1. Introduction A SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES IN BANACH SPACES

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1 Commun. Optim. Theory 2016 (2016), Article ID 20 Copyright c 2016 Mathematical Research Press. A SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES IN BANACH SPACES JONG KYU KIM 1, SALAHUDDIN 2, 1 Department of Mathematics Education, Kyungnam University, Changwon, , Korea 2 Department of Mathematics, Jazan University, Jazan, KSA Abstract. The aim of this work is to suggest a new system of nonconvex variational inequalities in a q-uniformly smooth Banach space and establish an equivalence relation between this system and fixed point problems. By using the equivalence formulation, we construct a new perturbed projection iterative algorithm with mixed errors for finding a solution set of a system of nonconvex variational inequalities. Also, we prove the convergence theorem of the suggested iterative sequences generated by the algorithm. Keywords. Nonconvex variational inequality; Iterative sequence; Convergence analysis; Mixed error; q-uniformly smooth Banach space. 1. Introduction Variational inequalities were introduced by Stampacchia [1] provided us with a powerful tool to study a wide class of problems arising in mechanics, physics, optimization and control theory, linear programming, economics and engineering sciences, see [2, 3, 4, 5] and the reference therein. In recent years, several authors studied different type of systems of variational inequalities and suggested iterative algorithms to find the approximate solutions of such systems, see [6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and the reference therein. We remark here that the almost Corresponding author addresses: jongkyuk@kyungnam.ac.kr (J.K. Kim), salahuddin12@mailcity.com (Salahuddin) Received August 3,

2 2 JONG KYU KIM, SALAHUDDIN all results concerning the system of solutions of iterative schemes for solving the system of variational inequalities and related problems are being considered in the setting of convex sets. Consequently the techniques are based on the projection of operators over convex sets, which may not hold in general, when the sets are nonconvex. It is known that the unified prox-regular sets are nonconvex and included the convex sets as special cases, see [16, 17, 18, 19, 20] and the reference therein. Inspired by the recent work going on this fields, see for example, [21, 22, 23, 24, 25, 26, 27, 28], we introduced and studied a new system of nonconvex variational inequalities in q- uniformly smooth Banach spaces. We established the equivalence between the system of nonconvex variational inequalities and the fixed point problems. By using the equivalence formulation, we construct a perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system. Also, we prove the convergence of the iterative algorithms under suitable conditions. 2. Preliminaries Let X be a real Banach space with the dual space X and let, be the dual pairing between X and X. Let CB(X) denote the family of all nonempty closed bounded subset of X. The generalized duality mapping J q : X 2 X is defined by J q (x) = { f X : x, f = x q, f = x q 1 }, x X, where q > 1 is a constant. In particular J 2 is a usual normalized duality mapping. It is known that in general J q (x) = x q 2 J 2 (x) for all x 0 and J q is single-valued if X is strictly convex. In the sequel, we always assume that X is a real Banach space such that J q is a single valued. If X is a Hilbert space then J q becomes the identity mapping on X. The modulus of smoothness of X is the function ρ X : [0, ) [0, ) is defined by ρ X (t) = sup{ 1 ( x + y + x y ) 1 : x 1, y t}. 2

3 A SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES IN BANACH SPACES 3 A Banach space X is called uniformly smooth if ρ X (t) lim = 0. t 0 t X is called q-uniformly smooth (q > 1) if there exists a constant c > 0 such that ρ X (t) < ct q, t > 0. Note that J q is a single-valued if X is uniformly smooth. Concerned with the characteristic inequalities in q-uniformly smooth Banach spaces, Xu [29] proved the following results. Lemma 2.1. [29] The real Banach space X is q-uniformly smooth if and only if there exists a constant c q > 0 such that for all x,y X x + y q x q + q y,j q (x) + c q y q. For a nonempty closed subset K of X, we denote by d K ( ) or d(,k) the usual distance function to K, that is, d K (u) = inf u v. v K Definition 2.2. The proximal normal cone of K at a point u X K is given by N P K(u) = {ζ X : u P K (u + αζ ) for some α > 0}, where P K (u) is the set of all projection of u onto K, that is, P K (u) = {v K : d K (u) = u v }. Lemma 2.3. [3] Let K be a nonempty closed subset of X. Then ζ NK P (u) if and only if there exists a constant α = α(ζ,u) > 0 such that ζ,v u α v u 2, v K. Lemma 2.4. [3] Let K be a nonempty closed and convex subset in X. Then ζ NK P (u) if and only if ζ,v u 0, v K.

