Variational Inclusion Governed by

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1 Filomat 3:0 07, Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: Variatioal Iclusio Govered by αβ-h.,.,.,.-mixed Accretive Mappig Sajeev Gupta,a, Shamshad Husai b, Vishu Naraya Mishra c a Departmet of Ecoomic Scieces Idia Istitute of Techology Kapur, Kapur-0806, Idia a Departmet of Mathematics, Shri Ram Murti Smarak College of Egieerig ad Techology, Bareilly-430, Idia, b Departmet of Applied Mathematics, Aligarh Muslim Uiversity, Aligarh-000, Idia, c Departmet of Mathematics, Idira Gadhi Natioal Tribal Uiversity, Lalpur, Amarkatak, , Idia Abstract. I this paper, we look ito a ew cocept of accretive mappigs called αβ-h.,.,.,.-mixed accretive mappigs i Baach spaces. We exted the cocept of proximal-poit mappigs coected with geeralized m-accretive mappigs to the αβ-h.,.,.,.-mixed accretive mappigs ad discuss its characteristics like sigle-valuable ad Lipschitz cotiuity. Some illustratio are give i support of αβ-h.,.,.,.-mixed accretive mappigs. Sice proximal poit mappig is a powerful tool for solvig variatioal iclusio. Therefore, As a applicatio of itroduced mappig, we costruct a iterative algorithm to solve variatioal iclusios ad show its covergece with acceptable assumptios.. Itroductio Variatioal iequality theory is providig mathematical models to some problems make a appearace i optimizatio ad cotrol, ecoomics, ad egieerig scieces. May heuristic has bee widely used these applicatios of variatioal iequalities, e.g., we refer to see [8], [0]-[],[4]. The proximal-poit mappig techique is a importat powerful tool to study varitioal iequalities ad their geeralizatio. Firstly, Huag ad Fag [6] ivestigated the geeralized m-accretive mappig ad defied its proximalpoit mappig i Baach spaces. Sice the a umber of mathematicia preseted various classes of 00 Mathematics Subject Classificatio. 47J0, 49J40, 49J53 Keywords. αβ-h.,.,.,.-mixed accretive mappig, Proximal-poit mappig, Variatioal iclusio, Iterative algorithm. Received: 0 Jue 06; Accepted: 05 May 07 Commuicated by Ljubomir Ćirić / Draga S. Djordjević Correspodig author: Sajeev Gupta a Curret address First author is supported by Departmet of Atomic Eergy, Natioal Board for Higher Mathematics, Mumbai, Idia, grat o. DAE Ref. No. /40/3/04/R&DII/6469, dated May 3, 04. addresses: guptasamp@gmail.com Sajeev Gupta,, s husai68@yahoo.com Shamshad Husai, vishuarayamishra@gmail.com Vishu Naraya Mishra

2 S. Gupta et al. / Filomat 3:0 07, geeralized m-accretive mappigs, see for examples [5, 7], [0, ]. Su et al. [] preseted a ew class of M-mootoe mappig i Hilbert spaces. I the past few days, Zou ad Huag [4], Kazmi et al. [4, 5] ivestigated H.,.-accretive mappigs, Ahmad et. al ivestigated H.,.-cocoercive mappig [] ad Husai ad Gupta [7] ivestigated H.,.,.,.-mixed cocoercive mappigs i Baach Hilbert spaces, a atural extesio of m-accretive M-mootoe mappig ad focussed o variatioal iclusios ivolvig these mappigs. I recet past, the techiques through differet classes of proximal-poit mappigs have bee developed to work o the existece of solutios ad to aalyze covergece ad stability of iterative algorithms for several classes of variatioal iclusios, see for example [, 4], [7]-[8], [0, ], [4]. Very recetly, Luo ad Huag [8] itroduced ad studied a class of B-mootoe ad Kazmi et al. [4] itroduced ad studied a class of geeralized H.,.-accretive mappigs i Baach spaces which is geeralizatio of H-mootoe mappigs [5]. They showed its proximal-poit mappig properties coected with B-mootoe ad geeralized H.,.-accretive mappig. This work is motivated ad ispired by the research works metioed above. We look ito a ew otio of αβ-h.,.,.,.-mixed accretive mappigs ad give its proximal-poit mappig. Further, we will discuss its characteristics that is sigle-valued as well as Lipschitz cotiuity. As a applicatio, we attempt to solve geeralized set-valued variatioal iclusios i real q-uiformly smooth Baach spaces. By usig the proximal-poit mappig techique, we costruct a iterative algorithm ad prove its covergece with acceptable assumptios. The results preseted i this paper ca be viewed as a extesio ad geeralizatio of some kow results [, 7], [4]-[6], [8, 4]. Some illustratios are give i support of itroduced results.. Prelimiaries Let us cosider a real Baach space E with orm. ad topological dual space E. We use ier product.,. deote the dual pair betwee E ad E ad E be the power set of E. Defiitio.. [3] A mappig J q : E E, where q >, is said to be geeralized duality mappig, if it is give as J q u = { f E : u, f = u q, f = u q }, u E. If J is the usual ormalized duality mappig o E, give as J q u = u q J u u 0 E. If E X, a real Hilbert space, the J becomes idetity mappig o X. Defiitio.. [3] A Baach space E is called smooth if for every u E with u =, there exists a uique f E such that f = f u =. The modulus of smoothess of E is a fuctio ρ E : [0, [0,, defied by { } ρ E t = sup u + v + u v : u, v t. Defiitio.3. [3] A Baach space E is called i uiformly smooth if ρ E t lim t 0 t = 0; ii q-uiformly smooth, for q >, if there exists c > 0 such that ρ E t c t q, t [0,. Note that J q is sigle-valued if E is uiformly smooth.

