Abstract
A new general iterative method for finding a common element of the set of solutions of variational inequality and the set of common fixed points of a countable family of nonexpansive mappings is introduced and studied. A strong convergence theorem of the proposed iterative scheme to a common fixed point of a countable family of nonexpansive mappings and a solution of variational inequality of an inverse strongly monotone mapping are established. Moreover, we apply our main result to obtain strong convergence theorems for a countable family of nonexpansive mappings and a strictly pseudocontractive mapping, and a countable family of uniformly kstrictly pseudocontractive mappings and an inverse strongly monotone mapping. Our main results improve and extend the corresponding result obtained by Klineam and Suantai (J Inequal Appl 520301, 16 pp, 2009).
Mathematics Subject Classification (2000): 47H09, 47H10
Keywords:
countable family of nonexpansive mappings; variational inequality; inverse strongly monotone mapping; strictly pseudocontractive mapping; countable family of uniformly kstrictly pseudocontractive mappings1 Introduction
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. In this paper, we always assume that a bounded linear operator A on H is strongly positive, that is, there is a constant
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on H:
where F is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. A mapping B of C into H is called monotone if 〈Bx  By, x  y〉 ≥ 0 for all x, y ∈ C. The variational inequality problem is to find x ∈ C such that 〈Bx, y  x〉 ≥ 0 for all y ∈ C. The set of solutions of the variational inequality is denoted by VI(C, B). A mapping B of C to H is called inverse strongly monotone if there exists a positive real number β such that 〈x  y, Bx  By〉 ≥ β Bx  By^{2 }for all x, y ∈ C.
Starting with an arbitrary initial x_{0 }∈ H, define a sequence {x_{n}} recursively by
It is proved by Xu [1] that the sequence {x_{n}} generated by (1.2) converges strongly to the unique solution of the minimization problem (1.1) provided the sequence {α_{n}} satisfies certain conditions.
On the other hand, Moudafi [2] introduced the viscosity approximation method for nonexpansive mappings. Let f be a contraction on H. Starting with an arbitrary initial x_{0 }∈ H, define a sequence {x_{n}} recursively by
where {σ_{n}} is a sequence in (0, 1). It is proved by Moudafi [2] and Xu [3] that under certain appropriate conditions imposed on {σ_{n}}, the sequence {x_{n}} generated by (1.3) strongly converges to the unique solution x* in C of the variational inequality
Recently, Marino and Xu [4] combined the iterative method (1.2) with the viscosity approximation method (1.3) and considered the following general iteration process:
and proved that if the sequence {α_{n}} satisfies appropriate conditions, the sequence {x_{n}} generated by (1.4) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem
where h is a potential function for γ f (i.e., h'(x) = γ f(x) for x ∈ H).
Chen, Zhang and Fan [5] introduced the following iterative process: x_{0 }∈ C,
where {α_{n}} ⊂ (0, 1) and {λ_{n}} ⊂ [a, b] for some a, b with 0 < a < b < 2β.
They proved that under certain appropriate conditions imposed on {α_{n}} and {λ_{n}}, the sequence {x_{n}} generated by (1.5) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping (say
Klineam and Suantai [6] modify the iterative methods (1.4) and (1.5) by proposing the following general iterative method: x_{0 }∈ C,
where P_{C }is the projection of H onto C, f is a contraction, A is a strongly positive linear bounded operator, B is a βinverse strongly monotone mapping, {α_{n}} ⊂ (0, 1) and {λ_{n}} ⊂ [a, b] for some a, b with 0 < a < b < 2β. They noted that when A = I and γ = 1, the iterative scheme (1.6) reduced to the iterative scheme (1.5).
Wangkeeree, Petrot and Wangkeeree [7] introduced the following iterative process:
where {α_{n}}and{β_{n}} ⊂ (0, 1) and T_{n }is a countable family of nonexpansive mappings, f is a contraction, and A is a strongly positive linear bounded operator. They proved that under certain appropriate conditions imposed on {α_{n}}, {β_{n}} and {T_{n}}, the sequence {x_{n}} converges strongly to
In this paper, motivated and inspired by Klineam and Suantai [6], we introduced the following iteration to find some solutions of variational inequality and fixed points of countable family of nonexpansive mappings in a Hilbert spaces H: x_{0 }∈ C,
where P_{C }is the projection of H onto C, f is a contraction, A is a strongly positive linear bounded operator, T_{n }is a countable family of nonexpansive mappings of C into itself, B is a βinverse strongly monotone mapping, {α_{n}} ⊂ (0, 1), and {λ_{n}} ⊂ [a, b] for some a, b with 0 < a < b < 2β.
