2.2) we know that limn→∞||u n - x*|| = 0 and limn→∞||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.

Declarations

Acknowledgements

The authors would like to thank the referees for useful comments and suggestions. This study was supported by the Natural Science Foundation of Sichuan Province (No. 08ZA008).

Authors’ Affiliations

(1)
Department of Mathematics, South West University of Science and Technology, Mianyang Sichuan, 621010, China
(2)
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, 650221, China
(3)
Department of Mathematics Education, Rins Gyeongsang National University, Jinju, 660-701, Korea
(4)
Department of Mathematics Education, Kyungnam University Masan, Kyungnam, 631-701, Korea

References

  1. Alber YaI, Guerre-Delabriere S: On the projection methods for fixed point problems. Analysis 2001, 21: 17–39.MathSciNetView ArticleGoogle Scholar
  2. Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal: Theory Methods Appl 2001, 47: 2683–2693. 10.1016/S0362-546X(01)00388-1MathSciNetView ArticleGoogle Scholar
  3. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Am Math Soc 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleGoogle Scholar
  4. Alber YaI, Chidume CE, Zegeye H: Approximating fixed points of total asymptotically nonex-pansive mappings. Fixed Point Theory Appl 2006, 10673: 20.MathSciNetGoogle Scholar
  5. Chidume CE, Ofoedu EU: A new iteration process for approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. Int J Math Math Sci 2009, 615107: 17.MathSciNetGoogle Scholar
  6. Chidume CE, Ofoedu EU: Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. J Math Anal Appl 2007, 333: 128–141. 10.1016/j.jmaa.2006.09.023MathSciNetView ArticleGoogle Scholar
  7. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleGoogle Scholar
  8. Marino G, Xu HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal Appl 2007, 329: 336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleGoogle Scholar
  9. Qihou L: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal: Theory Methods Appl 1996, 26: 1835–1842. 10.1016/0362-546X(94)00351-HView ArticleGoogle Scholar
  10. Kim TH, Xu HK: Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions. Nonlinear Anal: Theory Methods Appl 2008, 68: 2828–2836. 10.1016/j.na.2007.02.029MathSciNetView ArticleGoogle Scholar
  11. Censor Y, Elfving T: A multi-projection algorithm using Bregman projections in a product space. Numer Algorithms 1994, 8: 221–239. 10.1007/BF02142692MathSciNetView ArticleGoogle Scholar
  12. Byrne C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Problem 2002, 18: 441–453. 10.1088/0266-5611/18/2/310MathSciNetView ArticleGoogle Scholar
  13. Censor Y, Bortfeld T, Martin B, Trofimov A: A unified approach for inversion problem in intensity-modulated radiation therapy. Phys Med Biol 2006, 51: 2353–2365. 10.1088/0031-9155/51/10/001View ArticleGoogle Scholar
  14. Censor Y, Elfving T, Kopf N, Bortfeld T: The multiple-sets split feasibility problem and its applications. Inverse Problem 2005, 21: 2071–2084. 10.1088/0266-5611/21/6/017MathSciNetView ArticleGoogle Scholar
  15. Censor Y, Motova XA, Segal A: Pertured projections and subgradient projections for the multiple-setssplit feasibility problem. J Math Anal Appl 2007, 327: 1244–1256. 10.1016/j.jmaa.2006.05.010MathSciNetView ArticleGoogle Scholar
  16. Xu HK: A variable Krasnosel'skii-Mann algorithm and the multiple-sets split feasibility problem. Inverse Problem 2006, 22: 2021–2034. 10.1088/0266-5611/22/6/007View ArticleGoogle Scholar
  17. Yang Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Problem 2004, 20: 1261–1266. 10.1088/0266-5611/20/4/014View ArticleGoogle Scholar
  18. Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Problem 2005, 21: 1791–1799. 10.1088/0266-5611/21/5/017View ArticleGoogle Scholar
  19. Aoyama K, Kimura W, Takahashi W, Toyoda M: Approximation of common fixed points of acountable family of nonexpansive mappings on a Banach space. Nonlinear Anal: Theory Methods Appl 2007,67(8):2350–2360. 10.1016/j.na.2006.08.032MathSciNetView ArticleGoogle Scholar
  20. Moudafi A: The split common fixed point problem for demi-contractive mappings. Inverse Prob-lem 2010,26(055007):6.MathSciNetGoogle Scholar
  21. Xu HK: Iterative methods for split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problem 2010,26(105018):17.Google Scholar
  22. Censor Y, Segal A: The split common fixed point problem for directed operators. J Convex Anal 2009, 16: 587–600.MathSciNetGoogle Scholar
  23. Censor Y, Gibali A, Reich S: Algorithms for the split variational inequality problem. Numer Algorithm (accepted)Google Scholar

Copyright

Advertisement