3 S. Gupta et al. / Filomat 3:0 07, Lemma.4. [3] Let E be a real uiformly smooth Baach space. The E is q-uiformly smooth if ad oly if there exists c q > 0 such that, for all u, v E, u + v q u q + q v, J q u + c q v q. I order to proceed our ext step, we write basic importat cocepts ad defiitios, which will be used i this work. Lemma.5. A mappig f : E E is said to be i ξ-strogly accretive with ξ > 0, if f x f y, J q x y ξ x y q, x, y E; ii µ-cocoercive with µ > 0, if f x f y, J q x y µ f x f y q, x, y E; iii γ-relaxed cocoercive with γ > 0, if f x f y, J q x y γ f x f y q, x, y E; iv β-lipschitz cotiuous with β > 0, if f x f y β x y, x, y E; v α-expasive with α > 0, if f x f y α x y, x, y E; if α =, the it is expasive. Defiitio.6. [7] Let H : E E E E E, ad A, B, C, D : E E be the sigle-valued mappigs. The i HA,., C,. is said to be µ, γ -strogly mixed cocoercive regardig A, C with µ, γ > 0, if HAx, u, Cx, u HAy, u, Cy, u, J q x y µ Ax Ay q + γ x y q, x, y, u E; ii H., B,., D is said to be µ, γ -relaxed mixed cocoercive regardig B, D with µ, γ > 0, if Hu, Bx, u, Dx Hu, By, u, Dy, J q x y µ Bx By q + γ x y q, x, y, u E; iii HA, B, C, D is said to be symmetric mixed cocoercive regardig A, C ad B, D if HA,., C,. is µ, γ -strogly mixed cocoercive regardig A, C ad H., B,., D is µ, γ -relaxed mixed cocoercive regardig B, D; iv HA, B, C, D is said to be τ-mixed Lipschitz cotiuous regardig A, B, C ad D with τ > 0, if HAx, Bx, Cx, Dx HAy, By, Cy, Dy τ x y, x, y E. Defiitio.7. [8] Let S : E E ad M : E E E be the set-valued mappig. The i S is said to be accretive if u v, J q x y 0 x, y E, u Sx, v Sy; ii S is said to be strictly accretive if u v, J q x y > 0 x, y E, u Sx, v Sy;

4 S. Gupta et al. / Filomat 3:0 07, ad equality holds if ad oly if x = y. iii S is said to be µ -strogly accretive with µ > 0, if u v, J q x y µ x y q x, y E, u Sx, v Sy; iv S is said to be γ -relaxed accretive with γ > 0, if u v, J q x y γ x y q x, y E, u Sx, v Sy; v M f,. is said to be α-strogly accretive regardig f with α > 0, if u v, J q x y α x y q x, y, w E, u M f x, w v M f y, w; vi M., is said to be β-relaxed accretive regardig with β > 0, if u v, J q x y β x y q x, y, w E, u Mw, x v Mw, y; vii M.,. is said to be αβ-symmetric accretive regardig f ad if M f,. is α-strogly accretive regardig f ad M., is β-relaxed accretive regardig with α β ad α = β if ad oly if x = y. 3. αβ-h.,.,.,.-mixed Accretive Mappigs Firstly we cosider the followig assumptios, the we will itroduce αβ-h.,.,.,.-mixed accretive mappigs ad its proximal-poit mappig. Later we will discuss the properties of its proximal poit mappig properties. Let H : E E E E E, f, : E E ad A, B, C, D : E E be sigle-valued mappigs ad M : E E E be a set-valued mappig. Assumptio a : Let H is symmetric mixed cocoercive regardig A, C ad B, D. Assumptio a : Let A is α -expasive ad B is β -Lipschitz cotiuous. Defiitio 3.. Let assumptio a holds, the M is said to be αβ-h.,.,.,.-mixed accretive regardig A, C, B, D ad f, if i M is αβ-symmetric accretive regardig f ad ; ii H.,.,.,. + ρm f, E = E, for all ρ > 0. The followig example illustrate the Defiitios.6 ad 3.. Example 3.. Let q = ad E = R with usual ier product defied by x, x, y, y = x y + x, y. Let A, B, C, D : R R be defied by 4x 3x x Ax =, Bx =, Cx =, Dx = 4x 3x x x x, x = x, x R. Suppose that H : R R R R R is defied by HAx, Bx, Cx, Dx = Ax + Bx + Cx + Dx.

