Research Article Iterative Schemes for Zero Points of Maximal Monotone Operators and Fixed Points of Nonexpansive Mappings and Their Applications

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1 Hindawi Publihing Corporation Fixed Point Theory and Application Volume 2008, Article ID , 12 page doi: /2008/ Reearch Article Iterative Scheme for Zero Point of Maximal Monotone Operator and Fixed Point of Nonexpanive Mapping and Their Application Li Wei 1 and Yeol Je Cho 2 1 School of Mathematic and Statitic, Hebei Univerity of Economic and Buine, Shijiazhuang , China 2 Department of Mathematic Education and the RINS, Gyeongang National Univerity, Chinju , South Korea Correpondence hould be addreed to Yeol Je Cho, yjcho@gnu.ac.kr Received 16 Augut 2007; Accepted 25 November 2007 Recommended by Maimo Furi Two iterative cheme for finding a common element of the et of zero point of maximal monotone operator and the et of fixed point of nonexpanive mapping in the ene of Lyapunov functional in a real uniformly mooth and uniformly convex Banach pace are obtained. Two trong convergence theorem are obtained which extend ome previou work. Moreover, the application of the iterative cheme are demontrated. Copyright q 2008 L. Wei and Y. J. Cho. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. 1. Introduction and preliminarie In thi paper, we will preent two iterative cheme with error which are proved to be trongly convergent to a common element of the et of zero point of maximal monotone operator and the et of fixed point of nonexpanive mapping with repect to the Lyapunov functional in real uniformly mooth and uniformly convex Banach pace. Moreover, it i hown that ome reult propoed by Martinez-Yane and Xu in 1 and Solodov and Svaiter in 2 are pecial cae of our. Finally, we will demontrate the application of our iterative cheme on both finding the minimizer of a proper convex and lower emicontinuou function and olving the variational inequalitie. Let E be a real Banach pace with norm and let E be it dual pace. The normalized duality mapping J : E 2 E i defined a follow: Jx { x E : x, x x 2 x 2} x E, 1.1

2 2 Fixed Point Theory and Application where x, x denote the value of x E at x E. We ue ymbol and w to repreent trong and weak convergence in E or in E, repectively. A multivalued operator T : E 2 E with domain D T {x E : Tx/ } and range R T {Tx : x D T } i aid to be monotone if x 1 x 2,y 1 y 2 0for all x i D T and y i Tx i, i 1, 2. A monotone operator T i aid to be a maximal monotone if R J rt E for all r>0. For a monotone operator T, wedenotebyt 1 0 {x D T :0 Tx} the et of zero point of T. For a ingle-valued mapping S : E E, wedenotebyfix S {x E : Sx x} the et of fixed point of S. Lemma 1.1 ee 3, 4. The duality mapping J ha the following propertie. 1 If E i a real reflexive and mooth Banach pace, then J : E E i ingle-valued. 2 For all x E and λ R, J λx λjx. 3 If E i a real uniformly convex and uniformly mooth Banach pace, then J 1 : E E i alo a duality mapping. Moreover, J : E E and J 1 : E E are uniformly continuou on each bounded ubet of E or E, repectively. Lemma 1.2 ee 4. Let E be a real mooth and uniformly convex Banach pace and let T : E 2 E be a maximal monotone operator. Then T 1 0 i a cloed and convex ubet of E and the graph of T, G T, w i demicloed in the following ene: for all {x n } D T with x n x in E and for all yn Tx n with y in E, x D T and y Tx. y n Definition 1.3. Let E be a real mooth and uniformly convex Banach pace and let T : E 2 E be a maximal monotone operator. For all r > 0, define the operator Qr T : E E by Qr T x J rt 1 Jx for all x E. Definition 1.4 ee 5. LetE be a real mooth Banach pace. Then the Lyapunov functional ϕ : E E R i defined a follow: ϕ x, y x 2 2 x, j y y 2 x, y E, j y Jy. 1.2 Lemma 1.5 ee 5. Let E be a real reflexive, trictly convex and mooth Banach pace, let C be a nonempty cloed and convex ubet of E, and let x E. Then there exit a unique element x 0 C uch that ϕ x 0,x min {ϕ z, x : z C}. Define the mapping Q C of E onto C by Q C x x 0 for all x E. Q C i called the generalized projection operator from E onto C. ItieaytoeethatQ C i coincident with the metric projection P C in a Hilbert pace. Lemma 1.6 ee 5. Let E be a real reflexive, trictly convex and mooth Banach pace, let C be a nonempty cloed and convex ubet of E, and let x E. Then, for all y C, ϕ y, Q C x ) ϕ Q C x, x ) ϕ y, x. 1.3 Lemma 1.7 ee 6. Let E be a real mooth and uniformly convex Banach pace and let {x n } and {y n } be two equence of E. Ifeither{x n } or {y n } i bounded and ϕ x n,y n 0 a n,then x n y n 0 a n.

