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Iterative algorithms approach to a general system of nonlinear variational inequalities with perturbed mappings and fixed point problems for nonexpansive semigroups
Journal of Inequalities and Applications volume 2012, Article number: 133 (2012)
Abstract
In this paper, we introduce new iterative algorithms for finding a common element of the set of solutions of a general system of nonlinear variational inequalities with perturbed mappings and the set of common fixed points of a one-parameter nonexpansive semigroup in Banach spaces. Furthermore, we prove the strong convergence theorems of the sequence generated by these iterative algorithms under some suitable conditions. The results obtained in this paper extend the recent ones announced by many others.
Mathematics Subject Classification (2010): 47H09, 47J05, 47J25, 49J40, 65J15
1 Introduction
Variational inequality theory has been studied widely in several branches of pure and applied sciences. Indeed, applications of variational inequalities span as diverse disciplines as differential equations, time-optimal control, optimization, mathematical programming, mechanics, finance, and so on (see, e.g., [1, 2] for more details). Note that most of the variational problems include minimization or maximization of functions, variational inequality problems, quasivariational inequality problems, decision and management sciences, and engineering sciences problems. For more details, we recommend the reader [3–8, 29–31].
Let X be a real Banach space, and X* be its dual space. The duality mapping is defined by
where 〈·, ·〉 denotes the duality pairing between X and X*. If X : = H is a real Hilbert space, then J = I where I is the identity mapping. It is well known that if X is smooth, then J is single-valued, which is denoted by j (see [9]).
Let C be a nonempty closed and convex subset of X and T be a self-mapping of C. We denote → and ⇀ by strong and weak convergence, respectively. Recall that a mapping T : C → C is said to be L-Lipschitzian if there exists a constant L > 0 such that
If 0 < L < 1, then T is a contraction and if L = 1, then T is a nonexpansive mapping. We denote by Fix(T) the set of all fixed points set of the mapping T, i.e., Fix(T) = {x ∈ C : Tx = x}.
A mapping F : C → X is said to be accretive if there exists j(x - y) ∈ J(x - y) such that
A mapping F : C → X is said to be strongly accretive if there exists a constant η > 0 and j(x - y) ∈ J(x - y) such that
Remark 1.1. If X : = H is a real Hilbert space, accretive and strongly accretive mappings coincide with monotone and strongly monotone mappings, respectively.
Let H be a real Hilbert space, whose inner product and norm are denoted by〈·, ·〉 and ||·||, respectively. Let A be a strongly positive bounded linear operator on H, that is, there exists a constant such that
Remark 1.2. From the definition of operator A, we note that a strongly positive bounded linear operator A is a ||A||-Lipschitzian and η-strongly monotone operator.
Let C be a nonempty closed and convex subset of a real Banach space X. Recall that the classical variational inequality is to find x* ∈ C such that
where Ψ: C → X is a nonlinear mapping and j(x - x*) ∈ J(x - x*). The set of solution of variational inequality is denoted by VI(C, Ψ). If X : = H is a real Hilbert space, then (1.2) reduces to find x* ∈ C such that
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonex-pansive mapping on a real Hilbert space H
where C is the fixed point set of a nonexpansive mapping T on H and u is a given point in H.
In 2001, Yamada [10] introduced a hybrid steepest descent method for a nonexpansive mapping T as follows:
where F is a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and . He proved that if {λ n } satisfying appropriate conditions, then the sequence {x n }generated by (1.5) converges strongly to the unique solution of variational inequality
In 2006, Marino and Xu [11] introduced and considered the following general iterative method:
where A is a strongly positive bounded linear operator on a real Hilbert space H. They, proved that, if the sequence {α n } of parameters satisfies appropriate conditions, then the sequence {x n } generated by (1.7) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem
where C is the fixed point set of a nonexpansive mapping T and h is a potential function for γf (i.e., h'(x) = γf(x) for all x ∈ H).
