System of implicit nonconvex variationl inequality problems: A projection method approach

Transcription

1 Avalable onlne at J. Nonlnear Sc. Appl. 6 (203), Research Artcle System of mplct nonconvex varatonl nequalty problems: A projecton method approach K.R. Kazm a,, N. Ahmad b, S.H. Rzv a a Department of Mathematcs, Algarh Muslm Unversty, Algarh, Inda b Department of Mathematcs, Al-Jouf Unversty, P.O. Box 204, Skaka, Kngdom of Saud Araba Abstract In ths paper, we consder a new system of mplct nonconvex varatonal nequalty problems n settng of prox-regular subsets of two dfferent Hlbert spaces. Usng projecton method, we establsh the equvalence between the system of mplct nonconvex varatonal nequalty problems and a system of relatons. Usng ths equvalence formulaton, we suggest some teratve algorthms for fndng the approxmate soluton of the system of mplct nonconvex varatonal nequalty problems and ts specal case. Further, we establsh some theorems for the exstence and teratve approxmaton of the solutons of the system of mplct nonconvex varatonal nequalty problems and ts specal case. The results presented n ths paper are new and dfferent form the prevously known results for nonconvex varatonal nequalty problems. These results also generalze, unfy and mprove the prevously known results of ths area. Keywords: System of mplct nonconvex varatonal nequalty problems, prox-regular set, projecton method, teratve algorthm. 200 MSC: Prmary 47J0; Secondary 49J40, 90C33.. Introducton In 985, Pang [25] showed that a varety of equlbrum models, for example, the traffc equlbrum problem, the spatal equlbrum problem, the Nash equlbrum problem and the general equlbrum programmng problem can be unformly modelled as a varatonal nequalty defned on the product sets. He decomposed the orgnal varatonal nequalty nto a system of varatonal nequaltes and dscuss the convergence of method of decomposton for system of varatonal nequaltes. Later, t was notced that varatonal nequalty over product sets and the system of varatonal nequaltes both are equvalent, see Correspondng author Emal addresses: krkazm@gmal.com (K.R. Kazm), nahmadamu@gmal.com (N. Ahmad), shujarzv07@gmal.com (S.H. Rzv) Receved

2 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), for applcatons [3,, 2, 25]. Snce then many authors, see for example [2, 0,, 3, 8, 2] studed the exstence theory of varous classes of system of varatonal nequaltes by explotng fxed-pont theorems and mnmax theorems. Recently, a number of teratve algorthms based on projecton method and ts varant forms have been developed for solvng varous systems of varatonal nequaltes, see for nstance [7, 2, 5, 6, 9, 27]. It s well known that the projecton method and ts varant forms based on projecton operator over convex set are mportant tools for the study of exstence and teratve approxmaton of solutons of varous classes (systems) of varatonal nequalty problems n the convexty settngs, but these may not be applcable n general, when the sets are nonconvex. To overcome the dffcultes that rse from the nonconvexty of underlyng sets, the propertes of projecton operators over unformly prox-regular sets are used. In recent years, Bounkhel et al. [6], Moudaf [20], Wen [28], Kazm et al. [7], Noor [[23, 24, 22] and the relevant references cted theren], Almohammady et al. [], Balooee et al. [4] suggested and analyzed teratve algorthms for solvng some classes (systems) of nonconvex varatonal nequalty problems n the settng of unformly prox-regular sets. On the other hand, to the best of our knowledge, the study of teratve algorthms for solvng the systems of varatonal nequalty problems consdered n [7, 27] n nonconvex settng has not been done so far. Motvated and nspred by research gong on n ths area, we ntroduce a system of mplct nonconvex varatonal nequalty problems (n short, SINVIP) defned on the unformly prox-regular sets n dfferent two Hlbert spaces. SINVIP s dfferent from those consdered n [, 4, 6, 6, 20, 22, 23, 24, 28] and ncludes the new and known systems of nonconvex (convex) varatonal nequalty problems as specal cases. Usng the propertes of projecton operator over unformly prox-regular sets, we suggest some teratve algorthms for fndng the approxmate soluton of SINVIP and ts an mportant specal case. Further, we establsh some theorems for the exstence and teratve approxmaton of the solutons of the SINVIP and ts specal case. The results presented n ths paper are dfferent form the prevously known results for nonconvex varatonal nequalty problems. These results also generalze, unfy and mprove the prevously known results of ths area. The methods presented n ths paper extend, unfy and mprove the methods presented n [, 4, 6, 6, 4, 20, 22, 23, 24, 28]. 2. Prelmnares Let H be a real Hlbert space whose norm and nner product are denoted by and,, respectvely. Let K be a nonempty closed set n H, not necessarly convex. Frst, we recall the followng well-known concepts from nonlnear convex analyss and nonsmooth analyss, see [5, 8, 9, 26]. Defnton 2.. The proxmal normal cone of K at u H s gven by N P (K; u) := {ξ H : u P K (u + αξ)}, α > 0 s a constant and P K s projecton operator of H onto K, that s, P K (u) = {u K : d K (u) = u u }, d K (u) s the usual dstance functon to the subset K, that s, d K (u) = nf v u. v K The proxmal normal cone N P (K; u) has the followng characterzaton.