4 4 JONG KYU KIM, SALAHUDDIN Definition 2.5. Let f : X R be a locally Lipschitz continuous mapping with constant τ near a given point x X, that is, for some ε > 0, one has f (y) f (z) τ y z, y,z B(x;ε), where B(x;ε) denotes the open ball of radius r > 0 and centered at x. The generalized directional derivative of f at x in the direction v, denoted by f o (x;v) is defined as follows f o (x;v) = limsup y x t 0 f (y +tv) f (y) t where y is a vector in X and t is a positive scalar. Definition 2.6. The tangent cone T K (x) to K at a point x K is defined as follows T K (x) = {v X : d o K(x;v) = 0}. The normal cone of K at x by polarity with T K (x) is defined as follows N K (x) = {ζ : ζ,v 0, v T K (x)}. The Clarke normal cone NK C (x) is given by N C K(x) = co [ N P K(x) ], where co(s) mean the closure of the convex hull of S. It is clear that NK P(x) NC K (x). The converse is not true in general. Note that NC K (x) is always closed and convex, where as NK P (x) is always convex but may not be closed, see [3, 17, 30, 31, 32]. Definition 2.7. [30] For any r (0,+ ], a subset K r of X is called the normalized uniformly prox-regular (or uniformly r-prox-regular) if every nonzero proximal normal to K r can be realized by an r-ball, that is, for all x K r and all 0 ζ N P K r (x) with ζ = 1, ζ,x x 1 2r x x 2, x K r. Lemma 2.8. [3] A closed set K X is convex if and only if it is proximally smooth of radius r for every r > 0.

5 A SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES IN BANACH SPACES 5 Proposition 2.9. [32], Let r > 0 and let K r be a nonempty closed and uniformly r-prox-regular subset of X. Set Then the following statements are hold: (a) For all x U (r),p Kr (x) /0; U (r) = {u X : 0 d Kr (u) < r}. (b) For all r (0,r), P Kr is a Lipschitz continuous mapping with constant U (r ) = {u X : 0 d Kr (u) < r }; (c) The proximal normal cone is closed as a set-valued mapping. r r r on From Proposition 2.9 (c) we have N C K r (x) = N P K r (x). Therefore we define N Kr (x) = N C K r (x) = N P K r (x) for such a class of sets. Definition Let X be a q-uniformly smooth Banach space. A single-valued mapping p : X X is called (i) accretive if p(x) p(y), j q (x y) 0, x,y X, (ii) β-strongly accretive if there exists a constant β > 0 such that p(x) p(y), j q (x y) β x y q, x,y X, (iii) relaxed ρ-strongly accretive if there exists a constant ρ > 0 such that p(x) p(y), j q (x y) ρ x y q, x,y X, (iv) σ-lipschitz continuous if there exists a constant σ > 0 such that p(x) p(y) σ x y, x,y X. Definition Let p : X X be a single-valued mapping and let T : X X 2 X be a set valued mapping. Then T is said to be (i) accretive if u v, j q (x y) 0, x,x,y,y X,u T (x,y),v T (x,y ),