Recently, Tian [12] combined the iterative method (1.7) with the Yamada's method (1.5) and considered the general iterative method for a nonexpansive mapping T as follows:
Then, he proved that the sequence {x n } generated by (1.10) converges strongly to the unique solution of variational inequality
Let Ψ1, Ψ2 : C → X be two mappings. Yao et al. [7] considered the following problem of finding (x*, y*) ∈ C × C such that
which is called a general system of nonlinear variational inequalities in Banach spaces, where ρ1 > 0 and ρ2 > 0 are two constants. In particular, if ρ1 = 1 and ρ2 = 1 then problem (1.12) reduces to problem of finding (x*, y*) ∈ C × C such that
which is defined by Yao et al. [13].
Very recently, Yao et al. [7] introduced an iterative algorithm for solving the problem (1.12). To be more precise, they proved the following theorem.
Theorem YLKY[7]Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X and let Q C be a sunny nonexpansive retraction from X onto C. Let the mappings Ψ1, Ψ2 : C → X be α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let A : C → X be a strongly positive linear bounded operator with coefficient . Let Ω: = VI(C, Ψ1) ∩ VI(C, Ψ2). For given x0 ∈ C, let the sequence {x n } be generated by
Suppose that {α n } and {β n } are sequences in [0, 1] satisfying the following conditions:
(C1) lim n→∞ α n = 0 and ;
(C2) 0 < lim inf n→∞ β n ≤ lim sup n→∞ β n < 1.
Then, {x n } converges strongly to which solves the variational inequality (1.12).
On the other hand, motivated and inspired by the idea of Tian [12] and Yao et al. [7], we consider and introduce the following system of variational inequalities in Banach spaces: Let C be a nonempty closed and convex subset of a Banach space X. Let Ψ i , Φ i : C → X (i = 1, 2) be a mapping. First, we consider the following problem of finding (x*, y*) ∈ C × C such that
which is called a general system of nonlinear variational inequalities with perturbed mapping in Banach spaces, where ρ1 > 0 and ρ2 > 0 are two constants. In particular, if Φ1 = Φ2 = 0 then problem (1.15) reduces to problem (1.12). Further, if Φ1 = Φ2 = 0 and ρ1 = ρ2 = 1 then problem (1.15) reduces to problem (1.13). Second, we introduce iterative algorithms (3.15) below for finding a common element of the set of solutions of a general system of nonlinear variational inequalities with perturbed mappings (1.15) and the set of common fixed points of a one-parameter nonexpansive semigroup in Banach spaces. Furthermore, we show that our iterative algorithm converges strongly to a common element of the two aforementioned sets under some suitable conditions. Our results extend the main result of Tian [12] and Yao et al. [7] and the methods of the proof in this paper are also new and different.
2 Preliminaries
Let U = {x ∈ X : ||x|| = 1}. A Banach space X is said to be strictly convex if for all x, y ∈ U with x ≠ y. A Banach space X is called uniformly convex if for each ε > 0 there is a δ > 0 such that for x, y ∈ X with ||x||, ||y|| ≤ 1 and ||x - y|| ≥ ε, ||x + y|| ≤ 2(1 - δ) holds. The modulus of covexity of X defined by
for all ε ∈ [0 2]. It is known that every uniformly convex Banach space is strictly convex and reflexive [9]. The norm of X is said to be Gâteaux differentiable if the limit
exists for each x, y ∈ U. In this case X is smooth. The norm of X is said to be Fréchet differentiable if for each x ∈ U, the limit (2.1) is attained uniformly for y ∈ U. The norm of X is called uniformly Fréchet differentiable if the limit (2.1) is attained uniformly for x, y ∈ U. It is well known that (uniform) Fréchet differentiability of the norm of X implies (uniform) Gâteaux differentiability of the norm of X.
Let ρ X : [0, ∞) → [0, ∞) be the modulus of smoothness of X defined by
A Banach space X is said to be uniformly smooth if as t→ 0. Suppose that q > 1, then X is said to be q-uniformly smooth if there exists c > 0 such that ρ X (t) ≤ ctq . It is easy to see that if X is q-uniformly smooth, then q ≤ 2 and X is uniformly smooth. It is well known that X is uniformly smooth if and only if the norm of X is uniformly Fré chet differentiable and hence the norm of X is Fré chet differentiable, in particular, the norm of X is Fré chet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are L p , where p > 1. More precisely, L p is min{p, 2}-uniformly smooth for every p > 1.
