with an Application for Solving a System of Variational Inclusions 1

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1 Thai Journal of Mathematics Volume (03) Number : ISSN H(, )-Co-Accretive Mapping with an Application for Solving a System of Variational Inclusions Rais Ahmad,, B.S. Lee and Mohd. Akram Department of Mathematics Aligarh Muslim University, Aligarh-000, India raisain 3@rediffmail.com (R. Ahmad) akramkhan 0@rediffmail.com (Mohd. Akram) Department of Mathematics Kyungsung University, Busan , Republic of Korea bslee@ks.ac.kr Abstract : In this paper, we define a new class of H(, )-co-accretive mappings which are the sum of symmetric cocoercive mappings and symmetric accretive mappings. We prove that the resolvent operator associated with H(, )-coaccretive mapping is single-valued and Lipschitz continuous. Furthermore, we apply these new results to solve a system of variational inclusions in real q-uniformly smooth Banach spaces. Our results are extensions and improvements of some known results existing in the literature. An illustrative example is also given. Keywords : H(, )-co-accretive mapping; system; cocoercive; variational inclusion. 00 Mathematics Subject Classification : 47H09; 49J40; 47H7. The first author is supported by Department of Science and Technology, Government of India under grant no.sr/s4/ms:577/09. Corresponding author. Copyright c 03 by the Mathematical Association of Thailand. All rights reserved.

2 4 Thai J. Math. (03)/ R. Ahmad et al. Introduction Variational inequalities and variational inclusions are interesting and important mathematical problems and have been studied intensively in the recent past, since they have wide applications in mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium and engineering sciences, etc. (see, for example [ 7]). The resolvent operator technique for solving variational inequalities and variational inclusions is interesting and important. For more details of this area, we refer to [8 ]. In 008, Zou and Huang [3] introduced and studied H(, )-accretive mapping and its resolvent operator in Banach spaces. Very recently, Ahmad et al. [4] introduced and studied H(, )-cocoercive mapping and its resolvent operator in real Hilbert spaces. They also gave some examples to illustrate their results. Keeping in view the recent interesting developments of this area, we define a new mapping called H(, )-co-accretive mapping in Banach spaces. We define the resolvent operator associated with the H(, )-co-accretive mapping and prove that it is single-valued and Lipschitz continuous. Finally, we apply these new concepts to solve a system of variational inclusions and an example is given. Preliminaries Let E be a real Banach space with its norm, E be the topological dual of E, d is the metric induced by the norm. Let, be the duality pairing between E and E, CB(E) (respectively, E ) be the family of all nonempty closed and bounded subsets (respectively, all nonempty subsets) of E and D(, ) be the Häusdorff metric on CB(E) defined by D(A, B) = max { sup x A d(x, B), sup y B } d(a, y), where A, B CB(E) and d(x, B) = inf y B d(x, y) and d(a, y) = inf x A d(x, y). Definition. ([5]). For q >, the mapping J q : E E defined by J q (x) = {f E : x, f = x q, x q = f }, x E, is called a generalized duality mapping. In particular, J is the usual normalized duality mapping on E. It is well known that J q (x) = x q J (x), x( 0) E. Also if E X, a real Hilbert space, then J becomes the identity mapping on X. Definition. ([5]). A Banach space E is called smooth if, for every x E with x =, there exists a unique f E such that f = f(x) =. The modulus of smoothness of E is the function ρ E : [0, ) [0, ), defined by { } x + y + x y ρ E (t) = sup : x, y E, x =, y = t.

3 H(, )-Co-Accretive Mapping with an Application for Solving Definition.3 ([5]). A Banach space E is said to be (i) uniformly smooth, if ρ E (t) lim = 0; t 0 t (ii) q-uniformly smooth, for q >, if there exists a constant C > 0 such that ρ E (t) Ct q, t [0, ). It is well known (see e.g., [6]) that { q uniformly smooth, if < q, L q (or l q ) is uniformly smooth, if q. Note that if E is uniformly smooth then J q is single-valued. Xu [5] proved the following important lemma. Lemma.4. Let q > be a real number and E be a smooth Banach space. Then E is q-uniformly smooth if and only if there exists a constant C q > 0 such that for every x, y E, x + y q x q + q y, J q (x) + C q y q. 3 H(, )-Co-Accretive Mapping Throughout the paper, unless otherwise specified, we take E to be a q-uniformly smooth Banach space. First, we recall the following definitions and results. Definition 3. ([3, 4]). A mapping g : E E is said to be (i) accretive, if (ii) strictly accretive, if g(x) g(y),j q (x y) 0, x,y E; g(x) g(y),j q (x y) > 0, x,y E, and the equality holds if and only if x = y; (iii) δ g -strongly accretive, if there exists a constant δ g > 0 such that g(x) g(y),j q (x y) δ g x y q, x,y E; (iv) relaxed-accretive, if there exists a constant β > 0 such that g(x) g(y),j q (x y) ( β) x y q, x,y E;