2 Preliminaries
Let H be a real Hilbert space with inner product 〈·,·〉 and norm  · , and let C be a closed convex subset of H. We write x_{n }⇀ x to indicate that the sequence {x_{n}} converges weakly to x, and x_{n }→ x implies that {x_{n}} converges strongly to x. For every point x ∈ H, there exists a unique nearest point in C, denoted by P_{C}x, such that x  P_{C}x ≤ x  y for all y ∈ C and P_{C}x is called the metric projection of H onto C. We know that P_{C }is a nonexpansive mapping of H onto C. It is also known that P_{C }satisfies 〈x  y, P_{C}x  P_{C}y〉 ≥ P_{C}x  P_{C}y^{2 }for every x, y ∈ H. Moreover, P_{C}x is characterized by the properties: P_{C}x ∈ C and 〈x  P_{C}x, P_{C}x  y〉 ≥ 0 for all y ∈ C. In the context of the variational inequality problem, this implies that
A setvalued mapping T : H → 2^{H }is called monotone if for all x, y ∈ H, f ∈ Tx and g ∈ Ty imply 〈x  y, f  g〉 ≥ 0. A monotone mapping T : H → 2^{H }is maximal if the graph G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f ) ∈ H × H, 〈x  y, f  g〉 ≥ 0 for every (y, g) ∈ G(T) implies f ∈ Tx. Let A be an inverse strongly monotone mapping of C into H, and let N_{C}v be the normal cone to C at v ∈ C, i.e., N_{C}v = {w ∈ H : 〈v  u, w〉 ≥ 0, ∀u ∈ C}, and define
Then, T is maximal monotone and 0 ∈ Tv if and only if v ∈ V I(C, A).
Lemma 2.1 Let C be a closed convex subset of a real Hilbert space H. Given x ∈ H and y ∈ C, then
(i) y = P_{C}x if and only if the inequality 〈x  y, y  z〉 ≥ 0 for all z ∈ C,
(ii) P_{C }is nonexpansive,
(iii) 〈x  y, P_{C}x  P_{C}y〉 ≥ P_{C}x  P_{C}y^{2 }for all x, y ∈ H,
(iv) 〈x  P_{C}x, P_{C}x  y〉 ≥ 0 for all x ∈ H and y ∈ C.
Lemma 2.2 [4]Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient
Lemma 2.3 [8]Assume {a_{n}} is a sequence of nonnegative real numbers such that
where {γ_{n}} ⊂ (0, 1) and {δ_{n}} is a sequence in ℝ such that
(i)
(ii) lim sup_{n→∞ }δ_{n }/γ_{n }≤ 0 or
Then, lim _{n→∞ }a_{n }= 0.
Lemma 2.4 [9]Let C be a closed convex subset of a real Hilbert space H, and let T : C → C be a nonexpansive mapping such that F(T) ≠ ∅. If a sequence {x_{n}} in C such that x_{n }⇀ z and x_{n } Tx_{n }→ 0, then z = Tz.
To deal with a family of mappings, the following conditions are introduced: Let C be a subset of a real Banach space E, and let
Lemma 2.5 [10]Let C be a nonempty and closed subset of a Banach space E and let {T_{n}} be a family of mappings of C into itself which satisfies the AKTTcondition. Then, for each x ∈ C, {T_{n}x} converges strongly to a point in C. Moreover, let the mapping T be defined by
Then, for each bounded subset B of C,
In the sequel, we will write ({T_{n}}, T ) satisfies the AKTTcondition if {T_{n}} satisfies the AKTTcondition, and T is defined by Lemma 2.5 with
3 Main results
In this section, we prove a strong convergence theorem for a countable family of nonexpansive mappings.
Theorem 3.1 Let C be a closed convex subset of a real Hilbert space H, and let B : C → H be a βinverse strongly monotone mapping, also let A be a strongly positive linear bounded operator of H into itself with coefficient
for all n = 0, 1, 2, ..., where {α_{n}} ⊂ (0, 1) and {λ_{n}} ⊂ (0, 2β ). If {α_{n}} and {λ_{n}} are chosen so that λ_{n }∈ [a, b] for some a, b with 0 < a < b < 2β ,
Suppose that ({T_{n}}, T ) satisfies the AKTTcondition. Then, {x_{n}} converges strongly to q ∈ F, where q = P_{F }(γ f + I  A)(q) which solves the following variational inequality:
Proof. First, we show that the sequence {x_{n}} is bounded. Consider the mapping I λ_{n}B. Since B is a βinverse strongly monotone mapping, we have that for all x, y ∈ C,
For 0 < λ_{n }< 2β, implies that k(I  λ_{n}B)x  (I λ_{n}B)y^{2 }≤ x  y^{2}.
