 American Journal of Oper ations Research, 2011, 1, 305-311 doi:10.4236/ajor.2011.14035 Published Online December 2011 (http://www.SciRP.org/journal/ajor) Copyright © 2011 SciRes. AJOR 305 ,H - - -Accretive Operators and Generalized Variational-Like Inclusions* Rais Ahmad, Mohammad Dilshad Department of Mat hematics, Aligarh Muslim University, Aligarh, India E-mail: raisain_123@rediffmail.com, mdilshaad@gmail.com Received August 17, 2011; revised September 20, 2011; acce pt ed September 30, 2011 Abstract In this paper, we generalize ,H -accretive operator introduced by Zou and Huang [1] and we call it ,H - - -accretive operator. We define the resolvent operator associated with ,H - - -accretive operator and prove its Lipschitz continuity. By using these concepts an iterative algorithm is suggested to solve a generalized variational-like inclusion problem. Some examples are given to justify the definition of ,H - - -accretive operator. Keywords: ,H - - -Accretive Operator, Variational-Like Inclusion, Resolvent Operator, Algorithm, Convergence 1. Introduction Variational inclusion problems have emerged as a po- werful tool for solving a wide class of unrelated problems occuring in various branches of physical, engineering, pure and applied sciences in a unified and general frame work. In 2001, Huang and Fang [2] firstly introduced the generalized -accretive mappings and gave the defini- tion of resolvent operator for the generalized -accre- tive mappings in Banach spaces. Also, they have shown some properties of their resolvent operator. Since then, Fang and Huang, Lan, Cho and Verma and others introduced and studied several generalized operators such as H-accretive, mm -H -accretive and ,A -ac- cretive mappings. For example, see [3-16] and references therein. In 2008, Zou and Huang [1] introduced -accre- tive operator, its resolvent operator and applied them to solve a variational inclusion problem in Banach spaces. In this paper, we generalized -accretive operator to - ,H ,H ,H - -accretive operator and define its resol- vent operator. Further, we prove the Lipschitz continuity of resolvent operator and apply these new concepts to solve a variational-like inclusion problem. Some example are constructed. 2. Preliminaries let X be a real Banach spaces with its dual * , , be the duality pairing between X and * and (respec- tively 2X CB X) denote the family of non-empty subsets (respectively, closed and bounded subsets) of X. The generalized duality mapping is defined * :2 X q JX by 1 *:, , qq q Jxf Xxfxfx , X , where is a constant. In particular, 2 >1q is the usual normalized duality mapping. It is known that, 1 2 q q xxJx for and q 0x is single- valued if X* is strictly convex. If X is a real Hilbert space, then 2 becomes the identity map ping on X. The modulus of smoothness of X is the function :0, 0, X defined by 1 sup1: 1, 2 Xtxyxyxy t . A Banach space X is called uniformly smooth, if 00 lim X t t t . is called -uniformly smooth, if there exists a constant such that q >0C *This work is supported by Department of Science and Technology, Government of India under gr a nt no. SR/S4/MS: 577/09. q XtCt , . 