6 6 JONG KYU KIM, SALAHUDDIN (ii) (κ,υ)-relaxed cocoercive with respect to p if there exists a constant (κ,υ) > 0 such that for all x,x,y,y X u v, j q (p(x) p(y)) κ u v q + υ p(x) p(y) q, u T (x,y),v T (x,y ), (iii) ξ D-Lipschitz continuous in the first variable if there exists a constant ξ > 0 such that for all x,x X D(T (x,y),t (x,y )) ξ x x, y,y X, where D is the Hausdorff pseudo-metric, that is, for any two nonempty subsets A and B of X D(A,B) = max { supd(x,b),supd(y,a) }. x A y B 3. A system of nonconvex variational inequalities In this section, we introduce a new system of nonconvex variational inequalities in a Banach space and investigated their relations. Let T i : X X CB(X) be the nonlinear set valued mappings and let g i,h i : X X be the nonlinear single valued mappings such that K r g i (X) for i = 1,2,3. For any constants η i > 0 (i = 1,2,3), we consider the problem of finding x,y,z X and u T 1 (y,x),v T 2 (z,y),w T 3 (x,z) such that h 1 (x),h 2 (y),h 3 (z) K r,g 1 (x ),g 2 (x ),g 3 (x ) K r and η 1 u + h 1 (x) g 1 (y),g 1 (x ) h 1 (x) + 1 2r g 1(x ) h 1 (x) 2 0, x X, η 2 v + h 2 (y) g 2 (z),g 2 (x ) h 2 (y) + 1 2r g 2(x ) h 2 (y) 2 0, x X, η 3 w + h 3 (z) g 3 (x),g 3 (x ) h 3 (z) + 1 2r g 3(x ) h 3 (z) 2 0, x X. (3.1) Problem (3.1) is called a system of nonconvex variational inequalities. Lemma 3.1. Let K r be a uniformly r-prox-regular set. Then problem (3.1) is equivalent to finding x,y,z X and u T 1 (y,x),v T 2 (z,y),w T 3 (x,z) such that h 1 (x),h 2 (y),h 3 (z) K r and 0 η 1 u + h 1 (x) g 1 (y) + NK P r (h 1 (x)), 0 η 2 v + h 2 (y) g 2 (z) + NK P r (h 2 (y)),

7 A SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES IN BANACH SPACES 7 0 η 3 w + h 3 (z) g 3 (x) + NK P r (h 3 (z)), (3.2) where NK P r (s) is the P-normal cone of K r at s in the sense of nonconvex analysis. Proof Let (x,y,z,u,v,w) with x,y,z X,h 1 (x),h 2 (y),h 3 (z) K r and u T 1 (y,x),v T 2 (z,y),w T 3 (x,z) be a solution set of system (3.1). Since the vector zero always belongs to any normal cone, if η 1 u + h 1 (x) g 1 (y) = 0, then 0 η 1 u + h 1 (x) g 1 (y) + NK P r (h 1 (x)). If η 1 u + h 1 (x) g 1 (y) 0, then for all x X with g 1 (x ) K r (η 1 u + h 1 (x) g 1 (y)),g 1 (x ) h 1 (x) 1 2r g 1(x ) h 1 (x) 2. (3.3) From Lemma 2.3, we obtain (η 1 u + h 1 (x) g 1 (y)) NK P r (h 1 (x)). Hence, we have 0 η 1 u + h 1 (x) g 1 (y) + NK P r (h 1 (x)). (3.4) Similarly, we have 0 η 2 v + h 2 (y) g 2 (z) + NK P r (h 2 (y)) (3.5) and 0 η 3 w + h 3 (z) g 3 (x) + NK P r (h 3 (z)). (3.6) Conversely, if (x,y,z,u,v,w) with x,y,z X,h 1 (x),h 2 (y),h 3 (z) K r and u T 1 (y,x),v T 2 (z,y),w T 3 (x,z) is a solution set of system (3.2), then from Definition 2.7, (x,y,z,u,v,w) with x,y,z X and u T 1 (y,x),v T 2 (z,y),w T 3 (x,z) and h 1 (x),h 2 (y),h 3 (z) K r is a solution set of system (3.1). The problem (3.2) is called a system of nonconvex variational inclusions.