3 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), Lemma 2.2. Let K be a nonempty closed subset of H. Then ξ N P (K; u) f and only f there exsts a constant α > 0 such that ξ, v u α v u 2, v K. Defnton 2.3. The Clarke normal cone, denoted by N C (K; u), s defned as N C (K; u) = co[n P (K; u)], coa means the closure of the convex hull of A. Polqun et al. [26] and Clarke et al. [8] have ntroduced and studed a class of nonconvex sets, whch are called unformly prox-regular sets. Ths class of unformly prox-regular sets has played an mportant role n many nonconvex applcatons such as optmzaton, dynamc systems and dfferental nclusons. In partcular, we have Defnton 2.4. For a gven r (0, ], a subset K of H s sad to be normalzed unformly r-prox-regular f and only f every nonzero proxmal normal to K can be realzed by any r-ball, that s, u K and 0 ξ N P (K; u) wth ξ =, one has ξ, v u 2r v u 2, v K. It s clear that the class of normalzed unformly prox-regular sets s suffcently large to nclude the class of convex sets, p-convex sets, C, submanfolds (possbly wth boundary) of H, the mages under a C, dffeomorphsm of convex sets and many other nonconvex sets, see [8, 9]. It s note that f r =, then unformly r-prox-regularty of K reduces to ts convexty. It s known that f K s a unformly r-prox-regular set, the proxmal normal cone N P (K; u) s closed as a set-valued mappng. Thus, we have N C (K; u) = N P (K; u). Now, let us state the followng proposton whch summarzes some mportant consequences of the unformly prox-regulartes: Proposton 2.5. Let r > 0 and let K r be a nonempty closed and unformly r-prox-regular subset of H. Set U r = {x H : d(x, K r ) < r}. () For all x U r, P Kr (x) ; () For all r (0, r), P Kr s Lpschtz contnuous wth constant 3. System of mplct nonconvex varatonal nequalty problems r r r on U r = {x H : d(x, K r ) < r }. Throughout the paper unless otherwse stated, we assume that for each {, 2}, K,r s a prox-regular subset of H. Let N : H H 2 H, g : H H be nonlnear mappngs. For any constant ρ > 0, we consder the problem of fndng (x, x 2 ) H H 2 such that (g (x ), g 2 (x 2 )) K,r K 2,r2 and ρ N (x, x 2 ), y g (x ) + 2r y g (x ) 2 0, y K,r. (3.) We call the problem (3.), a system of mplct nonconvex varatonal nequalty problems (SINVIP).