Definition 2.1. A one-parameter family from C into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions:
-
(i)
T(0)x = x for all x ∈ C;
-
(ii)
T(s + t)x = T(s)T(t)x for all x ∈ C and s, t > 0;
-
(iii)
for each x ∈ C the mapping t ↦ T(t)x is continuous;
-
(iv)
||T(t)x - T(t)y|| ≤ ||x - y|| for all x, y ∈ C and t > 0.
Remark 2.2. We denote by the set of all common fixed points of , that is . We know that is nonempty if C is bounded [14].
Now, we present the concept of a uniformly asymptotically regular semigroup [15–17].
Definition 2.3. Let C be a nonempty closed and convex subset of a Banach space be a continuous operator semigroup on C. Then is said to be uniformly asymptotically regular (in short, u.a.r.) on C if for all h ≥ 0 and any bounded subset B of C such that
The nonexpansive semigroup {σ t : t > 0} defined by the following lemma is an example of u.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in [15].
Lemma 2.4. [18]Let C be a nonempty closed and convex subset of a uniformly convex Banach space X, B be a bounded closed and convex subset of C. If we denoteis a nonexpansive semi-group on C such that. For all h ≥ 0, the set, then
Example 2.5. The set {σ t : t > 0} defined by Lemma 2.4 is u.a.r. nonexpansive semigroup. In fact, it is obvious that {σ t : t > 0} is a nonexpansive semigroup. For each h > 0, we have
By Lemma 2.4, we obtain that
Let D be a nonempty subset of C. A mapping Q : C → D is said to be sunny[19] if
whenever Qx + t(x - Qx) ∈ C for x ∈ C and t ≥ 0. A mapping Q : C → D is said to be retraction if Qx = x for all x ∈ D. Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. It is well known that if X : = H is a real Hilbert space, then a sunny nonexpansive retraction Q C is coincident with the metric projection from X onto C. The following lemmas concern the sunny nonex-pansive retraction.
Lemma 2.6. [19]Let C be a closed and convex subset of a smooth Banach space X. Let D be a nonempty subset of C. Let Q : C → D be a retraction and let J be the normalized duality mapping on X. Then the following are equivalent:
-
(a)
Q is sunny and nonexpansive.
-
(b)
||Qx - Qy||2 ≤ 〈x - y, J(Qx - Qy)〉, ∀x, y ∈ C.
-
(c)
〈x - Qx, J(y - Qx)〉 ≤ 0, ∀x ∈ C, y ∈ D.
Lemma 2.7. [20]If X is a strictly convex and uniformly smooth Banach space and if T : C → C is a nonexpansive mapping having a nonempty fixed point set Fix(T), then the set Fix(T) is a sunny nonexpansive retraction of C.
Let ℕ be the set of positive integers and let l∞ be the Banach space of bounded valued functions on ℕ with supremum norm. Let LIM be a linear continuous functional on l∞ and let x = (a1, a2,...) ∈ l∞ . Then sometimes, we denote by LIM n (a n ) the value of LIM(x). We know that there exists a linear continuous functional LIM on l∞ such that LIM = LIM(1) = 1 and LIM(a n ) = LIM(an+1) for each x = (a1, a2,...) ∈ l∞ . Such a LIM is called a Banach limit. Let LIM be a Banach limit. Then
Let l∞ be a Banach space of all bounded real-valued sequences. A Banach limit LIM n [9] is a linear continuous functional on l∞ such that
for each x = (a1, a2,...) ∈ l∞ . Specially, if a n → a, then LIM(x) = a[9].
In order to prove our main results, we need the following lemmas.
Lemma 2.8. [21]Let X be a real 2-uniformly smooth Banach space with the best smoothness constant K > 0. Then the following inequality holds:
Lemma 2.9. [22]In a real Banach space X, the following inequality holds:
where j(x + y) ∈ J(x + y).
Lemma 2.10. [23]Let {x n } and {l n } be bounded sequences in a Banach space X and let {β n } be a sequence in [0, 1] with 0 < lim inf n→∞ β n ≤ lim sup n→∞ β n < 1. Suppose xn+1= (1 - β n )l n + β n x n for all integers n ≥ 0 and lim sup n→∞ (||ln+1- l n || - ||xn+1- x n ||) ≤ 0. Then, lim n→∞ ||l n - x n || = 0.