3 L. Wei and Y. J. Cho 3 Lemma 1.8 ee 7. Let E be a real reflexive, trictly convex and mooth Banach pace and let T : E 2 E be a maximal monotone operator with T 1 0 /. Then for all x E, y T 1 0 and r>0, one ha ϕy, Q T r x) ϕq T r x, x) ϕy, x). Lemma 1.9 ee 5. Let E be a real mooth Banach pace, let C be a convex ubet of E, letx E, and let x 0 C. Thenϕ x 0,x inf {ϕ z, x : z C} ifandonlyif z x 0,Jx 0 Jx 0 for all z C. Definition Let E be a real Banach pace. Then S : E E i aid to be nonexpanive with repect to the Lyapunov functional if ϕ Sx, Sy ϕ x, y for all x, y E. Remark If E i a real Hilbert pace H, thens i a nonexpanive mapping in the uual ene: Sx Sy x y for all x, y H. Lemma Let E be a real mooth and uniformly convex Banach pace. If S : E E i a mapping which i nonexpanive with repect to the Lyapunov functional, then Fix S i a convex and cloed ubet of E. Proof. In fact, we only need to prove the cae that Fix S /. For all x, y Fix S and t 0, 1, let z tx 1 t y.thenwehave ϕ z, Sz t x 2 2 x, JSz Sz 2) 1 t y 2 2 y, JSz Sz 2) t x 2 1 t y 2 z 2 tϕ x, Sz 1 t ϕ y, Sz t x 2 1 t y 2 z 2 tϕ x, z 1 t ϕ y, z t x 2 1 t y 2 z 2 ϕ z, z 0. By uing Lemma 1.7, we know that z Sz, which implie that Fix S i a convex ubet of E. For all x n Fix S uch that x n x, it follow that ϕ Sxn,Sx ϕ x n,x 0. Lemma 1.7 implie that Sx n Sx a n.sox Fix S. 2. Strong convergence theorem Throughout thi ection, we aume that E i a real uniformly mooth and uniformly convex Banach pace, S : E E i nonexpanive with repect to the Lyapunov functional and weakly equentially continuou and T : E 2 E i a maximal monotone operator with T 1 0 Fix S /. 1.4 Theorem 2.1. The equence {x n } generated by the following cheme: x 0 E, r 0 > 0, y n Qr T ) xn n e n, Jz n α n Jx n ) 1 α n Jyn, u n Sz n, H n { v E : ϕ ) ) v, u n ϕ v, zn αn ϕ ) ) )} v, x n 1 αn ϕ v, xn e n, W n { z E : z x n,jx 0 Jx n 0 }, x 0 n 0, 2.1

