Onjai-uea and Phuengrattana Fixed Point Theory and Applications (2015) 2015:16 DOI 10.1186/s13663-015-0265-x
RESEARCH
Open Access
A hybrid iterative method for common solutions of variational inequality problems and fixed point problems for single-valued and multi-valued mappings with applications Nawitcha Onjai-uea1 and Withun Phuengrattana1,2* *
Correspondence:
[email protected] 1 Department of Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand 2 Research Center for Pure and Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand
Abstract In this article, we propose a new iterative method for approximating a common element of the set of common fixed points of a finite family of k-strictly pseudononspreading single-valued mappings, the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems in Hilbert spaces. Furthermore, we prove that the proposed iterative method converges strongly to a common element of the above three sets, and we also apply our results to complementarity problems. Finally, we give two numerical examples to support our main result. MSC: 46C05; 47H09; 47H10 Keywords: fixed point; strictly pseudononspreading mappings; multi-valued mappings; variational inequality problems; Hilbert spaces
1 Introduction Let D be a nonempty subset of a real Hilbert space X. Let CB(D) denote the families of nonempty closed bounded subsets of D. The Hausdorff metric on CB(D) is defined by H(A, B) = max sup dist(x, B), sup dist(y, A) x∈A
for A, B ∈ CB(D),
y∈B
where dist(x, D) = inf{x – y : y ∈ D}. Let T : D → CB(D) be a multi-valued mapping. An element x ∈ D is said to be a fixed point of T if x ∈ Tx. The set of fixed points of T will be denoted by F(T). A multi-valued mapping T : D → CB(D) is called (i) nonexpansive if H(Tx, Ty) ≤ x – y for all x, y ∈ D; (ii) quasi-nonexpansive if F(T) = ∅ and H(Tx, Tp) ≤ x – p
for all x ∈ D and p ∈ F(T);
© 2015 Onjai-uea and Phuengrattana; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
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(iii) L-Lipschitzian if there exists L > such that H(Tx, Ty) ≤ Lx – y
for all x, y ∈ D.
It is clear that every nonexpansive multi-valued mapping T with F(T) = ∅ is quasinonexpansive. It is known that if T is a quasi-nonexpansive multi-valued mapping, then F(T) is closed. In general, the fixed point set of a quasi-nonexpansive multi-valued mapping T is not necessary to be convex. In the next lemma, we show that F(T) is convex under the assumption that Tp = {p} for all p ∈ F(T). The proof of this fact is very easy, therefore we omit it. Lemma . Let D be a nonempty closed convex subset of a real Hilbert space X. Assume that T : D → CB(D) is a quasi-nonexpansive multi-valued mapping. If Tp = {p} for all p ∈ F(T), then F(T) is convex. The fixed point theory of multi-valued mappings is much more complicated and harder than the corresponding theory of single-valued mappings. However, some classical fixed point theorems for single-valued mappings have already been extended to multi-valued mappings; see [, ]. The recent fixed point results for multi-valued mappings can be found in [–] and the references cited therein. For a single-valued case, a mapping t : D → D is called nonexpansive if tx – ty ≤ x – y for all x, y ∈ D. An element x ∈ D is called a fixed point of t if x = tx. Recall that a singlevalued mapping t : D → D is said to be nonspreading [, ] if tx – ty ≤ x – y + x – tx, y – ty for all x, y ∈ D. In , Kurokawa and Takahashi [] obtained a weak mean ergodic theorem of Baillon’s type for nonspreading single-valued mappings in Hilbert spaces. They also proved a strong convergence theorem for this class of single-valued mappings using an idea of mean convergence in Hilbert spaces. Later in , Osilike and Isiogugu [] introduced a new class of nonspreading type of mappings, which is more general than the class studied in [], as follows: A single-valued mapping t : D → D is called k-strictly pseudononspreading if there exists k ∈ [, ) such that tx – ty ≤ x – y + k (I – t)x – (I – t)y + x – tx, y – ty
for all x, y ∈ D.
Obviously, every nonspreading mapping is k-strictly pseudononspreading. Osilike and Isiogugu proved weak and strong convergence theorems for this mapping in Hilbert spaces. They also provided a property of a k-strictly pseudononspreading mapping as follows. Lemma . ([]) Let D be a nonempty closed convex subset of a real Hilbert space X, and let t : D → D be a k-strictly pseudononspreading mapping. If F(t) = ∅, then it is closed and convex. Many researchers studied the existence and convergence theorems of those singlevalued mappings in both Hilbert spaces and Banach spaces (e.g., see [–]).
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The problem of finding common fixed points has been extensively studied by mathematicians. To deal with a fixed point problem of a family of nonlinear mappings, several ways have appeared in the literature. For example, in , Atsushiba and Takahashi [] introduced a new mapping, called W -mapping, for finding a common fixed point of a finite family of nonexpansive mappings. This mapping is defined as follows. Let {ti }N i= be a finite family of nonexpansive mappings of D into itself. Let W : D → D be a mapping defined by U = β t + ( – β )I, U = β t U + ( – β )I, U = β t U + ( – β )I, .. . UN– = βN– tN– UN– + ( – βN– )I, W = UN = βN tN UN– + ( – βN )I, where I is the identity mapping of D and {βi }N i= is a sequence in (, ). This mapping is called the W -mapping generated by t , t , . . . , tN and β , β , . . . , βN . They also proved that if X is a strictly convex Banach space, then F(W ) = N i= F(ti ). In , Kangtunyakarn and Suantai [] introduced a new concept of the S-mapping for finding a common fixed point of a finite family of nonexpansive mappings as follows: Let {ti }N i= be a finite family of nonexpansive mappings of D into itself. Let S : D → D be a mapping defined by V = δ t + δ I + δ I, V = δ t V + δ V + δ I, V = δ t V + δ V + δ I, .. . VN– = δN– tN– VN– + δN– VN– + δN– I, S = VN = δN tN VN– + δN VN– + δN I, j
j
j
where I is the identity mapping of D and δj = (δ , δ , δ ) ∈ [, ]×[, ]×[, ], j = , , . . . , N , j j j where δ + δ + δ = for all j = , , . . . , N . This mapping is called the S-mapping generated by t , t , . . . , tN and δ , δ , . . . , δN . They proved the following lemma important for our results. Lemma . Let D be a nonempty closed convex subset of a strictly convex Banach space X. N Let {ti }N i= be a finite family of nonexpansive mappings of D into itself with i= F(ti ) = ∅, j j j j j j j and let δj = (δ , δ , δ ) ∈ [, ] × [, ] × [, ], j = , , . . . , N , where δ + δ + δ = , δ ∈ (, ) j j for all j = , , . . . , N – , δN ∈ (, ], and δ , δ ∈ [, ) for all j = , , . . . , N . Let S be the Smapping generated by t , t , . . . , tN and δ , δ , . . . , δN . Then S is a nonexpansive mapping and F(S) = N i= F(ti ).