4 44 Thai J. Math. (03)/ R. Ahmad et al. (v) Lipschitz continuous, if there exists a constant λ g > 0 such that g(x) g(y) λ g x y, x,y E; (vi) η expansive, if there exists a constant η > 0 such that g(x) g(y) η x y, x,y E; if η =, then it is expansive. (vii) cocoercive, if there exists a constant µ > 0 such that g(x) g(y),j q (x y) µ g(x) g(y) q, x,y E; (viii) relaxed-cocoercive, if there exists a constant γ > 0 such that g(x) g(y),j q (x y) ( γ) g(x) g(y) q, x,y E. Definition 3.. A multi-valued mapping G : E CB(E) is said to be D Lipschitz continuous, if for any x,y E, there exists a constant λ DG > 0 such that D(G(x),G(y)) λ DG x y. Definition 3.3 ([4]). Let H : E E E and A,B : E E be mappings. Then (i) H(A, ) is said to be cocoercive with respect to A, if there exists a constant µ > 0 such that H(Ax,u) H(Ay,u),J q (x y) µ Ax Ay q, x,y,u E; (ii) H(,B) is said to be relaxed-cocoercive with respect to B, if there exists a constant γ > 0 such that H(u,Bx) H(u,By),J q (x y) ( γ ) Bx By q, x,y,u E; (iii) H(A, ) is said to be r -Lipschitz continuous with respect to A, if there exists a constant r > 0 such that H(Ax,u) H(Ay,u),J q (x y) r x y, x,y,u E;

5 H(, )-Co-Accretive Mapping with an Application for Solving (iv) H(,B) is said to be r -Lipschitz continuous with respect to B, if there exists a constant r > 0 such that H(u,Bx) H(u,By),J q (x y) r x y, x,y,u E; (v) H(A,B) is said to be symmetric cocoercive with respect to A and B, if H(A, ) is cocoercive with respect to A and H(,B) is relaxedcocoercive with respect to B. Example 3.4. Let E = R, with an inner product defined by (x,x ),(y,y ) = x y x y x y + x y. Let A,B : R R be mappings defined by ( A(x,x ) = 3 x + 3 x, 3 x + ) 3 x, (x,x ) R, ( B(y,y ) = y y, y ) y, (y,y ) R. Let H : R R R be a mapping defined by H(Ax,Bx) = Ax + Bx, x R. Then for any u R, it is easy to verify that and H(Ax,u) H(Ay,u),x y 3 Ax Ay. H(u,Bx) H(u,By),x y ( ) Bx By. Thus H(A,B) is symmetric cocoercive with respect to A and B. Definition 3.5 ([7]). Let f,g : E E be the mappings and M : E E E be a multi-valued mapping. Then (i) M(f, ) is said to be strongly accretive with respect to f, if there exists a constant α > 0 such that u v,j q (x y) α x y q, x,y,w E and u M(f(x),w),v M(f(y),w); (ii) M(, g) is said to be relaxed-accretive with respect to g, if there exists a constant β > 0 such that u v,j q (x y) ( β) x y q, x,y,w E and u M(w,g(x)), v M(w,g(y));