Lemma 2.11. [24]Let C be a closed and convex subset of a strictly convex Banach space X. Let T1and T2be two nonexpansive mappings from C into itself with Fix(T1) ∩ Fix(T2) = Ø. Define a mapping S by
where δ is a constant in (0, 1). Then S is nonexpansive and Fix(S) = Fix(T1) ∩ Fix(T2).
Lemma 2.12. [9]Let C be a closed and convex subset of a reflexive Banach space X. Let μ be a proper convex lower semicontinuous function of C into (-∞, ∞] and suppose that μ(x n ) → ∞ as ||x n || → ∞. Then, there exists z ∈ C such that μ(z) = infx∈C{μ(x): x ∈ C}.
Lemma 2.13. [25]Let C be a nonempty closed convex subset of a smooth Banach space X and letbe a u.a.r. nonexpansive semigroup on C suchand at least there exists a T(h) which is demicompact. Then, for each x ∈ C, there exists a sequence {T(t k ): t k > 0, k ∈ℕ } ⊂ {T(h): h > 0} such that {T (t k )x} converges strongly to some point in, where lim k→∞ t k = ∞.
Lemma 2.14. [26]Let C be a nonempty closed and convex subset of a real uniformly convex Banach space and T : C → C be a nonexpansive mapping such that Fix(T) ≠ Ø . Then, I - T is demiclosed at zero.
Lemma 2.15. [27]Let X be a real smooth and uniformly convex Banach space and let r > 0. Then there exists a strictly increasing, continuous and convex function g : [0, 2r] → ℝ such that g(0) = 0 and
Lemma 2.16. [28]Assume that {a n } is a sequence of nonnegative real numbers such that
where {σ n } is a sequence in (0, 1) and {δ n } is a sequence in ℝ such that
-
(i)
;
-
(ii)
or .
Then, lim n→∞ a n = 0.
Lemma 2.17. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth Banach space X. Let F : C → X be a κ-Lipschitzian and η-strongly accretive operator with constants κ, η > 0. Letand τ = μ(η - μκ2K2). Then, for each, the mapping S : C → C defined by S : = (I - tμF) is contractive with a constant 1 - tτ.
Proof. Since and . This implies that 1 - tτ ∈ (0, 1). From Lemma 2.8, for all x, y ∈ C, we have
It follows that
Hence, we have S : = (I - tμF) is contractive with a constant 1 - tτ. This proof is complete.
Lemma 2.18. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let the mappings Ψ, Φ: C → H be-inverse strongly accretive and-inverse strongly accretive, respectively. Then, we have
In particular, if, then I - ρ(Ψ + Φ) is nonexpansive.
Proof. By the convexity of ||·||2 and Lemma 2.8, for all x, y ∈ C, we have
It is clear that, if , then I - ρ(Ψ + Φ) is nonexpansive. This proof is complete.
Lemma 2.19. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let Ψ i : C → H (i = 1, 2) be-inverse-strongly accretive and Φ i : C → H (i = 1, 2) be -inverse-strongly accretive. Let G: C → C be the mapping defined by
Ifand, then G : C → C is nonexpansive.
Proof. By Lemma 2.18, for all x, y ∈ C, we have
which implies that G : C → C is nonexpansive. This proof is complete.
Lemma 2.20. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let Ψ i : C → H (i = 1, 2) be -inverse-strongly accretive and Φ i : C → H (i = 1, 2) be -inverse-strongly accretive. For given(x*, y*) ∈ C × C is a solution of the problem (1.15) if and only if x* ∈ Fix(G) and y* = Q C (x* - ρ2(Ψ2 + Φ2)x*), where G is the mapping defined as in Lemma 2.19.
Proof. Let (x*, y*) ∈ C × C be a solution of the problem (1.15). Then, we can rewrite (1.15) as
From Lemma 2.6(c), we can deduce that (2.2) is equivalent to
This proof is complete.