8 8 JONG KYU KIM, SALAHUDDIN 4. Main results Lemma 4.1. For i = 1,2,3, let T i,g i,h i,η i be the same as in system (3.1). Then (x,y,z,u,v,w) with x,y,z X,h 1 (x),h 2 (y),h 3 (z) K r and u T 1 (y,x),v T 2 (z,y),w T 3 (x,z) is a solution set of system (3.1) if and only if h 1 (x) = P Kr [g 1 (y) η 1 u], h 2 (y) = P Kr [g 2 (z) η 2 v], h 3 (z) = P Kr [g 3 (x) η 3 w], (4.1) where P Kr is the projection of X onto the uniformly r-prox-regular set K r. Proof Let (x,y,z,u,v,w) with x,y,z X,h 1 (x),h 2 (y),h 3 (z) K r and u T 1 (y,x), v T 2 (z,y), w T 3 (x,z) be a solution set of system (3.1). From Lemma 3.1, we have 0 η 1 u + h 1 (x) g 1 (y) + NK P r (h 1 (x)), 0 η 2 v + h 2 (y) g 2 (z) + NK P r (h 2 (y)), 0 η 3 w + h 3 (z) g 3 (x) + NK P r (h 3 (z)), (4.2) which implies that g 1 (y) η 1 u (I + NK P r )(h 1 (x)), g 2 (z) η 2 v (I + NK P r )(h 2 (y)), g 3 (x) η 3 w (I + NK P r )(h 3 (z)). (4.3) Therefore, we have h 1 (x) = P Kr [g 1 (y) η 1 u], h 2 (y) = P Kr [g 2 (z) η 2 v], h 3 (z) = P Kr [g 3 (x) η 3 w], (4.4) where I is an identity mapping and P Kr = (I + NK P r ) 1. The converse part of this lemma is also trivial. This completes the proof.

9 A SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES IN BANACH SPACES 9 Remark 4.2. Inequality (4.1) can be written as follows: where η i > 0,i = 1,2,3 are constants. p = g 1 (y) η 1 u, h 1 (x) = P Kr (p), q = g 2 (z) η 2 v, h 2 (y) = P Kr (q), t = g 3 (x) η 3 w, h 3 (z) = P Kr (t), (4.5) The fixed point formulation (4.5) enables us to construct the following perturbed iterative algorithm with mixed errors. Algorithm 4.3. Let T i,g i,h i,η i > 0,i = 1,2,3 be the same as in system (3.1) such that h i : X X is an onto operator for each i = 1,2,3. For arbitrary chosen initial points (p 0,q 0,t 0 ) X X X, compute the iterative sequences {(x n,y n,z n,u n,v n,w n )} n=0 by using and h 1 (x n ) = P Kr (p n ), p n+1 = (1 α n )p n + α n (g 1 (y n ) η 1 u n + e n ) + r n, h 2 (y n ) = P Kr (q n ), q n+1 = (1 α n )q n + α n (g 2 (z n ) η 2 v n + c n ) + s n, h 3 (z n ) = P Kr (t n ), t n+1 = (1 α n )t n + α n (g 3 (x n ) η 3 w n + d n ) + l n, (4.6) u n T 1 (y n,x n ); u n u n+1 (1 + (1 + n) 1 ) D(T 1 (y n,x n ),T 1 (y n+1,x n+1 )), v n T 2 (z n,y n ); v n v n+1 (1 + (1 + n) 1 ) D(T 2 (z n,y n ),T 2 (z n+1,y n+1 )), w n T 3 (x n,z n ); w n w n+1 (1 + (1 + n) 1 ) D(T 3 (x n,z n ),T 3 (x n+1,z n+1 )), where initial points u 0 T 1 (y 0,x 0 ),v 0 T 2 (z 0,y 0 ),w 0 T 3 (x 0,z 0 ) are chosen arbitrary, 0 < α n 1 is a parameter and {e n } n=0,{c n} n=0,{d n} n=0,{r n} n=0,{s n} n=0 and {l n} n=0 are six sequences in X to take into account of a possible inexact computation of the resolvent operator satisfying the following conditions: n=1 lim e n = lim c n = lim d n = lim r n = lim s n = lim l n = 0, n n n n n n e n e n 1 <, r n r n 1 <, n=1 n=1 c n c n 1 <, s n s n 1 <, n=1 n=1 d n d n 1 <, l n l n 1 <. (4.7) n=1

