A GENERALIZED MEAN PROXIMAL ALGORITHM FOR SOLVING GENERALIZED MIXED EQUILIBRIUM PROBLEMS (COMMUNICATED BY MARTIN HERMANN)

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1 Bulleti of Mathematical Aalysis ad Applicatios ISSN: , URL: Volume 7 Issue 1 (2015), Pages 1-11 A GENERALIZED MEAN PROXIMAL ALGORITHM FOR SOLVING GENERALIZED MIXED EQUILIBRIUM PROBLEMS (COMMUNICATED BY MARTIN HERMANN) A.M. SADDEEK Abstract. I a recet paper [14], the fixed poit ad resolvet techiques have bee used to give a iterative algorithm for solvig geeralized mixed equilibrium problem (GMEP). I this paper, a exteded iterative algorithm, called geeralized mea proximal algorithm (GMPA) is preseted ad its covergece to the solutio of GMEP with θ-pseudomoote ad δ- Lipschitz cotiuous mappigs i Hilbert spaces is proved. Our approach is based o the basic otios of geeralized resolvets ad fixed poits. The results improve ad exted importat recet results. 1. Itroductio The geeralized mixed equilibrium problem (GMEP) is oe of the most geeral problems appearig i oliear aalysis. It is well recogized that the GMEP icludes variatioal iequalities, complemetarity problems, saddle poit problems, optimizatio, Nash equilibria problems, ad umerous problems i physics, mechaics ad ecoomics as special cases. For details, refer to [19]. Very recetly, Kazmi et al. [14] dealt with the aalysis of iterative algorithms (which exted ad improve the iterative methods give i [18], [23]) for solvig the GMEP i real Hilbert spaces by usig fixed poit ad resolvet methods. Further, uder the θ-pseudomootoicity ad δ-lipschitz cotiuity coditios for mappigs, they proved the weak covergece of the sequeces geerated by these iterative algorithms to the solutio of GMEP. Ispired ad motivated by recet research works i this filed, this paper itroduces a exteded iterative algorithm, called geeralized mea proximal algorithm (GMPA); moreover, it presets the covergece of the iterative sequece geerated by this algorithm to the solutio of GMEP with θ-pseudomootoe ad δ-lipschitz cotiuous mappigs i Hilbert spaces. The results obtaied i this paper exted the recetly correspodig results Mathematics Subject Classificatio. 49J30, 47J25; 65K15,65J15. Key words ad phrases. Geeralized mea proximal algorithm; geeralized mixed equilibrium problem; geeralized resolvet mappig; θ-pseudomootoe mappig; δ- Lipschitz cotiuous mappig; covergece; Hilbert space; fixed poit. c 2015 Uiversiteti i Prishtiës, Prishtië, Kosovë. Submitted October 28, Published December 12,

2 2 A.M. SADDEEK 2. Iequalities ad basic cocepts Let H be a real Hilbert space with ier product, ad correspodig orm. Let K be a oempty closed ad covex set i H, ad let 2 K deote the family of oempty subsets of K. We deote by R the filed of real umbers ad by N the set of all positive itegers. Let T : K K be a oliear mappig. The set of all fixed poits of T is deoted by F(T ). Let f : K K R ad g : H H R be oliear fuctios. We deote by dom(f) (resp. dom(g)) the domai of f (resp. the domai of g ). The iterior of the domai of f is deoted by it dom(f). The GMEP (see, for example, [14]) for f, g ad T is to fid u K such that f(u, v) + T u, v u + g(u, v) g(u, u) 0, v K. (2.1) As special cases of the problem (2.1), we have the followig problems: 1. If we cosider f(u, v) = 0 ad g(u, v) g(v), u, v K, the we get the followig mixed variatioal iequality: Fid u K such that T u, v u + g(v) g(u) 0, v K, (2.2) which was origially cosidered ad studied by Duvaut et al. [10]. 2. If we cosider T = 0 ad g = 0, the we get the followig equilibrium problem: Fid u K such that f(u, v) 0, v K, (2.3) which has bee iitialy itroduced by Blum et al. [3]. 