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Advances in Pure Mathematics, 2012, 2, 139-148 http://dx.doi.org/10.4236/apm.2012.23021 Published Online May 2012 (http://www.SciRP.org/journal/apm) Stable Perturbed Algorithms for a New Class of Generalized Nonlinear Implicit Quasi Variational Inclusions in Banach Spaces Salahuddin, Mohammad Kalimuddin Ahmad Department of Mathematics, Aligarh Muslim University, Aligarh, India Email: salahuddin12@mailcity.com, ahmad_kalimuddin@yahoo.co.in Received August 11, 2011; revised October 11, 2011; accepted October 20, 2011 ABSTRACT In this work, a new class of variational inclusion involving T-accretive operators in Banach spaces is introduced and studied. New iterative algorithms for stability for their class of variational inclusions and its convergence results are established. Keywords: T-Accretive Operators; Variational Inclusions; Iterative Algorithms; Stability Conditions; Convergence; Strong Accretivity; Banach Spaces 1. Introduction Variational inequality theory provides us with a simple, natural, general and unified framework for studying a wide range of unrelated problems arising in mechanics, physics, optimization and control theory nonlinear pro- gramming, economics, transportation, equilibrium and engineering sciences. In recent years, variational inequality has been ex- tended and generalized in different direction. A useful and important generalization of the variational inequality is called variational inclusions see [1-7]. Suppose E is a real Banach space with dual space E*, norm . and dual pairing .,. :2 , 2E is the family of all nonempty subsets of E, CB(E) is the family of all non- empty closed bounded subset of E and the generalized duality mapping E q JE is defined by ** 1 * :, , q q ** , , J ufEu fu f fu uE where q > 1 is a constant. In particular, 2 J is the usual normalized duality mapping. It is known that, in general 2 2 q q J uu 0 Ju for all and q u J is simple single valued, if E* is strictly convex. The modulus of smoothness of E is the function , 0, :0 E defined by 1 sup 1: 2 Etuvuv 1, .uvt A Banach space E is called uniformly smooth if 0 lim 0 E t t t >0 . E is called q-uniformly smooth, if there such that exists a constant ,1. q Ettq >0 q c,uvE Note that Jq is single valued, if E is uniformly smooth. Xu and Roach [8] and Xu [9] proved the following re- sults. Lemma 1.1. Let E be a real uniformly smooth Banach space. Then E is q-uniformly smooth if and only if there exists a constant such that for all ,. qq q qq uvu qvJu cv :TE E :2 Definition 1.1. [10] Let be a single-valued operator and E ME be a multivalued operator. M is said to be T-accretive if M is accretive and TMEE >0 hold for all . Remark 1.1. 1) From [11] it is easily establish that if TI (the identity map on E), then the definition of I- accretive operator is that of m-accretive operator. 2) Example 2.1 in [11] shows that an m-accretive operator need not be T-accretive for some T. Let , :TE E:NE EE :2 be two single valued mappings. Let E MEE tE ., :2 be a set-valued mapping such that for each fixed , E MtE ,,: be a T-accretive operator. For given f gp EE uE are map- pings, consider the following problem of finding such that 0,,, f uNuuMpugu (1) C opyright © 2012 SciRes. APM SALAHUDDIN, M. K. AHMAD 140 which is called the generalized nonlinear implicit quasi variational inclusions. Special Cases: 1) If E is a Hilbert space, then problem (1) is equiva- lent to finding such that uE ,, Dom ., 0, puM gu f uNuu Mpugu , (2) which is called the generalized nonlinear implicit quasi variational inclusions, considered by Ding [12] and Fang et al. [13]. 2) If M ut ,utE uE Mu for all , then problem (2) is equivalent to finding such that Dom 0, pu M fu Nu ,u Mpu :2 (3) where E ME , is a maximal monotone mapping. The problem (3) was considered by Huang [14]. 