So, the mapping I  λ_{n}B is nonexpansive.
Put y_{n }= P_{C}(x_{n } λ_{n}Bx_{n}) for all n ≥ 0. Let u ∈ F. Then u = P_{C}(u  λ_{n}Bu).
From P_{C }is nonexpansive implies that
Since I  λ_{n}B is nonexpansive, we have that y_{n } u ≤ x_{n } u. Then
Since A is strongly positive linear bounded operator, we have
By contraction of f, we have
It follows from induction that
Therefore, {x_{n}} is bounded, so are {y_{n}}, {T_{n}y_{n}}, {Bx_{n}}, and {f (x_{n})}.
Next, we show that x_{n+1 } x_{n} → 0 and y_{n } T_{n}y_{n} → 0 as n → ∞.
Since P_{C }is nonexpansive, we also have
Since I  λ_{n}B is nonexpansive, we have
So we obtain
where L = max{sup_{n∈ℕ }AT_{n1}y_{n  1}, sup_{n∈ℕ }γ f (x_{n  1})} and M = sup{Bx_{n1} : n∈ℕ}.
Since {T_{n}} satisfies the AKTTcondition, we get that
From condition (C3), (C4) and by Lemma 2.3, we have x_{n+1 } x_{n} → 0.
For u ∈ F and u = P_{C}(u  λ_{n}Bu), we have
So, we obtain
where
Since α_{n }→ 0 and x_{n+1 } x_{n} → 0, we obtain Bx_{n } Bu → 0 as n → ∞.
Further, by Lemma 2.1, we have
So, we have
which implies
Since α_{n }→ 0, x_{n+1 } x_{n} → 0, and Bx_{n } Bu → 0, we obtain x_{n } y_{n} → 0 as n → ∞.
Next, we have
Since α_{n }→ 0 and {f (x_{n})}, {AT_{n}y_{n}} are bounded, we have x_{n+1 } T_{n}y_{n} → 0 as n → ∞. Since
it implies that x_{n } T_{n}y_{n} → 0 as n → ∞. Since
we obtain x_{n } T_{n}x_{n} → 0 as n → ∞. Moreover, from
it follows that y_{n } T_{n}y_{n} → 0 as n → ∞.
By y_{n } x_{n} → 0, T_{n}y_{n } x_{n} → 0 and Lemma 2.5, we have
Hence, lim_{n→∞ }Tx_{n } x_{n} = 0. Observe that P_{F }(γ f +I  A) is a contraction.
By Lemma 2.2, we have that
Then, Banach's contraction mapping principle guarantees that P_{F }(γ f +I  A) has a unique fixed point, say q ∈ H. That is, q = P_{F }(γ f + I  A)q. By Lemma 2.1, we obtain
Choose a subsequence
As
Since y_{n } T_{n}y_{n} → 0, we obtain
where N_{C}v is normal cone to C at v ∈ C, that is N_{C}v = {w ∈ H : 〈v  u, w〉 ≥ 0, ∀u ∈ C}. Then S is a maximal monotone. Let (v, w) ∈ G(S). Since w  Bv ∈ N_{C}v and y_{n }∈ C, we have 〈v  y_{n}, w  Bv〉 ≥ 0. On the other hand, by Lemma 2.1 and from y_{n }= P_{C}(x_{n } λ_{n}Bx_{n}), we have
Hence,
This implies 〈v  p, w〉 ≥ 0. Since S is maximal monotone, we have p ∈ S ^{1}0 and hence p ∈ V I(C, B). We obtain that p ∈ F. By (3.1), we have 〈(γ f  A)q, p  q〉 ≤ 0. It follows that
Finally, we prove x_{n }→ q. By y_{n } u ≤ x_{n } u and Schwarz inequality, we have
Since {x_{n}}, {f (x_{n})} and {T_{n}y_{n}} are bounded, we can take a constant η > 0 such that
for all n ≥ 0. It follows that
where β_{n }= 2〈T_{n}y_{n } q, γ f(q)  Aq〉 +ηα_{n}. By lim sup_{n→∞}〈(γ f  A)q, T_{n}y_{n } q〉 ≤ 0, we get lim sup _{n→∞ }β_{n }≤ 0. By Lemma 2.3 and (3.2), we can conclude that x_{n }→ q. This completes the proof. ■
Corollary 3.2 Let C be a closed convex subset of a real Hilbert space H, and let B : C → H be a βinverse strongly monotone mapping, also let f : C → C be a contraction with coefficient α(0 < α < 1). Let {T_{n}} be a countable family of nonexpansive mappings from a subset C into itself with
for all n = 0, 1, 2, ..., where {α_{n}} ⊂ (0, 1) and {λ_{n}} ⊂ (0, 2β ). If {α_{n}} and {λ_{n}} are chosen so that λ_{n }∈ [a, b] for some a, b with 0 < a < b < 2β,