3. If we take T = 0, the we get the followig geeralized equilibrium problem: Fid u K such that f(u, v) + g(u, v) g(u, u) 0, v K, (2.4) which was itroduced ad studied by may authors (see, for example, [24]). 4. If we take g = 0, the we get the followig mixed equilibrium problem: Fid u K such that f(u, v) + T u, v u 0, v K, (2.5) which was studied by Moudafi et al. [20]. 5. If f = 0 ad g = 0, the we get the followig classical variatioal iequality: Fid u K such that T u, v u 0, v K, (2.6) which was first suggested ad studied by Hartma et al. [12] i fiite dimesioal spaces. Now, we preset the basic cocepts that will be eeded i the sequel. A multivalued mappig A : K 2 K is said to be mootoe mappig if for each u, v K ad w A(u), w A(v), we have w w, u v 0. A mootoe mappig A is said to be maximal mappig if its graph G(A) = {(u, w) K K : w A(u)} is ot properly cotaied i the graph of ay other mootoe mappig A : K 2 K (see, for example, [25]). For a give multivalued maximal mootoe mappig A : K 2 K ad a costat

3 A GENERALIZED MEAN PROXIMAL ALGORITHM FOR SOLVING 3 r > 0, the resolvet mappig (or the proximal mappig) associated with A (see, for example, [11]) is the sigle-valued mappig Jr A : K K defied by J A r (u) = (I + ra) (u), (2.7) for ay u K, where I is the idetity mappig. It is kow (see, for example, [4]) that a mootoe mappig A is maximal if ad oly if its resolvet mappig Jr A is defied everywhere. Moreover, it is well kow (see, for example, [11]) that the resolvet is firmly oexpasive, that is, J A r (u) J A r (v), u v J A r (u) J A r (v) 2, u, v K. Clearly, every firmly oexpasive is oexpasive, that is, J A r (u) J A r (v) u v, u, v K. For every r > 0 we set A r = 1 r (I J A r ). It is called the Yosida approximatio. It is kow (see, for example, [1], [25]) that A r (u) AJ A r (u) for all u K ad F(J A r ) = A 0. Thus the problems of existece ad approximatio of zeros of maximal mootoe mappigs ca be formulated as the correspodig problems of fixed poits of firmly oexpasive mappigs. For the bifuctios f : K K R ad g : H H R, let us assume that the followig coditios hold. (C 1 ) for each fixed v K, u f(u, v) is a proper, covex ad lower semicotiuous, (C 2 ) for each fixed v H, u g(u, v) is a proper, covex ad lower semicotiuous, (C 3 ) for all u K, f(u, v) + f(v, u) 0, (C 4 ) there exists u dom f(, v) dom g(, v) such that, either f(, v) or g(, v) is cotiuous at u. (C 5 ) g is skew symmetric, i.e., g(u, u) g(u, v) g(v, u) + g(v, v) 0, u, v H. Let (f + g)(, v) = f(, v) + g(, v) deote the subdifferetial of fuctio (f + g)(, v). It is well kow (see, for example, [1]) that if (it dom f(, v)) dom g(, v) φ, the the subdifferetial (f + g)(, v) is a maximal mootoe mappig with respect to the first argumet, so we deote by J f(,v)g(,v) r (u) = (I + r (f + g)(, v)) (u), r > 0, (2.8) the resolvet mappig associated with (f + g)(, v) for ay fixed v K. O the other had, Kazmi et al. [14] used the auxiliary priciple techique to characterize the resolvets (defied by this techique) as firmly oexpasive mappigs. Related to the GMEP (2.1), Kazmi et al. [14] cosidered the followig, called the oliear resolvet equatios (see, for example, [21], [22]): Fid ω K such that T Jr f(,v)g(,v) ω + Ar f(,v)g(,v) ω = 0, (2.9)