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Applications of W -mappings and S-mappings for fixed point problems can be found in [–]. Let B : D → X be a nonlinear mapping. The variational inequality problem is to find a point u ∈ D such that Bu, v – u ≥
for all v ∈ D.
(.)
The set of solutions of (.) is denoted by VI(D, B). A mapping B : D → X is called φ-inverse strongly monotone [] if there exists a positive real number φ such that x – y, Bx – By ≥ φBx – By
for all x, y ∈ D.
Variational inequality theory, which was first introduced by Stampacchia [] in , emerged as an interesting and fascinating branch of applicable mathematics with a wide range of applications in economics, industry, network analysis, optimizations, pure and applied sciences etc. In recent years, much attention has been given to developing efficient iterative methods for treating solution problems of variational inequalities (e.g., see [– ]). In , Takahashi and Toyoda [] introduced an iterative method for finding a common element of the set of fixed points of nonexpansive single-valued mappings and the set of solutions of variational inequalities for φ-inverse strongly monotone mappings in Hilbert spaces. Recently, by using the concept of S-mapping, Kangtunyakarn [] introduced a new method for finding a common element of the set of fixed points of k-strictly pseudononspreading single-valued mappings and the set of solutions of variational inequality problems in Hilbert spaces. Question A How can we construct an iteration process for finding a common element of the set of common fixed points of a finite family of k-strictly pseudononspreading singlevalued mappings, the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems? In the recent years, the problem of finding a common element of the set of fixed points of single-valued mappings and multi-valued mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many researchers. However, no researchers have studied the problem of finding a common element of three sets, i.e., the set of common fixed points of a finite family of single-valued mappings, the set of common fixed points of a finite family of multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems. In this article, motivated by [] and the research described above, we propose a new hybrid iterative method for finding a common element of the set of a common fixed point of a finite family of k-strictly pseudononspreading single-valued mappings, the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems in Hilbert spaces and provide an affirmative answer to Question A.
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2 Preliminaries In this section, we give some useful lemmas for proving our main results. Let D be a nonempty closed convex subset of a real Hilbert space X. Let PD be the metric projection of X onto D, i.e., for x ∈ X, PD x satisfies the property x – PD x = miny∈D x – y. It is well known that PD is a nonexpansive mapping of X onto D. Lemma . ([]) Let X be a Hilbert space, let D be a nonempty closed convex subset of X, and let B be a mapping of D into X. Let u ∈ D. Then, for λ > , u = PD (I – λB)u
⇐⇒
u ∈ VI(D, B).
Lemma . ([]) Let D be a nonempty closed convex subset of a real Hilbert space X, and let PD : X → D be the metric projection. Given x ∈ X and z ∈ D, then z = PD x if and only if the following holds: x – z, y – z ≤ for all y ∈ D. Lemma . ([]) Let D be a nonempty closed convex subset of a real Hilbert space X, and let PD : X → D be the metric projection. Then the following inequality holds: y – PD x + x – PD x ≤ x – y
for all x ∈ X and y ∈ D.
Lemma . ([]) Let X be a real Hilbert space. Then x – y = x – x, y + y
for all x, y ∈ X.
Lemma . ([]) Let X be a Hilbert space. Let x , x , . . . , xN ∈ X and α , α , . . . , αN be real numbers such that N i= αi = . Then N N αi x i = αi xi – αi αj xi – xj . i=
i=
≤i,j≤N
Lemma . ([]) Let D be a nonempty closed convex subset of a real Hilbert space X. Given x, y, z ∈ X and b ∈ R, the set u ∈ D : y – u ≤ x – u + z, u + b is closed and convex. Lemma . ([]) In a strictly convex Banach space X, if x = y = λx + ( – λ)y for all x, y ∈ X and λ ∈ (, ), then x = y. The following lemma obtained by Kangtunyakarn [] is useful for our results.
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Lemma . Let D be a nonempty closed convex subset of a Hilbert space X. Let t : D → D be a k-strictly pseudononspreading mapping with F(t) = ∅. Then F(t) = VI(D, I – t). Remark . ([]) From Lemmas . and ., we have
F(t) = F PD I – λ(I – t)
for all λ > .
3 Main results In this section, we prove a strong convergence theorem which solves the problem of finding a common element of the set of common fixed points of a finite family of k-strictly pseudononspreading single-valued mappings, the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems in Hilbert spaces. Before starting the main theorem of this section, we need to prove the following useful lemma in Hilbert spaces. Lemma . Let D be a nonempty closed convex subset of a real Hilbert space X, and let {ti }N i= be a finite family of k-strictly pseudononspreading single-valued mappings of D into itself such that N i= F(ti ) = ∅. Let Ri : D → D be defined by Ri x = PD (I – λ(I – ti ))x for all x ∈ D, λ ∈ (, – k), and i = , , . . . , N . Suppose that β , β , . . . , βN are real numbers such that < βi < for all i = , , . . . , N – and < βN ≤ . Let W be the W -mapping generated by R , R , . . . , RN and β , β , . . . , βN . Then the following hold: (i) W is quasi-nonexpansive; N (ii) F(W ) = N i= F(ti ) = i= F(Ri ). Proof (i) For each x ∈ D and z ∈
N
i= F(ti ),
Ri x – z = PD I – λ(I – ti ) x – z
≤ I – λ(I – ti ) x – z = (x – z) – λ(I – ti )x
= x – z – λ x – z, (I – ti )x + λ I – ti .