4 4 Fixed Point Theory and Application converge trongly to Q T 1 0 Fix S x 0 provided i {α n } 0, 1 i a equence uch that α n 1 δ, for ome δ 0, 1 ; ii {r n } 0, i a equence uch that inf n 0 r n > 0; iii {e n } E i a equence uch that e n 0 a n. Proof. We plit the proof into five tep. Step 1. Both H n and W n are cloed and convex ubet of E. Noting the fact that ϕ v, z n ) αn ϕ v, x n ) 1 αn ) ϕ v, xn e n ) zn 2 αn xn 2 1 αn ) xn e n 2 2 v, Jzn α n Jx n 1 α n ) J xn e n ), 2.2 ϕ v, u n ) ϕ v, zn ) z n 2 u n 2 2 v, Jz n Ju n, we can eaily know that H n i a cloed and convex ubet of E. It i obviou that W n i alo a cloed and convex ubet of E. Step 2. T 1 0 Fix S H n W n for each nonnegative integer n. To oberve thi, take p T 1 0 Fix S. From the definition of the maximal monotone operator, we know that there exit y 0 E uch that y 0 Q T r 0 x 0 e 0. It follow from Lemma 1.8 that ϕ p, y 0 ϕ p, x 0 e 0.Then ϕ p, u 0 ) ϕ p, z0 ) α0 ϕ p, x 0 ) 1 α0 ) ϕ p, y0 ) α0 ϕ p, x 0 ) 1 α0 ) ϕ p, x0 e 0 ), 2.3 which implie that p H 0. On the other hand, it i clear that p W 0 E. Thenp H 0 W 0 and therefore x 1 Q H0 W 0 x 0 are well defined. Suppoe that p H n 1 W n 1 and x n i well defined for ome n 1. Then there exit y n E uch that y n Q T r n x n e n. Lemma 1.8 implie that ϕ p, y n ϕ p, x n e n.thu ϕ ) ) p, u n ϕ p, zn αn ϕ ) ) ) p, x n 1 αn ϕ p, yn ϕ ) p, z n αn ϕ ) ) ) p, x n 1 αn ϕ p, yn 2.4 which implie that p H n. It follow from Lemma 1.9 that p x n,jx 0 Jx n p Q Hn 1 W n 1 x 0,Jx 0 JQ Hn 1 W n 1 x 0 0, 2.5 which implie that p W n. Hence x 0 i well defined. Then, by induction, the equence generated by 2.1 i well defined and T 1 0 Fix S H n W n for each n 0.

5 L. Wei and Y. J. Cho 5 Step 3. {x n } i a bounded equence of E. In fact, for all p T 1 0 Fix S H n W n, it follow from Lemma 1.6 that ϕ p, Q Wn x 0 ) ϕ QWn x 0,x 0 ) ϕ p, x0 ). 2.6 From the definition of W n and Lemma 1.5 and 1.9, we know that x n Q Wn x 0, which implie that ϕ p, x n ϕ x n,x 0 ϕ p, x 0. Therefore, {x n } i bounded. Step 4. ω x n T 1 0 Fix S, whereω x n denote the et coniting all of the weak limit point of {x n }. From the fact x n Q Wn x 0, x n 1 W n and Lemma 1.6,wehave ϕ x n 1,x n ) ϕ xn,x 0 ) ϕ xn 1,x 0 ) 2.7 Therefore, lim n ϕ x n,x 0 exit. Then ϕ x n 1,x n 0, which implie from Lemma 1.7 that x n 1 x n 0an. Since xn 1 H n,wehave ϕ x n 1,u n ) ϕ xn 1,z n ), ϕ x n 1,z n ) αn ϕ x n 1,x n ) 1 αn ) ϕ xn 1,x n e n ) Notice that ϕ x n 1,x n e n ) ϕ xn 1,x n ) x n e n 2 x n 2 2 x n 1,Jx n J x n e n ) Since J : E E i uniformly continuou on each bounded ubet of E and e n 0, we know from 2.10 that ϕ x n 1,x n e n 0, which implie that ϕ x n 1,z n 0by 2.9. Moreover, 2.8 implie that ϕ x n 1,u n 0an.UingLemma 1.7, we know that x n 1 z n 0, x n 1 u n 0an. Since both J : E E and J 1 : E E are uniformly continuou on bounded ubet, we have x n y n 0an. From Step 3, we know that ω xn /. Then, for all q ω x n, there exit a ubequence of {x n }, for implicity, we till denote it by {x n } w w w w uch that x n q a n.therefore,un q, zn q and yn q a n. Since S : E E i weakly continuou and u n Sz n,thenq Fix S. From the iterative cheme 2.1, weknow that there exit v n Ty n uch that r n v n J x n e n Jy n.thenv n 0an. Lemma 1.2 implie that q T 1 0. Step 5. x n q Q T 1 0 Fix S x 0 a n. Let {x ni } be any ubequence of {x n } which i weakly convergent to q T 1 0 Fix S. Since x 0 and q T 1 0 Fix S H n W n,wehaveϕ x n 1,x 0 ϕ q,x 0.Then it follow that ϕ x n,q ) ϕ x n,x 0 ) ϕ x0,q ) 2 x n x 0,Jq Jx 0 ϕ q,x 0 ) ϕ x0,q ) 2 x n x 0,Jq Jx 0, 2.11