4 4 A.M. SADDEEK where T : K K is a give mappig, Ar f(,v)g(,v) The equatios (2.9) ca be writte as which ca be modeled by the equatio = 1 f(,v)g(,v) r (I Jr ), r > 0. ω Jr f(,v)g(,v) ω + rt Jr f(,v)g(,v) ω = 0, (2.10) where S is a oliear mappig of K ito itself defied by u = Su, (2.11) Su = u γ[ω Jr f(,v)g(,v) ω + rt Jr f(,v)g(,v) ω], γ > 0. (2.12) Here ω = u rt u, that is, u is a fixed poit of S. It has bee show i [14] that the GMEP (2.1) ad equatios of the type (2.9) have the same set of solutios. Let θ : K K R be a give real-valued fuctio. Accordig to [8], [15] the mappig T : K K is said to be (i) θ-pseudomootoe if for all u, v K, we have implies T u, v u + θ(u + v) 0 T v, v u + θ(u + v) 0; (ii) δ-pseudo-cotractive if for all u, v K, there exists a costat δ > 0 such that T u T v, u v δ u v 2 ; (iii) δ-lipschitz cotiuous if for all u, v K, there exists a costat δ > 0 such that T u T v δ u v. It should be poited out that the class of δ-pseudo-cotractive (resp. pseudocotractive, i.e., δ = 1 i (ii)) mappigs is larger that of δ-lipschitz cotiuous (resp. oexpasive, i.e., Lipschitz cotiuous with costat δ = 1 ) mappigs, the coverse, however, is false (see, for example, [9]). 3. Iterative algorithms Let T : K K be a self mappig ad let B = [a,j ] be a lower triagular matrix with oegative etries, zero colum limits, ad row sums 1. For ay u 0 K, the sequece {u } give by u +1 = T ū, ū = a,j u j (3.1) is called the Ma iterative process (see, for example, [17]). Especially, if ū = u (i.e., a,j = 1 whe j = ad a,j = 0 for all j ), the u is called the Picard iterative sequece. Most of the research i this filed has focused o the followig special Ma iterative sequece {u }: j=0 u 0 K, u +1 = (1 γ )u + γ T u, 0, (3.2)

5 A GENERALIZED MEAN PROXIMAL ALGORITHM FOR SOLVING 5 where γ is a sequece i (0, 1) satisfyig the coditios (i) γ 0 = 1, (ii) 0 < γ < 1 for > 0, ad (iii) γ =. Recetly, Combettes et al. [6] itroduced the followig so called geeralized Ma iterative algorithm: =0 u 0 K, u +1 = (1 γ )ū + γ (S ū + e ), 0, (3.3) where {S } is a sequece of mappigs of K ito itself, e K is the error made i the computatio of S ū, ad γ (0, 2). Motivated by (3.3), Saddeek [26] itroduced the followig so called mea proximal algorithm: u 0 K, R u +1 = (1 γ )R ū + γ (J R,A r (R (ū )) + e ), 0, (3.4) where Jr R,A = (R + r A) (i.e., R -resolvets (or geeralized resolvets) of A), A : K 2 K is a maximal mootoe mappig, {R } is a sequece of liear, positive defiite ad self-adjoit mappigs of K ito itself, {e } is a bouded sequece of elemets of K itroduced to take ito accout possible iexact computatio of Jr R,A (R (ū )), 0 < r <, ad 0 < a γ b < 2. If R = I ad Jr A = S, the the algorithm (3.4) is reduced to the geeralized Ma iterative algorithm (3.3). If A = f, where f : K R is a proper, covex ad lower semicotiuous fuctio o K, the algorithm (3.4) collapses to the followig algorithm (see, for example, [13], [25]): where u 0 K, R u +1 = (1 γ )R ū + γ (J R,f r (R (ū )) + e ), 0, (3.5) Jr R,f (R (ū )) = (R + r f) (R (ū )) = Arg mi {f(z) + 1 z ū 2 }, z K 2r ad {e } is a bouded sequece of elemets of K itroduced to take ito accout possible iexact computatio of Jr R,f (R (ū )). Based o the resolvet mappig (2.8) ad the fixed poit formulatio (2.11), Kazmi et al. [14] exteded the iterative methods give i ([18], [23]) ad they itroduced the followig iterative algorithm for the GMEP (2.1): u 0 K, u +1 = Su, 0, (3.6) where u Su is defied by (2.12). This iterative approach requires the restrictive assumptios that T must be θ- pseudomootoe ad δ- Lipschitz cotiuous to esure the weak covergece. We will ow itroduce ad aalyze the followig GMPA for solvig the GMEP (2.1) with T = T R. Our geeralized algorithm is based o the basic otios of fixed poits ad geeralized resolvets of f + g. Let {γ } be a sequece i (0, 2) ad {r } a sequece i (0, ). Give a iitial u 0 K, defie {u } by R u +1 = R ū γ [ω Jr R,f,g ω + r T R Jr R,f,g ω + e ], 0, (3.7) where ω = ū r T R ū ad J R,f,g r = (R + r (f + g)).