5 S. Gupta et al. / Filomat 3:0 07, I additio, let f, : R R be defied by 5x 3 f x = x x, x = x x + 5x 3 4 x x, x = x, x R. ad M : R R R is defied by M f x, x = f x x, x = x, x R. The, costats i Defiitio.6 ad 3. havig values µ, γ = 4,, µ, γ = 3,, τ = 4 α = 5 ad β = 7 4. It shows that H is symmetric mixed cocoercive regardig A, C ad B, D, M is symmetric accretive regardig f ad, ad H is mixed Lipschitz cotiuous regardig A, B, C ad D. Further, it ca be obtaied easily that [HA, B, C, D+ρ M f, ]R = R. Thus M is αβ-mixed accretive with respect to A, B, C, D ad f,. Remark 3.3. i If HA, B, C, D = HA, B, the αβ-h.,.,.,.-mixed accretive mappig reduces to geeralized H.,.-accretive mappig cosidered i [6]. ii If HA, B, C, D = B, the αβ-h.,.,.,.-mixed accretive mappig reduces to geeralized B-mootoe mappig cosidered i [8]. iii If HA, B, C, D = HA, B, M.,. = M ad M is accretive, the αβ-h.,.,.,.-mixed accretive mappig reduces to H.,.-accretive mappig cosidered i [4]. iv If E is Hilbert space, M f, = M ad M is m-relaxed mootoe, the αβ-h.,.,.,.-mixed accretive mappig reduces to H.,.,.,.-mixed cocoercive mappig cosidered i [7]. Sice αβ-h.,.,.,.-mixed accretive mappig is a geeralizatio of the maximal accretive mappig, it is logical that they have similar properties. The ext result guaratee this suppositio. Propositio 3.4. Let M be a αβ-h.,.,.,.-mixed accretive mappig regardig A, C, B, D ad f,. assumptios a ad a hold with α > β, µ > µ, α > β ad γ, γ > 0. If the followig iequality u v, J q x y 0, satisfied for all v, y GraphM f,, implies u, x M f,, where GraphM f, = {u, x E E : u, x M f x, x}. If Proof. Assume o the cotrary that there exists u 0, x 0 GraphM f, such that u 0 v, J q x 0 y 0, y, v GraphM f,. By defiitio of αβ-h.,.,.,.-mixed accretive, we kow that H.,.,.,. + ρ M f, E = E, holds for all ρ > 0. So there exists u, x GraphM f, such that HAx, Bx, Cx, Dx + ρu = HAx 0, Bx 0, Cx 0, Dx 0 + ρu 0 E. Now, ρ u 0 ρ u = HAx, Bx, Cx, Dx HAx 0, Bx 0, Cx 0, Dx 0 E. ρ u 0 ρ u, J q x 0 x = HAx 0, Bx 0, Cx 0, Dx 0 HAx, Bx, Cx, Dx, Sice M is αβ-symmetric accretive regardig f ad, we obtai α β x 0 x q ρ u 0 u, J q x 0 x J q x 0 x. = HAx 0, Bx 0, Cx 0, Dx 0 HAx, Bx, Cx, Dx, J q x 0 x = HAx 0, Bx 0, Cx 0, Dx 0 HAx, Bx 0, Cx, Dx 0, J q x 0 x HAx, Bx 0, Cx, Dx 0 HAx, Bx, Cx, Dx, J q x 0 x.