3 Main results
Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let F : C → X be a κ-Lipschitizian and η-strongly accretive operator with constants κ, η > 0, V : C → C be an L-Lipschitzian mapping with a constant L ≥ 0 and T : C → C be a nonexpansive mapping with Fix(T) = Ø. Let and 0 ≤ γL < τ, where τ = μ(η - μκ2K2). For each , consider the mapping S t : C → C defined by
It is easy to see that S t is contractive. Indeed, from Lemma 2.17, for all x, y ∈ C, we have
Hence S t is contractive. By the Banach contraction principle, S t has a unique fixed point, denoted by x t , which uniquely solves the fixed point equation
Lemma 3.1. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let F : C → X be a κ-Lipschitizian and η-strongly accretive operator with constants κ, η > 0, V : C → C be an L-Lipschitzian mapping with a constant L ≥ 0 and T : C → C be a nonexpansive mapping with Fix(T) = Ø. Letand 0 ≤ γL < τ, where τ = μ(η - μκ2K2). Then the net {x t } defined by (3.1) converges strongly toas t → 0, whereis the unique solution of the variational inequality
Proof. We observe that
It follows that
First, we show the uniqueness of a solution of the variational inequality (3.2). Suppose that , are solution of (3.2), then
and
Adding up (3.4) and (3.5), we have
Note that (3.3) implies that and the uniqueness is proved. Below, we use to denote the unique solution of (3.2).
Next, we show that {x t } is bounded. Without loss of generality, we may assume that . Take p ∈ Fix(T). From Lemma 2.17, we have
It follows that
Hence, {x t } is bounded, so are {V x t } and {FTx t }. Assume {t n } ⊂ (0, 1) is such that t n → 0 as n → ∞. Set . Define a mapping ϕ : C → ℝ by
where LIM n is a Banach limit on l∞ . Note that X is reflexive and ϕ is continuous, convex functional and ϕ (x) → ∞ as ||x|| → ∞. From Lemma 2.12, there exists z ∈ C such that ϕ(z) = infx∈Cϕ(x). This implies that the set
Observe that
For all z ∈ C, we have
which implies that T(K) ⊂ K; that is, K is invariant under T. Since X is a uniformly smooth Banach space, it has the fixed point property for nonexpansive mapping T. Then, there exists such that . Since is also minimization of ϕ over C, it follows that x ∈ C and t ∈ (0, 1),
From Lemma 2.9, we have
Taking Banach limit over n ≥ 1, then
which in turn implies that
and hence
Again since X is a uniformly smooth Banach space, we have that the duality mapping j is norm-to-norm uniformly continuous on a bounded subset of C (see [9], Lemma 1), letting t → 0, we obtain
In particular
Set x t = Q C y t , where y t = tγVx t + (I - tμF)Tx t . Notice that and . For , by Lemma 2.6(c), we have
Thus, we have
which implies that
It follows from (3.7) that
This implies that . Hence, there exists a subsequence of {x n } such that as i → ∞. From (3.6) and Lemma 2.14, we get .
Next, we show that solves the variational inequality (3.2). We note that
which derives that
Note that I - T is accretive (i.e., 〈(I - T )x - (I - T )y, j(x - y)〉 ≥ 0, for x, y ∈ C). For all v ∈ Fix(T), it follows from (3.9) and Lemma 2.6(c) that
where M = supn ≥ 1{μκ||x t - v||} and . Now, replacing t in (3.10) with t n and taking the limit as n → ∞, we noticing that for , we obtain . Hence is the solution of the variational inequality (3.2). Consequently, by uniqueness. Therefore, as t → 0. This completes the proof.
Lemma 3.2. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let F : C → X be a κ-Lipschitizian and η-strongly accretive operator with constants κ, η > 0, V : C → C be an L-Lipschitzian mapping with a constant L ≥ 0 and T : C → C be a nonexpansive mapping with Fix(T) = Ø. Letand 0 ≤ γL < τ, where τ = μ(η - μκ2K2). Assume that the net {x t } defined by (3.1) converges strongly toas t → 0. Suppose that {x n } is bounded and lim n→∞ ||x n - Tx n || = 0. Then
Proof. Notice that x t = Q C y t , where y t = tγVx t + (I - tμF)Tx t . We note that
It follows from Lemma 2.6(c) that
which in turn implies that
Since x n - Tx n → 0 as n → ∞, taking the upper limit as n → ∞ firstly, and then as t → 0 in (3.12), we have
On the other hand, we note that
Taking the upper limit as n → ∞, we have
Since X is a uniformly smooth Banach space, we have that the duality mapping j is norm-to-norm uniformly continuous on bounded subset of C (see [9], Lemma 1), then
Then, from (3.13) and (3.14), we have
This completes the proof.