6 46 Thai J. Math. (03)/ R. Ahmad et al. (iii) M(f,g) is said to be symmetric accretive with respect to f and g, if M(f, ) is strongly accretive with respect to f and M(, g) is relaxedaccretive with respect to g. Now we define the following H(, )-co-accretive mapping. Definition 3.6. Let A,B,f,g : E E and H : E E E be singlevalued mappings. Let M : E E E be a multi-valued mapping. The mapping M is said to be H(, )-co-accretive with respect to A,B,f and g, if H(A,B) is symmetric cocoercive with respect to A and B, M(f,g) is symmetric accretive with respect to f and g and (H(A,B)+λM(f,g))(E) = E, for all λ > 0. Definition 3.7. Let A,B,f,g : E E and H : E E E be singlevalued mappings. Let M : E E E be an H(, )-co-accretive mapping with respect to A,B,f and g. The resolvent operator R H(, ) : E E is defined by R H(, ) (u) = [H(A,B) + λm(f,g)] (u), u E and λ > 0. In the rest of the paper, whenever we mention that M is H(, )-coaccretive mapping, we mean that H(A, B) is symmetric cocoercive with respect to A and B with constants µ and γ, respectively and M(f,g) is symmetric accretive with respect to f and g with constants α and β, respectively. Next, we prove that the resolvent operator is single-valued and Lipschitz continuous. Theorem 3.8. Let A,B,f,g : E E and H : E E E be singlevalued mappings. Let M : E E E be an H(, )-co-accretive mapping with respect to A,B,f and g. Let A be η-expansive and B be σ-lipschitz continuous and let α > β, µ > γ and η > σ. Then the resolvent operator is single-valued. R H(, ) (u) = [H(A,B) + λm(f,g)] (u), u E, λ > 0, Proof. For any given u E, let x,y [H(A,B)+λM(f,g)] (u). It follows that H(A(x),B(x)) + u λm(f(x),g(x)), and H(A(y),B(y)) + u λm(f(y),g(y)).

7 H(, )-Co-Accretive Mapping with an Application for Solving Since M is H(, )-co-accretive with respect to A,B,f and g, we have (α β) x y q H(A(x),B(x)) + u ( H(A(y),B(y)) + u),j q (x y) = H(A(x),B(x)) ( H(A(y),B(y))),J q (x y) = H(A(x),B(x)) H(A(y),B(x)),J q (x y) H(A(y),B(x)) H(A(y),B(y)),J q (x y) ( µ) A(x) A(y) q + γ B(x) B(y) q. (3.) Since A is η-expansive and B is σ-lipschitz continuous, thus (3.) becomes which implies that 0 (α β) x y q µη q x y q + γσ q x y q, 0 [(α β) + (µη q γσ q )] x y q 0. Since α > β,µ > γ and η > σ, it follows that x = y and so the resolvent operator defined by [H(A,B)+λM(f,g)] is single-valued. This completes the proof. Theorem 3.9. Let A,B,f,g : E E and H : E E E be single-valued mappings. Suppose M : E E E is an H(, )-co-accretive mapping with respect to A,B,f and g. Let A be η-expansive and B be σ-lipschitz continuous such that α > β,µ > γ and η > σ. Then the resolvent operator R H(, ) : E E is Lipschitz continuous with constant θ, that is, R H(, ) (v) θ u v, u,v E and λ > 0, where θ = λ(α β)+(µη q γσ q ). Proof. Let u,v be any given points in E. It follws that and and so R H(, ) (u) = [H(A,B) + λm(f,g)] (u) R H(, ) (v) = [H(A,B) + λm(f,g)] (v) (u H(A(RH(, ) (u)),b(rh(, ) λ (u)))) M(f(RH(, ) (u)),g(rh(, ) (u))) and

8 48 Thai J. Math. (03)/ R. Ahmad et al. (v H(A(RH(, ) (v)),b(rh(, ) λ (v)))) M(f(RH(, ) (v)),g(rh(, ) (v))). Since M is symmetric accretive with respect to f and g, we have (α β) R H(, ) (u) RH(, ) (v) q (u H(A(RH(, ) (u)),b(rh(, ) λ (u)))) (v H(A(RH(, ) (v)),b(rh(, ) λ (v)))), J q (R H(, ) (v)) λ u v,j q(r H(, ) (v)) λ H(A(RH(, ) (u)),b(rh(, ) (u))) H(A(R H(, ) (v)),b(rh(, ) (u))), J q (R H(, ) (v)) λ H(A(RH(, ) (v)),b(rh(, ) (u))) H(A(R H(, ) (v)),b(rh(, ) (v))), J q (R H(, ) (v)). Since H is symmetric cocoercive with respect to A and B, A is η-expansive and B is σ-lipschitz continuous, we have or (α β) R H(, ) (u) RH(, ) (v) q λ u v,j q(r H(, ) (v)) λ (µηq γσ q ) R H(, ) (v) q, (α β) + λ (µηq γσ q ) R H(, ) (v) q It follows that u v,j q (R H(, ) (v)) λ u v,j q(r H(, ) (v)). [λ(α β) + (µη q γσ q )] R H(, ) (v) q,