4 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), Some specal cases of SINVIP (3.): () If g = I, the dentty operator on H, then SINVIP (3.) reduces to the system of nonconvex varatonal nequalty problems (n short, SNVIP) of fndng (x, x 2 ) K,r K 2,r2 such that whch appears to be new. ρ N (x, x 2 ), y x + 2r y x 2 0, y K,r, (3.2) (2) If r = +, that s, K,r = K, the convex subset of H, and g = I, then SINVIP (3.) reduces the system of varatonal nequalty problems of fndng (x, x 2 ) K,r K 2,r2 such that N (x, x 2 ), y x 0, y K, (3.3) whch has been consdered and studed n [2, 27] usng dfferent approaches. The followng defntons are needed n the proof of man result. Defnton 3.. A nonlnear mappng g : H H s sad to be k -strongly monotone f there exsts a constant k > 0 such that g (x ) g (y ), x y k x y 2, x, y H. Defnton 3.2. Let N : H H 2 H be a nonlnear mappng. Then N s sad to be () δ -strongly monotone n the frst argument f there exsts a constant δ > 0 such that N (x, x 2 ) N (y, x 2 ), x y δ x y 2, x, y H, x 2 H 2 ; () (L (N,), L (N,2))-mxed Lpschtz contnuous f there exst constants L (N,), L (N,2) > 0 such that N (x, x 2 ) N (y, y 2 ) L (N,) x y + L (N,2) x 2 y 2 2, x, y H, x 2, y 2 H 2. Smlarly to Defnton 3.2(), we can defne the strongly monotoncty of N 2 : H H 2 H 2 n the second argument. Frst we prove the followng techncal Lemmas. Lemma 3.3. SINVIP (3.) s equvalent to the followng system of mplct nonconvex varatonal nclusons of fndng (x, x 2 ) H H 2 such that (g (x ), g 2 (x 2 )) K,r K 2,r2 and 0 ρ N (x, x 2 ) + N p K,r (g (x )), (3.4) for ρ > 0, =, 2, N p K,r (u) denotes the proxmal normal cone of K,r at u n the sense of nonconvex analyss and 0 s the zero vector n H. Proof. Let (x, x 2 ) H H 2 wth (g (x ), g 2 (x 2 )) K,r K 2,r2 be a soluton of SINVIP (3.). If N (x, x 2 ) = 0, evdently the ncluson (3.4) follows for each =, 2. If N (x, x 2 ) 0, from (3.) and Lemma 2.2, we get the ncluson (3.4) for each =, 2. Converse part drectly follows from Defnton 2.4.

5 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), Lemma 3.4. (x, x 2 ) H H 2 wth (g (x ), g 2 (x 2 )) K,r K 2,r2 s a soluton of SINVIP (3.) f and only f t satsfes the system of relatons ] g (x ) = P K,r [g (x ) ρ N (x, x 2 ), (3.5) for ρ > 0, =, 2, P K,r s the projecton operator of H onto the prox-regular set K,r. Proof. The proof s drectly followed from Lemma 3.3 and from the fact that P K,r = (I + N p K,r ). The alternatve formulaton (3.5) of SINVIP (3.) enable us to suggest and analyze the followng teratve algorthm for solvng SINVIP (3.). Iteratve algorthm 3.5. For each =, 2, gven z 0 H, compute the teratve sequences {z n }, {xn } defned by the teratve schemes: for all n = 0,, 2,..., and ρ > 0. g (x n ) = P K,r (z n ), (3.6) z n+ = g (x n ) ρ N (x n, x n 2 ), (3.7) Remark 3.6. We observe that {g (x n )} n=0 K,r for each =, 2. Takng g = I n Iteratve algorthm 3.5, we have the followng teratve algorthm for solvng SNVIP (3.2). Iteratve algorthm 3.7. For each =, 2, gven z 0 H, compute the teratve sequences {z n }, {xn } defned by the teratve schemes: for all n = 0,, 2,..., and ρ > 0. x n = P K,r (z n ), z n+ = x n ρ N (x n, x n 2 ), Now, we prove some theorems