10 10 JONG KYU KIM, SALAHUDDIN Theorem 4.4. Let T i,g i,h i,η i,i = 1,2,3 be the same as in system (3.1) such that (i) T i is a relaxed (κ i,υ i )-cocoercive with constant κ i,υ i > 0 and ξ i D-Lipschitz continuous mapping in the first variable, (ii) h i is a β i -strongly accretive and σ i -Lipschitz continuous mapping, (iii) g i is a µ i -Lipschitz continuous and relaxed ρ i -strongly accretive mapping. If the constants η i > 0 satisfying the following conditions: r(ω 1 + ϕ 1 ) (r r )(1 π 2 ) < 1, r(ω 2 + ϕ 2 ) (r r )(1 π 3 ) < 1, r(ω 3 + ϕ 3 ) (r r < 1, (4.8) )(1 π 1 ) for each i = 1,2,3 Ω 1 + ϕ 1 1 π 2 < 1, Ω 2 + ϕ 2 1 π 3 < 1, Ω 3 + ϕ 3 1 π 1 < 1, ϕ i = q 1 qη i (κ i ξ q i υ i ) + c q η q i ξ q i, Ω i = 1 q + qρ i + c q µ q i π i = q 1 qβ i + c q σ q i < 1, where r (0,r), then there exists x,y,z X with h 1 (x ),h 2 (y ),h 3 (z ) K r and u T 1 (y,x ), v T 2 (z,y ), w T 3 (x,z ) such that (x,y,z,u,v,w ) is a solution of (3.1) and sequences {(x n,y n,z n,u n,v n,w n )} n=0 generated by Algorithm 4.3 converges strongly to (x,y,z,u,v,w ), respectively. Proof. From (4.6), we have p n+1 p n (1 α n ) p n p n 1 + α n ( y n y n 1 (g 1 (y n ) g 1 (y n 1 )) + y n y n 1 η 1 (u n u n 1 ) + e n e n 1 ) + r n r n 1. (4.9) Since g 1 is a relaxed ρ 1 -strongly accretive and µ 1 -Lipschitz continuous mapping, we have y n y n 1 (g 1 (y n ) g 1 (y n 1 )) q = y n y n 1 q q g 1 (y n ) g 1 (y n 1 ), j q (y n y n 1 ) + c q g 1 (y n ) g 1 (y n 1 ) q y n y n 1 q + qρ 1 y n y n 1 q + µ q 1 c q y n y n 1 q (1 + qρ 1 + µ q 1 c q) y n y n 1 q.

11 A SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES IN BANACH SPACES 11 Therefore, we have y n y n 1 (g 1 (y n ) g 1 (y n 1 )) 1 q + qρ 1 + c q µ q 1 y n y n 1. (4.10) Since T 1 is a ξ 1 D-Lipschitz continuous mapping with constant ξ 1 > 0, we have u n u n 1 ξ 1 (1 + n 1 ) D(T 1 (y n,x n ),T 1 (y n 1,x n 1 )) ξ 1 (1 + n 1 ) y n y n 1. (4.11) Again T 1 is a relaxed (κ 1,υ 1 )-cocoercive and ξ 1 D-Lipschitz continuous mapping in the first variable, we get y n y n 1 η 1 (u n u n 1 ) q y n y n 1 q qη 1 u n u n 1, j q (y n y n 1 ) + c q η q 1 u n u n 1 q y n y n 1 q qη 1 ( κ 1 u n u n 1 q + υ 1 y n y n 1 q ) + c q η q 1 ξ q 1 (1 + n 1 ) q y n y n 1 q y n y n 1 q + (qη 1 κ 1 ξ q 1 (1 + n 1 ) q qη 1 υ 1 ) y n y n 1 q + c q η q 1 ξ q 1 (1 + n 1 ) q y n y n 1 q (1 qη 1 (κ 1 ξ q 1 (1 + n 1 ) q υ 1 ) + c q η q 1 ξ q 1 (1 + n 1 ) q ) y n y n 1 q. Therefore, we have y n y n 1 η 1 (u n u n 1 ) q 1 qη 1 (κ 1 ξ q 1 (1 + n 1 ) q υ 1 ) + c q η q 1 ξ q 1 (1 + n 1 ) q y n y n 1. (4.12) From (4.9), (4.10) and (4.12), we get ( q p n+1 p n (1 α n ) p n p n 1 + α n 1 + qρ 1 + c q µ q 1 + q 1 qη 1 (κ 1 ξ q 1 (1 + n 1 ) q υ 1 ) + c q η q 1 ξ q 1 (1 + n 1 ) q ) y n y n 1 Using the similar computation from (4.9)-(4.13), ( we have q q n+1 q n (1 α n ) q n q n 1 + α n 1 + qρ 2 + c q µ q 2 +α n e n e n 1 + r n r n 1. (4.13) + q 1 qη 2 (κ 2 ξ q 2 (1 + n 1 ) q υ 2 ) + c q η q 2 ξ q 2 (1 + n 1 ) q ) z n z n 1