(.)
By ti is k-strictly pseudononspreading, we have
ti x – ti z ≤ x – z + k (I – ti )x – (I – ti )z + (I – ti )x, (I – ti )z = x – z + k (I – ti )x .
(.)
Since
ti x – ti z = I – (I – ti ) x – I – (I – ti ) z
= (x – z) – (I – ti )x – (I – ti )z
= x – z – x – z, (I – ti )x + (I – ti )x , it follows by (.) that
( – k)(I – ti )x ≤ x – z, (I – ti )x .
(.)
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Therefore, by (.), we have Ri x – z ≤ x – z – ( – k)λ(I – ti )x + λ (I – ti )x = x – z – λ( – k – λ)(I – ti )x ≤ x – z . This implies that Ri x – z ≤ x – z
for all i = , , . . . , N.
(.)
Let j ∈ {, , . . . , N}, we get Uj x – z = βj Rj Uj– x + ( – βj )x – z ≤ βj Rj Uj– x – z + ( – βj )x – z ≤ βj Uj– x – z + ( – βj )x – z. So, we have Wx – z = UN x – z ≤ βN UN– x – z + ( – βN )x – z
≤ βN βN– UN– x – z + ( – βN– )x – z + ( – βN )x – z = βN βN– UN– x – z + ( – βN βN– )x – z .. . ≤ βN βN– · · · β U x – z + ( – βN βN– · · · β )x – z
≤ βN βN– · · · β β x – z + ( – β )x – z + ( – βN βN– · · · β )x – z = x – z. Thus, W is a quasi-nonexpansive mapping. N (ii) Since N i= F(ti ) ⊂ F(W ) is trivial, it suffices to show that F(W ) ⊂ i= F(ti ). To show N this, we suppose that p ∈ F(W ) and z ∈ i= F(ti ). Then we have p – z = Wp – z = βN (RN UN– p – z) + ( – βN )(p – z) ≤ βN RN UN– p – z + ( – βN )p – z ≤ βN UN– p – z + ( – βN )p – z ≤ βN βN– RN– UN– p – z + ( – βN βN– )p – z .. .
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≤ βN βN– · · · β R U p – z + ( – βN βN– · · · β )p – z ≤ βN βN– · · · β U p – z + ( – βN βN– · · · β )p – z ≤ βN βN– · · · β β R U p – z + ( – βN βN– · · · β β )p – z ≤ βN βN– · · · β β U p – z + ( – βN βN– · · · β β )p – z ≤ βN βN– · · · β β β R p – z + ( – βN βN– · · · β β β )p – z ≤ p – z.
(.)
This shows that p – z = βN βN– · · · β β (R p – z) + ( – β )(p – z) + ( – βN βN– · · · β )p – z. Thus, p – z = β (R p – z) + ( – β )(p – z). Again by (.), we have p – z = R p – z = β (R p – z) + ( – β )(p – z). This implies by Lemma . that R p = p and hence U p = p. Again by (.), we get p – z = βN βN– · · · β β (R U p – z) + ( – β )(p – z) + ( – βN βN– · · · β )p – z, and hence p – z = β (R U p – z) + ( – β )(p – z).
(.)
By (.), we get p – z = R U p – z. From U p = p and (.), we have p – z = R p – z = β (R U p – z) + ( – β )(p – z). This implies by Lemma . that R p = p and hence U p = p. By continuing this process, we can conclude that Ri p = p and Ui p = p for all i = , , . . . , N – . Since p – RN p ≤ p – Wp + Wp – RN p = p – Wp + ( – βN )p – RN p,
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which yields that p = RN p since p ∈ F(W ). Hence p = Ri p for all i = , , . . . , N and thus p∈ N i= F(Ri ). From Remark ., we have
F(ti ) = F PD I – λ(I – ti ) = F(Ri ) for all i = , , . . . , N. N N This implies that N i= F(Ri ) = i= F(ti ), and hence p ∈ i= F(ti ). Therefore, F(W ) = N F(t ). This completes the proof. i i= We now prove our main theorem. Theorem . Let D be a nonempty closed convex subset of a real Hilbert space X. Let {ti }N i= be a finite family of continuous and k-strictly pseudononspreading mappings of D into itself, let {Ti }N i= be a finite family of quasi-nonexpansive and L-Lipschitzian mappings N from D into CB(D) with Ti p = {p} for all i = , , . . . , N , p ∈ N i= F(Ti ), and let {Bi }i= be a finite family of φi -inverse strongly monotone mappings from D into X. Let Ri : D → D be defined by Ri x = PD (I – λ(I – ti ))x for all x ∈ D, λ ∈ (, ), and i = , , . . . , N . Suppose that β , β , . . . , βN are real numbers such that < βi < for all i = , , . . . , N – and < βN ≤ . Let W : D → D be the W -mapping generated by R , R , . . . , RN and β , β , . . . , βN . Let Gi : D → D be defined by Gi x = PD (I – ηBi )x for all x ∈ D, η ∈ (, φi ), and i = , , . . . , N . j j j j j j j Suppose δj = (δ , δ , δ ) ∈ [, ] × [, ] × [, ], j = , , . . . , N , where δ + δ + δ = , δ ∈ j j (, ) for all j = , , . . . , N – , δN ∈ (, ], and δ , δ ∈ [, ) for all j = , , . . . , N . Let S : D → D be the S-mapping generated by G , G , . . . , GN and δ , δ , . . . , δN . Assume that N N F := N i= F(ti ) ∩ i= F(Ti ) ∩ i= VI(D, Bi ) = ∅. Let x ∈ D with C = D, and let {xn }, {yn }, and {zn } be sequences defined by yn = αn() xn + αn() Wxn + αn() Sxn , zn = γn() yn +
N
γn(i) qn(i) ,
qn(i) ∈ Ti yn ,
i=
(.)