Theorem 3.3. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let F : C → X be a κ-Lipschitizian and η-strongly accretive operator with constants κ, η > 0, V : C → C be an L-Lipschitzian mapping with a constant L ≥ 0. Let and 0 ≤ γL < τ, where τ = μ(η - μκ2K2). Let be a u.a.r. nonexpansive semigroup from C into itself such that and least there exists a T(h) which is demicompact. Let Ψ i : C → X (i = 1, 2) be -inverse-strongly accretive and Φ i : C → X (i = 1, 2) be-inverse-strongly accretive. Assume that, where G is defined as in Lemma 2.19. For given x1 ∈ C, let {x n } be a sequence defined by
where for all i= 1, 2. Suppose that {α n } and {β n } are sequences in [0, 1] and {t n } is a sequence in (0, ∞) satisfying the following conditions:
(C1) lim n→∞ α n = 0 and;
(C2) 0 < lim inf n→∞ β n ≤ lim sup n→∞ β n < 1;
(C3) tn+1= h + t n , for all h > 0 and lim n→∞ t n = ∞.
Then, the sequence {x n } defined by (3.15) converges strongly to as n → ∞, which is the unique solution of the variational inequality
and is the solution of the problem(1.15), where .
Proof. Note that from the condition (C 1), we assume without loss generality, that for all n ≥ 1. First, we show that {x n } is bounded. Take . It follows from Lemma 2.19 that
Put y* = Q C (x* - ρ2(Ψ2 + Φ2)x*), then x* = Q C (y* - ρ1(Ψ1 + Φ1)y*). From (3.15), we observe that
Set u n = Q C [α n γVx n + (I - α n μF)T(t n )y n ]. From Lemma 2.17, we have
It follows that
By induction, we have
Hence, {x n } is bounded, so are {z n }, {y n } and {u n }.
Next, we show that ||xn+1- x n || → 0 as n → ∞. We observe that
Now, we can take a constant M > 0 such that
Then, we have
Combining (3.17) and (3.18), we obtain
Since {T(h): h > 0} is a u.a.r. nonexpansive semigroup and lim n→∞ t n = ∞, then for all h > 0, and for any bounded subset B of C containing {x n } and {y n }, we obtain that
Consequently, it follows from the conditions (C 1), (C 2) and (3.19) that
Hence, by Lemma 2.10, we obtain that
Consequently, we have
We observe that
From (3.21) and (3.23), we have
Next, we show that lim n→∞ ||x n - T(h)x n || = 0, ∀h > 0. For all x*, y* ∈ Ω, by the convexity of || · ||2 and Lemma 2.8, we have
In a similar way, we can get
Substituting (3.25) into (3.26), we have
Set u n = Q C v n , where v n = α n γVx n + (I - α n μF)T(t n )y n . From Lemma 2.6(c), we have
It follows that
By the convexity of || · ||2 and (3.28), we have
Substituting (3.27) into (3.29), we have
which in turn implies that
Since lim inf n→∞ (1 - β n ) > 0, , for all i = 1, 2, ||xn+1- x n || → 0 and α n → 0, we have
Let r1 = supn ≥ 1{||z n - y*||, ||y n - x*||}. By Lemma 2.6(b) and Lemma 2.15, we have
which in turn implies that
Let r2 = supn ≥ 1{||x n - x*||, ||z n - y*||}. Again, by Lemma 2.6(b) and Lemma 2.15, we have
which in turn implies that
Substituting (3.32) into (3.31), we obtain
And, then substituting (3.33) into (3.29), we obtain
which in turn implies that
Since lim inf n→∞ (1 - β n ) > 0, ||xn+1- x n || → 0, α n → 0 and (3.30), we have
It follows from the properties of g1 and g2 that
Consequently, we have
On the other hand, we observe that
From (3.24) and (3.34), we obtain that
For all h > 0, we note that
Since {T(h): h > 0} is a u.a.r. nonexpansive semigroup, it follows from (3.20) and (3.35) that
Since {T(h): h > 0} is a u.a.r. nonexpansive semigroup, by Lemma 2.13, for each x ∈ C, there exists a sequence {T(t k ): t k > 0, k ∈ ℕ} {T(h): h > 0} such that {T(t k )x} converges strongly to some point in , where t k → ∞ as k → ∞. Define a mapping T : C → C by
By [25, Remark 3.4], we see that the mapping T is nonexpansive such that . From (3.36), we obtain that
Define a mapping W : C → C by
where δ is a constant in (0, 1). By Lemma 2.11, we see that the mapping W is nonexpansive such that
Notice that
From (3.34) and (3.37), we obtain that
Next, we show that
where and x t is the unique fixed point of the contraction mapping T t : C → C given by
By Lemma 3.1, we have , which solves the variational inequality
By (3.21) and Lemma 3.2, we obtain that
Finally, we show that as n → ∞. Notice that u n = Q C v n , where v n = α n γV x n + (I - α n μF)T(t n )y n . Then, from Lemma 2.6(c), we have
It follows from (3.40) that
Put σ n : = (τ - γL)α n (1 - β n ) and . Then (3.41) re-duces to formula
It is easily seen that and (using 3.39)
Hence, by Lemma 2.16, we conclude that as n → ∞. This completes the proof.