12 12 JONG KYU KIM, SALAHUDDIN +α n c n c n 1 + s n s n 1. (4.14) Again using the similar computation from ( (4.9)-(4.14), we have q t n+1 t n (1 α n ) t n t n 1 + α n 1 + qρ 3 + c q µ q 3 + q 1 qη 3 (κ 3 ξ q 3 (1 + n 1 ) q υ 3 ) + c q η q 3 ξ q 3 (1 + n 1 ) q ) x n x n 1 +α n d n d n 1 + l n l n 1. (4.15) On the other hand, by using (4.6) and Proposition 2.9, we find that x n x n 1 x n x n 1 (h 1 (x n ) h 1 (x n 1 )) + h 1 (x n ) h 1 (x n 1 ) x n x n 1 (h 1 (x n ) h 1 (x n 1 )) + P Kr (p n ) P Kr (p n 1 ) x n x n 1 (h 1 (x n ) h 1 (x n 1 )) + r r r p n p n 1. Since h 1 is β 1 -strongly accretive and σ 1 -Lipschitz continuous, we have (4.16) x n x n 1 (h 1 (x n ) h 1 (x n 1 )) q = x n x n 1 q q h 1 (x n ) h 1 (x n 1 ), j q (x n x n 1 ) + c q h 1 (x n ) h 1 (x n 1 ) q x n x n 1 q qβ 1 x n x n 1 q + c q σ q 1 x n x n 1 q (1 qβ 1 + c q σ q 1 ) x n x n 1 q. Therefore, we have x n x n 1 (h 1 (x n ) h 1 (x n 1 )) q 1 qβ 1 + c q σ q 1 x n x n 1. (4.17) Substituting (4.17) in (4.16), we obtain x n x n 1 q 1 qβ 1 + c q σ q 1 x n x n 1 + r r r p n p n 1, (4.18) which implies that Similarly, we have x n x n 1 y n y n 1 r (r r )(1 q 1 qβ 1 + c q σ q 1 ) p n p n 1. (4.19) r (r r )(1 q 1 qβ 2 + c q σ q 2 ) q n q n 1, (4.20)