6 6 A.M. SADDEEK Remark 3.1. (i) If r = r, γ = γ, R = I, T = T, e = 0 ad ū = u for all 0, the the GMPA (3.7) reduces to algorithm 4.1 i Kazmi et al. [14]. R, the the GMPA collapses to algorithm (24) i Saddeek [26]. Heceforth, l 1 (resp. l+) 1 deotes the class of summable sequeces i R (resp. R + ). We ow recall the followig defiitios ad lemma for our mai results: Defiitio 3.1. (see [6]) A matrix B is cocetratig if every sequece {µ } i R + such that (ii) If T = 0, g = 0, ad I J R,f r is replaced by J R,f, r ( (ε ) l 1 +) µ +1 µ + ε N coverges. Defiitio 3.2. (see, for example, [7, p. 40]). A subset K of a Hilbert space H is said to be boudedly compact if every bouded sequece i K has a subsequece covergig to a poit i K. Lemma 3.1. (see [16, Theorem 3.5.4]). Let {µ } be a sequece i R. The µ µ µ µ. 4. Covergece aalysis I this sectio, we first exted Theorem 4.1 of Kazmi et al. [14] to the case of two sequeces of mappigs. The o a Hilbert space, we establish the covergece aalysis of the GMPA (3.7) to the solutio of the GMEP (2.1) with T = T R. For this purpose, we defie, as i Saddeek [26], T u R by T u R = sup T u, η, η 0 K η R where 2 R =, R = R,. If η η R, oe ca easily show that T u R T u. Theorem 4.1. Let K be a oempty closed ad covex subset of a real Hilbert space H. Let f : K K R ad g : H H R be oliear bifuctios such that (it dom(f) dom(g) φ) ad the coditios (C 1 ) (C 5 ) hold. Let {R } ad {T } be two sequeces of mappigs of K ito itself such that for each N (i) R is liear, positive defiite ad self adjoit, (ii) T R is θ-pseudomootoe, where θ(u, v) = f(u, v) + g(u, v) g(u, u) u, v K, (iii) T R is δ-pseudocotractive. Let ũ K be a solutio of the followig GMEP The f(u, v) + T R u, v u + g(u, v) g(u, u) 0, v K. (4.1) u ũ, R u r (T R u T ω ) (1 r δ) R u 2, u K, (4.2) R where R u = u Jr R,f,g (u r T R u) ad ω = u r T R u, provided η η R η 0 K. Proof. Note that for each N, R is ivertible ad R is liear because of coditio (i). Sice for each N, the mappig T R is θ-pseudomootoe, the for all ũ, ỹ K T R ũ, ỹ ũ + θ(ũ + ỹ) 0

7 A GENERALIZED MEAN PROXIMAL ALGORITHM FOR SOLVING 7 implies where i.e., T R ỹ, ỹ ũ + θ(ũ + ỹ) 0, θ(ũ + ỹ) = f(ũ, ỹ) + g(ũ, ỹ) g(ũ, ũ), f(ũ, ỹ) + T R ỹ, ỹ ũ + g(ũ, ỹ) g(ũ, ũ) 0, ỹ K. This together with coditio (C 3 ) implies that f(ỹ, ũ) + T R ỹ, ỹ ũ + g(ũ, ỹ) g(ũ, ũ) 0, ỹ K. (4.3) Now takig ỹ = u R u i (4.3), we have f(u R u, ũ)+ T R (u R u), (u R u) ũ +g(ũ, u R u) g(ũ, ũ) 0. (4.4) Now, let ũ be a solutio