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Fixed Point Theory and Applications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems Fixed Point Theory and Applications 2011, 2011:101 doi:10.1186/1687-1812-2011-101 Yonghong Yao (yaoyonghong@yahoo.cn) Yeol Je Cho (yjcho@gsnu.ac.kr) Yeong-Cheng Liou (simplex_liou@hotmail.com) ISSN 1687-1812 Article type Research Submission date 3 November 2010 Acceptance date 20 December 2011 Publication date 20 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/101 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Fixed Point Theory and Applications go to http://www.fixedpointtheoryandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Yao et al. ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems Yonghong Yao1 , Yeol Je Cho∗2 and Yeong-Cheng Liou3 1 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, People’s Republic of China 2 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea 3 Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan ∗ Corresponding author: yjcho@gsnu.ac.kr. E-mail addresses: YY:yaoyonghong@yahoo.cn Y-CL:simplex liou@hotmail.com. Abstract In this paper, we show the hierarchical convergence of the following implicit 1
double-net algorithm: λs 1 xs,t = s[tf (xs,t ) + (1 − t)(xs,t − µAxs,t )] + (1 − s) T (ν )xs,t dν, ∀s, t ∈ (0, 1), λs 0 where f is a ρ-contraction on a real Hilbert space H , A : H → H is an α-inverse strongly monotone mapping and S = {T (s)}s≥0 : H → H is a nonexpansive semi- group with the common ﬁxed points set F ix(S ) = ∅, where F ix(S ) denotes the set of ﬁxed points of the mapping S , and, for each ﬁxed t ∈ (0, 1), the net {xs,t } con- verges in norm as s → 0 to a common ﬁxed point xt ∈ F ix(S ) of {T (s)}s≥0 and, as t → 0, the net {xt } converges in norm to the solution x∗ of the following variational inequality:   ∗ x ∈ F ix(S );      Ax∗ , x − x∗ ≥ 0, ∀x ∈ F ix(S ). Keywords: ﬁxed point; variational inequality; double-net algorithm; hierarchical convergence; Hilbert space. MSC(2000): 49J40; 47J20; 47H09; 65J15. 1 Introduction In nonlinear analysis, a common approach to solving a problem with multiple solutions is to replace it by a family of perturbed problems admitting a unique solution and to obtain a particular solution as the limit of these perturbed solutions when the perturbation vanishes. In this paper, we introduce a more general approach which consists in ﬁnding a particular part of the solution set of a given ﬁxed point problem, i.e., ﬁxed points which 2
solve a variational inequality. More precisely, the goal of this paper is to present a method for ﬁnding hierarchically a ﬁxed point of a nonexpansive semigroup S = {T (s)}s≥0 with respect to another monotone operator A, namely, Find x∗ ∈ F ix(S ) such that Ax∗ , x − x∗ ≥ 0, ∀x ∈ F ix(S ). (1.1) This is an interesting topic due to the fact that it is closely related to convex pro- gramming problems. For the related works, refer to [1–19]. This paper is devoted to solve the problem (1.1). For this purpose, we propose a double-net algorithm which generates a net {xs,t } and prove that the net {xs,t } hierar- chically converges to the solution of the problem (1.1), that is, for each ﬁxed t ∈ (0, 1), the net {xs,t } converges in norm as s → 0 to a common ﬁxed point xt ∈ F ix(S ) of the nonexpansive semigroup {T (s)}s≥0 and, as t → 0, the net {xt } converges in norm to the unique solution x∗ of the problem (1.1). 2 Preliminaries Let H be a real Hilbert space with inner product ·, · and norm · , respectively. Recall a mapping f : H → H is called a contraction if there exists ρ ∈ [0, 1) such that f (x) − f (y ) ≤ ρ x − y , ∀x, y ∈ H. A mapping T : C → C is said to be nonexpansive if Tx − Ty ≤ x − y , ∀x, y ∈ H. 3