9 H(, )-Co-Accretive Mapping with an Application for Solving u v R H(, ) (v) q [λ(α β) + (µη q γσ q )] R H(, ) (v) q. So where R H(, ) (v) θ u v, θ = λ(α β) + (µη q γσ q ). This completes the proof. Remark 3.0. Definitions 3.6, 3.7 and Theorems 3.8, 3.9 generalize the corresponding concepts and results in [3, 6, 8 0]. Now we consider our main results. 4 System of Variational Inclusions In this section, we apply H(, )-co-accretive mapping for solving a system of variational inclusions and we assume that for each i =,,E i are q i -uniformly smooth Banach spaces with norm i. Let A,B,f,g : E E ;A,B,f,g : E E be nonlinear mappings. Let S,H : E E E and T,H : E E E be nonlinear mappings and let M : E E E be an H (, )-co-accretive mapping with respect to A,B,f and g and M : E E E be an H (, )-co-accretive mapping with respect to A,B,f and g. Let P : E E and G : E E be multi-valued mappings. We consider the following system of variational inclusions. Find (x,y) E E,u P(x) and v G(y) such that θ S(x,v) + M (f (x),g (x)); θ T(u,y) + M (f (y),g (y)), (4.) where θ and θ are the zero vectors of E and E, respectively. Note that for suitable choices of mappings A,B,A,B,f,g,f,g,S, H,T,H,M,M,P,G and the spaces E,E, the system of variational inclusions (4.) reduces to various systems of variational inclusions (inequalities) existing in the literature.

10 40 Thai J. Math. (03)/ R. Ahmad et al. Lemma 4.. For any given (x,y) E E,u P(x),v G(y),(x,y,u,v) is a solution of the system of variational inclusions (4.) if and only if (x,y,u,v) satisfies x = R H (, ) λ,m (, ) [H (A,B )(x) λ S(x,v)]; y = R H (, ) λ,m (, ) [H (A,B )(y) λ T(u,y)], where λ,λ are two constants, R H (, ) λ,m (, ) (x) = [H (A,B )+λ M (f,g )] (x) and R H (, ) λ,m (, ) (y) = [H (A,B ) + λ M (f,g )] (y), x E,y E. Proof. Proof is an immediate consequence of definitions of R H (, ) λ,m (, ) and R H (, ) λ,m (, ). Based on Lemma 4., we define the following iterative Algorithm for solving the system of variational inclusions (4.). Algorithm 4.. For any given (x 0,y 0 ) E E,u 0 P(x 0 ),v 0 G(y 0 ), compute (x n,y n ) E E,u n P(x n ) and v n G(y n ) such that x n+ = R H (, ) λ,m (, ) [H (A,B )(x n ) λ S(x n,v n )]; y n+ = R H (, ) λ,m (, ) [H (A,B )(y n ) λ T(u n,y n )]; u n P(x n ), u n u n+ D(P(x n ),P(x n+ )); v n G(y n ), v n v n+ D(G(y n ),G(y n+ )); where n = 0,,,... and λ,λ > 0 are two constants. Theorem 4.3. For each i =,, let E i be q i -uniformly smooth Banach spaces, A i be η i -expansive mappings and B i be σ i -Lipschitz continuous mappings. Let S,H : E E E be mappings such that H is r - Lipschitz continuous with respect to A and r -Lipschitz continuous with respect to B ;S is δ S -strongly accretive in the first argument and λ S,λ S - Lipschitz continuous in the first and second arguments, respectively. Let T,H : E E E be mappings such that H is r 3 -Lipschitz continuous with respect to A and r 4 -Lipschitz continuous with respect to B ;T is δ T - strongly accretive in the second argument and λ T,λ T -Lipschitz continuous in the first and second arguments, respectively. Let P : E CB(E ) is D-Lipschitz continuous with constant λ DP and G : E CB(E ) is D- Lipschitz continuous with constant λ DG. Suppose that M : E E E