1q  R. AHMAD ET AL. 306 Note that q is single-valued if is uniformly smooth. The following inequality in -uniformly smooth Banach spaces has been proved by Xu [17]. q Lemma 2.1. Let be a real uniformly smooth Banach space. Then is -uniformly smooth if and only if there exists a constant such that for all q>0 q C , yX, , qq qq q yxqyJxCy . Definition 2. 1. Let ,: BX X and ,: XX X be the single-valued mappings. i) is said to be -accretive, if , ,0 q AxAyJx y , , yX; ii) is said to be strictly -accretive, if is - accretive and equality holds if and only if x = y; iii) A， is said to be -strongly -accretive with respect to , if there exists a constant >0 such that ,,, , ,,; q q, Ax uHAy uJxyxy xyu X iv) , B is said to be -relaxed -accretive with respect to , if there exists a constant B>0 such that ,,, , ,,; q q, uBxH uByJxyxy xyu X v) is said to 1-Lipschitz continuous with respect to , if there exists a constant such that (,)H Ar 1>0r 1 ,, Ax uHAy urxy,,, yu X. In a similar way, we can define the Lipschitz continuity of the mapping with respect to . ,H B vi) is said to be -Lipschitz continuous, if there exists a constant >0 such that , yxy, , yX. Definition 2.2. Let ,:NXXX :X MX X be the single- valued mappings. Let be multi-valued mapping. 2 i) is said to be -accretive, if , ,0 q uvJxy ,, yX zX , , , for each fixed ,uMxz vM ,yz ; ii) M is said to be strictly -accretive, if M is - accretive and equality holds if and only if y; iii) N is said to -relaxed t -accretive in the first argument, if there exists a constant such that >0t ,,, , ,,; q q NxuNyu Jxytxy xyu X iv) N is said to be -Lipschitz continuous in the first argument, if there exists a constant >0 such that ,,NxuNyuxy , ,, yu X. Similarly, we can define the Lipschitz continuity of N in the second argument. 3. ,H - - -Accretive Operator In this section, we generalize -accretive operator [1] and call it ,H ,H - - -accretive operator and discuss some of its properties. Definition 3. 1. Let ,, : BX X , ,: XX X :X MX X be the single-valued mappings. Let be a multi-valued mapping. M is said to be 2 ,H - - -accretive operator with respect to mappings and , if for each fixed , BzX , z is -accretive in the first argument and ,, AB M z XX . Remark 3.1. If x , X and >0 , ,MM and , yx y then ,H - - -accretive operator reduces to -accretive opera- tor, which was introduced and studied by Zou and Huang [1]. ,H Example 3.1. Let . Let , 0Ax sinBx x , , Ax ByAxBy and 22 , xz xz, X and for each fixed zX . Let ,,2 xzMxzx x and ,2 y xy . Then 2 ,,,,22, 2 0, y MxzMyzxyx y xy which means that , z is -accretive in the first argument. Also, for any X, it follows from above that ,, , 0sin22sin, , A BMzxHAx BxMx z xx x which means that ,, ABM z is surjective. Thus M is ,H - - -accretive operator with respect to mappings and . B Example 3.2. Let X, A, B, H, and M are same as in Example 3.1. Let 2 ,2 z Mxz e. Then , 22 ,, , sin, xz , A BMzxHAx BxMx z xe which shows that 0, , ABM zX , that is ,, AB M z is not surjective, hence M is Copyright © 2011 SciRes. AJOR  307 R. AHMAD ET AL. not - ,H - -accretive operator with respect to