Cn+ = p ∈ Cn : zn – p ≤ yn – p ≤ xn – p , xn+ = PCn+ x ,
n ∈ N,
where {αn() }, {αn() }, {αn() }, {γn(i) } (i = , , . . . , N ) are sequences in (, ) satisfying the following conditions: (i) αn() + αn() + αn() = , limn→∞ αn() = , and < a ≤ αn() , αn() < ; (i) (ii) < b ≤ γn(i) < for all i = , , . . . , N and N i= γn = . Then {xn }, {yn }, and {zn } converge strongly to u = PF x . Proof We shall divide our proof into steps. Step . We show that PCn+ x is well defined for every x ∈ X. Let x, y ∈ X. Since Bi is a φi -inverse strongly monotone mapping and η ∈ (, φi ), for i = , , . . . , N , we get that Gi x – Gi y = PC (I – ηBi )x – PC (I – ηBi )y ≤ (x – y) – η(Bi x – Bi y)
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≤ x – y – ηx – y, Bi x – Bi y + η Bi x – Bi y ≤ x – y – ηφi Bi x – Bi y + η Bi x – Bi y = x – y – η(φi – η)Bi x – Bi y ≤ x – y . This shows that Gi = PD (I – ηBi ) is a nonexpansive mapping for all i = , , . . . , N . By Lemma ., the closedness and convexity of F(Gi ), we have that VI(D, Bi ) = F(PD (I –ηBi )) = F(Gi ) is closed and convex for all i = , , . . . , N . So, N i= VI(D, Bi ) is closed and convex. N By Lemmas . and ., we also know that i= F(Ti ) and N i= F(ti ) are closed and con N N F(t ) ∩ F(T ) ∩ VI(D, B ) is also closed and convex. By vex. Hence, F := N i i i i= i= i= Lemma ., we observe that Cn is closed and convex. Let p ∈ F . Since Gi is nonexpansive and ti is k-strictly pseudononspreading for all i = , , . . . , N , it implies by Lemmas . and . that p ∈ F(S) and p ∈ F(W ). So, we have N () (i) (i) γn qn – p zn – p = γn yn + i=
≤ γn() yn – p +
N
γn(i) qn(i) – p
i=
≤ γn() yn – p +
N
γn(i) H(Ti yn , Ti p)
i=
≤ γn() yn – p +
N
γn(i) yn – p
i=
= yn – p = αn() xn + αn() Wxn + αn() Sxn – p ≤ αn() xn – p + αn() Wxn – p + αn() Sxn – p ≤ xn – p. This shows that p ∈ Cn+ and hence F ⊂ Cn+ ⊂ Cn . Therefore, PCn+ x is well defined. Step . We show that limn→∞ xn = q for some q ∈ D. Since F is a nonempty closed convex subset of a real Hilbert space X, there exists a unique ν ∈ F such that ν = PF x . From xn = PCn x and xn+ ∈ Cn+ ⊂ Cn , for all n ∈ N, we get that xn – x ≤ xn+ – x for all n ∈ N. On the other hand, by F ⊂ Cn , we obtain that xn – x ≤ ν – x
for all n ∈ N.
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This implies that {xn } is bounded and nondecreasing. So, limn→∞ xn – x exists. For m > n ∈ N, we have xm = PCm x ∈ Cm ⊂ Cn . It implies by Lemma . that xm – xn ≤ xm – x – xn – x
for all n ∈ N.
Since limn→∞ xn – x exists, it implies that {xn } is a Cauchy sequence. Hence, there exists an element q ∈ D such that limn→∞ xn = q. Step . We show that q ∈ N i= F(Ti ). From Step , we have lim xn+ – xn = .
(.)
n→∞
Since xn+ ∈ Cn+ , we get that zn – xn ≤ zn – xn+ + xn+ – xn ≤ xn – xn+ + xn+ – xn = xn+ – xn and yn – xn ≤ yn – xn+ + xn+ – xn ≤ xn – xn+ + xn+ – xn ≤ xn+ – xn . This implies by (.) that lim zn – xn =
(.)
lim yn – xn = .
(.)
n→∞
and
n→∞
Thus, limn→∞ zn = q and limn→∞ yn = q. Let p ∈ F . By Lemma . and the definition of zn , for each j = , , . . . , N , we have N () (i) (i) γn qn – p zn – p = γn yn +
i=
N
() = γn (yn – p) + γn(i) qn(i) – p i=
≤ γn() yn – p +
N
γn(i) qn(i) – p – γn() γn(j) qn(j) – yn
i=
≤ γn() yn – p +
N i=
γn(i) H(Ti yn , Ti p) – γn() γn(j) qn(j) – yn
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≤ γn() yn – p +
N i=
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γn(i) yn – p – γn() γn(j) qn(j) – yn
– yn ≤ xn – p – γ () γ (j) q(j) – yn . = yn – p
– γn() γn(j) qn(j) n
n
n
By condition (ii), it implies that b qn(j) – yn ≤ γn() γn(j) qn(j) – yn ≤ xn – p – zn – p
≤ xn – zn xn – zn + zn – p . Thus, by (.), we have lim qn(j) – yn = for all j = , , . . . , N.