13 and A SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES IN BANACH SPACES 13 z n z n 1 It follows from (4.13) and (4.20) that where r (r r )(1 q 1 qβ 3 + c q σ q 3 ) t n t n 1. (4.21) p n+1 p n (1 α n ) p n p n 1 + α n r(ω 1 + ϕ 1 ) (r r )(1 π 2 ) q n q n 1 +α n e n e n 1 + r n r n 1, (4.22) ϕ 1 = q 1 qη 1 (κ 1 ξ q 1 (1 + n 1 ) q υ 1 ) + c q η q 1 ξ q 1 (1 + n 1 ) q, and From (4.14) and (4.21), we obtain where Ω 1 = q 1 + qρ 1 + c q µ q 1 π 2 = q 1 qβ 2 + c q σ q 2. q n+1 q n (1 α n ) q n q n 1 + α n r(ω 2 + ϕ 2 ) (r r )(1 π 3 ) t n t n 1 +α n c n c n 1 + s n s n 1, (4.23) ϕ 2 = q 1 qη 2 (κ 2 ξ q 2 (1 + n 1 ) q υ 2 ) + c q η q 2 ξ q 2 (1 + n 1 ) q, and Again from (4.15) and (4.19), we obtain where Ω 2 = q 1 + qρ 2 + c q µ q 2 π 3 = q 1 qβ 3 + c q σ q 3. t n+1 t n (1 α n ) t n t n 1 + α n r(ω 3 + ϕ 3 ) (r r )(1 π 1 ) p n p n 1 +α n d n d n 1 + l n l n 1, (4.24) ϕ 3 = q 1 qη 3 (κ 3 ξ q 3 (1 + n 1 ) q υ 3 ) + c q η q 3 ξ q 3 (1 + n 1 ) q, Ω 3 = q 1 + qρ 3 + c q µ q 3

14 14 JONG KYU KIM, SALAHUDDIN and π 1 = 1 q qβ 1 + c q σ q 1. Now we define on X X X by (x,y,z) = x + y + z, (x,y,z) X X X. It is obvious that (X X X, ) is a Banach space. Applying (4.22)-(4.24), we have (p n+1,q n+1,t n+1 ) (p n,q n,t n ) = (1 α n ) (p n,q n,t n ) (p n 1,q n 1,t n 1 ) + α n χ(n) (p n,q n,t n ) (p n 1,q n 1,t n 1 ) where +α n (e n,c n,d n ) (e n 1,c n 1,d n 1 ) + (r n,s n,l n ) (r n 1,s n 1,l n 1 ), (4.25) { } r(ω1 + ϕ 1 ) χ(n) = max (r r )(1 π 2 ), r(ω 2 + ϕ 2 ) (r r )(1 π 3 ), r(ω 3 + ϕ 3 ) (r r. (4.26) )(1 π 1 ) Let χ(n) χ as n, where { } r(ω1 + ϕ 1 ) χ = max (r r )(1 π 2 ), r(ω 2 + ϕ 2 ) (r r )(1 π 3 ), r(ω 3 + ϕ 3 ) (r r. (4.27) )(1 π 1 ) From condition (4.8), we know that 0 χ < 1. Then for χ = 1 2 (χ + 1) (χ,1) there exists n 0 1 such that χ(n) = χ for each n n 0. Thus it follows from (4.25) that for each n n 0, (p n+1,q n+1,t n+1 ) (p n,q n,t n ) = (1 α n ) (p n,q n,t n ) (p n 1,q n 1,t n 1 ) + α n χ (p n,q n,t n ) (p n 1,q n 1,t n 1 ) +α n (e n,c n,d n ) (e n 1,c n 1,d n 1 ) + (r n,s n,l n ) (r n 1,s n 1,l n 1 ) (1 α n (1 χ)) [ (1 α n (1 χ)) (p n 1,q n 1,t n 1 ) (p n 2,q n 2,t n 2 ) +α n (e n 1,c n 1,d n 1 ) (e n 2,c n 2,d n 2 ) + (r n 1,s n 1,l n 1 ) (r n 2,s n 2,l n 2 ) ] +α n (e n,c n,d n ) (e n 1,c n 1,d n 1 ) + (r n,s n,l n ) (r n 1,s n 1,l n 1 ) = (1 α n (1 χ)) 2 (p n 1,q n 1,t n 1 ) (p n 2,q n 2,t n 2 ) +α n [ (1 αn (1 χ)) (e n 1,c n 1,d n 1 ) (e n 2,c n 2,d n 2 ) + (e n,c n,d n ) (e n 1,c n 1,d n 1 ) ]