Remark 3.4. Note that Lemma 2.17 and Lemma 2.18 play an important role in the proof of Theorem 3.3. These are proved in the framework of the more general uniformly convex and 2-uniformly smooth Banach space. Lemma 2.17 is quite similar to the result of Yamada [10] which is obtained in a Hilbert space but we extended that result to a Banach space.
Remark 3.5. Theorem 3.3 extends the main result of Yao et al. [7] in the following ways:
-
(i)
A general system of variational inequalities (1.12) containing two inverse-strongly accre-tive mappings are extends to a general system of nonlinear variational inequalities (1.15) containing perturbed mappings.
-
(ii)
Theorem 3.3 for finding an element (G is defined as in Lemma 2.19) is more general the one of finding elements of of Yao et al. [7].
Furthermore, our method of the proof is very different from that in [7, Theorem 3.7] because it can be applied to solving the problem of finding a common element of the set of common fixed points of a one-parameter nonexpansive semigroup and the set of solutions of a general system of nonlinear variational inequalities containing perturbed mappings.
From Theorem 3.3, Lemma 2.4 and Example 2.5, we have the following result.
Corollary 3.6. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let F : C → X be a κ-Lipschitizian and η-strongly accretive operator with constants κ, η > 0, V : C → C be an L-Lipschitzian mapping with a constant L ≥ 0. Let and 0 ≤ γL < τ, where τ = μ(η - μκ2K2). Let be a u.a.r. nonexpansive semigroup from C into itself such that and least there exists a T(h) which is demicompact. Let Ψ i : C → X (i = 1, 2) be -inverse-strongly accretive and Φ i : C → X (i = 1, 2) be-inverse-strongly accretive. Assume that, where G is defined as in Lemma 2.19. For given x1 ∈ C, let {x n } be a sequence defined by
wherefor all i = 1, 2. Suppose that {t n } is a positive real divergent sequence such that , and{α n }, {β n } are sequences in [0, 1] satisfying the following conditions:
(C1) lim n→∞ α n = 0 and;
(C2) 0 < lim inf n→∞ β n ≤ lim sup n→∞ β n < 1.
Then, the sequence {x n } defined by (3.42) converges strongly to as n → ∞, which is the unique solution of the variational inequality
andis the solution of the problem (1.15), where .
References
Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and their Applications. Academic Press Inc., New York; 1980.
Noor MA: General variational inequalities and nonexpansive mappings. J Math Anal Appl 2007, 331: 810–822. 10.1016/j.jmaa.2006.09.039
Yao Y, Shahzad N: New methods with perturbations for non-expansive mappings in Hilbert spaces. Fixed Point Theory Appl 2011, 2011: 79. 10.1186/1687-1812-2011-79
Yao Y, Shahzad N: Strong convergence of a proximal point algorithm with general errors. Optim Lett 2012, 6: 621–628. 10.1007/s11590-011-0286-2
Yao Y, Liou YC, Chen CP: Algorithms construction for nonexpansive mappings and inverse-strongly monotone mappings. Taiwan J Math 2011, 15: 1979–1998.