the mappings A and B. Theorem 3.1. Let , AB be -strongly -ac- cretive with respect to A, -relaxed -accretive with respect to B, > . Let M be an - ,H - - accretive operator with respect to mappings A and B. Then the operator is single- 1 ,,HABM z valued for each fi xed . zX Proof. For any given u and , let zX 1 , ,, yHABMz u . Then ,, Ax BxuM x z ,, , Ay ByuMy z . Since , z is -accretive in the first argument, we have 0, , ,,, ,,,, ,,, ,, ,,,, ,,,, 0. q q q q q qq q Ax Bx Bx Bx By Bx Bx uHAy ByuJxy HAxHAyBy Jxy HAxHAy BxHAy Bx HAyJ xy HAxHAyBx Jxy H AyH AyByJxy xy xyxy Since > , we have y and so is single-valued. This com- pletes the proof. ,B 1 ,HA z M Definition 3.2. Let , AB be -strongly - accretive with respect to A and -relaxed -accretive with respect to B and > . Let M be an ,H - - -accretive operator with respect to mappings A and B. Then for each fixed , the resolvent operator is defined by zX , ,: Mz R ,H HXX ,() ,, Mz RuHABMz 1u uX, . Theorem 3.2. Let , AB be -strongly -ac- cretive with respect to A, -relaxed -accretive with respect to B, > and is -Lipschitz continuous. Let is a -:2H X MX X , - -accretive operator with respect to mappings A and B. Then the resolvent operator is , ,: H Mz RX X1q -Lips- chitz continuous i.e., 1 ,, ,, ()Ru ,uv ()q HH Mz Mz RRv uv , and each fixed . XzX Proof. Let ,uv X , then by definition of resolvent operator, it follows that 1 , ,,, H Mz RuHABMz u v , and 1 , ,,, H Mz RvHABMz . Then ,, ,, , , , ,, HH Mz Mz H Mz uHAR uBR u MRuz and ,, ,, , , , ,. HH Mz Mz H Mz vHARv BRv MRv z Let , , , H Mz Pu Ru , , H Mz Pv Rv Since , z is -accretive in the first argument, we have ,, ,0 q uHAPuBPuvH APvBPv JPuPv , ,, , ,,,. q q uvJ PuPvHAPuBPu HAPvBPvJPu Pv It follows that 1 ,,, ,,, , ,,, , ,,, , q q q q q qq uvPu PvuvJPuPv HA PuBPuHA PvBPv JPuPv HAPu BPuHAPv BPu JPuPv HAPv BPuHAPvBPv JPuPv Pu PvPu PvPu Pv q 1 1qq q uvPu PvPu Pv 1q PuPvu v i.e. 1 ,, ,, q HH Mz Mz RuRv u v . This completes the proof. Copyright © 2011 SciRes. AJOR  R. AHMAD ET AL. 308 4. An Application for Solving Generalized Variational-Like Inclusions In this section, we apply ,H - - -accretive operator for solving generalized variational-like inclusions. Let ,, :STGXCB X ,,: be the multi-valued map- pings. BXX , ,,: NXXX :2MX X be sin- gle-valued mappings. Suppose be a multi-valued mapping such that X is ,H - - - accretive operator. We consider the following problem of finding X , uSx, vTx and and zGx 0, ,Nuv Mxz . (4.1) Problem (4.1) is called generalized variational-like inclusion pro blem. Below are some special cases of our problem: i) If is real Hilbert space and , z is maximal monotone operator then a problem similar to (4.1) was introduced and studied by Huang [18 ]. ii) If , is single-valued and identity mapping and and 0TG N S ,= N ,MM then our problem reduces to the problem considered by Bi et al. [19], that is find such that uX 0Nu Mu . It is clear that for suitable choices of operators in- volved in the formulation of problem (4.1), one can obtain many variational-like inclusions studied in recent past. Lemma 4.1. Let be a -uniformly smooth Ba- nach space. q X,,GSTCB ,: :X