Strong convergence theorems of nonlinear operator equations for countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings with applications
https://doi.org/10.1186/1687-1812-2012-69
© chang et al; licensee Springer. 2012
Received: 16 January 2012
Accepted: 30 April 2012
Published: 30 April 2012
Abstract
The purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping which contains many kinds of mappings as its special cases, and then by using the hybrid shrinking technique to propose an iterative algorithm for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of common fixed points for a countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the article not only generalize some recent results from single-valued mappings to multi-valued mappings, but also improve and extend the main results of Homaeipour and Razani.
2000 AMS Subject Classification: 47J06; 47J25.
Keywords
1. Introduction
In the sequel, we use F(T ) to denote the set of fixed points of a mapping T , and use
and to denote the set of all real numbers and the set of all nonnegative real numbers, respectively. We denote by x
n
→ x and x
n
⇀ x the strong convergence and weak convergence of a sequence {x
n
}, respectively.
- (I)If A ≡ 0, the problem (1.1) is equivalent to finding u ∈ C such that(1.3)
- (II)If Θ ≡ 0, the problem (1.1) is equivalent to finding u ∈ C such that(1.4)
which is called the mixed variational inequality of Browder type (VI)[2].
exists for all x, y ∈ U. X is said to be uniformly smooth if the above limit is attained uniformly in x, y ∈ U.
- (i)
If X is uniformly smooth, then X is reflexive and the normalized duality mapping J is uniformly continuous on each bounded subset of X;
- (ii)
If X is a reflexive and strictly convex Banach space, then J -1 is norm-weak-continuous;
- (iii)
If X is a smooth, strictly convex, and reflexive Banach space, then J is single-valued, one-to-one and onto;
- (iv)
A Banach space X is uniformly smooth if and only if X* is uniformly convex;
- (v)
Each uniformly convex Banach space X has the Kadec-Klee property, i.e., for any sequence {x n } ⊂ X, if x n ⇀ x ∈ X and ||x n || → ||x||, then x n → x.
- (a)
ϕ (x, ∏ C y) + ϕ (∏ C y, y) ≤ ϕ (x, y) for all x ∈ C and y ∈ X;
- (b)If x ∈ X and z ∈ C, then
- (c)
For x, y ∈ X, ϕ(x, y) = 0 if and only if x = y.
In the sequel, we denote by 2 C the family of all nonempty subsets of C.
- (1)A point p ∈ C is said to be an asymptotic fixed point of T, if there exists a sequence {x n } in C such that {x n } converges weakly to p and
- (2)
A multi-valued mapping T : C → 2 C is said to be relatively nonexpansive [3], if
- (a)
F(T ) ≠ Ø;
- (b)
ϕ (p, w) ≤ ϕ (p, x), ∀x ∈ C, w ∈ Tx, p ∈ F(T)
- (c)
.
- (2)A multi-valued mapping T : C → 2 C is said to be quasi-ϕ-asymptotically nonexpansive if F(T ) ≠ Ø and there exists a real sequence {k n } ⊂ [1, ∞) with k n → 1 such that(1.8)
- (3)A multi-valued mapping T : C → 2 C is said to be ({ν n }, {μ n },ζ)-total quasi-ϕ-asymptotically nonexpansive, if F(T) ≠ Ø and there exist nonnegative real sequences {ν n }, {μ n } with ν n → 0, μ n → 0 (as n → ∞) and a strictly increasing continuous function with ζ (0) = 0 such that for all x ∈ C, p ∈ F(T )(1.9)
- (4)A total quasi-ϕ-asymptotically nonexpansive multi-valued mapping T : C → 2 C is said to be uniformly L-Lipschitz continuous if there exists a constant L > 0 such that
- (5)
A multi-valued mapping T : C → 2 C is said to be closed if, for any sequences {x n } and {w n } in C with w n ∈ T (x n ), if x n → x and w n → y, then y ∈ Tx.