6 S. Gupta et al. / Filomat 3:0 07, Sice assumptio a holds, we have from 3 α β x 0 x q µ Ax 0 Ax q γ x 0 x q + µ Bx 0 Bx q γ x 0 x q. Sice assumptio a holds, we have from 4 α β x 0 x q µ α q x 0 x q γ x 0 x q + µ β q x 0 x q γ x 0 x q = [µ α µ qβ q + γ + γ ] x 0 x 0 α β x 0 x q [µ α q µ β q + γ + γ ] x 0 x q 0 l + κ x 0 x q 0, where l = µ α q µ β q + γ + γ ad κ = α β, which gives x 0 = x sice α > β, µ > µ, α > β, ad γ, γ > 0. By, we have u 0 = u, a cotradictio. This complete the proof. Theorem 3.5. Let M be a αβ-h.,.,.,.-mixed accretive mappig regardig A, C, B, D ad f,. If assumptios a ad a hold with α > β, µ > µ, α > β ad γ, γ > 0, the HA, B, C, D + ρm f, is sigle-valued. 4 Proof. For ay give x E, let u, v HA, B, C, D + ρm f, x. It follows that { HAu, Bu, Cu, Du + x ρm f, u, HAv, Bv, Cv, Dv + x ρm f, v. Sice M is αβ-symmetric accretive with respect to f ad, we have α β u v q ρ HAu, Bu, Cu, Du + x HAv, Bv, Cv, Dv + x, J qu v α β u v q HAu, Bu, Cu, Du + x HAv, Bv, Cv, Dv + x, J q u v = HAu, Bu, Cu, Du HAv, Bv, Cv, Dv, J q u v = HAu, Bu, Cu, Du HAv, Bu, Cv, Du, J q u v HAv, Bu, Cv, Du HAv, Bv, Cv, Dv, J q u v. 5 Sice assumptio a holds, we have from 5 ρα β u v q µ Au Av q γ u v q + µ Bu Bv q γ u v q. 6 Sice assumptio a holds, we have from 6 ρα β u v q µ α q u v q γ u v q + µ β q u v q γ u v q = [µ α q µ β q + γ + γ ] u v q 0 α β u v q µ α q µ β q + γ + γ u v q 0 l + ρκ u v q 0, where l = µ α q µ β q + γ + γ ad κ = α β. Sice α > β, µ > µ, α > β ad γ, γ > 0, it follows that u v 0. This implies that u = v ad so HA, B, C, D + ρm f, is sigle-valued.

7 S. Gupta et al. / Filomat 3:0 07, Defiitio 3.6. Let M be a αβ- H.,.,.,.-mixed accretive mappig regardig A, C, B, D ad f,. If assumptios a ad a hold with α > β, µ > µ, α > β ad γ, γ > 0, the the proximal-poit mappig R H.,.,.,. ρ, M.,. : E E is defied by R H.,.,.,. ρ, M.,. u = HA, B, C, D + ρm f, u, u E. 7 Now we prove that the proximal-poit mappig defied by 7 is Lipschitz cotiuous. Theorem 3.7. Let M : E E E be a αβ- H.,.,.,.-mixed accretive mappig with respect to A, C, B, D ad f,. If assumptios a ad a hold with α > β, µ > µ, α > β ad γ, γ > 0, the the proximal-poit mappig R H.,.,.,. : E E is ρ, M.,. l+ρκ-lipschitz cotiuous, that is, R H.,.,.,. u R H.,.,.,. v u v, u, v E. ρ, M.,. ρ, M.,. l + ρκ Proof. For give poits u, v E, It proceed from Defiitio 3.6 that R H.,.,.,. ρ, M.,. u = HA, B, C, D + ρm f, u, R H.,.,.,. ρ, M.,. v = HA, B, C, D + ρm f, v. Let w = R H.,..,. ρ, M.,. u ad w = R H.,..,. ρ, M.,. v. ρ u H A w, B w, C w, D w M f w, w ρ v H A w, B w, C w, D w M f w, w. Sice M is αβ-symmetric accretive with respect to f ad, we have ρ u HAw, Bw, Cw, Dw v HAw, Bw, Cw, Dw, J q w w α β w w q, ρ u v HAw, Bw, Cw, Dw + HAw, Bw, Cw, Dw, J q w w α β w w q, which implies u v, J q w w HAw, Bw, Cw, Dw HAw, Bw, Cw, Dw, J q w w + ρα β w w q. Now, we have u v w w q u v, w w HAw, Bw, Cw, Dw HAw, Bw, Cw, Dw, J q w w + ρα β w w q = HAw, Bw, Cw, Dw HAw, Bw, Cw, Dw, J q w w + HAw, Bw, Cw, Dw HAw, Bw, Cw, Dw, J q w w + ρα β w w q. Sice assumptio a holds, we have u v w w q µ Aw Aw q + γ w w q µ Bw Bw q + γ w w q +ρα β w w q.