(.)
n→∞
For each i = , , . . . , N , we get
dist(q, Ti q) ≤ q – yn + yn – qn(i) + dist qn(i) , Ti q ≤ q – yn + yn – qn(i) + H(Ti yn , Ti q) ≤ q – yn + yn – qn(i) + Lyn – q = ( + L)q – yn + yn – qn(i) . Since limn→∞ yn = q, it implies by (.) that dist(q, Ti q) = for all i = , , . . . , N. This shows that q ∈ Ti q for all i = , , . . . , N , and hence q ∈ Step . We show that q ∈ N i= VI(D, Bi ). For p ∈ F , we have
N
i= F(Ti ).
yn – p ≤ αn() xn – p + αn() Wxn – p + αn() Sxn – p – αn() αn() Wxn – Sxn ≤ xn – p – αn() αn() Wxn – Sxn . This implies by condition (i) that a Wxn – Sxn ≤ αn() αn() Wxn – Sxn ≤ xn – p – yn – p
≤ xn – yn xn – p + yn – p . Then, by (.), we have lim Wxn – Sxn = .
n→∞
(.)
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Since yn – Wxn = αn() xn + αn() Wxn + αn() Sxn – Wxn = αn() (xn – Wxn ) + αn() (Sxn – Wxn ) ≤ αn() xn – Wxn + αn() Sxn – Wxn , it follows by condition (i) and (.) that lim yn – Wxn = .
(.)
n→∞
From (.) and (.), we obtain that xn – Wxn ≤ xn – yn + yn – Wxn →
as n → ∞.
(.)
Since yn – xn = αn() (Wxn – xn ) + αn() (Sxn – xn ) and < a < αn() < , we get aSxn – xn ≤ αn() Sxn – xn ≤ yn – xn + αn() Wxn – xn . This implies by (.) and (.) that lim Sxn – xn = .
(.)
n→∞
Since xn → q ∈ D as n → ∞, it follows by (.) and the nonexpansiveness of S that Sq – q ≤ Sq – Sxn + Sxn – xn + xn – q ≤ xn – q + Sxn – xn →
as n → ∞.
This shows that q ∈ F(S). Since PD (I – ηBi )x = Gi x for all x ∈ D and i = , , . . . , N , by Lemma ., we have VI(D, Bi ) = F(PD (I – ηBi )) = F(Gi ) for all i = , , . . . , N . By Lemma ., we obtain
F(S) =
N i=
F(Gi ) =
N
VI(D, Bi ).
i=
Thus, q ∈ N i= VI(D, Bi ). Step . We show that q ∈ N i= F(ti ). Since ti is continuous for all i = , , . . . , N , it follows that PD (I – λ(I – ti )) is continuous for all i = , , . . . , N . So, W is continuous. This implies by xn → q that Wxn → Wq as n → ∞. Then, by (.), we have Wq – q ≤ Wq – Wxn + Wxn – xn + xn – q → as n → ∞. This shows that q ∈ F(W ). By Lemma ., we have q ∈ Step . Finally, we show that q = u = PF x .
N
i= F(ti ).
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Since xn = PCn x and F ⊂ Cn , we obtain x – xn , xn – p ≥ for all p ∈ F .
(.)
Taking limits in the above inequality, we get x – q, q – p ≥
for all p ∈ F .
This shows that q = PF x = u. By Step to Step , we conclude that {xn }, {yn }, and {zn } converge strongly to u = PF x . This completes the proof. As a direct consequence of Theorem ., we have the following two corollaries. Corollary . Let D be a nonempty closed convex subset of a real Hilbert space X. Let {ti }N i= be a finite family of continuous and k-strictly pseudononspreading mappings of D into itself, and let {Ti }N i= be a finite family of quasi-nonexpansive and L-Lipschitzian mappings from D into CB(D) with Ti p = {p} for all i = , , . . . , N , p ∈ N i= F(Ti ). Let Ri : D → D be defined by Ri x = PD (I – λ(I – ti ))x for all x ∈ D, λ ∈ (, ), and i = , , . . . , N . Suppose that β , β , . . . , βN are real numbers such that < βi < for all i = , , . . . , N – and < βN ≤ . Let W : D → D be the W -mapping generated by R , R , . . . , RN and β , β , . . . , βN . Assume N that F := N i= F(ti ) ∩ i= F(Ti ) = ∅. Let x ∈ D with C = D, and let {xn }, {yn }, and {zn } be sequences defined by
yn = αn() + αn() xn + αn() Wxn , zn = γn() yn +
N
γn(i) qn(i) ,
qn(i) ∈ Ti yn ,
i=
Cn+ = p ∈ Cn : zn – p ≤ yn – p ≤ xn – p , xn+ = PCn+ x ,
n ∈ N,
where {αn() }, {αn() }, {αn() }, {γn(i) } (i = , , . . . , N) are sequences in (, ) satisfying the following conditions: (i) αn() + αn() + αn() = , limn→∞ αn() = , and < a ≤ αn() , αn() < ; (i) (ii) < b ≤ γn(i) < for all i = , , . . . , N and N i= γn = . Then {xn }, {yn }, and {zn } converge strongly to u = PF x . Proof Let Bi x = for all x ∈ D and i = , , . . . , N in Theorem .. Then we obtain that Sxn = xn for all n ∈ N. Therefore the conclusion follows. Corollary . Let D be a nonempty closed convex subset of a real Hilbert space X. Let t : D → D be a continuous and k-strictly pseudononspreading mapping, let T : D → CB(D) be a quasi-nonexpansive and L-Lipschitzian mapping with Tp = {p} for all p ∈ F(T), and let B : D → X be a φ-inverse strongly monotone mapping. Assume that F := F(t) ∩ F(T) ∩ VI(D, B) = ∅. Let x ∈ D with C = D, and let {xn }, {yn }, and {zn } be sequences defined by
yn = αn() xn + αn() PD I – λ(I – t) xn + αn() PD (I – ηB)xn ,
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zn = γn() yn + γn() qn , qn ∈ Tyn , Cn+ = p ∈ Cn : zn – p ≤ yn – p ≤ xn – p , xn+ = PCn+ x ,
n ∈ N,
where λ ∈ (, ), η ∈ (, φ), and {αn() }, {αn() }, {αn() }, {γn() }, {γn() } are sequences in (, ) satisfying the following conditions: (i) αn() + αn() + αn() = , limn→∞ αn() = , and < a ≤ αn() , αn() < ; (ii) < b ≤ γn() , γn() < and γn() + γn() = . Then {xn }, {yn }, and {zn } converge strongly to u = PF x . Proof In Theorem ., put N = , t = t, T = T, B = B, β = , and δ = . Then W = PD (I – λ(I – t)) and S = PD (I – ηB). Hence, we obtain the desired result from Theorem .. Remark . It is known that the class of k-strictly pseudononspreading mappings contains the classes of nonexpansive mappings and nonspreading mappings. Thus, Lemma ., Theorem ., Corollaries . and . can be applied to these classes of mappings.