6 6 Fixed Point Theory and Application which yield lim up ϕ x ni,q ) ϕ q ),x 0 ϕ x0,q ) 2 q x 0,Jq Jx 0 n 2 q q, Jq Jx Hence ϕ x ni,q 0ai. It follow from Lemma 1.7 that x ni q a i. Thi mean that the whole equence {x n } converge weakly to q and each weakly convergent ubequence of {x n } converge trongly to q. Therefore, x n q Q T 1 0 Fix S x 0 a n. Remark 2.2. If E i reduced to a real Hilbert pace H and S I, thenq T r n equal to J T r n I r n T 1. So the iterative cheme 2.1 i reduced to the following one introduced by Yane and Xu in 1 : x 0 H choen arbitrarily, y n α n x n 1 α n J T r n x n e n, H n { v H : yn v 2 xn v α n x n v, e n en 2}, 2.13 W n { z H : z x n,x 0 x n 0 }, x n 1 P Hn W n x 0, n 0. They proved that, if T 1 0 /, then the equence {x n } generated by 2.13 converge trongly to P T 1 0x 0 provided i {α n } 0, 1 i a equence uch that α n 1 δ for ome δ 0, 1 ; ii {r n } 0, i a equence uch that inf n r n > 0; iii {e n } E i a equence uch that e n 0. Remark 2.3. If E i reduced to a real Hilbert pace H, α n 0, e n 0andS I, then 2.1 include the following iterative cheme introduced by Solodov and Svaiter in 2 : x 0 H, 0 v n 1 r n yn x n ), vn Ty n, H n { z H : z y n,v n 0 }, 2.14 W n { z H : z x n,x 0 x n 0 }, x n 1 P Hn W n x 0, n 0. They proved that, if T 1 0 / and lim inf n r n converge trongly to P T 1 0x 0. > 0, then the equence generated by 2.14

7 L. Wei and Y. J. Cho 7 Corollary 2.4. Suppoe that E and S are the ame a thoe in Theorem 2.1. Fori 1, 2,...,m,let T i : E 2 E be maximal monotone operator. Denote D: m i 1 T 1 i 0 FixS) and uppoe that D /. Then the equence {x n } generated by x 0 E, r 0,i > 0, i 1, 2,...,m, y n,i Q T ) i r n,i xn e n, i 1, 2,...,m, Jz n,i α n,i Jx n ) 1 α n,i Jyn,i, i 1, 2,...,m, u n,i Sz n,i, i 1, 2,...,m, H n,i { v E : ϕ ) ) v, u n,i ϕ v, zn,i αn,i ϕ ) ) )} v, x n 1 αn,i ϕ v, xn e n, m H n : H n,i, i 1 W n { z E : z x n,jx 0 Jx n 0 }, x 0 n 0, i 1, 2,...,m, 2.15 converge trongly to Q D x 0 provided i {α n,i } 0, 1 i a equence uch that α n,i 1 δ, for ome δ 0, 1, i 1, 2,...,m and n 0; 1, 2,..., ii {r n,i } 0, i a equence uch that inf n 0 r n,i > 0 for i 1, 2,...,m; iii {e} E i a equence uch that e n 0 a n. Similar to the proof of Theorem 2.1, we have the following reult. Theorem 2.5. The equence {x n } generated by x 0 E, r 0 > 0, y n Qr T ) xn n e n, Jz n α n Jx 0 ) 1 α n Jyn, u n Sz n, H n { v E : ϕ ) ) v, u n ϕ v, zn αn ϕ ) ) )} v, x 0 1 αn ϕ v, xn e n, W n { z E : z x n,jx 0 Jx n 0 }, x 0 n 0, 2.16 converge trongly to Q T 1 0 Fix S x 0 provided i {α n } 0, 1 i a equence uch that α n 0 a n ; ii {r n } 0, i a equence uch that inf n 0 r n > 0; iii {e n } Ei a equence uch that e n 0a n.