of GMEP (4.1), hece I particular for ũ = J R,f,g r f(j R,f,g r u, v)+ 1 f(ũ, v) + T R ũ, v ũ + g(ũ, v) g(ũ, ũ) 0. (4.5) Jr R,f,g r u, u = J R,f,g r u u, v J R,f,g r u r T R Jr R,f,g u, we have u +g(j R,f,g r u, v) g(jr R,f,g u, Jr R,f,g u) 0. (4.6) Puttig v = Jr R,f,g (u r T R u) = u R u i (4.5), we obtai f(ũ, u R u) + T R ũ, (u R u) ũ + g(ũ, u R u) g(ũ, ũ) 0. (4.7) Settig u := u r T R u, Jr R,f,g (u) := Jr R,f,g (u r T R u) = u R u ad v = ũ i (4.6), we obtai f(u R u, ũ)+ 1 r u R u (u r T R u), ũ (u R u) +g(u R u, ũ) g(u R u, u R u) 0. Now, addig (4.4) ad (4.8) ad usig (C 5 ), we obtai (4.8) R u r T R u + r T R (u R u), (u R u) ũ 0. (4.9) This together with the δ-pseudocotractivity of T implies that R u r (T R u T R (u R u)), u ũ R u r (T R u T R (u R u)), R u R u 2 R r T R u T R (u R u), R u R u 2 r R T R u T R (u R u), u (u R u) (1 r δ) R u 2, R the required result (4.2). Theorem 4.2. Let all assumptios i Theorem 4.1 be satisfied, except for coditio (iii) let it be replaced by the δ-lipschitz cotiuous coditio. Let ũ be a solutio of GMEP (4.1). The the iterative sequece {u } geerated by (3.7), where e H, { e R } l1,, a (0, 1), u 0 K, ad B is cocetratig, coverges weakly to ũ. If i additio dom (R (I r T R ) + r (f + g)(i r T R )) is boudedly compact, the {u } coverges strogly to ũ. r δ < 1, lim r = r > 0, 0 < a γ < 2(1 rδ) a (1+r δ) 2

8 8 A.M. SADDEEK Proof. Let ũ be a solutio of GMEP (4.1). By usig (3.7), ad the covexity of R, we get for 0 u +1 ũ 2 R = ū ũ γ R ( ū ũ γ R = ([ ū ũ 2 R 2γ R (ω J R,f,g r (ω J R,f,g r ω + r T R Jr R,f,g ω ) + e ) 2 R ω + r T R Jr R,f,g ω ) R + γ R e R ) 2 r ω + r T R Jr R,f,g ω ), ū ũ R (ω J R,f,g + γ R 2 (ω Jr R,f,g ω + r T R Jr R,f,g ω ) 2 R ] γ R e R ) 2 = ([ ū ũ 2 R 2γ ω Jr R,f,g ω + r T R Jr R,f,g ω, ū ũ + γ 2 ω J R,f,g r ω + r T R J R,f,g r ω 2 ] 1 R 2 + γ e R )2 = ([ ū ũ 2 R 2γ R ū r (T R ū T R (ū R ū )), ū ũ + γ 2 R ū r (T R u T R ( u R ū ) 2 R ] γ e R )2 = ([ ū ũ 2 R 2γ R ū r (T R ū T R (ū R ū )), ū ũ + γ[ 2 R ū 2 2r R R ū, T R ū T R (ū R ū ) R + r T 2 R u T R ( u R ū ) 2 R ] γ e R )2 ([ ū ũ 2 R 2γ R ū r (T R ū T R (ū R ū )), ū ũ + γ[ 2 R ū 2 + 2r R + r T 2 R u T R ( u R ū R R ū ) 2 R T R ū T R (ū R ū ) R ] γ e R )2. (4.10) As ay δ-lipschitz cotiuous mappig is δ -pseudocotractive, it results by (4.2), ad (4.10) that u +1 ũ 2 R ( ū ũ R γ [2(1 r δ) γ (1 + r δ) 2 ] R ū 2 R + γ e R )2. (4.11) This, together with (3.1) (takig ito accout γ < 2(1 rδ) (1+r δ) 2 that u +1 ũ R j=0 ] 