8 Sice assumptio a holds, we have S. Gupta et al. / Filomat 3:0 07, u v w w q [µ α q µ β q + γ + γ ] w w q + ρα β w w q l + ρκ w w q, where l = µ α q µ β q + γ + γ ad κ = α β. Hece, that is u v w w q l + ρκ w w q, R H.,.,.,. u R H.,.,.,. v u v, u, v E. ρ, M.,. ρ, M.,. l + ρκ This completes the proof. 4. A Applicatio of αβ-h.,.,.,.-mixed Accretive Mappigs. Here we shall show that the αβ-h.,.,.,.-mixed accretive mappig uder acceptable assumptios ca be used as a powerful tool to solve variatioal iclusio problem i Baach space. Let S, T : E CBE be the set-valued mappigs, ad let f, : E E, A, B, C, D : E E, F : E E E ad H : E E E E E be sigle-valued mappigs. Suppose that set-valued mappig M : E E E be a αβ- H.,.,.,.-mixed accretive mappig regardig A, C, B, D ad f,. We cosider the followig geeralized set-valued variatioal iclusio: for give λ E, fid u E, v Su ad w Tu such that λ Fv, w + M f u, u. 8 If S, T : E E be sigle-valued mappigs ad M.,. = ρn., where ρ > 0 is a costat, the the problem 8 reduces to the followig problem: fid u E such that λ FSu, Tu + ρnu. 9 If M is a A, η-accretive mappig, the the problem 9 was itroduced ad studied by La et al. [7]. If ρ =, λ = 0 ad FSu, Tu = Tu for all u E, where T : E E is a sigle-valued mappig, the the problem 9 reduces to the followig problem: fid u E such that 0 Tu + Nu. 0 If N is a H.,.-accretive mappig, the the problem 0 was studied by Zou ad Huag [4]; ad N is a geeralized m-accretive mappig, the the problem 0 was studied by Bi et al. [4]. If E is a Hilbert space ad N is a H-mootoe mappigs, the the problem 0 was itroduced ad studied by Fag ad Huag [5] ad icludes may variatioal iequalities iclusios ad complemetarity problems as special cases. For example, see [0, ]. Lemma 4.. Let u E, v Su ad w Tu is a solutio of problem 8 if ad oly if u E, v Su ad w Tu satisfies the followig relatio: u = R H.,.,.,. ρ,m.,. [HAu, Bu, Cu, Du ρfv, w + ρλ], ρ > 0.

9 Proof. Observe that for ρ > 0, λ Fw, v + M f u, u S. Gupta et al. / Filomat 3:0 07, [HAu, Bu, Cu, Du ρfv, w + ρλ] HAu, Bu, Cu, Du + ρm f u, u [HAu, Bu, Cu, Du ρfv, w + ρλ] HA, B, C, D + ρm f, u u = HA, B, C, D + ρm f, [HAu, Bu, Cu, Du ρfv, w + ρλ] u = R H.,.,.,. ρ M.,. [HAu, Bu, Cu, Du ρfv, w + ρλ]. Remark 4.. We ca rewrite the equality as: z = HAu, Bu, Cu, Du ρfv, w + ρλ, u = R H.,.,.,. ρ, M.,. z. By usig the result of Nadler [9], this fixed poit formulatio allow us to costruct the iterative algorithm as give below: Algorithm 4.3. For ay give z 0 E, we ca choose u 0 E such that sequeces {u }, {v } ad {w } satisfy u = R H.,.,.,. z ρ, M.,., v Su, v v DSu, Su +, w Tu, w w DTu, Tu +, z + = HAu, Bu, Cu, Du ρfv, w + ρλ + e, e j e j ϖ j <, ϖ 0,, lim e = 0, j= where ρ > 0 is a costat, λ E is ay give elemet ad e E is a error to take ito accout a possible iexact computatio of the proximal-poit mappig poit for all 0, ad D.,. is the Hausdorff metric o CBE. Next, we eed the followig defiitios which will be used to state ad prove the mai result. Defiitio 4.4. A set-valued mappig G : E CBE is said to be D-Lipschitz cotiuous with costat l > 0, if DGx, Gy l x y, x, y E. Defiitio 4.5. Let S, T : E E be the set-valued mappigs, A, B, C, D : E E, F : E E E ad H : E E E E E be sigle-valued mappigs. The i F is said to be σ-strogly accretive regardig S ad HA, B, C, D i the first compoet with costat σ > 0, if Fv,. Fv,., J q HAu, Bu, Cu, Du HAv, Bv, Cv, Dv σ HAu, Bu, Cu, Du HAv, Bv, Cv, Dv q, u, v E ad v Su, v Sv; ii F is said to be δ-strogly accretive regardig T ad HA, B, C, D i the secod compoet with δ > 0, if F., w F., w, J q HAu, Bu, Cu, Du HAv, Bv, Cv, Dv δ HAu, Bu, Cu, Du HAv, Bv, Cv, Dv q, u, v E ad w Tu, w Tv;