concernng the exstence and teratve approxmaton of solutons of SINVIP (3.) and SNVIP (3.2). Theorem 3.8. For each {, 2}, let K,r be closed and unformly prox-regular subsets of real Hlbert space H ; let N : H H 2 H be δ -strongly monotone n th argument and (L (N,), L (N,2))-mxed Lpschtz contnuous and let g : H H be k -strongly monotone and contnuous. For each {, 2} and j {, 2}\{}, f the constant ρ > 0 satsfes the followng condton: { M < ρ < mn M +, ψ, := (b k a e ) 2 b 2 ( e2 )( a2 ) b 2 ( e2 ) M := b k a e b 2 ( e2 ) ; a := µ φ ; ψ j < ; b := µ b L (Nj,) 2k + 3 ; µ := r + N (x n, xn 2 ), φ := b ρ j L (Nj,); r r r ; r } + N (x n+, (3.8) 2 )

6 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), e := ρ 2 2ρ k 2k + 3 2δ + L 2 (N,) ; b k > a e + b ( e 2 )( a2 ) [, 0); r (0, r ); r (0, ]. Then the sequences {x n } and {zn } generated by Iteratve algorthm 3.5 converge strongly to x, z H, respectvely, (x, x 2 ) H H 2 wth g (x ) K,r s a soluton of SINVIP (3.). Proof. From (3.6), we have z n+ = g (x n+ ) g (x n ) ρ (N (x n+ 2 ) N (x n, x n 2 )) g (x n+ ) g (x n ) ρ (x n+ x n ) + ρ N (x n+ 2 ) N (x n, x n 2 ) (x n+ x n ). (3.9) Snce N s δ -strongly monotone n th argument and (L (N,), L (N,2))-mxed Lpschtz contnuous, we have the followng estmates: N (x n+ 2 ) N (x n, x n 2 ) (x n+ x n ) N (x n+ 2 ) N (x n 2 ) (x n+ x n ) + N (x n 2 ) N (x n, x n 2 ) 2δ + L 2 (N,) xn+ x n + L (N,2) x n+ 2 x n 2 2. (3.0) (N 2 (x n+ 2 ) N 2 (x n, x n 2 ) (x n+ 2 x n 2 ) 2 (N 2 (x n+ 2 ) N 2 (x n+, x n 2 ) (x n+ 2 x n 2 ) 2 + (N 2 (x n+, x n 2 ) N 2 (x n, x n 2 ) 2 2δ 2 + L 2 (N 2,2) xn+ 2 x n L (N2,) x n+ x n. (3.) Next t follows from the condtons (3.8) on ρ that z n K,r for all n =, 2,.... Hence usng the k -strongly monotoncty of g, Proposton 2.5 and (3.6), we have the followng estmates. x n+ x n 2 g (x n+ ) g (x n ) 2 2 g (x n+ ) g (x n ) + x n+ x n, x n+ x n or x n+ x n = g (x n+ ) g (x n ) 2 2 g (x n+ ) g (x n ), x n+ x n 2 x n+ x n, x n+ x n. µ 2 z n+ z n 2 (2k + 2) x n+ x n 2. µ 2k + 3 zn+ z n, (3.2) µ = r r. r g (x n+ ) g (x n ) ρ (x n+ x n ) 2 g (x n+ ) g (x n ) 2 2ρ g (x n+ ) g (x n ), x n+ x n + ρ 2 x n+ x n 2 µ 2 z n+ z n 2 + (ρ 2 2ρ k ) x n+ x n 2. (3.3)

7 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), From (3.2) and (3.3), we have g (x n+ ) g (x n ) ρ ((x n+ x n ) µ Now, from (3.9)-(3.) and (3.4), we have z n+ µ [ µ + ρ 2k ρ2 2ρ k 2k ρ2 2ρ k 2k + 3 z n+ z n ( ) 2δ + L 2 (N,) z n+ zn+ z n. (3.4) z n + µ jl (N,j) 2kj + 3 zn+ j z n j j ], (3.5) for each {, 2} and j {, 2}\{}. Defne norm on H H 2 by (x, x 2 ) = (H H 2, ) s a Banach space. It follows from (3.5) that 2 = x for all (x, x 2 ) H H 2. We note that (z n+2, z n+2 2 ) (z n+, z n+ 2 ) = 2 = z n+ θ := max{θ, θ 2 }; θ := µ [p + b (ρ e + ρ j d j )]; [ p := b := + ρ2 2ρ k 2k + 3 2k + 3 ; θ (z n+, z n+ 2 ) (z n, z n 2 ), (3.6) ] 2 ; e := 2δ + L 2 (N,) ; d j := L (Nj,). From condton (3.8), we have 0 < θ <. Hence t follows from (3.6) that {(z n, zn 2 )} s a Cauchy sequence n H H 2. Assume that (z n, zn 2 ) (z, z 2 ) n H H 2 as n, that s, for each =, 2, z n z n H as n. Hence, we observe from (3.2) that {x n } s also a Cauchy sequence and hence assume that x n x n H as n for each =, 2. Further