11 H(, )-Co-Accretive Mapping with an Application for Solving... 4 be H (, )-co-accretive with respect to A,B,f and g with α > β,µ > γ and η > σ ;M : E E E be H (, )-co-accretive with respect to A,B,f and g with α > β,µ > γ and η > σ. Assume that there exist constants λ,λ > 0 satisfying the following condition: θ q L + θ λ λ T λ DP < ; θ q L + θ λ λ S λ DG <, (4.) where θ = λ (α β ) + (µ η q γ σ q ),θ = λ (α β ) + (µ η q γ σ q ); L = [(r + r ) q λ q δ S + λ q λ S (r + r ) q + λ q λ S + λ q Cq λ q S ]; L = [(r 3 + r 4 ) q λ q δ T + λ q λ T (r 3 + r 4 ) q + λ q λ T + λ q Cq λ q T ]. Then (x,y) E E,u P(x),v G(y) is a solution of the system of variational inclusions (4.) and the sequences {x n }, {y n }, {u n } and {v n } generated by the Algorithm 4. converge strongly to x,y,u and v, respectively. Proof. From Algorithm 4. and Theorem 3.9, we have x n+ x n where = R H (, ) λ,m (, ) [H (A (x n ),B (x n )) λ S(x n,v n )] R H (, ) λ,m (, ) [H (A (x n ),B (x n )) λ S(x n,v n )] θ H (A (x n ),B (x n )) H (A (x n ),B (x n )) λ (S(x n,v n ) S(x n,v n ) + S(x n,v n ) S(x n,v n )) θ H (A (x n ),B (x n )) H (A (x n ),B (x n )) λ (S(x n,v n ) S(x n,v n ) + θ λ S(x n,v n ) S(x n,v n ), θ = λ (α β ) + (µ η q γ σ q ). (4.3) Since H is r -Lipschitz continuous with respect to A and r -Lipschitz continuous with respect to B,S is δ S -strongly accretive in the first argument and λ S -Lipschitz continuous in the first argument and using Lemma

12 4 Thai J. Math. (03)/ R. Ahmad et al..4, we have H (A (x n ), B (x n )) H (A (x n ), B (x n )) λ (S(x n, v n ) S(x n, v n )) q H (A (x n ), B (x n )) H (A (x n ), B (x n )) q λ q S(x n, v n ) S(x n, v n ), J q [H (A (x n ), B (x n )) where H (A (x n ), B (x n ))] + λ q C q S(x n, v n ) S(x n, v n ) q H (A (x n ), B (x n )) H (A (x n ), B (x n )) q λ q S(x n, v n ) S(x n, v n ), J q (x n x n ) λ q S(x n, v n ) S(x n, v n ), J q [H (A (x n ), B (x n )) H (A (x n ), B (x n ))] J q (x n x n ) + λ q C q S(x n, v n ) S(x n, v n ) q (r + r ) q x n x n q λ q δ S x n x n q + λ q S(x n, v n ) S(x n, v n ) [ H (A (x n ), B (x n )) H (A (x n ), B (x n )) q + x n x n q ] + λ q Cq λ q S x n x n q (r + r ) q x n x n q λ q δ S x n x n q + λ q λ S x n x n [(r + r ) q x n x n q + x n x n q ] + λ q Cq λ q S x n x n q = L x n x n q, (4.4) L = [(r + r ) q λ q δ S + λ q λ S (r + r ) q + λ q λ S + λ q Cq λ q S ]. Since S is λ S -Lipschitz continuous in the second argument and G is D- Lipschitz continuous with constant λ DG, by using Algorithm 4., we have S(x n,v n ) S(x n,v n ) λ S v n v n λ S D(G(y n ),G(y n )) λ S λ DG y n y n. (4.5) Due to (4.4) and (4.5), (4.3) becomes x n+ x n θ q L x n x n + θ λ λ S λ DG y n y n. (4.6)