1 2 ad r δ < 1, implies a,j u j ũ R + 2 e R. (4.12) Sice B is cocetratig ad { e R l1 }, it results that lim u ũ R = ρ(ũ). The, it follows from Lemma 3.1 ad (4.12) that lim ū ũ R = ρ(ũ) <. Moreover, the sequece {ɛ } defied by ɛ = σ e R 4 sup 0 u +1 ũ R < lies i l e 2 R, where σ = Usig the covexity of 2 R ad the restrictios o γ ad r, from (4.11) we obtai R ū 2 1 R a 2 [ ū ũ 2 R [ u +1 ũ R γ e R ]2 ] (4.13) 1 a 2 [ ū ũ 2 R u +1 ũ 2 R + 2γ u +1 ũ R e R 1 a 2 [ a,j u j ũ 2 R u +1 ũ 2 R + ɛ ]. j=0 + γ2 e 2 ] R

9 A GENERALIZED MEAN PROXIMAL ALGORITHM FOR SOLVING 9 Sice { ū ũ 2 R } coverges, the it follows from Lemma 3.1 that a,j u j ũ 2 R u +1 ũ 2 R 0. j=0 It therefore follows from (4.13) that lim R ū R = 0. (4.14) Further, sice T R is δ-lipschitz cotiuous ad u +1 ū R = γ R 2( R (ω J R,f,g r (ω J R,f,g = 2( ω J R,f,g r r = 2( ω Jr R,f,g = 2( the from (4.14) we obtai ω + r T R Jr R,f,g ω + e ) R ω + r T R Jr R,f,g ω ) R + R e R ) ω + r T R J R,f,g r ω + r T R J R,f,g r R ū + r (T R J R,f,g r ω R R + e R ω R ω T R ū ) R 2( R ū R + r R ū R + e R ), + e R ) ) + e R ) lim u +1 ū R = 0. (4.15) As {u } is bouded, it results that there exists a subsequece of {u }, deoted by {u k }, which coverges to some ṽ K. Followig the argumets i the proof of ([2, Corollary 6.1]), we ca show that lim R k u k = 0. (4.16) k Therefore ṽ F(J R,f,g r (I r T R )) (i.e., ṽ is a solutio of GMEP (4.1) (see [14, Lemma 3.3]). Suppose there are two weak limit poits of {ū }, say ṽ ad ω. As above, we have ṽ ad ω are solutios of GMEP (4.1) ad that lim v ṽ R = ρ(ṽ), lim v ω R = ρ( ω). (4.17) Similar to the proof of [6, Theorem 3.5], we ca show that the sequeces { v 2 R 2 v, ṽ R } ad { v 2 R 2 v, ω R } coverge ad therefore so does { v, ṽ ω R }. This ad the weak covergece of { v }, imply that ṽ ω 2 R = 0 ad, hece, ṽ = ω. Thus, all the weak limit poits of { v } coicide ad hece, the sequece { v } coverges weakly to the solutio ũ of GMEP (4.1). This together with (4.15) yields u ũ. (4.18) Sice dom (R (I r T R ) + r (f + g)(i r T R )) is boudedly compact, it follows by (4.16) ad [5, Theorem 6.9] that the set of strog cluster poits of the sequece {ū } is oempty. The rest of the argumet ow follows exactly as i the proof of secod assertio of Theorem 3.5 i [6] to yield that {u } coverges strogly to ũ, completig the proof of the Theorem. Remark 4.1. As was metioed i Remark 3.1, our results improve ad exted may importat recet results (for example, [6, 14,18 23,26]).