3) If , M ut E uE ut for each t, then prob- lem (2) is equivalent to finding such that , , Dom ., ,, , pu gu f uNuuvpu pu gu vgu R tE (4) where such that for each :EE , is a proper convex lower semi- continuous function with ., :tER ., .t 0f uE ., ,, u M puu :TE E :2 Range Domp (5) The problem (4) was considered by Ding [15] for g to be an identity mapping. 4) If and g is the identity mapping, then prob- lem (1) is equivalent to finding such that Dom 0, pu M Nuu (6) which is called the generalized strongly nonlinear im- plicit quasi variational inclusions, considered by Shim et al. [16]. Remark 1.2. For a suitable choice of f, g, p, N, M and the space E, a number of classes of variational inequali- ties, complementarity problems and the variational in- clusions can be obtained as special cases of the general- ized nonlinear implicit quasivariational inclusions (1). Let be a strictly monotone operator and E MEE , . be a T-accretive operator. Fang and Huang [11] defined the resolvent operator 1 (., ) ,., Mt T J vTMt v vE :TE E :2 (7) By Theorem 2.2 in [11], we know that if is a strictly accretive operator and E MEE (., ) ,: Mt T is a T-accretive operator, then the operator J EE :TE E >0 is a single valued. From the proof of Theorem 2.3 in [11], it is easy to obtain the following result. Lemma 1.2. [10] Let be a strictly accre- tive operator with constant and for each fixed tE , :2 E MEE (., ) ,: Mt T be a T-accretive operator then the operator J EE is Lipschitz continuous with constant 1, i.e., (., )(., ) ,, 1,. Mt Mt TT J uJvuv uvE 2 qqq q aba b 2max,2 max, 2. qq qq qq q a babab ab (8) Lemma 1.3. Let a and b be two nonnegative real numbers. Then . (9) Proof. Definition 1.2. Let n M 0,1, 2,n and M be a maximal mono- tone mappings for . The sequence n M G n is said to be graph converges to M (write M M) if for every , there exists a sequences ,uvGraph M ,n uvGraph Mn uun vv n n nn such that and as . Lemma 1.4. [3] Let M and M be the maximal mono- tone mappings for G n . Then 0,1, 2,n M M , n MM if and only if J uJu uE (10) 1 M JIM . and for every >0 , where n, n and b Lemma 1.5. Let a n be three se- quences of nonnegative numbers satisfying the following condition. There exists a positive integers such that c 0 n 10 1, for nnnnnn atabtcnn (11) 0,1 n t 0 n n t , where lim 0 n nb 0 n n c , and 0 n an. Then as . 0 inf :ann 0. n. Then Proof. Let >0. Sup- pose that >0 n a Then 0.nn for all It follows from (11), that 1 11 22 nnnnnn nnnnn aattbc abttc . nn0 n,n 10 nn (12) for all 0 Since b as there exists such that 1 1,forall. 2n bnn Copyright © 2012 SciRes. APM SALAHUDDIN, M. K. AHMAD 141 Combining (11) and (12), we have ,0,if; q TuTv Juvuv 1 aa 1 2 nn nn tc 1 nn for all , which implies that 1 1 2nn nn ta 11 . n nn c 0 This is a contradiction. Therefore, 0 j n a j and so there exists a subsequence such that as . It follows from (11) that n a j n a 1 j jjjj nnn btc .j j1k E :2 nn aa and so as A simple induction leads to as for all and this means that as . n This completes the proof. 10 j n a 0 0 jk n a a :TE n Lemma 1.6. Let be a strictly accretive op- erator and for a fixed t, E E MEE , be a T-accretive operator in the first variable. If u is a solu- tion of the problem (1) if and only if (., ) , Mgu T g uJTpu f u Nuu >0 where is a constant and 1 ., .Mgu uE , , , 0, ,. pu gu u Tpu uu Nu u ,:Tg E E (., ) , Mgu T JT Proof. is a solution of (1) (., ) , 0, 0, 0 , ., Mgu T fuNu uMpugu fuNu uM TpufuNu Mpugu Tpufu N TMgupu puJTpuf u 2. Existence and Uniqueness Theorems In this section, we show the existence and uniqueness of solutions for the problem (1) in terms of Lemma 1.6. Definition 2.1. Let E be a real uniformly smooth Ba- nach space and be