Denote the set of ﬁxed points of the mapping T by F ix(T ). Recall also that a family S := {T (s)}s≥0 of mappings of H into itself is called a nonexpansive semigroup if it satisﬁes the following conditions: (S1) T (0)x = x for all x ∈ H ; (S2) T (s + t) = T (s)T (t) for all s, t ≥ 0; (S3) T (s)x − T (s)y ≤ x − y for all x, y ∈ H and s ≥ 0; (S4) for all x ∈ H , s → T (s)x is continuous. We denote by F ix(T (s)) the set of ﬁxed points of T (s) and by F ix(S ) the set of all common ﬁxed points of S , i.e., F ix(S ) = s≥0 F ix(T (s)). It is known that F ix(S ) is closed and convex ([20], Lemma 1). A mapping A of H into itself is said to be monotone if Au − Av, u − v ≥ 0, ∀u, v ∈ H, and A : C → H is said to be α-inverse strongly monotone if there exists a positive real number α such that Au − Av, u − v ≥ α Au − Av 2 , ∀u, v ∈ H. It is obvious that any α-inverse strongly monotone mapping A is monotone and 1 α -Lipschitz continuous. Now, we introduce some lemmas for our main results in this paper. Lemma 2.1. [21] Let H be a real Hilbert space. Let the mapping A : H → H be α-inverse 4
strongly monotone and µ > 0 be a constant. Then, we have 2 2 + µ(µ − 2α) Ax − Ay 2 , (I − µA)x − (I − µA)y ≤ x−y ∀x, y ∈ H. In particular, if 0 ≤ µ ≤ 2α, then I − µA is nonexpansive. Lemma 2.2. [22] Let C be a nonempty bounded closed convex subset of a Hilbert space H and {T (s)}s≥0 be a nonexpansive semigroup on C . Then, for all h ≥ 0, t t 1 1 lim sup T (s)xds − T (h) T (s)xds = 0. t t t→∞ x∈C 0 0 Lemma 2.3. [23] (Demiclosedness Principle for Nonexpansive Mappings) Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C be a nonexpansive mapping with F ix(T ) = ∅. If {xn } is a sequence in C converging weakly to a point x ∈ C and {(I − T )xn } converges strongly to a point y ∈ C , then (I − T )x = y . In particular, if y = 0, then x ∈ F ix(T ). Lemma 2.4. Let H be a real Hilbert space. Let f : H → H be a ρ-contraction with coeﬃcient ρ ∈ [0, 1) and A : H → H be an α-inverse strongly monotone mapping. Let µ ∈ (0, 2α) and t ∈ (0, 1). Then, the variational inequality    ∗ x ∈ F ix(S );  (2.1)     tf (z ) + (1 − t)(I − µA)z − z, x∗ − z ≥ 0, ∀z ∈ F ix(S ),  is equivalent to its dual variational inequality    ∗  x ∈ F ix(S ); (2.2)     tf (x∗ ) + (1 − t)(I − µA)x∗ − x∗ , x∗ − z ≥ 0, ∀z ∈ F ix(S ).  5
Proof. Assume that x∗ ∈ F ix(S ) solves the problem (2.1). For all y ∈ F ix(S ), set x = x∗ + s(y − x∗ ) ∈ F ix(S ), ∀s ∈ (0, 1). We note that tf (x) + (1 − t)(I − µA)x − x, x∗ − x ≥ 0. Hence, we have tf (x∗ + s(y − x∗ )) + (1 − t)(I − µA)(x∗ + s(y − x∗ )) − x∗ − s(y − x∗ ), s(x∗ − y ) ≥ 0, which implies that tf (x∗ + s(y − x∗ )) + (1 − t)(I − µA)(x∗ + s(y − x∗ )) − x∗ − s(y − x∗ ), x∗ − y ≥ 0. Letting s → 0, we have tf (x∗ ) + (1 − t)(I − µA)(x∗ ) − x∗ , x∗ − y ≥ 0, which implies the point x∗ ∈ F ix(S ) is a solution of the problem (2.2). Conversely, assume that the point x∗ ∈ F ix(S ) solves the problem (2.2). Then, we have tf (x∗ ) + (1 − t)(I − µA)x∗ − x∗ , x∗ − z ≥ 0. Noting that I − f and A are monotone, we have (I − f )z − (I − f )x∗ , z − x∗ ≥ 0 and Az − Ax∗ , z − x∗ ≥ 0. 6