- (6)A countable family of multi-valued mappings is said to be uniformly ({ν n }, {μ n }, ζ)-total quasi-ϕ-asymptotically nonexpansive, if and there exist nonnegative real sequences ({ν n }, {μ n } with ν n → 0, μ n → 0 and a strictly increasing continuous function with ζ(0) = 0 such that for all x ∈ C,(1.10)
- (1)Every quasi-ϕ-asymptotically nonexpansive multi-valued mapping must be a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping. In fact, taking ζ(t) = t, t ≥ 0, k n = ν n + 1 and μ n = 0, then (1.6) can be rewritten as
- (2)
The class of quasi-ϕ-asymptotically nonexpansive multi-valued mappings contains properly the class of quasi-ϕ-nonexpansive multi-valued mappings as a subclass, but the converse is not true.
- (3)
The class of quasi-ϕ-nonexpansive multi-valued mappings contains properly the class of relatively nonexpansive multi-valued mappings as a subclass, but the converse is not true.
Example 1.6 Now we give some examples of single-valued and multi-valued total quasi-ϕ-asymptotically nonexpansive mappings.
(1) Single-valued total quasi- ϕ -asymptotically nonexpansive mapping.
where {a i } is a sequence in (0, 1) such that . It is proved in [4] that T is total quasi-ϕ-asymptotically nonexpansive.
(2) Multi-valued total quasi- ϕ -asymptotically nonexpansive mappings.
It is easy to see that F (T ) = {0}, therefore F(T) is nonempty.
- (I)if f(x) > 1, ∀x ∈ I, then for any g ∈ T(f), we have a ≤ f(x) - g(x) ≤ b. Hence for any p ∈ F(T ) = {0} we have
- (II)
For any , there exists some such that .
- (1)If , then we have . By (1.13), for any p ∈ F(T) = {0}, we have
- (2)If there exists x 1 ∈ I such that , then by the definition of T , we have . Since , and so g = 0. Hence for any p ∈ F(T), by (1.13) we have
for any f ∈ D, g ∈ T n (f), n ≥ 1, p ∈ F(T). This shows that T : C → 2 C is a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping.
- (1)If f(x) > 1, ∀x ∈ I, since {f n } converges uniformly to f, then there exists n 0 ≥ 1 such that f n (x) > 1, ∀x ∈ I, ∀n ≥ n 0. By the definition of T, we have(1.16)
- (2)
If there exists some point x 2 ∈ I such that 0 ≤ f (x 2) ≤ 1, then T(f) = {0}. Since {f n } converges uniformly to f, then there exists a positive integer n 2 such that 0 ≤ f n (x 2) ≤ 1, ∀n ≥ n 2. By the definition of T, this implies that T(f n ) = 0, ∀n ≥ n 2. Since g n ∈ T(f n ), this implies that g n = 0, ∀n ≥ n 2. Since g n → g, g = 0. Therefore g ∈ T(f).
These show that T is a closed mapping.
Concerning the weak and strong convergence of iterative sequences to approximate a common element of the set of solutions for a generalized MEP, the set of solutions for variational inequality problems, and the set of common fixed points for single-valued relatively non-expansive mappings, single-valued quasi-ϕ-nonexpansive mappings, single-valued quasi-ϕ-asymptotically nonexpansive mappings and single-valued total quasi-ϕ-asymptotically non-expansive mappings have been studied by many authors in the setting of Hilbert or Banach spaces (see, for example, [4–21] and the references therein). Very recently, in 2011, Homaeipour and Razani [3] introduced the concept of multi-valued relatively nonexpansive mappings and proved some weak and strong convergence theorems to approximation a fixed point for a single relatively nonexpansive multi-valued mapping in a uniformly convex and uniformly smooth Banach space X which improve and extend the corresponding results of Matsushita and Takahashi [5].
Motivated and inspired by the researches going on in this direction, the purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping which contains multi-valued relatively nonexpansive mappings and many other kinds of mappings as its special cases, and then by using the hybrid shirking iterative algorithm for finding a common element of the set of solutions for a generalized MEP, the set of solutions for variational inequality problems, and the set of common fixed points for a countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the article not only generalize the corresponding results of [4–21] from single-valued mappings to multi-valued mappings, but also improve and extend the main results of Homaeipour and Razani [3]. The method given in this article is quite different from that one adopted in [3].
2. Preliminaries
In order to prove our main results, the following conclusions and notations will be needed.
Lemma 2.1[8] Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed convex set of X. Let {x n } and {y n } be two sequences in C such that x n → p and ϕ(x n , y n ) → 0, where ϕ is the function defined by (1.1), then y n → p.