two single valued op- erators; T is said to be 1) Accretive if , q TuTv Juv0,,u vE or, equivalently 2 ,0TuTv Juv,,;u vE 2) Strictly accretive if T is accretive and >0r 3) Strongly accretive if there exists a constant such that ,,, q q TuTvJu vru vuvE or, equivalently 2 2 ,, ,;TuTvJ u vruvuvE 4) Lipschitz continuous if there exists a constant s > 0 such that ,,;TuTvsuvuvE >0 5) Strongly accretive with respect to g if there exists a constant such that ,,, q q TuTvJgu gvgu gvuvE or, equivalently 2 2 ,,,.TuTvJ gu gvgu gvuvE :NE E E : Definition 2.2. Let and g EE be the maps, then 1) .,.N >0 is said to be strongly accretive with respect to first argument if there exists a constant such that ,.,. ,,, q q NuNvJu vu vuvE or, equivalently 2 2 ,.,. ,,,;NuNvJu vuvuvE >0 2) N is said to relaxed accretive with respect to g if there exists a constant such that ,.,. ,, ,; q q NuNvJgu gvgu gv uv E >0 3) N is Lipschitz continuous in first argument if there exists a constant such that ,.,., ,.NuNvu vuvE :E Theorem 2.1. Let E be a q-uniformly smooth Banach space and TE be a strongly accretive and Lipschitz continuous with positive constants and respectively. Let be the strongly accretive and Lipschitz continuous with positive constants :pE E and respectively. Let be Lipschitz con- tinuous with positive constants ,:fgE E and respectively. Let :NE EE 1 p >0 be relaxed accretive with respect to in the first and second arguments with constants and >0 respectively, where 1 is defined by :pE E puT puTpu 1 for all uE . Assume that N is Lipschitz continuous with respect to first and second argument with constants >0 and Copyright © 2012 SciRes. APM SALAHUDDIN, M. K. AHMAD Copy APM 142 >0 right © 2012 SciRes. respectively. Let :2 E MEE be a T-accre- tive operator with second argument. If there exist con- stants >0 and >0 such that for all ,, ,uvw E (., )(., ,, Mgu M TT )gv J wJ w gugv (13) and 11,QP (14) where 1 11 q q 1 2 1. q qqqqqq qq q Pq C * uE . Then the problem (1) has a unique solution Proof. By Lemma 1.6, it is enough to show that the mapping EE* uE has a unique fixed point F : where F is defined as follows. (., ) ,, Mgu T F uupuJTpufuNuu q C Qq and uE (15) . From (13) and (15), we have for all ., ., , , ., ., ,, ., ., ,, , , ,, ,, Mgu Mgv TT Mgu Mgu TT Mgu Mgv TT FuFvupu JTpufuNuuvpvJTpvfvNvv uvpupvJT pufuNuuJT pvfvNvv JT vNvvJTpvfvNvv u vpupv 1 pvf (16) 1 ,, 1,(,) . T puT pvNuuNvvfufvgugv uvpupvT puT pvNuuNvvfufvuv Now since p is strongly accretive and Lipschitz continuous, we have , 1 1. qqq qq qq q q q q q q q q q uvpu pvuvqpu pvJuvCpu pv uvq uvCuv qC uv uvpu pvqCuv (17) By the strong accretivity of p with constant , we have 1, 1 q q qq pupvu vpupvJu vpupvJu vu v u vpupv that is ,, .NuuNvuuv (20) ,,Nvu Nvv pupvu v . (18) .u v (21) Since f is Lipschitz continuous, we get Similarly, by the strong accretivity of T with constant we have f .ufvuv (22) T puT pvpupv By . (19) Since :NE EE is a relaxed accretive with re- nd second argumenspect to p in the first a >0 -Lipschitz continuity of N with respect to first ent and t with constant argum -Lipschitz continuity of N with respect nd arguments, we have and >0 re have spectively, from (18) and (19), we to seco SALAHUDDIN, M. K. AHMAD 143 ,,, q q NJT puT pv T Tpv and similarly q q q qq uuNvu pu pu pv uv (23) ,,, . q q qq NvuNvvJTpuT pv uv ,p (24) From (23) and (24), Lipschitz continuity of T, Lemma 1.1 and Lemma 1.3, we have ,, ,,, ,, 2 q q q q qq qqq q qqqqq qqq q q qqq qqqqq TpuTpvNuuNvv Tpu Tpvq NuuNvvJTpu TpvCNuuNvv TpuTpvuvqu vCu v qCuv (25) Now from (16), (17), (22) and (25), we have 2. q q 1 11 12 1, q q qqq qqqqqq qq F uFvqCqCuv Q Puvuv (26) where 1 1q q q QqC and 1 2 q qqqqqq qq q Pq C n 1, nn and 1 QP x x . From (14), we know that 0< <1 . Therefore, there uE such that