Suppose that ({T_{n}}, T ) satisfies the AKTTcondition. Then {x_{n}} converges strongly to q ∈ F, where q = P_{F }(γ f + I  A)(q) which solves the following variational inequality:
Proof. Taking A = I and γ = 1 in Theorem 3.1, we get the results. ■
4 Applications
In this section, we apply the iterative scheme (1.8) and Theorem 3.1 for finding a common fixed point of countable family of nonexpansive mappings and strictly pseudocontractive mapping and inverse strongly monotone mapping.
A mapping T : C → C is called strictly pseudocontractive if there exists k with 0 ≤ k < 1 such that
If k = 0, then T is nonexpansive. Put B = I  T, where T : C → C is a strictly pseudocontractive mapping with k. Then, B is ((1  k)/2)inverse strongly monotone and B ^{1 }(0) = F(T). Hence, for all x, y ∈ C,
Conversely, since H is a real Hilbert space, we have
Thus, we have
Theorem 4.1 Let C be a closed convex subset of a real Hilbert space H, and let A be a strongly positive linear bounded operator of H into itself with coefficient
for all n = 0, 1, 2, ..., where {α_{n}} ⊂ [0, 1) and {λ_{n}} ⊂ [0, 1  β). If {α_{n}} and {λ_{n}} are chosen so that λ_{n }∈ [a, b] for some a, b with 0 < a < b < 1  β,
Suppose that ({T_{n}}, T) satisfies the AKTTcondition. Then, {x_{n}} converges strongly to q ∈ F, such that
Proof. Put B = I  S, then B is ((1  k)/2)inverse strongly monotone and F(S) = V I(C, B) and P_{C}(x_{n } λ_{n}Bx_{n}) = (1  λ_{n})x_{n }+λ_{n}Sx_{n}. Therefore, by Theorem 3.1, the conclusion follows. ■
Lemma 4.2 [9]Let T : C → H be a kstrictly pseudocontractive, then
(i) the fixed point set F(T) of T is closed convex so that the projection P_{F(T) }is well defined;
(ii) define a mapping S : C → H by
If μ ∈ [k, 1), then S is a nonexpansive mapping such that F(T) = F(S).
A family of mappings
Let {T_{n }: C → C} be a countable family of uniformly kstrictly pseudocontractions. Let
Corollary 4.3 Let C be a closed convex subset of a real Hilbert space H, and let B : C → H be a βinverse strongly monotone mapping, also let A be a strongly positive linear bounded operator of H into itself with coefficient
for all n = 0, 1, 2, ..., where {α_{n}} ⊂ (0, 1) and {λ_{n}} ⊂ (0, 2β). If {α_{n}} and {λ_{n}} are chosen so that λ_{n }∈ [a, b] for some a, b with 0 < a < b < 2β,
Then, {x_{n}} converges strongly to q ∈ F, where q = P_{F }(γ f + I  A)(q) which solves the following variational inequality:
Proof. Let {T_{n}} be a countable family of uniformly kstrictly pseudocontractions from a subset C into itself. Set S_{n }= μI + (1  μ)T_{n }where μ ∈ [k, 1). By Lemma 4.2, we have S_{n }is nonexpansive and F (S_{n}) = F (T_{n}). Therefore, by Theorem 3.1, the conclusion follows. ■
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
AB study and researched nonlinear analysis and also wrote this article. SS participated in the process of the study and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the Centre of Excellence in Mathematics for financial support under the project RG153022. The first author is also supported by the Graduate School, Chiang Mai University, Thailand.
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