Lemma 2.2 Let X and C be as in Lemma 2.1. Let T : C → 2 C be a closed and ({ν n }, {μ n }, ζ)-total quasi-ϕ-asymptotically nonexpansive multi-valued mapping. If μ1 = 0, then the fixed point set F (T) of T is a closed and convex subset of C.
By Lemma 1.2(c), p = u. Hence, p ∈ Tp. This implies that F (T ) is a closed set in C.
By Lemma 2.1, we have u n → q (as n → ∞). This implies that un+1→ q (as n → ∞). Since un+1∈ Tu n and T is closed, we have q ∈ Tq, i.e., q ∈ F(T).
This completes the proof of Lemma 2.2.
For solving the generalized MEP, let us assume that the function is convex and lower semi-continuous, the nonlinear mapping A : C → X* is continuous and monotone, and the bifunction satisfies the following conditions:
(A1) Θ(x, x) = 0, ∀x ∈ C.
(A2) Θ is monotone, i.e., Θ(x, y) + Θ(y, x) ≤ 0, ∀x, y ∈ C.
(A3) lim sup t ↓0 Θ(x + t(z - x), y) ≤ Θ(x, y), ∀x, y, z ∈ C.
(A4) The function y ↦ Θ (x, y) is convex and lower semicontinuous.
- (i)
- (ii)[13] Define a mapping T r : X → C by
- (a)
T r is single-valued;
- (b)T r is a firmly nonexpansive-type mapping, i.e., ∀z, y ∈ X,
- (c)
F(T r ) = EP(Θ) = F(T r );
- (d)
EP(Θ) is closed and convex;
- (e)
ϕ(q, T r (x)) + ϕ(T r (x), x) ≤ ϕ(q, x), ∀q ∈ F(T r ).
- (i)There exists u ∈ C such that ∀y ∈ C(2.6)
- (ii)If we define a mapping K r : C → C by(2.7)
- (a)
K r is single-valued;
- (b)K r is a firmly nonexpansive-type mapping, i.e., ∀z, y ∈ X
- (c)
F(K r ) = Ω = F(K r );
- (d)
Ω is a closed convex set of C;
- (e)
ϕ (p, K r (z)) + ϕ (K r (z), z) ≤ ϕ (p, z), ∀p ∈ F(K r ), z ∈ X.
Remark 2.6 It follows from Lemma 2.4 that the mapping K r : C → C defined by (2.6) is a relatively nonexpansive mapping. Thus, it is quasi-ϕ-nonexpansive.
3. Main results
In this section, we shall use the hybrid iterative algorithm to find a common element of the set of solutions of a generalized MEP, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasi-ϕ-asymptotically nonexpansive multi-valued mappings. For the purpose we give the following hypotheses:
(H1) X is a uniformly smooth and strictly convex Banach space with Kadec-Klee property and C is a nonempty closed convex subset of X;
(H2) is a bifunction satisfying the conditions (A1)-(A4), A : C → X* is a continuous and monotone mapping, and is a lower semi-continuous and convex function.
(H3) is a countable family of closed and uniformly ({ν n }, {μ n }, ζ)-total quasi-ϕ-asymptotically nonexpansive multi-valued mappings and for each i = 1, 2, . . . , T i is uniformly L i -Lipschitzian with μ1 = 0.
We have the following
- (i)
for each n ≥ 0, ;
- (ii)
lim infn→∞ β n ,0, β ni > 0 for any i ≥ 1;
- (iii)
0 ≤ α n ≤ α < 1 for some α ∈ (0, 1).
If is nonempty and
is a bounded subset of C, then the sequence {x
n
} converges strongly to ∏
G
x0.
Now we divide the proof of Theorem 3.1 into six steps.
(I)
and C
n
are closed and convex for each n ≥ 0.
In fact, it follows from Lemma 2.2 that F(T
i
), i ≥ 1 is closed and convex subsets of C. Therefore
is a closed and convex subsets in C.
is closed and convex. Therefore C n is closed and convex for each n ≥ 0.
(II) {x n } is bounded and {ϕ (x n , x0)} is a convergent sequence.
