Research Article Generalized Equilibrium Problem with Mixed Relaxed Monotonicity

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1 e Scieific World Joural, Aricle ID , 4 pages hp://dx.doi.org/ /2014/ Research Aricle Geeralized Equilibrium Problem wih Mixed Relaxed Moooiciy Haider Abbas Rizvi, 1 Adem KJlJçma, 2 ad Rais Ahmad 1 1 Deparme of Mahemaics, Aligarh Mus Uiversiy, Aligarh , Idia 2 Deparme of Mahemaics ad Isiue for Mahemaical Research, Uiversii Pura Malaysia, Selagor, Malaysia Correspodece should be addressed o Adem Kılıçma; akilic@upm.edu.my Received 14 Jue 2014; Acceped 1 July 2014; Published 10 July 2014 Academic Edior: M. Mursalee Copyrigh 2014 Haider Abbas Rizvi e al. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial work is properly cied. We exed he cocep of relaxed α-moooiciy o mixed relaxed α-β-moooiciy. The cocep of mixed relaxed α-βmoooiciy is more geeral ha may exisig coceps of moooiciies. Fially, we apply his cocep ad well kow KKMheory o obai he soluio of geeralized equilibrium problem. 1. Iroducio Geeralized moooiciies provide a way of fidig parameer moves ha yield moooiciy of model soluios ad allow sudyig he moooiciy of fucios or subse of variables. I rece pas, may researchers have proposed may impora geeralizaios of moooiciy such as pseudomoooiciy, relaxed moooiciy, relaxed α- β-moooiciy, quasimoooiciy, ad semimoooiciy; see [1 3]. Karamardia ad Schaible [4] iroduced various kids of moooe mappigs which i he case of gradie mappigs are relaed o geeralized covex fucios. For more deails, we refer o [5 7]. May problems of pracical ieres i opimizaio, ecoomics, ad egieerig ivolve equilibrium i heir descripio.theechiquesivolvedihesudyofequilibrium problems are applicable o a variey of diverse areas ad proved o be producive ad iovaive. Blum ad Oeli [8] ad Noor ad Oeli [9] have show ha he mahemaical programmig problem ca be viewed as special realizaio of absrac equilibrium problems. Ispired ad moivaed by he rece developme of equilibrium problems ad heir soluios mehods, i his paper, we exed he cocep of relaxed α-moooiciy o mixed relaxed α-β-moooiciy. Fially, his cocep is applied wih KKM-heory o solve a geeralized equilibrium problem. The resuls of his paper ca be viewed as geeralizaio of may kow resuls; see [10 13]. 2. Preiaries Le K be a oempy subse of real Baach space X. Leφ: K K Rbe a real-valued fucio ad le f:k K R be a equilibrium fucio; ha is, f(x, x) = 0,forallx K. We cosider he followig geeralized equilibrium problem: fid x Ksuch ha f (x,y) +φ(x,y) φ(x, x) 0, y K. (1) Problem (1) has bee sudied by may auhors i differe seigs; see, for isace, [14]. If φ 0, he he problem (1) reducesoheclassical equilibrium problem, ha is, o fid x Ksuch ha f(x, y) 0, wih f (x, x) =0, y K. (2) Problem (2) was iroduced ad sudied by Blum ad Oeli [8]. Weeedhefollowigdefiiioadresulsihesequel. Defiiio 1. A real-valued fucio defied o a covex subse K of X is said o be hemicoiuous if 0 +f(x+(1 ) y) = f (y), for each x, y K. (3)