4 Applications to complementarity problems In this section, we apply our results to complementarity problems in Hilbert spaces. Let D be a nonempty closed convex cone in a real Hilbert space X, i.e., a nonempty closed set with rD + sD ⊂ D for all r, s ∈ [, ∞). The polar of D is the set D∗ = y ∈ X : x, y ≥ , ∀x ∈ D . Let B : D → X be a nonlinear mapping. The complementarity problem is to find an element u ∈ D such that Bu ∈ D∗
and
u, Bu = .
(.)
The set of solutions of (.) is denoted by CP(D, B). A complementarity problem is a special case of a variational inequality problem. The following lemma indicates the equivalence between the complementarity problem and the variational inequality problem. The proof of this fact can be found in []; for convenience of the readers, we include the details. Lemma . Let D be a nonempty closed convex cone in a real Hilbert space X, and let D∗ be the polar of D. Let B be a mapping of D into X. Then VI(D, B) = CP(D, B). Proof Let x ∈ VI(D, B). Then we have u – x, Bx ≥ for all u ∈ D.
(.)
In particular, if u = , we have x, Bx ≤ . If u = λx with λ > , we have λx – x, Bx = (λ –)x, Bx ≥ and hence x, Bx ≥ . Therefore, x, Bx = . Next, we show that Bx ∈ D∗ .
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To show this, suppose not. Then there exists u ∈ D such that u , Bx < . By (.), we obtain u – x, Bx ≥ . So, we get > u , Bx ≥ x, Bx = . This is a contradiction. Thus, Bx ∈ D∗ . So, x ∈ CP(D, B). Conversely, let x ∈ CP(D, B). Then we have Bx ∈ D∗
and
x, Bx = .
For any u ∈ D, we get u – x, Bx = u, Bx – x, Bx = u, Bx ≥ . Thus, x ∈ VI(D, B). This completes the proof.
Theorem . Let D be a nonempty closed convex cone in a real Hilbert space X. Let {ti }N i= be a finite family of continuous and k-strictly pseudononspreading mappings of D into itself, let {Ti }N i= be a finite family of quasi-nonexpansive and L-Lipschitzian mappings N from D into CB(D) with Ti p = {p} for all i = , , . . . , N , p ∈ N i= F(Ti ), and let {Bi }i= be a finite family of φi -inverse strongly monotone mappings from D into X. Let Ri : D → D be defined by Ri x = PD (I – λ(I – ti ))x for all x ∈ D, λ ∈ (, ), and i = , , . . . , N . Suppose that β , β , . . . , βN are real numbers such that < βi < for all i = , , . . . , N – and < βN ≤ . Let W : D → D be the W -mapping generated by R , R , . . . , RN and β , β , . . . , βN . Let Gi : D → D be defined by Gi x = PD (I – ηBi )x for all x ∈ D, η ∈ (, φi ), j j j and i = , , . . . , N . Suppose δj = (δ , δ , δ ) ∈ [, ] × [, ] × [, ], j = , , . . . , N , where j j j j j j δ + δ + δ = , δ ∈ (, ) for all j = , , . . . , N – , δN ∈ (, ], and δ , δ ∈ [, ) for all j = , , . . . , N . Let S : D → D be the S-mapping generated by G , G , . . . , GN and δ , δ , . . . , δN . N N Assume that F := N i= F(ti ) ∩ i= F(Ti ) ∩ i= CP(D, Bi ) = ∅. Let x ∈ D with C = D, and let {xn }, {yn }, and {zn } be sequences defined by yn = αn() xn + αn() Wxn + αn() Sxn , zn = γn() yn +
N
γn(i) qn(i) ,
qn(i) ∈ Ti yn ,
i=
Cn+ = p ∈ Cn : zn – p ≤ yn – p ≤ xn – p , xn+ = PCn+ x ,
n ∈ N,
where {αn() }, {αn() }, {αn() }, {γn(i) } (i = , , . . . , N) are sequences in (, ) satisfying the following conditions:
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(i) αn() + αn() + αn() = , limn→∞ αn() = , and < a ≤ αn() , αn() < ; (i) (ii) < b ≤ γn(i) < for all i = , , . . . , N and N i= γn = . Then {xn }, {yn }, and {zn } converge strongly to u = PF x . Proof By Lemma ., we get that VI(D, Bi ) = CP(D, Bi ) for all i = , , . . . , N.
Then we obtain the result.
5 Numerical results In this section, we give two numerical examples to support our main result. Example . We consider the nonempty closed convex subset D = [, ] of the real Hilbert space R. Define two single-valued mappings t and t on [, ] as follows: t x = – x,
t x = –
x.
Also we define two multi-valued mappings T and T on [, ] as follows: T x =
x x , ,
x T x = , .
For i = , , let Bi : [, ] → [, ] ⊂ R be defined by Bi x =
x i.