Thus, it follows that t (I − f )z − (I − f )x∗ , z − x∗ + (1 − t)µ Az − Ax∗ , z − x∗ ≥ 0, which implies that tf (z ) + (1 − t)(I − µA)z − z, x∗ − z ≥ tf (x∗ ) + (1 − t)(I − µA)x∗ − x∗ , x∗ − z ≥ 0. This implies that the point x∗ ∈ F ix(S ) solves the problem (2.1). This completes the proof. 3 Main results In this section, we ﬁrst introduce our double-net algorithm and then prove a strong convergence theorem for this algorithm. Throughout, we assume that (C1) H is a real Hilbert space; (C2) f : H → H is a ρ-contraction with coeﬃcient ρ ∈ [0, 1), A : H → H is an α-inverse strongly monotone mapping, and S = {T (s)}s≥0 : H → H is a nonexpansive semigroup with F ix(S ) = ∅; (C3) the solution set Ω of the problem (1.1) is nonempty; (C4) µ ∈ (0, 2α) is a constant, and {λs }0
For any s, t ∈ (0, 1), we deﬁne the following mapping λs 1 x → Ws,t x := s[tf (x) + (1 − t)(x − µAx)] + (1 − s) T (ν )xdν. λs 0 We note that the mapping Ws,t is a contraction. In fact, we have λs 1 Ws,t x − Ws,t y = s[tf (x) + (1 − t)(x − µAx)] + (1 − s) T (ν )xdν λs 0 λs 1 −s[tf (y ) + (1 − t)(y − µAy )] − (1 − s) T (ν )ydν λs 0 ≤ st f (x) − f (y ) + s(1 − t) (x − µAx) − (y − µAy ) λs 1 +(1 − s) (T (ν )x − T (ν )y )dν λs 0 ≤ stρ x − y + s(1 − t) x − y + (1 − s) x − y = [1 − (1 − ρ)st] x − y , which implies that Ws,t is a contraction. Hence, by Banach’s Contraction Principle, Ws,t has a unique ﬁxed point, which is denoted xs,t ∈ H , that is, xs,t is the unique solution in H of the ﬁxed point equation xs,t = s[tf (xs,t ) + (1 − t)(xs,t − µAxs,t )] (3.1) λs 1 + (1 − s) T (ν )xs,t dν, ∀s, t ∈ (0, 1). λs 0 Now, we give some lemmas for our main result. Lemma 3.1. For each ﬁxed t ∈ (0, 1), the net {xs,t } deﬁned by (3.1) is bounded. Proof. Taking any z ∈ F ix(S ), since I − µA is nonexpansive (by Lemma 2.1), it follows 8
from (3.1) that xs,t − z λs 1 = s[tf (xs,t ) + (1 − t)(I − µA)xs,t ] + (1 − s) T (ν )xs,t dν − z λs 0 λs 1 ≤ s tf (xs,t ) + (1 − t)(I − µA)xs,t − z + (1 − s) T (ν )xs,t dν − z λs 0 ≤ s t f (xs,t ) − f (z ) + t f (z ) − z + (1 − t) (I − µA)xs,t − (I − µA)z +(1 − t) (I − µA)z − z + (1 − s) xs,t − z ≤ s[tρ xs,t − z + t f (z ) − z + (1 − t) xs,t − z + (1 − t)µ Az ] +(1 − s) xs,t − z = [1 − (1 − ρ)st] xs,t − z + st f (z ) − z + s(1 − t)µ Az . This implies that 1 xs,t − z ≤ (t f (z ) − z + (1 − t)µ Az ) (1 − ρ)t 1 ≤ max{ f (z ) − z , µ Az }. (1 − ρ)t Thus, it follows that, for each ﬁxed t ∈ (0, 1), {xs,t } is bounded and so are the nets {f (xs,t )} and {(I − µA)xs,t }. This completes the proof. Lemma 3.2. xs,t → xt ∈ F ix(S ) as s → 0. 1 Proof. For each ﬁxed t ∈ (0, 1), we set Rt := max{ f (z ) − z , µ Az }. It is clear (1−ρ)t that, for each ﬁxed t ∈ (0, 1), {xs,t } ⊂ B (p, Rt ), where B (p, Rt ) denotes a closed ball with the center p and radius Rt . Notice that λs 1 T (ν )xs,t dν − p ≤ xs,t − p ≤ Rt . λs 0 9