Yao Y, Chen R, Liou YC: A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem. Mathematical and Computer Modelling 2012, 55: 1506–1515. 10.1016/j.mcm.2011.10.041
Yao Y, Liou Y-C, Kang SM, Yu Y: Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces. Nonlinear Anal Theory Method Appl 2011, 74: 6024–6034. 10.1016/j.na.2011.05.079
Yao Y, Liou Y-C, Kang SM: Two-step projection methods for a system of variational inequality problems in Banach spaces. J Glob Optim doi: 10.1007/s10898–011–9804–0
Takahashi W: Nonlinear functional analysis. Yokohama Publishers, Yokohama; 2000.
Yamada I: The hybrid steepest descent method for the variational inequality problems over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their applications. Studies in Computational Mathematics. Volume 8. Edited by: Butnariu, D, Censor, Y, Reich, S. North-Holland, Amsterdam; 2001:473–504.
Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J Math Anal Appl 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028
Tian M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal Theory Methods Appl 2010, 73: 689–694. 10.1016/j.na.2010.03.058
Yao Y, Noor MA, Noor KI, Liou YC: Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl Math 2010, 110: 1211–1224. 10.1007/s10440-009-9502-9
Browder FE: Nonexpansive nonlinear operators in a Banach space. Proc Natl Acad Sci USA 1965, 54(4):1041–1044. 10.1073/pnas.54.4.1041
Aleyner A, Censor Y: Best approximation to common fixed points of a semigroup of nonexpansive operator. Nonlinear Convex Anal 2005, 6(1):137–151.
Aleyner A, Reich S: An explicit construction of sunny nonexpansive retractions in Banach spaces. Fixed Point Theory Appl 2005, 2005(3):295–305.
Benavides TD, Acedo GL, Xu HK: Construction of sunny nonexpansive retractions in Banach spaces. Bull Austr Math Soc 2002, 66(1):9–16. 10.1017/S0004972700020621
Chen R, Song Y: Convergence to common fixed point of nonexpansive semigroup. J Comput Appl Math 2007, 200: 566–575. 10.1016/j.cam.2006.01.009
Reich S: Asymptotic behavior of contractions in Banach spaces. J Math Anal Appl 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3
Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions. Topol Methods Nonlinear Anal 1993, 2: 333–342.
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal Theory Methods Appl 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K
Lu LS: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces. J Math Anal Appl 1995, 194: 114–125. 10.1006/jmaa.1995.1289
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequence for one-parameter nonexpansive semigroup without Bochner integrals. J Math Anal Appl 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017
Bruck RE: Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans Am Math Soc 1973, 179: 251–262.
Nan Li X, Gu JS: Strong convergence of modified Ishikawa iteration for nonexpansive semigroup in Banach space. Nonlinear Anal Theory Methods Appl 2010, 73: 1085–1092. 10.1016/j.na.2010.04.040
Browder FE: Nonexpansive nonlinear operators in a Banach space. Proc Natl Acad Sci USA 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041
Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim 2002, 13: 938–945. 10.1137/S105262340139611X
Xu HK: Iterative algorithms for nonlinear operators. J Lond Math Soc 2002, 66(1):240–256. 10.1112/S0024610702003332
Jung JS: Some algorithms for finding fxed points and solutions of variational inequalities. Abstr Appl Anal 2012, 2012: 16. (Article ID 153456)
Cholamjiak P: A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces. Fixed Point Theory Appl 2009, 2009: 18. (Article ID 719360)
Cholamjiak P, Suantai S: A new hybrid algorithm for variational inclusions, generalized equilibrium problems and a finite family of quasi-nonexpansive mappings. Fixed Point Theory Appl 2009, 2009: 20. (Article ID 350979)
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The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No.55000613).
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PS and PK conceived the idea, designed the research and wrote the article; PS conducted the research. Both authors have read and approved the final manuscript.
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Sunthrayuth, P., Kumam, P. Iterative algorithms approach to a general system of nonlinear variational inequalities with perturbed mappings and fixed point problems for nonexpansive semigroups. J Inequal Appl 2012, 133 (2012). https://doi.org/10.1186/1029-242X-2012-133
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DOI: https://doi.org/10.1186/1029-242X-2012-133