Let W be the W -mapping generated by R , R and β , β , where Ri x = P[,] (I – (I – ti ))x for all i = , , and β = β = . Let S be the S-mapping generated by G , G and δ , δ , where Gi x = P[,] (I – Bi )x for all i = , , and δ = δ = ( , , ). Let {xn }, {yn }, and {zn } , αn() = n– , αn() = n– , γn() = + n , γn() = n– , be generated by (.), where αn() = n n n n γn() = n– for all n ∈ N. Then the sequences {x }, {y }, and {z } converge strongly to , n n n n where {} = i= F(ti ) ∩ i= F(Ti ) ∩ i= VI([, ], Bi ). Solution. It is easy to see that t , t are k-strictly pseudononspreading, T , T are quasinonexpansive and Lipschitzian, and B , B are inverse strongly monotone. Obviously, {Ti }i= satisfies the condition Ti p = {p} for all i = , , p ∈ i= F(Ti ) since i= F(Ti ) = {}. From the definitions of these mappings, we get that i=
F(ti ) ∩
i=
F(Ti ) ∩
VI [, ], Bi = {}.
i=
For every n ∈ N, αn() = n , αn() = n– , αn() = n– , γn() = + n , γn() = n– , γn() = n– . n n n n Then the sequences {αn() }, {αn() }, {αn() }, {γn() }, {γn() }, and {γn() } satisfy all the conditions of Theorem .. For any arbitrary x ∈ C = [, ], it follows by (.) that ≤ z ≤ y ≤ x ≤ . Then we have
z + y C = p ∈ C : |z – p| ≤ |y – p| ≤ |x – p| = , .
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Since
z +y
Page 18 of 23
≤ x , we get
x = PC x =
z + y .
By continuing this process, we obtain that zn + yn , Cn+ = p ∈ Cn : |zn – p| ≤ |yn – p| ≤ |xn – p| = , and hence xn+ = PCn+ x =
zn + yn .
Now, we have the following algorithm: x ∈ [, ], n – n – xn + Wxn + Sxn , n n n n – () n – () + yn + q + q , zn = n n n n n zn + yn , n ∈ N. xn+ = yn =
qn(i) ∈ Ti yn , i = , ,
(.)
Since W is the W -mapping generated by R , R and β , β , where Ri x = P[,] (I – (I – ti ))x for all i = , and β = β = , and S is the S-mapping generated by G , G and δ , δ , where Gi x = P[,] (I – Bi )x for all i = , and δ = δ = ( , , ), we obtain that Wx = Put qn() =
x,
yn
Sx =
and qn() =
yn .
, x ,
for all x ∈ [, ].
Then we rewrite (.) as follows:
x ∈ [, ], , , + xn , yn = , ,,n + yn , zn = ,n zn + yn , n ∈ N. xn+ =
(.)
Using algorithm (.) with the initial point x = ., we have numerical results in Table . From Table , we see that the sequences {xn }, {yn }, and {zn } converge to . We ob serve that x = . is an approximation of the common element in i= F(ti ) ∩ i= F(Ti ) ∩ i= VI([, ], Bi ) with accuracy at significant digits. Next, we give the numerical example to support our main theorem in a two-dimensional space of real numbers.
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Table 1 The values of the sequences {xn }, {yn }, and {zn } in Example 5.1 n 1 2 3 4 5 . .. 16 .. . 23 24 25 26 27
xn
yn
zn
4.5000000 2.5792991 1.4384368 0.7948310 0.4371686 . .. 0.0005599 .. . 0.0000079 0.0000043 0.0000023 0.0000013 0.0000007
3.7807316 2.1441268 1.1914900 0.6572003 0.3610815 . .. 0.0004611 .. . 0.0000065 0.0000035 0.0000019 0.0000010 0.0000006
1.3778666 0.7327468 0.3981720 0.2171369 0.1184806 . .. 0.0001484 .. . 0.0000021 0.0000011 0.0000006 0.0000003 0.0000002
Example . Let x = (x() , x() ), y = (y() , y() ) ∈ R , and let the inner product ·, · : R × R → R be defined by x, y = x() y() + x() y() and the usual norm · : R → R be defined by x = (x() ) + (x() ) . We consider the nonempty closed convex subset D = [, ] × [, ] of the real Hilbert space R . Define three single-valued mappings t , t , and t on D as follows: () () t x = – x , – x ,
() () t x = – x , – x ,
() () t x = – x , – x .
Define three multi-valued mappings T , T , and T on D as follows: x x T x = , ,
x T x = , ,
x x T x = , .
For i = , , , let Bi : D → D ⊂ R be defined by Bi x =
ix() ix() , .
Let W be the W -mapping generated by R , R , R and β , β , β , where Ri x = PD (I – (I – ti ))x for all i = , , , and β = β = β = . Let S be the S-mapping generated by G , G , G and δ , δ , δ , where Gi x = PD (I – Bi )x for all i = , , , and δ = δ = δ = ( , , ). Let () () () () () xn = (x() n , xn ), yn = (yn , yn ), and zn = (zn , zn ) and the sequences {xn }, {yn }, and {zn } be generated by (.), where αn() = n , αn() = n– , αn() = n– , γn() = + n , γn() = n– , n n n () () n– γn = γn = n for all n ∈ N. Then the sequences {xn }, {yn }, and {zn } converge strongly to (, ), where {(, )} = i= F(ti ) ∩ i= F(Ti ) ∩ i= VI(D, Bi ). Solution. It is obvious that t , t , t are k-strictly pseudononspreading, T , T , T are quasi-nonexpansive and Lipschitzian, and B , B , B are inverse strongly monotone. Also, {Ti }i= satisfies the condition Ti p = {p} for all i = , , , p ∈ i= F(Ti ) since i= F(Ti ) = {(, )}. Obviously, i= F(ti ) ∩ i= F(Ti ) ∩ i= VI(D, Bi ) = {(, )}. For every n ∈ N, αn() = n , αn() = n– , αn() = n– , γn() = + n , γn() = n– , γn() = γn() = n– . Then n n n n () () () () () () () the sequences {αn }, {αn }, {αn }, {γn }, {γn }, {γn } and {γn } satisfy all the conditions
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of Theorem .. Now, we get the following algorithm: x ∈ D, n – n – xn + W xn + Sxn , n n n n – () n – () n – () + yn + q + q + q , zn = n n n n n n n yn =
qn(i) ∈ Ti yn , i = , , , ()
() () () () Cn+ = p = p() , p() ∈ Cn : zn() – y() + yn n p + zn – yn p
() () ()
()
() () () () + yn – zn – zn ≤ , zn – x() n p + zn – xn p
() () ()
() () + x() + xn – zn – zn ≤ , y() n n – xn p
() () () () () + y() + x() + xn – yn – yn ≤ , n – xn p n xn+ = PCn+ x ,
(.)
n ∈ N.