the contnuty of N, g and P K,r and Iteratve algorthm 3.5 mply that ] g (x ) = P K,r [g (x ) ρ N (x, x 2 ). Now t follows from Lemma 3.4 that (x, x 2 ) H H 2 wth g (x ) K,r (3.). Ths completes the proof. s a soluton of SINVIP Theorem 3.9. For each {, 2} and j {, 2}\{}, let K,r, N and g be same as n Theorem 3.8, and let the constant ρ > 0 satsfy the condton: { M < ρ < mn M +, ψ, r + N (x n, xn 2 ), δ 2 L2 (N,) ( a2 ) := ; a := (2 + φ ); φ := ρ j L µ b (N,); L 2 (N,) r } + N (x n+, (3.7) 2 )

8 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), M := b := δ L 2 (N,) ; ψ < ; δ > L µ j b j d (N,) ( a 2 ); a2 < ; 2k + 3 ; µ := r r r ; r (0, r ); r (0, ]. Then the sequences {x n } and {zn } generated by Iteratve algorthm 3.5 converge strongly to x, z H, respectvely, (x, x 2 ) H H 2 wth g (x ) K,r s a soluton of SINVIP (3.). Proof. From (3.6), we have z n+ g (x n+ ) g (x n ) (x n+ x n ) + (x n+ x n 2 ) ρ (N (x n+ 2 ) N (x n, x n 2 )). (3.8) Usng the same arguments used n proof of Theorem 3.8, we have the followng estmates: (x n+ x n ) ρ (N (x n+ 2 ) N (x n, x n 2 )) 2ρ δ + ρ 2 L 2 (N,) xn+ x n + ρ L (N,2) x n+ 2 x n 2 2. (3.9) (x n+ 2 x n 2 ) ρ 2 (N 2 (x n+ 2 ) N 2 (x n, x n 2 )) 2 2ρ 2 δ 2 + ρ 2 L 2 (N 2,2) xn+ 2 x n ρ 2 L (N2,) x n+ x n. (3.20) and x n+ x n µ b z n+ z n, (3.2) µ = r r r ; b := 2k + 3. g (x n+ ) g (x n ) (x n+ x n ) 2µ b z n+ z n. (3.22) Now, from (3.7)-(3.22), we have z n+ 2µ b z n+ z n + µ b e z n+ z n + ρ µ j b j L (N,j) zj n+ zj n j, (3.23) e := 2ρ δ + ρ 2 L2 (N,) for each {, 2} and j {, 2}\{}. It follows from (3.23) that. (z n+2, z n+2 2 ) (z n+, z n+ 2 ) θ (z n+, z n+ 2 ) (z n, z n 2 ), (3.24) θ := max{θ, θ 2 } and θ := b µ [2 + e + ρ j d j )]; d j := L (Nj,). From condton (3.7), we have 0 < θ <. Hence t follows from (3.24) that {(z n, zn 2 )} s a Cauchy sequence n H H 2. The rest of the proof s same as the proof of Theorem 3.8. Ths s completes the proof. In the followng theorem, we have relaxed the condton of strongly monotoncty from N.

9 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), Theorem 3.0. For each {, 2}, let K,r be closed and unformly prox-regular subsets of real Hlbert space H ; let N be (L (N,), L (N,2))-mxed Lpschtz contnuous and let g be k -strongly monotone and contnuous. For each {, 2} and j {, 2}\{}, f the constant ρ > 0 satsfes the condton: { 0 < ρ < mn, ψ, r + N (x n, xn 2 ), r } + N (x n+, (3.25) 2 ) := [ ] ( + φ ) > 0; ψ < ; φ := ρ j L L (N,) µ b µ j b j L (Nj,) (N,j) b := 2k + 3 ; µ := r r r ; r (0, r ); r (0, ]. Then the sequences {x n } and {zn } generated by Iteratve algorthm 3.5 converge strongly to x, z H, respectvely, (x, x 2 ) H H 2 wth g (x ) K,r s a soluton of SINVIP (3.). Proof. Snce N s (L (N,), L (N,2))-mxed Lpschtz contnuous, we have z n+ g (x n+ ) g (x n ) + ρ N (x n+ 2 ) N (x n, x n 2 ) ( g (x n+ ) g (x n ) + ρ L (N,) x n+ x n + L (N,2) x n+ 2 x n 2 2 ). Hence t follows from (3.2) that z n+ µ b z n+ z n + ρ L (N,)µ b z n+ z n + ρ L (N,2)µ 2 b 2 z n+ 2 z n 2 2, (3.26) whch gves µ = r r r ; b := 2k + 3, (z n+2, z n+2 2 ) (z n+, z n+ 2 ) θ (z n+, z n+ 2 ) (z n, z n 2 ), (3.27) θ := max{θ, θ 2 } and θ := b µ ( + ρ L (N,) + ρ j L (Nj,j)). From condton (3.25), we have 