Moreover, we observe that if x ∈ B (p, Rt ), then T (s)x − p ≤ T (s)x − T (s)p ≤ x − p ≤ Rt , that is, B (p, Rt ) is T (s)-invariant for all s ∈ (0, 1). From (3.1), it follows that λs 1 T (τ )xs,t − xs,t ≤ T (τ )xs,t − T (τ ) T (ν )xs,t dν λs 0 λs λs 1 1 + T (τ ) T (ν )xs,t dν − T (ν )xs,t dν λs λs 0 0 λs 1 + T (ν )xs,t dν − xs,t λs 0 λs λs 1 1 ≤ T (τ ) T (ν )xs,t dν − T (ν )xs,t dν λs λs 0 0 λs 1 +2 xs,t − T (ν )xs,t dν λs 0 λs 1 ≤ 2s tf (xs,t ) + (1 − t)(xs,t − µAxs,t ) − T (ν )xs,t dν λs 0 λs λs 1 1 + T (τ ) T (ν )xs,t dν − T (ν )xs,t dν . λs λs 0 0 By Lemma 2.2, for all 0 ≤ τ < ∞ and ﬁxed t ∈ (0, 1), we deduce lim T (τ )xs,t − xs,t = 0. (3.2) s→0 Next, we show that, for each ﬁxed t ∈ (0, 1), the net {xs,t } is relatively norm-compact as s → 0. In fact, from Lemma 2.1, it follows that 2 2 + µ(µ − 2α) Axs,t − Az 2 . xs,t − µAxs,t − (z − µAz ) ≤ xs,t − z (3.3) 10
By (3.1), we have 2 xs,t − z = st f (xs,t ) − f (z ), xs,t − z + st f (z ) − z, xs,t − z +s(1 − t) (I − µA)xs,t − (I − µA)z, xs,t − z +s(1 − t) (I − µA)z − z, xs,t − z λs 1 +(1 − s) T (ν )xs,t dν − z, xs,t − z λs 0 ≤ st f (xs,t ) − f (z ) xs,t − z + st f (z ) − z, xs,t − z +s(1 − t) (I − µA)xs,t − (I − µA)z xs,t − z − s(1 − t)µ Az, xs,t − z λs 1 +(1 − s) T (ν )xs,t dν − z xs,t − z λs 0 2 ≤ stρ xs,t − z + st f (z ) − z, xs,t − z − s(1 − t)µ Az, xs,t − z 2 +s(1 − t) (I − µA)xs,t − (I − µA)z xs,t − z + (1 − s) xs,t − z 2 ≤ stρ xs,t − z + st f (z ) − z, xs,t − z − s(1 − t)µ Az, xs,t − z s(1 − t) 2 + xs,t − z 2 ) + (1 − s) xs,t − z 2 . + ( (I − µA)xs,t − (I − µA)z 2 This together with (3.3) imply that 2 xs,t − z 2 ≤ stρ xs,t − z + st f (z ) − z, xs,t − z − s(1 − t)µ Az, xs,t − z s(1 − t) 2 2 + xs,t − z 2 ) + (1 − s) xs,t − z 2 + ( xs,t − z + µ(µ − 2α) Axs,t − Az 2 2 ≤ [1 − (1 − ρ)st] xs,t − z + st f (z ) − z, xs,t − z −s(1 − t)µ Az, xs,t − z , 11
which implies that 2 xs,t − z (3.4) 1 ≤ tf (z ) + (1 − t)(I − µA)z − z, xs,t − z , ∀z ∈ F ix(S ). (1 − ρ)t Assume that {sn } ⊂ (0, 1) is such that sn → 0 as n → ∞. By (3.4), we obtain immedi- ately that 2 xsn ,t − z (3.5) 1 ≤ tf (z ) + (1 − t)(I − µA)z − z, xsn ,t − z , ∀z ∈ F ix(S ). (1 − ρ)t Since {xsn ,t } is bounded, without loss of generality, we may assume that, as sn → 0, {xsn ,t } converges weakly to a point xt . From (3.2) and Lemma 2.3, we get xt ∈ F ix(S ). Further, if we substitute xt for z in (3.5), then it follows that 1 2 xsn ,t − xt ≤ tf (xt ) + (1 − t)(I − µA)xt − xt , xsn ,t − xt . (1 − ρ)t Therefore, the weak convergence of {xsn ,t } to xt actually implies that xsn ,t → xt strongly. This has proved the relative norm-compactness of the net {xs,t } as s → 0. Now, if we take the limit as n → ∞ in (3.5), we have 2 xt − z 1 ≤ tf (z ) + (1 − t)(I − µA)z − z, xt − z , ∀z ∈ F ix(S ). (1 − ρ)t In particular, xt solves the following variational inequality:     xt ∈ F ix(S );      tf (z ) + (1 − t)(I − µA)z − z, xt − z ≥ 0, ∀z ∈ F ix(S ),  12