10 S. Gupta et al. / Filomat 3:0 07, iii F is said to be ɛ -Lipschitz cotiuous i the first compoet with ɛ > 0, if Fu, v Fv, v ɛ u v, u, v, v E; iv F is said to be ɛ -Lipschitz cotiuous i the secod compoet with ɛ > 0, if Fv, u Fv, v ɛ u v, u, v, v E. Next, we fid the covergece of iterative algorithm for geeralized set-valued variatioal iclusio 8. Theorem 4.6. Let us cosider the problem 8 ad assume that i S ad T are l ad l D-Lipschitz cotiuous, respectively; ii HA, B, C, D is τ-mixed Lipschitz cotiuous regardig A, B, C ad D; iii F is is σ-strogly accretive regardig S ad HA, B, C, D i the first compoet ad δ-strogly accretive regardig T ad HA, B, C, D i the secod compoet; iv F is is ɛ, ɛ -Lipschitz cotiuous i the first ad secod compoet, respectively; v 0 < q τ q + c q ρ q ɛ l + ɛ l q ρqσ + δτ q < l + ρκ; where l = µ α q µ β q + γ + γ ad κ = α β, ad α > β, µ > µ, α > β ad γ, γ, ρ > 0. The problem 8 has a solutio u, v, w, where u E, v Su ad w Tu, ad the iterative sequeces {u }, {v } ad {w }, geerated by Algorithms 4.3 coverges strogly to u, v ad w, respectively. Proof. Usig the Lipschitz cotiuity of S ad T, it follows from Algorithms 4.3 such that v + v + DSu +, Su + l u + u, w + w + DTu +, Tu + l u + u, for = 0,,,... From ad Theorem 3.7, we have u + u R H.,.,.,. ρ, M.,. z + R H.,.,.,. ρ, M.,. z = Now, we estimate z + z by usig Algorithms 4.3, we have z + z = [HAu, Bu, Cu, u ρfv, w + ρλ + e ] By Lemma.4, we have [HAu, Bu, Cu, u ρfv, w + ρλ + e ] HAu, Bu, Cu, u HAu, Bu, Cu, u l + ρκ z + z. 5 + ρfv, w ρfv, w + e e. 6 HAu, Bu, Cu, Du HAu, Bu, Cu, Du ρfv, w Fv, w q HAu, Bu, Cu, Du HAu, Bu, Cu, Du q + c q ρ q Fv, w Fv, w q ρq Fv, w Fv, w, J q HAu, Bu, Cu, Du HAu, Bu, Cu, Du. From ii, we get HAu, Bu, Cu, Du HAu, Bu, Cu, Du τ u u. 8 7

11 By Algorithm 4.3, ad coditios i ad iv, we get S. Gupta et al. / Filomat 3:0 07, Fv, w Fv, w Fv, w Fv, w + Fv, w Fv, w Usig coditios iii, we get ɛ v v + ɛ w w ɛ + DSu, Su + ɛ + DTu, Tu ɛ l + + ɛ l + u u. 9 Fv, w Fv, w, J q HAu, Bu, Cu, Du HAu, Bu, Cu, Du Fv, w Fv, w, J q HAu, Bu, Cu, Du HAu, Bu, Cu, Du + Fv, w Fv, w, J q HAu, Bu, Cu, Du HAu, Bu, Cu, Du σ + δ HAu, Bu, Cu, Du HAu, Bu, Cu, Du q σ + δτ q u u q. 0 From 7-9, we have HAu, Bu, Cu, Du HAu, Bu, Cu, Du ρfv, w Fv, w q τ q + c q ρ q ɛl + + ɛ l + q ρqσ + δτ q u u. Combiig 5, 6 ad, we have u + u R H.,.,.,. ρ, M.,. z + R H.,.,.,. ρ, M.,. z ϕ u u + l + ρκ e e, where Let ϕ = ϕ = l + ρκ l + ρκ q q τ q + c q ρ q ɛ l + + ɛ l + q ρqσ + δτ q. 3 τ q + c q ρ q ɛ l + ɛ l q ρqσ + δτ q. 4 Sice ϕ ϕ as. By, we kow that 0 < ϕ < ad hece there exist 0 > 0 ad ϕ 0 0, such that ϕ ϕ 0 for all 0. Therefore, by, we have 5 implies that u + u ϕ 0 u u + u + u ϕ 0 0 u 0 + u 0 + l + ρκ e e l + ρκ where t = e e for all 0. Hece, for ay m > 0, we have u m u m u p+ u p p= m p= j= ϕ p 0 0 u 0 + u 0 + ϕ j 0 t, 6 m ϕ p l + ρκ 0 p= p 0 j= t p j ϕ p j 0. 7