2 2 The Scieific World Joural Defiiio 2. Le f:k 2 X be a mulivalued mappig. The f is said o be a KKM-mappig if, for ay fiie subse {y 1,y 2,...,y } of K, co{y 1,y 2,...,y } f(y i), co deoes he covex hull. Lemma 3 (see [15]). Le K be a oempy subse of a opological vecor space X ad le f:k 2 X be a KKM-mappig. If f(y) is closed i X for all y Kadcompac for a leas oe y K,he y K f(y) =φ. Defiiio 4. Le X be a Baach space. A mappig f:x R is said o be lower semicoiuous a x 0 X,if f(x 0 ) if f(x ), (4) for ay sequece {x } of X such ha x x 0. Defiiio 5. Le X be a Baach space. A mappig f:x R is said o be weakly upper semicoiuous a x 0 X,if f(x 0 ) x sup f(x ), (5) for ay sequece {x } of X such ha x x 0. Now, we exed he defiiio of relaxed α-moooiciy [11]omixedrelaxedα-β-moooiciy. Defiiio 6. Amappigf:K K Ris said o be mixed relaxed α-β-moooe, if here exis mappigs α:k R wih α(x) = p α(x),forall>0ad β:k K R,such ha f (x, y) +f(y, x) α(y x) +β(x, y), α(y x) 0 [p + x, y K, (6) β(x,y+(1 ) x) ]=0, (7) ad p>1is a cosa. If β=0,hedefiiio6 reduces o he defiiio of geeralized relaxed α-moooe; ha is, f (x, y) + f (y, x) α (y x), x, y K, (8) α(y x) 0 [p ] =0, p>1is a cosa. (9) If α = 0,heDefiiio6 reduces o he defiiio of geeralized relaxed β-moooe; ha is, f (x, y) +f(y, x) β(x, y), x, y K, (10) β(x,y+(1 ) x) =0. (11) 0 If boh α=0=β,hedefiiio6 coicides wih he defiiio of moooiciy; ha is, f (x, y) +f(y, x) 0, x, y K. (12) Defiiio 7. Amappigφ:K K R {±0}is said o be o-diagoally covex if, for ay fiie subse {x 1,x 2,...,x } of K ad λ i 0 (i = 1,2,...,) wih λ i =1ad x= λ ix i,oehas λ i φ(x, x i ) 0. (13) 3. Exisece of Soluio for Geeralized Equilibrium Problem We esablish his secio wih he discussio of exisece of soluio for geeralized equilibrium problem by usig mixed relaxed α-β-moooiciy. Theorem 8. Suppose f:k K Ris mixed relaxed α-βmoooe, hemicoiuous i he firs argume ad covex i he secod argume wih f(x, x) = 0,for all x K.Le φ:k K R be covex i he secod argume. The, geeralized equilibrium problem (1) is equivale o he followig problem. Fid x Ksuch ha f(y,x) + φ (x, x) φ(x, y) α (y x) + β (x, y), α(x) = p α(x) ad p>1is a cosa. y K, (14) Proof. Suppose ha he geeralized equilibrium problem (1) admis a soluio; ha is, here exiss x Ksuch ha f(x, y) + φ (x, y) φ (x, x) 0, y K. (15) Sice f is mixed relaxed α-β-moooe, we have f (x,y) +f(y, x) α(y x) +β(x,y), y K, (16) f (y, x) α(y x) +β(x,y) f(x,y), (17) Addig φ(x, x) φ(x, x) o boh sides of (17), we have f(y,x) + φ (x, x) φ(x, y) α(y x) + β (x, y) [f (x, y) + φ (x, y) φ (x, x)] α(y x) + β (x, y), (18) Hece, x Kis a soluio of problem (14). Coversely, suppose ha x Kis a soluio of problem (14); ha is, f(y,x) + φ (x, x) φ(x, y) α (y x) + β (x, y), (19)

3 The Scieific World Joural 3 Le x = y+(1 )x, [0,1],ady K;heclearlyx K as K is covex. Thus from (17), we have f(x, x) + φ (x, x) φ(x, x ) α(x x) + β (x, x ). (20) Sice f is covex i he secod argume, we have 0=f(x,x ) f(x,y) + (1 ) f (x, x) (21) which implies ha [f(x, x) f (x,y)] f(x, x). (22) Also as φ is covex i he secod argume, we have φ(x, x ) φ(x, y) + (1 ) φ (x, x), φ (x, y) φ (x, x )+(1 ) φ (x, x), φ (x, x) φ(x, y) φ (x, x) φ(x, x ), (23) [φ (x, x) φ(x,y)] φ(x, x) φ(x,x ). (24) Addig (22) ad(24), we have I follows ha [f(x, x) f (x,y)+φ(x, x) φ(x, y)] f(x, x) + φ (x, x) φ(x, x ) α(x x) + β (x, x ). f(x, x) f (x,y)+φ(x, x) φ(x, y) α(x x) p α(y x) + β(x, x ) + β(x, x ), p > 1. (25) (26) Sice f is hemicoiuous i he firs argume, akig 0,wehave ha is, we have f (x, x) f(x, y) + φ (x, x) φ(x, y) 0; (27) f(x, y) + φ (x, y) φ (x, x) 0, y K. (28) Hece x Kis a soluio of geeralized equilibrium problem (1). Theorem 9. Le K be a oempy bouded closed covex subse of a real Baach space X.Lef: K K Rbe a mixed relaxed α-β-moooe, hemicoiuous i he firs argume, covex i he secod argume wih f(x, x) = 0, o-diagoally covex, ad lower semicoiuous. Le φ:k K Rbe covex i he secod argume, o-diagoally covex, ad lower semicoiuous; α:k Ris weakly upper semicoiuous ad β : K K R is weakly upper semicoiuous i he secod argume. The he mixed equilibrium problem (1) admis a soluio. Proof. Cosider a mulivalued mappig F:K 2 X such ha F(y)={x K:f(x, y) + φ (x, y) φ (x, x) 0}, (29) We show ha y K F(y) = φ; hais,x Kis a soluio of geeralized equilibrium problem (1). Our claim is ha F is a KKM-mappig. Suppose o corary ha is F is o a KKM-mappig; he here exiss afiiesubse{x 1,x 2,...,x } of K ad λ i 0 (i = 1,2,...) wih λ i =1such ha I follows ha x 0 = λ i x i f (x 0,x i ) +φ(x 0,x i ) φ(x 0,x 0 ) <0, Also we have F(y i ). (30) λ i [f (x 0,x i )+φ(x 0,x i ) φ(x 0,x 0 )]<0, for,2,...,, for,2,...,. (31) (32) which coradicsheo-diagoal covexiy of f ad φ.hece F is a KKM-mappig. Now cosider aoher mulivalued mappig G:K 2 X such ha G(y)={x K:f(y,x) + φ (x, x) φ(x, y) α(y x) + β (x, y)}, (33) We will show ha F(y) G(y), Foraygivey K, le x F(y);he f(x, y) + φ (x, y) φ (x, x) 0. (34) I follows from he mixed relaxed α-β-moooiciy of f ha f(y,x) + φ (x, x) φ(x, y) α(y x) + β (x, y) [f (x, y) + φ (x, y) φ (x, x)] α(y x) + β (x, y) ; (35) ha is, x G(y).Thus F(y) G(y) ad cosequely G is also KKM-mappig. Sice f ad φ boh are covex i he secod argume adlowersemicoiuous,husheybohareweaklylower semicoiuous. From weakly upper semicoiuiy of α, weakly upper semicoiuiy of β i he secod argume, ad he cosrucio of G, i is accessible o see ha G(y) is weakly closed for all y K.SiceK is closed, bouded, ad covex, i is weakly compac ad cosequely G(y) is weakly