0 < θ <. and hence (3.27) mples {(z n, zn 2 )} s a Cauchy sequence n H H 2. The rest of the proof s same as the proof of Theorem 3.8. Ths s completes the proof. Fnally we prove the followng theorem for the exstence and teratve approxmaton of solutons of SNVIP (3.2). Theorem 3.. For each {, 2} and j {, 2}\{}, let K,r be closed and unformly prox-regular subset of real Hlbert space H ; let N be δ -strongly monotone n th argument and (L (N,), L (N,2))-mxed Lpschtz contnuous. If ρ > 0 satsfes the followng condton: δ L 2 (N,) < ρ < mn L 2 (N,) { δ L 2 (N,) +, ψ, r r + N (x n, xn 2 ), + N (x n+ 2 ) := δ 2 L2 (N,) ( a2 ) ; δ > L (N,) ( a 2 ); a2 := φ ; µ a 2 < ; }, (3.28) φ := ρ j L (Nj,); ψ < ; µ := r µ j L (N,j) r r ; r (0, r ); r (0, ]. Then the sequences {x n } and {zn } generated by Iteratve algorthm 3.7 converge strongly to x, z H, respectvely, (x, x 2 ) K,r K 2,r2 s a soluton of SNVIP (3.2).

10 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), Proof. We easly observe that the condton (3.28) on ρ assures that z n K,r for all n =, 2,.... From Iteratve algorthm 3.7 and Proposton 2.5, we have µ = r r r and x n+ x n µ z n+ z n, (3.29) z n+ = x n+ x n ρ (N (x n+ 2 ) N (x n, x n 2 )). (3.30) Snce N s δ -strongly monotone n th argument and (L (N,), L (N,2))-mxed Lpschtz contnuous, from (3.9),(3.20),(3.29) and (3.30), we have e := z n+ µ e z n+ z n + ρ µ j L (N,j) zj n+ zj n j, (3.3) 2ρ δ + ρ 2 L2 (N,) for each {, 2} and j {, 2}\{}. It follows from (3.3) that (z n+2, z n+2 2 ) (z n+, z n+ 2 ) = 2 = z n+ θ := max{θ, θ 2 }; θ := µ (e + ρ j d j ); d j := L (Nj,). θ (z n+, z n+ 2 ) (z n, z n 2 ), (3.32) From condton (3.28), we have 0 < θ <. Hence t follows from (3.32) that {(z n, zn 2 )} s a Cauchy sequence n H H 2. The rest of the proof s same as the proof of Theorem 3.8. Ths s completes the proof. Remark 3.2. () The methods presented n ths paper extend and unfy the methods consdered n [4, 5, 6, 7, 8, 20, 2, 23, 24] to the systems of nonconvex varatonal nequalty problems defned on the product of two dfferent Hlbert spaces. () The methods presented n ths paper mprove the methods consdered n [2, 23, 24] n the sense that the contnuty of g s requred nstead of the Lpschtz contnuty. () One needs further research effort to extend the methods presented for solvng the systems of nonconvex varatonal nequalty problems nvolvng set-valued mappngs. References [] M. Almohammady, J. Balooee, Y.J. Cho, and M. Rooh, Iteratve algorthms for new class of extended general nonconvex set-valued varatonal nequaltes, Nonlnear Anal. TMA serees A 73(2) (200) [2] Q.H. Ansar and J.-C. Yao, A fxed pont theorem and ts applcatons to a system of varatonal nequaltes, Bull. Austral. Math. Soc. 59 (999) , 3 [3] J.P. Aubn, Mathematcal Methods of Game Theory and Economc, North-Holland, Amsterdam, 982. [4] J. Balooee, Y.J. Cho and M.K. Kang, Projecton methods and a new system of extended general regularzed nonconvex set-valued varatonal nequaltes, J. Appl. Math. 202 Artcle ID (202) 8 Pages. [5] S. Boralugoda and R.A. Polqun, Local ntegraton of prox-regular functons n Hlbert spaces, J. Convex Anal. 3 (2006) [6] M. Bounkhel, L. Tadj and A. Hamd, Iteratve schemes to solve nonconvex varatonal problems, J. Inequal. pure Appl. Math. 28(4) (2003) -4. [7] Y.J. Cho and X. Qn, Systems of generalzed nonlnear varatonal nequaltes and ts projecton methods, Nonlnear Anal. TMA seres A 69(2) (2003) [8] F.H. Clarke, Yu. S. Ledyaev, R.J. Stern and P.R. Wolensk, Nonsmooth Analyss and Control Theory, Sprnger, New York, , 2, 2