12 j= S. Gupta et al. / Filomat 3:0 07, Sice e j e j ϖ j <, ϖ 0, ad 0 < ϕ 0 <, it follows that u m u 0 as, ad so {u } is a Cauchy sequece i E. From 3 ad 4, it follows that {v } ad {w } are also Cauchy sequeces i E. Thus, there exist u, v ad w such that u u, v v ad w w as. I the sequel, we will prove that v Su. I fact, sice v Su, we have dv, Su v v + dv, Su v v + DSu, Su v v + ρ u u 0, as, which implies that dv, Su = 0. Sice Su CBE, it follows that v Su. Similarly, it is easy to see that w Tu. By the cotiuity of R H.,.,.,. ρ, M.,., A, B, C, D, S, T ad F ad Algorithms 4.3, we kow that u, v ad w satisfy u = R H.,.,.,. ρ,m.,. [HAu, Bu, Cu, Du ρfu, z + ρλ]. By Lemma 4., u, v, w is a solutio of the problem 8. This completes the proof The followig example shows that assumptios i to v of Theorem 4.6 are satisfied for variatioal iclusio problem 8. Example 4.7. Let q = ad E = R with usual ier product. i Let S, T : R R are idetity mappigs, the R, S are -Lipschitz cotiuous for =,. Let A, B, C, D : R R be defied by 0 Ax = x 5 0 x, Bx = x x 5 x, Cx =, Dx = x Suppose that H : R R R R R is defied by x HAx, By, Cx, Dy = Ax + Bx + Cx + Dx, x R. The, it is easy to cheek that x, x = x, x R. H.,.,.,. is 0, -strogly mixed cocoercive regardig A, C ad 5, -relaxed mixed cocoercive regardig B, D, ad A is -expasive for = 0, ad B is -Lipschitz cotiuous for = 4, 5. ii HA, B, C, D is 9 -mixed Lipschitz cotiuous regardig A, B, C ad D for = 9, 0. Let f, : R R be defied by f x = x 4 3 x x + x, x = x 3 4 x 3 4 x + 4 x, x = x, x, R. Suppose that M : R R R is defied by M f x, x = f x x, x = x, x, R. The, it is easy to check that M f, is -strogly accretive regardig f for =, 3 ad -relaxed accretive regardig for = 3, 4. Moreover, for ρ =, M is αβ-h.,.,.,.-mixed accretive regardig A, C, B, D ad f,.

13 Let F : R R R are defied by Fx, y = x 4 + y 5, x, y, R. The, it is easy to check that S. Gupta et al. / Filomat 3:0 07, iii F is is 9 -strogly accretive regardig S ad HA, B, C, D i the first compoet for = 30, 40 ad -strogly accretive with respect to T ad HA, B, C, D i the secod compoet for = 40, 50; 9 iv F is is -Lipschitz cotiuous i the first compoet for = 3, 4 ad secod compoet for = 4, 5. Therefore, for the costats l = l =, µ = 0, γ =, µ = 5, γ =, α = 0., β = 0., α = 0.5, β = 0.5, σ = 0.75, δ = 0.580, ɛ = 0.5, ɛ = 0., τ =.9, q =, l =.9, κ = Lipschitz cotiuous i the obtaied i i to v above, all the coditios of the Theorem 4.7 is satisfied for the geeralized mixed variatioal iclusio problem 8 for ρ = 0.35 ad c q =. Ackowledgemet: The first author is extremely grateful to Professor Joydeep Dutta, Departmet of Ecoomic Scieces, Idia Istitute of Techology, Kapur, Idia for iferetial ad productive criticism to prepare mauscript. We sicerely thak the reviewer for costructive ad valuable commets, which were of great help i revisig the mauscript. Accordigly, the revised mauscript has bee systematically improved with ew iformatio ad additioal iterpretatios. Refereces [] J.P. Aubi, A. Cellia Differetial iclusios, Spriger-Verlag, Berli, 984. [] R. Ahmad, M. Dilshad, M.M. Wog, J.C. Yao: H.,.-cocoercive operator ad a applicatio for solvig geeralized variatioal iclusios, Abstract ad Applied Aalysis, Article ID , -. [3] R. Ahmad, M. Rahama ad H.A. Rizvi: Graph covergece for H.,.