be multi-valued map- pings, BXX be single-valued mappings and : X be a mapping satisfying yxy : 0xX x and , where . Let = 0 ,, ker ker : NXXX :2 X X be the single-valued mappings. Let be a multi-valued mapping such that MX is - ,H - - accretive operator. Then ,,, uvz where X , , and is a solution of problem (4.1) if and only if xuS vT x x , zG ,, uvz satisfies , , =, H Mz , RHAxBxNu v . (4.2) Proof. Let ,,, uvz where X, uSx, and satisfies the Equation (4.2), i.e., vTx zGx , , =, H Mz , RHAxBxNu v , , . Using the definition of resolvent operator, we have 1 ,, , ,,, HABM zHAxBxNuv AxBxNuvHAx BxMxz 1 0, , 0,, (0), ,0, ,. Nuv Mxz NuvMxz NuvMuzNuvMxz This completes the proof. Based on Lemma 4.1, we define the following algori- thm. Algorithm 4.1. Let , , , G S T , , B , , N , and all are same as in Lemma 4.1. For any given 0 X , 00 xuS, 00 and xvT x 0 zG0 and 0<<1 , compute the sequences n , n u, n v and n z by the following iterative scheme: , 1,,, n H nnn Mz nn RHAxBxNuv ; (4.3) 1 11 , ,; nn n nnn nnn uSx u uSxSxxx 1 (4.4) 1 11 , ,; nn n nnnnnn vTx vvTx Txxx 1 (4.5) 1 11 , ,; nn n nnn nnn zGx zzGx Gxxx 1 (4.6) 0, 1,2,n , where , is the Hausdorff metric on CB X. Theorem 4.1. Let be -uniformly smooth Ba- nach space and q ,,: BXX ,, be the single-valued mappings. Let : NXXX X MX be the single-va- lued mappings and be multi-va- lued mappings. Suppose be a multi- valued mapping such that XX ,,GS :TCB :X2 is - ,H - -accre- tive operator with respect to mappings and . Assume that B i) yx y and ; ker 0 ii) , AB is -strongly -accretive with respect to and -relaxed -accretive with respect to ; B iii) ,H is 1-Lipschitz continuous in the first argument and -Lipschitz continuous in the second argument; r 2 r iv) ,N is 1 -Lipschitz continuous in the first argument and 2 -Lipschitz continuous in the second argument; v) ,N is -relaxed t -accretive in the first argument; vi) is -Lipschitz continuous; vii) , and are -Lipschitz continuous with constant STG , ST and G respectively; viii) ,, 1 ,, 1 HH nn Mz Mz nn RxRxzz , >0 , 1 , nn zzX ; Copyright © 2011 SciRes. AJOR  R. AHMAD ET AL. Copyright © 2011 SciRes. AJOR 309 ix) 111 121 1212 1 1 < qq G qqqqq qSSSqS q rrqt qrrC T . Then ,,, uvz where X , , and is a solution of problem (4.1), and the sequences uSx vT x zGx n , , and n u n v n z de- fined in Algorithm 3.1 converge strongly to , , and , respectively in u v z . Since (,) AB r is 1-Lipschitz continuous in the first argument and 2-Lipschitz continuous in the second argument, we have r Proof. Using Algorithm 4.1, Lipschitz continuity of resolvent operator and condition (viii), we have 1 , 1, ,11 11 , , , ,11 11 , ,, ,, ,(,) ,, n n n n H nnnnnn Mz Hnnnn Mz Hnn nn Mz Hnnnn Mz M xxRHAxBxNuv RHAxBxNuv RHAxBxNuv RHAxBxNuv R 1 ,11 11 , ,11 11 , 1 11 11 1 11 ,, ,, ,, ,, =,, n n Hnnnn z Hnnnn Mz q nn nn nnnnnn q nnn n HAx BxNuv RHAxBxNuv HAxBx Nuv 11 1 111 11211 2 ,, ,(,) ,, nnn n nnnn nn nn nn nnnn HAxBxHAxBx HAxBxHAx Bx HAx BxHAx Bx rx xrx xrrx x 1 and hence 11 121 ,, qq q nnn nnn HAxBxHAxBxr rxx . (4.9) 1 AxBxN uvzz H AxBxHAxBx 1 1 111 1 ,, ,, . nnnn q nn nn nn Nu vNuv Nu vNu v zz (4.7) Since S is -Lipschitz