8 8 Fixed Point Theory and Application Remark 2.6. If E i reduced to a real Hilbert pace H and S I, then the iterative cheme 2.16 i reduced to the following one, which i imilar to that in 1 : x 0 H choen arbitrarily, y n α n x 0 1 α n Jr T n x n e n, H n { v H : yn v 2 xn v 2 α x0 n 2 2 xn x 0,v ) 2 ) ) 1 α n xn v, e n 1 αn e n 2 α n x n 2}, W n { z H : } z x n,x 0 x n 0, x n 1 P Hn W n x 0 n Corollary 2.7. Suppoe that E, S, T i,andd are the ame a thoe in Corollary 2.4. IfD /, then the equence {x n } generated by x 0 E, r 0,i > 0, y n,i Q T ) i r n,i xn e n, Jz n,i α n,i Jx 0 ) 1 α n,i Jyn,i, u n,i Sz n,i, H n,i { v E : ϕ ) ) v, u n,i ϕ v, zn,i αn,i ϕ ) ) )} v, x 0 1 αn,i ϕ v, xn e n, m : H n,i, i 1, 2,...,m, H n i 1 W n { z E : z x n,jx 0 Jx n 0 }, x 0 n 0, 2.18 converge trongly to Q D x 0 provided i {α n,i } 0, 1 i a equence uch that α n,i 0 a n for i 1, 2,...,m; ii {r n,i } 0, i a equence uch that inf n 0 r n,i > 0 for i 1, 2,...,m; iii {e n } Ei a equence uch that e n 0a n. 3. Application to minimization problem Definition 3.1. Let f: E, be a proper convex and lower emicontinuou function. Then the ubdifferential f of f i defined by f z ) { v E : f y ) f z ) y z, v, y E } z E 3.1

9 L. Wei and Y. J. Cho 9 Theorem 3.2. Let E, S, {α n }, {r n },and{e n } be the ame a thoe in Theorem 2.1. Letf : E, be a proper convex and lower emicontinuou function. Let {x n } be the equence generated by x 0 E, r 0 > 0, { y n arg min f z 1 } z n 2 1 ) z,j xn e n, z E 2r n r n Jz n α n Jx n ) 1 α n Jyn, u n Sz n, H n { v E : ϕ ) ) v, u n ϕ v, zn αn ϕ ) ) )} v, x n 1 αn ϕ v, xn e n, W n { z E : z x n,jx 0 Jx n 0 }, x 0 n If f 1 0 FixS) /,then{x n } converge trongly to Q f 1 0 Fix S x 0. Proof. Since f: E, i a proper convex and lower emicontinuou function, the ubdifferential f of f i a maximal monotone operator from E into E. We alo know that { y n arg min f z 1 } zn 2 1 ) z,j xn e n 3.3 z E 2r n r n i equivalent to 0 f ) 1 y n Jy n 1 J x n e n ) 3.4 r n r n Thu we have y n Q f r n x n e n and o Theorem 2.1 implie that {x n } converge trongly to Q f 1 0 Fix S x 0 a n. Similarly, we have the following theorem. Theorem 3.3. Let E, S, {α n }, {r n },and{e n } be the ame a thoe in Theorem 2.5. Letf: E, be a proper convex and lower emicontinuou function. Let {x n } be the equence generated by x 0 E, r 0 > 0, { y n arg min f z 1 z n 2 1 } ) z,j xn e n, z E 2r n r n Jz n α n Jx 0 ) 1 α n Jyn, u n Sz n, H n { v E : ϕ ) ) v, u n ϕ v, zn αn ϕ ) ) )} v, x 0 1 αn ϕ v, xn e n, W n { z E : z x n,jx 0 Jx n 0 }, x 0 n If f 1 0 FixS) /,then{x n } converge trongly to Q f 1 0 Fix S x 0.