15 A SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES IN BANACH SPACES 15 + (1 α n (1 χ)) (r n 1,s n 1,l n 1 ) (r n 2,s n 2,l n 2 ) + (r n,s n,l n ) (r n 1,s n 1,l n 1 ). (1 α n (1 χ)) n n 0 (p n0 +1,q n0 +1,t n0 +1) (p n0,q n0,t n0 ) n n 0 + α n (1 α n (1 χ)) i=1 (e n (i 1),c n (i 1),d n (i 1) ) (e n i,c n i,d n i ) i=1 n n 0 + i=1 (1 α n (1 χ)) i 1 (r n (i 1),s n (i 1),l n (i 1) ) (r n i,s n i,l n i ). (4.28) Hence for any m n > n 0, we have (p m,q m,t m ) (p n,q n,t n ) m 1 j=n m 1 (p j+1,q j+1,t j+1 ) (p j,q j,t j ) j=n (1 α n (1 χ)) j n 0 (p n0 +1,q n0 +1,t n0 +1) (p n0,q n0,t n0 ) m 1 j n 0 + α n j=n i=1 (1 α n (1 χ)) i=1 (e n (i 1),c n (i 1),d n (i 1) ) (e n i,c n i,d n i ) m 1 j n 0 + j=n i=1 (1 α n (1 χ)) i=1 (r n (i 1),s n (i 1),l n (i 1) ) (r n i,s n i,l n i ). (4.29) Since (1 α n (1 χ)) (0,1), it follows from (4.7) and (4.29) that (p m,q m,t m ) (p n,q n,t n ) = p m p n + q m q n + t m t n 0 as n. Hence {p n },{q n } and {t n } are Cauchy sequences in X and so there exists p,q and t X such that p n p,q n q and t n t as n. By the inequalities (4.19), (4.20), (4.21), it follows that the sequences {x n },{y n } and {z n } are Cauchy sequences in X. Thus there exists x,y,z X such that x n x,y n y and z n z as n. Since for each i = 1,2,3,T i is a

16 16 JONG KYU KIM, SALAHUDDIN ξ i D-Lipschitz continuous mapping in the first variable, therefore it follows from (4.6) that u n u n+1 (1 + (1 + n) 1 ) D(T 1 (y n,x n ),T 1 (y n+1,x n+1 )) (1 + (1 + n 1 ))ξ 1 y n y n+1 0, as n, v n v n+1 (1 + (1 + n) 1 ) D(T 2 (z n,y n ),T 2 (z n+1,y n+1 )) (1 + (1 + n 1 ))ξ 2 z n z n+1 0, as n, (4.30) w n w n+1 (1 + (1 + n) 1 ) D(T 3 (x n,z n ),T 3 (x n+1,z n+1 )) (1 + (1 + n 1 ))ξ 3 x n x n+1 0, as n. Hence {u n },{v n },{w n } are Cauchy sequences in X and so there exists u,v,w X such that x n x,y n y and z n z as n. Further u n T 1 (y n,x n ) we have d(u,t 1 (y,x )) = inf{ u ς : ς T 1 (y,x )} u u n + d(u n,t 1 (y,x )) u u n + (1 + n 1 ) D(T 1 (y n,x n ),T 1 (y,x )) (4.31) u u n + ξ 1 y n y 0 as n. Hence d(u,t 1 (y,x )) = 0, therefore u n u T 1 (y,x ) X. Since v n T 2 (z n,y n ), by the similar proof of (4.31), we obtain d(v,t 2 (z,y )) v v n + ξ 2 z n z 0 as n. (4.32) Again, since w n T 3 (x n,z n ), we obtain d(w,t 3 (x,z )) w w n + ξ 3 x n x 0 as n. (4.33) The right side of inequalities (4.31), (4.32) and (4.33) tend to zero as n. Hence u T 1 (y,x ),v T 2 (z,y ),w T 3 (x,z ). Since the operators g 1,g 2 and g 3 are continuous, it follows from (4.6) and (4.7) that p = g 1 (y ) η 1 u, q = g 2 (z ) η 2 v, t = g 3 (x ) η 3 w. (4.34)

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