exists a unique F uu . This 3. truurbed iterative solving the problem (1) and tive se- ith er- completes the proof. Perturbed Algorithm In this section, we consct some new pert algorithms with errors for s and Stability prove the convergence and stability of the itera quences generated by the perturbed algorithms w rors. Definition 3.1. Let T be a self mapping of E and 1, nn x fTx define an iterative procedure which yields a sequence of point n x in E. Suppose that xE :Txx and n x converges to a fixed point x of T. Let n y E and let 1. nn n yfy ,T 1) If lim 0 n n thatn iterative procedure implies lim n yx , then the defined by fTx is said to be T-stable or stable with respect to T. 2) If n 0 n implies that lim n nyx , then the e iterativprocedure n x is lmost T-stable. stability results of iterative algorithms have been several a19]. As was sh he stabi of the theoretical and numerical interest. Remark 3.1. An iterative procedure n said to be a Some established by Ha uthors [17-own by rder and Hicks [20], the study on tlity is both x which is T- stable is almost T-stable and an iterative procedure n x which is almost T-stable need not be T-stable [21]. Algorithm 3.1. Let ,,,:fpgTE E and :NE EE be the five single valued mappings. Let n of M be the set-valued mapping from EE M into the power set of E such that for each tE, ., n M t and t are T-accretive mappings and M ., ., ., G n M tMt. For any given 0 uE, the per- turbed iterative sequence n u with errors is defined as follows: Copyright © 2012 SciRes. APM SALAHUDDIN, M. K. AHMAD 144 ., 1 1 nn Mgv nnnnn uuvpv , , nT nn nnnnnnT JTpvfv vuwpw T T , , , nnnnn n nnnnn n nnnn Nvvel pwfwNwwr pufuNuus ) n (27 ., 1 nn Mg w J ., , 1 nn Mgu nnnnnnT wu upuJ for 0,1,2,n, where n , n and are n three sequences in [0,1], n e, n r, n s and n l are four sequences in E satisfying the following condi- : tions 00 , nn nnn nn lim lim lim0 . n nn ser l (28) From Algorithm 3.1, we obtain the following algorithm for the problem (3). Algorithm 3.2. Let ,, : f pT EE and :NE EE be four single valued mappings. Let n M and M be T-accretive mappings from E into the power of E such that G n M M. For any given 0 uE , define the perturbed iterative sequences n u with errors as follows: 1, 1, , n n M nnnnnnTnnnnnnn nn n n uuvpvJTpvfvNvvel w puus 0, 1,n , , 1 1 n M nnn T M nnnnnnT vu wpJT wuupuJT , n nnnnn pwfwNwwr (29) n nnnn fuNu 2, , then n , , e, n n , n r, n n s and n l are same as Algorithm 3.1 Rem n nd M rithms . ark 3.2. For a suitable choice of ,,, ,TfgNM , Algorithm 3.1 reduces to several known Algo- [22-24] as special cases. Theorem 3.1. Let ,,, a f pgT and N be the same as in rem 2.1. Suppose that n M and M are set-valued ings from EE into the power set of E such that tE, ., n exists constants >0 and >0 s ,,uvz uch that for each E and 0n (., )Mu Theo mapp for each M t and ., M t are T-accretive ings and ., ., G n mapp M tMt. Assume that there (., ) ,, (.,)(., ) ,, n Mv TT n Mu Mv TT J zJ zuv zJzuv J (30) and the condition (13) holds. Let n y E and define a sequence n be a sequence in of real numbers as follows: 1 nn nnn xy zpz T ., 1, ., , ., 1, , , nn nn n Mgx n nT Mgz nTn nnnnn Mgy nn nnnnn xpxJT Ne JpzfzNzzr pyfyNyys (31) , 1 n nn nnnT zy ypyJT where n nn n n nnnn nnn pxfxxxl yy , n , n , n e, n r, n s and n l are sdefinedorithm 3.1. Then the following ho ame seq ld u s: ences in Alg que n defined by Algorithm 3.1 con- e unique solution u of the prob- lem 1) The se verges strongly (1). nce u to th 2) If nnnn with 0n lim 0 n n , then lim n nyu . 