This implies that {ϕ (x n , x0)} is bounded. By virtue of (1.6), we know that {x n } is bounded.
Therefore {ϕ(x n , x0)} is a convergent sequence.
(III)for all n ≥ 0.
i.e., u ∈ Cn+1and so for all n ≥ 0.
(IV) {x n } converges strongly to some point p* ∈ C.
(V) Now we prove that.
By the assumption that y ↦ H(x, y) is convex and lower semi-continuous, letting n → ∞ in (3.13), from (3.11) and (3.12), we have H(y, p*) ≤ 0, ∀y ∈ C.
Dividing both sides of the above equation by t, we have H(y t , y) ≤ 0, ∀y ∈ C. Letting t ↓ 0, from condition (A3), we have H(p*, y) ≤ 0, ∀y ∈ C, i.e., p* ∈ Ω, and .
(VI) we prove that.
In view of the definition of , from (3.14) we have p* = q. Therefore, . This completes the proof of Theorem 3.1.
The following theorems can be obtained from Theorem 3.1 immediately.
where and {α n } are sequences in 0[1] satisfying the conditions (i), (ii), (iii) in Theorem 3.1. If is a bounded subset of C, then {x n } converges strongly to .
Proof. Since is a countable family of closed and uniformly quasi-ϕ-asymptotically nonexpansive multi-valued mappings, by Remark 1.5(2), it is a countable family of closed and uniformly total quasi-ϕ-asymptotically nonexpansive multi-valued mappings with non-negative sequences {ν n = (k n - 1)}, {μ n = 0} and a strictly increasing and continuous function ζ(t) = t, t ≥ 0. Hence (as n → ∞). Therefore all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.3 can be obtained from Theorem 3.1 immediately.
where and {α n } are sequences in 0[1] satisfying the conditions (i), (ii), (iii) in Theorem 3.1. If , then {x n } converges strongly to .
Proof. Since is a countable family of closed quasi-ϕ-nonexpansive multi-valued mappings, by Remark 1.5(3), it is a countable of closed and uniformly quasi-ϕ- asymptotically nonexpansive multi-valued mappings with sequence {k
n
= 1}. Hence Therefore, the conditions appearing in Theorem 3.3: "
is a bounded subset in C" and "for each i ≥ 1, T
i
is uniformly L
i
-Lipschitz" is no use here. Therefore all conditions in Theorem 3.3 are satisfied. The conclusion of Theorem 3.4 can be obtained from Theorem 3.3 immediately.
Remark 3.5 Theorems 3.1, 3.3, and 3.4 not only generalize the corresponding results of Matsushita and Takahashi [5], Plubtieng and Ungchittrakool [6], Ceng et al. [9], Su et al. [10], Ofoedu and Malonza [11], Wang et al. [12], Chang et al. [4, 7, 8, 13, 17, 19, 20], Yao et al. [14], Zegeye et al. [15] and Nilsrakoo and Saejung [16] from single-valued mappings to multi-valued mappings, but also improve and extend the main results of Homaeipour and Razani [3] and the method adopted in this article is also different from that one adopted in [3].
4. Applications
In this section, we shall utilize the results presented in Section 3 to study some problems.
(I) Application to convex feasibility problem.
The "so called" convex feasibility problem for a family of mappings (where ω is a finite positive integer or +∞) is to finding a point in the nonempty intersection , where C i is a fixed point set of T i , i = 1, 2, . . . , ω.
converges strongly to a point , which is a solution of the convex feasibility problem for a countable family of closed and quasi-ϕ-nonexpansive multi-valued mappings where .
(II) Application to generalized MEP
converges strongly to a point p* = ∏Ωx0, which is a solution of the generalized MEP (1.1).
(III) Application to optimization problem
converges strongly to a point p* = ∏ K x0 which is a solution of the optimization problem min x∈C ψ(x), where K ⊂ C is the set of solutions to this optimization problem.
(IV) Application to the mixed variational inequality problem of Browder type
converges strongly to a point p* = ∏ Q x0 which is a solution of the mixed variational inequality of Browder type (1.4), where Q is the set of solutions to equation (1.4).
Declarations
Acknowledgements
This study was supported by the Natural Science Foundation of Yunnan Province (Grant No. 2011FB074).
Authors’ Affiliations
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