11 K.R. Kazm, N. Ahmad, S.H. Rzv, J. Nonlnear Sc. Appl. 6 (203), [9] F.H. Clarke, R.J. Stern and P.R. Wolensk, Proxmal smoothness and the lower C 2 property, J. Convex Anal. 2 (-2) (995) , 2 [0] C. Cohen and F. Chaplas, Nested monotony for varatonal nequaltes over product of spaces and convergence of teratve algorthms, J. Optm. Theory Appl. 59 (988) [] M. Ferrs and J.-S. Pang, Engneerng and economcs applcatons of complementarty problems, SIAM Rev. 39 (997) [2] Z. He, C. Chen and F. Gu, Vscosty approxmaton method for nonexpansve nonself-mappng and varatonal nequalty, J. Nonlnear Sc. Appl. (3) (2008) [3] G. Kassey and J. Kolumban, System of mult-valued varatonal nequaltes, Publ. Math. Debrecen 56 (2000) [4] K.R. Kazm, M.I. Bhat and Naeem Ahmed, An teratve algorthm based on M-proxmal mappngs for a generalzed mplct varatonal nclusons n Banach spaces, J. Comput. Appl. Math. 233 (2009) , 3.2 [5] K.R. Kazm and F.A. Khan, Iteratve approxmaton of a unque soluton of a system of varatonal-lke nclusons n real q-unformly smooth Banach spaces, Nonlnear Anal. TMA seres A 67(3) (2007) , 3.2 [6] K.R. Kazm, F.A. Khan and M. Shahzad, A system of generalzed varatonal ncluson nvolvng generalzed H(. )-accretve mappng n real q-unformly smooth Banach spaces, Appl. Math. Comput. 27(23) (20) , 3.2 [7] K.R. Kazm, F.A. Khan and M. Shahzad, Two step algorthm for solvng regularzed generalzed mxed varatonal nequalty problem, Bull. Korean Math. Soc. 47 (200) , 3.2 [8] I.V. Konnov, Relatvely monotone varatonal nequaltes over product sets, Oper. Res. Lett. 28 (200) 2-26., 3.2 [9] H.G. L, Generalzed fuzzy random set-valued mxed varatonal nclusons nvolvng random nonlnear (A ω, η ω)- accretve mappngs n Banach spaces, J. Nonlnear Sc. Appl. 3() (200) [20] A. Moudaf, Projecton methods for a system of nonconvex varatonal nequaltes, Nonlnear Anal. TMA seres A 7 (2009) , 3.2 [2] A. Nagurney, Network Economcs: A Varatonal Inequalty Approach, Kluwer Academc Publshers, Dordrecht, 993., 3.2 [22] M.A. Noor, Projecton methods for nonconvex varatonal nequaltes, Optm. Lett. 3 (2009) [23] M.A. Noor and E. Al-sad, Iteratve methods for general nonconvex varatonal nequaltes, Appl. Comput. Math. 0(2) (20) , 3.2 [24] M.A. Noor and K.I. Noor, New system of general nonconvex varatonal nequaltes, Appl. Math. E-Notes 0 (200) , 3.2 [25] J.-S. Pang, Asymmetrc varatonal nequaltes over product of sets: Applcatons and teratve methods, Math. Prog. 3 (985) [26] R.A. Polqun and R.T. Rockafellar, Prox-regular functons n varatonal analyss, Trans. Amercan Math. Soc. 348 (996) , 2 [27] R.U. Verma, Generalzed system of relaxed cocoercve varatonal nequaltes and projecton methods, J. Optm. Theory Appl. 2() (2004) , 3 [28] D.-J. Wen, Projecton methods for a generalzed system of nonconvex varatonal nequaltes wth dfferent operators, Nonlnear Anal. TMA seres A 73 (200)