-co-accretive mappig with over-relaxed proximal poit method for solvig a geeralized variatioal iclusio problem, Ira. J Math. Sci. Iform., 07, [4] Z.S. Bi, Z. Hart, Y.P. Fag, Sesitivity aalysis for oliear variatioal iclusios ivolvig geeralized m-accretive mappigs, Joural of Sichua Uiversity, 40003, [5] Y.-P. Fag ad N.-J. Huag, H-mootoe operator ad resolvet operator techique for variatioal iclusios, Appl. Math. Comput., , [6] N.-J. Huag ad Y.-P. Fag, Geeralized m-accretive mappigs i Baach spaces, Joural of Sichua Uiversity, 38400, [7] S. Husai ad S. Gupta, H.,.,.,.-mixed cocoercive operators with a applicatio for solvig variatioal iclusios i Hilbert spaces, J. Fuct. Space. Appl., Article ID , -3. [8] S. Husai, S. Gupta ad V.N. Mishra, Graph covergece for the H.,.-mixed mappig with a applicatio for solvig the system of geeralized variatioal iclusios, Fixed Poit Theory ad Applicatios, Article ID 30403, -. [9] S. Husai, S. Gupta ad V.N. Mishra, Geeralized H.,.,.-η-cocoercive operators ad geeralized set-valued variatioal-like iclusios, Joural of Mathematics, Article ID , -0. [0] S. Husai, H. Sahper ad S. Gupta, H.,.,.-η-proximal-poit mappig with a applicatio, Applied Aalysis i Biological ad Physical Scieces, the series Spriger Proceedigs i Mathematics ad Statistics, 8606, [] S. Husai ad S. Gupta, A resolvet operator techique for solvig geeralized system of oliear relaxed cocoercive mixed variatioal iequalities, Advaces i Fixed Poit Theory, 0, 8-8. [] S. Husai ad S. Gupta, Algorithm for solvig a ew system of geeralized variatioal iclusios i Hilbert spaces, Joural of Calculus of Variatios, Article ID , -8. [3] S. Husai, S. Gupta ad H. Sahper, Algorithm for solvig a ew system of geeralized oliear quasi-variatioal-like iclusios i Hilbert spaces, Chiese Joural of Mathematics, Article ID , -7. [4] K.R. Kazmi, N. Ahmad ad M. Shahzad, Covergece ad stability of a iterative algorithm for a system of geeralized implicit variatioal-like iclusios i Baach spaces, Appl. Math. Comput., 80, [5] K.R. Kazmi, M.I. Bhat ad N. Ahmad, A iterative algorithm based o M-proximal mappigs for a system of geeralized implicit variatioal iclusios i Baach spaces, J. Comput. Appl. Math., , [6] K.R. Kazmi, F.A. Kha ad M. Shahzad, A system of geeralized variatioal iclusios ivolvig geeralized H.,.-accretive mappig i real q-uiformly smooth Baach spaces, Appl. Math. Comput., 70,

14 S. Gupta et al. / Filomat 3:0 07, [7] H.-Y. La, Y. J. Cho, ad R. U. Verma, Noliear relaxed cocoercive variatioal iclusios ivolvig A, η-accretive mappigs i Baach spaces, Comput. Math. Appl., , [8] X.P. Luo, N.-J. Huag, A ew class of variatioal iclusios with B-mootoe operators i Baach spaces, J. Comput. Appl. Math., 33000, [9] S.B. Nadler, Multivalued cotractio mappig, Pacific J. Math., 30969, [0] J.-W. Peg, O a ew system of geeralized mixed quasi-variatioal-like iclusios with H, η-accretive operators i real q- uiformly smooth Baach spaces, Noliear Aal., 68008, [] J.-W. Peg, D.L. Zhu, A ew system of geeralized mixed quasi-vatiatioal iclusios with H, η-mootoe operators, J. Math. Aal. Appl., 37007, [] J. Su, L. Zhag, X. Xiao, A algorithm based o resolvet operators for solvig variatioal iequalities i Hilbert spaces, Noliear Aal. TMA , [3] H.K. Xu, Iequalities i Baach spaces with applicatios, Noliear Aal., 699, [4] Y.-Z. Zou ad N.-J. Huag, H.,.-accretive operator with a applicatio for solvig variatioal iclusios i Baach spaces, Appl. Math. Comput., 04008,