10 10 A.M. SADDEEK 5. Applicatio to miimizatio problem We cosider the problem of fidig a miimizer of the sum of two proper covex lower semicotiuous bifuctios. Theorem 5.1. Let f : K K R ad g : H H R be two proper covex lower semicotiuous bifuctios such that (it dom(f) dom(g) φ) ad the coditios (C 3 ) (C 5 ) hold. Let {R } be a sequece of liear, positive defiite ad self adjoit mappigs of K ito itself. Let u 0 K ad let {u } be a sequece geerated by R u +1 = R ū γ [ū Arg mi v K {(f + g)(ū, v) + 1 2r v ū 2 R } + e ], 0, (5.1) where r (0, ), γ (0, 2), ad { e R } satisfy lim r = r > 0, 0 < a γ < 2 ad { e R } l1. If B is cocetratig, dom (R + r (f + g)) is boudedly compact ad ( (f + g)) (0) φ, the {u } coverges strogly to v K, which is the miimizer of f + g. Proof. Puttig T = 0, N ad takig J R,f,g r (ū ) = Arg mi v K {(f + g)(ū, v) + 1 2r v ū 2 R } i Theorem 4.2. The desired coclusio follows immediately. Ackowledgmets. The author is extremely grateful to the referee for his/her valuable commets ad suggestios. Refereces [1] V. Barbu, Optimal Cotrol of Variatioal Iequalities, Pitma Publishig Limited, Lodo, [2] H. H. Bauschke, P. L. Combettes, A weak-to-strog covergece priciple for Fejer-mootoe methods i Hilbert spaces, Math. Oper. Res. 26 (2001) [3] E. Blum, W. Oettli, From optimizatio ad variatioal iequalities to equilibrium problems, Math. Stud. 63 (1994) [4] H. Brezis, Opérateurs Maximaux Mootoe et Semigroupes de Cotractios das les Espaces de Hilbert, North-Hollad, Amsterdam, [5] P. L. Combettes, Quasi-Fejéria aalysis of some optimizatio algorithms, i D. Butariu, Y. Cesor, S. Reich (Eds), Iheretly Parallel Algorithms for Feasibility ad Optimizatio, Elsevier, New York, [6] P. L. Combettes, T. Peae, Geeralized Ma iterates for costructig fixed poits i Hilbert spaces, J. Math. Aal. Appl. 275 (2) (2002) [7] M. M. Day, Normed Liear Spaces, Academic Press, New York, [8] K. Deimlig, Zeros of accretive operators, Mauscripta Math. 13 (1974) [9] K. Deimlig, Geometric Properties of Baach Spaces ad Noliear Iteratios, Lecture Notes i Mathematics, vol (2009) ISBN: [10] G. Duvaut, L. Lios, Les Iéquatios e Méchaique et e Physique, Duod, Paris, [11] D. Gabay, Applicatios of the method of multipliers to variatioal iequalities, i: M. Forti, R. Glowiski (Eds), Augmeted Lagragia Methods: Applicatios to the Numerical Solutio of Boudary Value Problems, North-Hollad, Amsterdam, [12] P. Hartma, G. Stampacchia, O some oliear elliptic differetial fuctioal equatios, Acta Math. 115 (1966) [13] S. Kamimura, W. Takahashi, Approximatig solutios of maximal mootoe operators i Hilbert spaces, J. Approx. Theory 106 (2000) [14] K. R. Kazmi, S. H. Rizvi, Iterative algorithms for geeralized equilibrium problems, J. Egypt. Math. Soc. 21 (2013) [15] K. R. Kazmi, A. Khaliq, A. Raouf, Iterative approximatio of solutio of geeralized mixed set-valued variatioal iequality problem, Math. Iequal. Appl. 10 (3) (2007)

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