3) If lim n nyu implies that lim 0 n n . Proof. Let uE be the unique solution of the prob- lem (1). It is easy to see that the conclusion (1) follows from the conclusion (2). Now we prove that (2) is true. It follows from Lemma 1.6 that n and Copyright © 2012 SciRes. APM SALAHUDDIN, M. K. AHMAD 145 ., , Mgu nn T uuupuJTpufu 1,.Nuu (32) From (27), (31) and (32), we have 1uu pu ., 11 , ., 1, , 1, 1, , nn Mgu nnnn T Mgx nnnnT nnnnnnn nnnTnnn nn yu yuupuJTpufuNuu yyxpxJTpxfxNxxel fNxx ., 1 nn n n Mgx nn yxpxJT px n x , ., ., ,, ., , 1 , nn nn Tn nn n nnn Mgu Mgx nTnnnn T nnn nnn Mgx nTn n JT uxpx pu JTpxfxNxxJTpuuNu x upxpu JTpxfx .,Mgu pu ,nn fuNuu el ,nn n u el 1yu yu ., , ., ., ,, ., ., ,, , , , , ,, 1 nn n nn n Mgx nnT Mgu Mgx nT T Mgu Mgu nTTnn n nnn Nx xJTpufuNuu JTpufuNuuJTpufuNuu J Tpufu NuuJTpufu Nuuel y * ,, 1 ,, 1 nn n n nnnn nn nnnnn nnn nnn n nnn n nnnnnnnn n nn nnn uxupxpu Tpx TpufxfuNxxNuu gx guGel y uxupxpu Tpx TpuNxxNuu fx fuxuGe l y uxupxpu ** ,, , nnn n nnnnnnnnn Tpx TpuNxxNuu xuxuG el (33) where ., ., ,, , , n Mgu Mgu nT T GJTpufuNuuJTpu fuN 0.uu (34) It follows from (17) and (25), that Copyright © 2012 SciRes. APM SALAHUDDIN, M. K. AHMAD 146 1, q qn q nn upxpu q C xu x , nn f xfuxu 1 . q qn x u (35) Substituting (35) into (33), we have ,, 2 qq qq qqqq nnn Tpx TpuNxxNuuqC , nn n n xu l 11 nnn nn nn yu yu Ge (36) where 1 nnn nnnnn xu yu zuG r (38) and again 1 1and 1q q q qqqqqqq q QP QqC Pq C 1 , 2. q q (37) From (14), at 0<<1 1 11 nnn nn nnnn nnnnnn nnnn zu yu yu Gs yu Gs yuG s (39) where 111 n we know th . Similarly, we have . From (38) and (39), we get 11 , nnnnnnnnnnn nnnnnnnnn nnnnnnnn 1 n x uyuyuGsGr yuGsG r yuGsG r (40) d (40), we get since 11 1 n . From (36) an 2 2 1 1 . 1 nnnnnnnnnnnnnnnn nnnnnnnnnnnnn yuG sGrG el GsG rG el 11 11 n nnn nn y uyu yu (41) 11. nnnnnn atabtc Let From the assumption, we know that n b, n a, 2 1 . nn nnnnnn nn n bG sGr Ge , , 1 1 nn nnnnn ayu clt (42) We can write (42) as follows: n c and n t satisfy the conditions of Lemma 1.5. This im- nd so n plies that 0 n a a y u . he condition (3). Suppose that Next, we prove t lim n nyu . It follows (28), (38) and (39) that n zu and n x u . From (31), we have ., 1, ., 1 , 1, 1, n nn Mgx nnnnnnnTnnnnnnn Mgx nnnnnnnnn Tnnnn yyxpxJTpxfxNxxel yuelyxpxJTpxfxN xx As in the proof of (36), we have . n u (43) ., , 1, . nn Mgx nnnnnTnnnn nn nn nn yxpxJTpxfxN xxu xuG (44) It follows from (43) and (44) that 1yu 11. nn nnnnnnnnn y ue lxuxuG (45) Copyright © 2012 SciRes. APM SALAHUDDIN, M. K. AHMAD 147 That lim 0 n n is implies th . This completes the proof. Theorem 3.2. Let f, p, N and T be the same as in Theo- ccretive mappings from E into the power set of E such that G n rem 3.1. Let n M and M be T-a M. M Assume that there exists constant >0 such that hold. Let (14) n n be a sequence in E and define y as follows: 1, , , , , 1, 1, nn nnnTnnn nnn n M nnnT nnnnnn n M nnnnnnTnnnnn xpxTxNxxel xy pzJTpzfzNzzr zyypyJTpyfyNyys (46) (1 ) nn nnn yy z n M J n pxf n where n , n , n , n e, n r, n s and n l same in Algorithm 3.2, then are 1) The sequence n u defineorithm 3.2, con- verges s d by Alg trongly to unique solution u* of the problem (3), 2) If nnnn with and 0 n n , 3) n yu implies that lim 0 n n lim n 0 n , then * * n yulim n lim n . ve or is to establish existence and zed nonlinear implicit quasi in Banach spaces. We de- ped the T-rrator with T-accretive map- by using thof Fang and Huang [11] and Pe] and proved that the problem (1) is equivalent to a fixed point problem. On the basis of fixed point formulation we suggested perturbed iterative algorithm with errors and by the theory of Hick and Harder [ proved the convergence and stability of iterativ quences generated by algorithms. 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Huang, “Mann and Ishikawa Type Perturbed Itera- tive Algorithms for Generalized Nonlinear Implicit Qua- sivariational Inclusions,” Computer Mathematics with Ap- plications, Vol. 35, No. 10, 1998, pp. 1-7. Copyright © 2012 SciRes. APM |