Since W is the W -mapping generated by R , R , R and β , β , β , where Ri x = PD (I – (I – ti ))x for all i = , , and β = β = β = , and S is the S-mapping generated by G , G , G and δ , δ , δ , where Gi x = PD (I – Bi )x for all i = , , and δ = δ = δ = ( , , ), we obtain that () , () , () , () Wx = x , x x , x , Sx = for all x ∈ D. , , , Put qn() =
yn ,
qn() =
yn ,
and qn() =
yn .
Then algorithm (.) becomes
x ∈ D, , ,, + x() , yn = ,, ,,n n ,, ,, + x() , ,, ,,,n n () zn = + y , + y() , ,n n ,n n
() () () () () Cn+ = p = p() , p() ∈ Cn : zn() – y() + yn n p + zn – yn p ()
() ()
() () () + y() – zn – zn ≤ , zn() – x() n n p + zn – xn p
() () ()
() () + x() + xn – zn – zn ≤ , y() n n – xn p
() () () () () + y() + x() + xn – yn – yn ≤ , n – xn p n xn+ = PCn+ x ,
(.)
n ∈ N.
() For any arbitrary (x() n , xn ) ∈ D, by algorithm (.), we see that (i) (i) ≤ zn(i) ≤ y(i) n ≤ xn ≤ x ≤ for all i = , , and n ∈ N.
The numerical results for the sequences {xn }, {yn }, and {zn } are shown in Table and Table .
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Table 2 The values of the sequences {xn }, {yn }, and {zn } with the initial point x1 = (7.5, 9.1) in Example 5.2 n 1 2 3 4 5 6 7 8 9 . .. 45 . .. 55 56 57 58 59 60 61 62
xn
yn
zn
(7.5000000,9.1000000) (4.6619648,5.6749079) (2.8183693,3.4332496) (1.6857031,2.0577916) (1.0092457,1.2223240) (0.5676300,0.7527995) (0.7174567,0.1573218) (0.3517952,0.4346565) (0.4298726,0.0764960) . .. (0.0000963,0.0000292) . .. (0.0000088,0.0000042) (0.0000033,0.0000065) (0.0000054,0.0000020) (0.0000022,0.0000036) (0.0000033,0.0000009) (0.0000015,0.0000020) (0.0000020,0.0000004) (0.0000010,0.0000011)
(6.3029396,7.6067234) (3.8840555,4.7013476) (2.3412722,2.8357215) (1.3983068,1.6970922) (0.8364468,1.0071576) (0.4701681,0.6199094) (0.5940218,0.1294942) (0.2911794,0.3576574) (0.3557172,0.0629290) . .. (0.0000795,0.0000240) . .. (0.0000072,0.0000034) (0.0000027,0.0000053) (0.0000044,0.0000017) (0.0000018,0.0000030) (0.0000030,0.0000008) (0.0000012,0.0000017) (0.0000016,0.0000003) (0.0000008,0.0000009)
(3.0687437,3.7035235) (1.7741233,2.1474385) (1.0459308,1.2668192) (0.6176583,0.7496374) (0.3669562,0.4418484) (0.2053231,0.2707153) (0.2585586,0.0563647) (0.1264280,0.1552922) (0.1541523,0.0272707) . .. (0.0000340,0.0000103) . .. (0.0000031,0.0000015) (0.0000012,0.0000023) (0.0000019,0.0000007) (0.0000008,0.0000013) (0.0000012,0.0000003) (0.0000005,0.0000007) (0.0000007,0.0000001) (0.0000004,0.0000004)
Table 3 The values of the sequences {xn }, {yn }, and {zn } with the initial point x1 = (0, 75.6) in Example 5.2 n 1 2 3 4 5 6 7 8 9 .. . 15 .. . 30 31 32 33 34 35 36 37
xn
yn
zn
(0.0000000,75.6000000) (0.0000000,46.9810256) (0.0000000,28.3496228) (0.0000000,16.9381045) (0.0000000,10.0697666) (0.0000000,5.9686106) (0.0000000,3.5306826) (0.0000000,2.0855624) (0.0000000,1.2306135) .. . (0.0000000,0.0512698) .. . (0.0000000,0.0000175) (0.0000000,0.0000103) (0.0000000,0.0000060) (0.0000000,0.0000035) (0.0000000,0.0000021) (0.0000000,0.0000012) (0.0000000,0.0000007) (0.0000000,0.0000004)
(0.0000000,63.1943178) (0.0000000,38.9211840) (0.0000000,23.4156103) (0.0000000,13.9691137) (0.0000000,8.2971795) (0.0000000,4.9149845) (0.0000000,2.9061647) (0.0000000,1.7161061) (0.0000000,1.0123560) .. . (0.0000000,0.0421428) .. . (0.0000000,0.0000144) (0.0000000,0.0000084) (0.0000000,0.0000049) (0.0000000,0.0000029) (0.0000000,0.0000017) (0.0000000,0.0000010) (0.0000000,0.0000006) (0.0000000,0.0000003)
(0.0000000,30.7677335) (0.0000000,17.7780617) (0.0000000,10.4605987) (0.0000000,6.1704195) (0.0000000,3.6400418) (0.0000000,2.1463806) (0.0000000,1.2649601) (0.0000000,0.7451208) (0.0000000,0.4387110) .. . (0.0000000,0.0181501) .. . (0.0000000,0.0000062) (0.0000000,0.0000036) (0.0000000,0.0000021) (0.0000000,0.0000012) (0.0000000,0.0000007) (0.0000000,0.0000004) (0.0000000,0.0000003) (0.0000000,0.0000001)
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements The authors would like to thank the referees for valuable suggestions on the research paper. Received: 27 October 2014 Accepted: 13 January 2015
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