continuous with constant S and using (4.4), we have 111 11 , . n nnn nnn nn Sn nn nSnn u uSx Sxxx xx xxxx 1 (4.10) Since, ,N is -relaxed t -accretive in the first argument and using (4.10), we have 11 11 ,,,, . nnnnqnn q qq n nnS nn Nu vNuvJu u tu utxx (4.11) As, ,N is 1 -Lipschitz continuous in the first argument, (,)H r is 1-Lipschitz continuous in the first argument and 2-Lipschitz continuous in the second argument and r is -Lipschitz continuous, we have Now, we estimate 11 1 11 11 111 11 ,, (,, ,,, ,,,, ,,,, ,,, , nnn n q nnnn q nnn nnn nnq nnnn nnqnnnn q qnn qnnnn n HAxBxHAx Bx Nu vNuv HAxBxHAx BxqNuv NuvJu uqNu v NuvJHAx BxHAxBx JuuCNuv Nuv HAx 11 11 1 111 1 11 ,, ,,,, , , , ,,,. q nnn nn nnq nnnn q nnnnn n qq nnqnnn n BxHAxBxqNuv NuvJu uqNu v NuvHAx BxHAxBx uuCNuvNu v 1 11 11 1 1 1 1112 1 1 11 1 1 11121 1 11 1 11 112 ,, ,, , nnnn qq nnn nnn q q nn nn q qnn q q n Snn nn q qn q Snn q nqn SS Nu vNuv HAxBxHAxBxuu uu rr xx uu xx rrxx xx rr 1 1. qq nn xx (4.12) Also 1 111 ,, nnn n n nnS nn Nu vNuv uu xx 1 (4.8)  R. AHMAD ET AL. 310 11 ,, q qq qn nnn nSn n Nu vNuvxx . 1 (4.13) Using (4.9) , (4.11), (4.12) and (4.13), (4.8) becomes 11 1 1 11 1 121 11 1 11 121 121 ,,,, q nnn nnnnn qq q nqn SS nn qq qn qS nn qq qq nn qn SS Sq HAxBxHAx BxNuvNuv qrrxx Cxx rr qtqrrC 12 1 1 q qq qn nn S nn rr x xqtx x 1 qq qn Snn xx 11 1 11 1 121 1211 ,,,, ()() nnn nnnnn q q qnnqqnqqn qSS SqSn H AxBxHAxBxNuvNuv rrqtqrrCxx . n (4.14) Since is ,N 2 -Lipschitz continuous in the second ad g the -Lipschitz continuity of T with rgument, an constant usin T and (4.5), we have 1 21 1 , nn n Tn nn n Nu vN xx xx 1121 211 , , nn nn n nn nn uvv v Tx Txxx 2. n Tnn xx 1 Also, using -Lipschitz continuity of G with constant (4.15) G and (4.6), we have 111 , n nnn nnn zzGx Gxx (4.16) Using (4.14),(4.15) and (4.16), (4.7 ) becomes 11 1 . nn Gn nn nGn n xx xxxx x 11 11 112 1121 21 , qqqq qq nn qnqn q nnSSS qS nn TGnn xxrrqtqrrC xx or 11 n nn nn xx x, where 11 11 121 121 2 . qqq qq nnnqn qSSSqS nn TG rr qtqrrC q qn Since 0< <1 , it follows that n , as where n 1111 1112 12 2 qqq qqqqq qSS SqST rrqt qrrC G . From (ix), it hat <1 follows t , and consequently n is a Cauchy sequence i. Since X is a Banach space, there exists n X X, such that n x as n. From (4.4), (4.5) and (4.6) of Algorithm 4.1, it follows that n u, n v and z all are Cauchy sequences in X, that is there exi and zXthat , n st ,u v such n n vv and n zz as n. Now, using continuity of operators S, T, G, uu the , B, , N , and M and by Algorithm 4.1, we have (,) (, ),, H Mz RHAxBxN uv . Noe l show t uSx w, wshalhat ,, , 0 as . n nn nSn duSxu uSx uuSxS uux xn du x Copyright © 2011 SciRes. AJOR  R. AHMAD ET AL. 311 This implies that , since . Similarly, pletes the 5. References [1] Y. Z. Zou and N. J. Huang, “-Accretive Operator ion for SolvinVariational Inclusions in Banach Spematics a 16 /j.amc.2008.07.024 ,0duSx , it follows that uS zGx Sx CBX ca proof. x . This comwe n prove that vTx, ,H g with an Applicat aces,” Applied Mathnd Computation, Vol. 204, No. 2, 2008, pp. 809-8. doi:10.1016 . J. Huang and Y. P. 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