4 4 The Scieific World Joural compac i K for all y K. Therefore, from Lemma 3 ad Theorem 8,we have y K F(y)= G(y) y K ha is, here exiss x Ksuch ha =φ; (36) f (x,y) +φ(x,y) φ(x, x) 0, y K. (37) Thus, he geeralized equilibrium problem (1) admis a soluio. Coflic of Ieress The auhors declare ha here is o coflic of ieress regardig he publicaio of his paper. [11] N. K. Mahao ad C. Nahak, Equilibrium problems wih geeralized relaxed moooiciies i Baach spaces, Opsearch,vol. 51, o. 2, pp , [12] R. U. Verma, O moooe oliear variaioal iequaliy problems, Commeaioes Mahemaicae Uiversiais Caroliae, vol. 39, o. 1, pp , [13] R. U. Verma, O geeralized variaioal iequaliies ivolvig relaxed Lipschiz ad relaxed moooe operaors, Mahemaical Aalysis ad Applicaios,vol.213,o.1,pp , [14] M. A. Noor, Auxiliary priciple echique for equilibrium problems, Opimizaio Theory ad Applicaios, vol. 122, o. 2, pp , [15] K. Fa, A geeralizaio of Tychooff s fixed poi heorem, Mahemaische Aale,vol.142,pp ,1961. Ackowledgmes The auhors express heir sicere haks o he referees for he careful ad deailed readig of he paper ad he very helpful suggesios ha improved he paper subsaially. The auhors also ackowledge ha his research was par of he research projec ad was parially suppored by Uiversii Pura Malaysia uder ERGS / Refereces [1] M. Bai, S. Zhou, ad G. Ni, Variaioal-like iequaliies wih relaxed η-α pseudomoooe mappigs i Baach spaces, Applied Mahemaics Leers,vol.19,o.6,pp ,2006. [2] Y. Che, O he semi-moooe operaor heory ad applicaios, Mahemaical Aalysis ad Applicaios, vol. 231,o.1,pp ,1999. [3] Y. P. Fag ad N. J. Huag, Variaioal-like iequaliies wih geeralized moooe mappigs i Baach spaces, Opimizaio Theory ad Applicaios, vol.118,o.2,pp , [4] S. Karamardia ad S. Schaible, Seve kids of moooe maps, Opimizaio Theory ad Applicaios,vol.66, o. 1, pp , [5] R. Ellaia ad A. Hassoui, Characerizaio of osmooh fucios hrough heir geeralized gradies, Opimizaio, vol.22,o.3,pp ,1991. [6] S. Karamardia, S. Schaible, ad J. Crouzeix, Characerizaios of geeralized moooe maps, Opimizaio Theory ad Applicaios, vol. 76, o. 3, pp , [7] S. Komlósi, Geeralized moooiciy ad geeralized covexiy, JouralofOpimizaioTheoryadApplicaios,vol.84, o. 2, pp , [8] E. Blum ad W. Oeli, From opimizaio ad variaioal iequaliies o equilibrium problems, The Mahemaics Sude, vol.63,o.1 4,pp ,1994. [9] M. A. Noor ad W. Oeli, O geeral oliear complemeariy problems ad quasi-equilibria, Le Maemaiche, vol.49, o. 2, pp , [10] S. Chag, B. S. Lee, ad Y. Che, Variaioal iequaliies for moooe operaors i oreflexive Baach spaces, Applied Mahemaics Leers,vol.8,o.6,pp.29 34,1995.

5 Advaces i Operaios Research Advaces i Decisio Scieces Applied Mahemaics Algebra Probabiliy ad Saisics The Scieific World Joural Ieraioal Differeial Equaios Submi your mauscrips a Ieraioal Advaces i Combiaorics Mahemaical Physics Complex Aalysis Ieraioal Mahemaics ad Mahemaical Scieces Mahemaical Problems i Egieerig Mahemaics Discree Mahemaics Discree Dyamics i Naure ad Sociey Fucio Spaces Absrac ad Applied Aalysis Ieraioal Sochasic Aalysis Opimizaio