or the equivalent dual variational inequality (see Lemma 2.4):     xt ∈ F ix(S );  (3.6)     tf (xt ) + (1 − t)(I − µA)xt − xt , xt − z ≥ 0, ∀z ∈ F ix(S ).  Notice that (3.6) is equivalent to the fact that xt = PF ix(S ) (tf + (1 − t)(I − µA))xt , that is, xt is the unique element in F ix(S ) of the contraction PF ix(S ) (tf +(1 − t)(I − µA)). Clearly, it is suﬃcient to conclude that the entire net {xs,t } converges in norm to xt ∈ F ix(S ) as s → 0. This completes the proof. Lemma 3.3. The net {xt } is bounded. Proof. In (3.6), if we take any y ∈ Ω, then we have tf (xt ) + (1 − t)(I − µA)xt − xt , xt − y ≥ 0. (3.7) By virtue of the monotonicity of A and the fact that y ∈ Ω, we have (I − µA)xt − xt , xt − y ≤ (I − µA)y − y, xt − y ≤ 0. (3.8) Thus, it follows from (3.7) and (3.8) that f (xt ) − xt , xt − y ≥ 0, ∀y ∈ Ω (3.9) and hence 2 xt − y ≤ xt − y, xt − y + f (xt ) − xt , xt − y = f (xt ) − f (y ), xt − y + f (y ) − y, xt − y 2 ≤ ρ xt − y + f (y ) − y, xt − y . 13
Therefore, we have 1 2 xt − y ≤ f (y ) − y, xt − y , ∀y ∈ Ω. (3.10) 1−ρ In particular, 1 xt − y ≤ f (y ) − y , ∀t ∈ (0, 1), 1−ρ which implies that {xt } is bounded. This completes the proof. Lemma 3.4. If the net {xt } converges in norm to a point x∗ ∈ Ω, then the point solves the variational inequality (I − f )x∗ , x − x∗ ≥ 0, ∀x ∈ Ω. (3.11) Proof. First, we note that the solution of the variational inequality (3.11) is unique. Next, we prove that ωw (xt ) ⊂ Ω, that is, if (tn ) is a null sequence in (0, 1) such that xtn → x weakly as n → ∞, then x ∈ Ω. To see this, we use (3.6) to get t µAxt , z − xt ≥ (I − f )xt , xt − z , ∀z ∈ F ix(S ). 1−t However, since A is monotone, we have Az, z − xt ≥ Axt , z − xt . Combining the last two relations yields that t µAz, z − xt ≥ (I − f )xt , xt − z , ∀z ∈ F ix(S ). (3.12) 1−t Letting t = tn → 0 as n → ∞ in (3.12), we get Az, z − x ≥ 0, ∀z ∈ F ix(S ), 14
which is equivalent to its dual variational inequality Ax , z − x ≥ 0, ∀z ∈ F ix(S ). That is, x is a solution of the problem (1.1) and hence x ∈ Ω. Finally, we prove that x = x∗ , the unique solution of the variational inequality (3.11). In fact, by (3.10), we have 1 2 xtn − x f (x ) − x , xtn − x , ∀x ∈ Ω. ≤ 1−ρ Therefore, the weak convergence to x of {xtn } implies that xtn → x in norm. Thus, if we let t = tn → 0 in (3.10), then we have f (x ) − x , y − x ≤ 0, ∀y ∈ Ω, which implies that x ∈ Ω solves the problem (3.11). By the uniqueness of the solution, we have x = x∗ and it is suﬃcient to guarantee that xt → x∗ in norm as t → 0. This completes the proof. Thus, by the above lemmas, we can obtain immediately the following theorem. Theorem 3.5. For each (s, t) ∈ (0, 1) × (0, 1), let {xs,t } be a double-net algorithm deﬁned implicitly by (3.1). Then, the net {xs,t } hierarchically converges to the unique solution x∗ of the hierarchical ﬁxed point problem and the variational inequality problem (1.1), that is, for each ﬁxed t ∈ (0, 1), the net {xs,t } converges in norm as s → 0 to a common ﬁxed point xt ∈ F ix(S ) of the nonexpansive semigroup {T (s)}s≥0 . Moreover, as t → 0, 15
the net {xt } converges in norm to the unique solution x∗ ∈ Ω and the point x∗ also solves the following variational inequality:    ∗ x ∈ Ω;      (I − f )x∗ , x − x∗ ≥ 0, ∀x ∈ Ω.  Acknowledgments Yonghong Yao was supported in part by Colleges and Universities Science and Technol- ogy Development Foundation (20091003) of Tianjin and NSFC 11071279. Yeol Je Cho was supported by the Korea Research Foundation Grant funded by the Korean Gov- ernment (KRF-2008-313-C00050). Yeong-Cheng Liou was supported in part by NSC 99-2221-E-230-006. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the ﬁnal manuscript. 16
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