10 10 Fixed Point Theory and Application 4. Application on olving the variational inequalitie Definition 4.1 ee 4. LetE be a real Banach pace. A ingle-valued operator A : E E i aid to be hemicontinuou if it i continuou along each line egment in E with repect to the weak topology of E. Definition 4.2. Let E be a real Banach pace and let C be a nonempty cloed and convex ubet of E. LetA : C E be a ingle-valued monotone operator which i hemicontinuou. Then a point u C i aid to be a olution of the variational inequality for A if y u, Au 0, y C. 4.1 We denote by VI C, A the et of all olution of the variational inequality for A. Definition 4.3. Let E be a real Banach pace and let C be a nonempty cloed and convex ubet of E.WedenotebyN C x the normal cone for C at a point x C,thati, N C x { x E : y x, x 0, y C }. In 8, it i proven that the operator T : E 2 E defined by Ax N C x, x C, Tx, x/ C, i a maximal monotone operator. It i eay to verify that T 1 0 VI C, A. Theorem 4.4. Let E, S be the ame a thoe in Theorem 2.1 and let C be a nonempty cloed and convex ubet of E. LetA : C E be a ingle-valued monotone operator which i hemicontinuou. Let {x n } be a equence generated by x 0 E, r 0 > 0, y n VI C, A 1 )) ) J J xn e n, r n Jz n α n Jx n ) 1 α n Jyn, u n Sz n, H n { v E : ϕ ) ) v, u n ϕ v, zn αn ϕ ) ) )} v, x n 1 αn ϕ v, xn e n, W n { z E : } z x n,jx 0 Jx n 0, x 0 n 0, 4.4 which converge trongly to Q VI C,A Fix S x 0 provided i {α n } 0, 1 i a equence uch that α n 1 δ, for ome δ 0, 1 ; ii {r n } 0, i a equence uch that inf n 0 r n > 0; iii {e n } Ei a equence uch that e n 0a n.

11 L. Wei and Y. J. Cho 11 Proof. Note that y n VI C, A 1rn )) ) J J xn e n y y n, Ay n 1 r n Jyn J x n e n )) 0 J x n e n ) rn Ty n Jy n y n J r n T ) 1 J xn e n ) Q T rn xn e n ), y C 4.5 where T itheameathatindefinition 4.3. Then the reult follow from Theorem 2.1. Similarly, we have the following reult. Theorem 4.5. Let E, C, S and A be the ame a thoe in Theorem 4.4.Let{x n } be a equence generated by x 0 E, r 0 > 0, y n VI C, A 1 )) ) J J xn e n, r n Jz n α n Jx 0 ) 1 α n Jyn, u n Sz n, H n { v E : ϕ ) )} v, u n ϕ v, zn αn ϕ ) ) )} v, x 0 1 αn ϕ v, xn e n, W n { z E : } z x n,jx 0 Jx n 0, x n 1 Q Hn Wn x 0 n 0, 4.6 which converge trongly to Q VI C,A Fix S x 0 provided i {α n } 0, 1 i a equence uch that α n 0 a n ; ii {r n } 0, i a equence uch that inf n 0 r n > 0; iii {e n } E i a equence uch that e n 0 a n. Remark 4.6. It will be intereting to conider imilar problem when a ingle mapping S i replaced by an amenable emigroup S of mapping that are nonexpanive with repect to the Lyapunov functional and to combine the iterative cheme for the fixed point et determined by a left regular equence of mean a demontrated in the recent work 9 with that of Theorem 2.1. Acknowledgment Thi paper i upported by the National Nature Science Foundation of China Grant no The author thank the reviewer for their good uggetion.

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