13 H(, )-Co-Accretive Mapping with an Application for Solving Again from Algorithm 4. and Theorem 3.9, we have y n+ y n = R H(, ) λ,m (, ) [H (A (y n ), B (y n )) λ T(u n, y n )] R H(, ) λ,m (, ) [H (A (y n ), B (y n )) λ T(u n, y n )] θ H (A (y n ), B (y n )) H (A (y n ), B (y n )) λ (T(u n, y n ) T(u n, y n ) + T(u n, y n ) T(u n, y n )) θ H (A (y n ), B (y n )) H (A (y n ), B (y n )) λ (T(u n, y n ) T(u n, y n ) + θ λ T(u n, y n ) T(u n, y n ). (4.7) Since H is r 3 -Lipschitz continuous with respect to A and r 4 -Lipschitz continuous with respect to B,T is δ T -strongly accretive in the second argument and λ T -Lipschitz continuous in the second argument, by using Lemma.4, we have H (A (y n ), B (y n )) H (A (y n ), B (y n )) λ (T(u n, y n ) T(u n, y n ) q H (A (y n ), B (y n )) H (A (y n ), B (y n )) q λ q T(u n, y n ) T(u n, y n ), J q [H (A (y n ), B (y n )) where H (A (y n ), B (y n ))] + λ q C q T(u n, y n ) T(u n, y n ) q H (A (y n ), B (y n )) H (A (y n ), B (y n )) q λ q T(u n, y n ) T(u n, y n ), J q (y n y n ) λ q T(u n, y n ) T(u n, y n ), J q [H (A (y n ), B (y n )) H (A (y n ), B (y n ))] J q (y n y n ) + λ q C q T(u n, y n ) T(u n, y n ) q (r 3 + r 4 ) q y n y n q λ q δ T y n y n q + λ q T(u n, y n ) T(u n, y n ) [ H (A (y n ), B (y n )) H (A (y n ), B (y n )) q + y n y n q ] + λ q Cq λ q T y n y n q (r 3 + r 4 ) q y n y n q λ q δ T y n y n q + λ q λ T y n y n [(r 3 + r 4 ) q y n y n q + y n y n q ] + λ q Cq λ q T y n y n q = L y n y n q, (4.8) L = [(r 3 + r 4 ) q λ q δ T + λ q λ T (r 3 + r 4 ) q + λ q λ T + λ q Cq λ q T ]. Since T is λ T -Lipschitz continuous in the first argument and P is D-

14 44 Thai J. Math. (03)/ R. Ahmad et al. Lipschitz continuous with constant λ DP, by using Algorithm 4., we have T(u n,y n ) T(u n,y n ) λ T u n u n λ T D(P(x n ),P(x n )) λ T λ DP x n x n. (4.9) Due to (4.8) and (4.9), (4.7) becomes y n+ y n θ q L y n y n + θ λ λ T λ DP x n x n. (4.0) Combining (4.6) and (4.0), we have x n+ x n + y n+ y n [θ q L + θ λ λ T λ DP ] x n x n where + [θ q L + θ λ λ S λ DG ] y n y n φ(θ)[ x n x n + y n y n ], (4.) φ(θ) = max[θ q L + θ λ λ T λ DP,θ q L + θ λ λ S λ DG ]. From (4.), it follows that 0 < φ(θ) < and so (4.) implies that {x n } and {y n } are both Cauchy sequences. Thus there exist x,y E such that x n x and y n y as n. Now we prove that u n u P(x) and v n v G(y). In fact, it follows from the D-Lipschitz continuities of P,G and Algorithm 4. that and u n+ u n D(P(x n+ ),P(x n )) λ DP x n+ x n (4.) v n+ v n D(G(y n+ ),G(y n )) λ DG y n+ y n. (4.3) From (4.) and (4.3), we know that {u n } and {v n } are also Cauchy sequences in E. Thus there exist u,v E such that u n u and v n v as n. Furthermore, d(u,p(x)) u u n + d(u n,p(x)) u u n + D(P(x n ),P(x)) u u n + λ DP x n x 0, as n,

15 H(, )-Co-Accretive Mapping with an Application for Solving which implies that d(u,p(x)) = 0. Since P(x) CB(E), it follows that u P(x). Similarly, we can show that v G(y). By continuity of H,H,A,A, B,B,M,M,S,T,P,G,R H (, ) λ,m (, ),RH (, ) λ,m (, ) and Algorithm 4., we have x = R H (, ) λ,m (, ) [H (A,B )(x) λ S(x,v)], y = R H (, ) λ,m (, ) [H (A,B )(y) λ T(u,y)]. By Lemma 4., (x,y,u,v) is a solution of problem (4.). This completes the proof. Corollary 4.4. Let P,G be the single-valued, identity mappings, M : E E E be generalized H (, )-accretive mapping, M : E E E be generalized H (, )-accretive mapping and all other conditions are same as in Theorem 4.3, then one can obtain Theorem 6. of Kazmi et al. [7] without uniqueness. Acknowledgement : The authors would like to thank the referees for their comments and suggestions on the manuscript. References [] R.P. Agarwal, Y.J. Cho, N.J. Huang, Sensitivity analysis for strongly nonlinear quasi- variational inclusions, Appl. Math. Lett. 3 (6) (000) 9 4. [] R.P. Agarwal, N.J. Huang, Y.J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings, J. Inequal. Appl. 7 (6) (00) [3] R. Ahmad, A.H. Siddiqi, Zubair Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 63 (005) [4] R. Ahmad, Q.H. Ansari, An iterative algorithm for generalized nonlinear variational inclusions, Appl. Math. Lett. 3 (000) 3 6. [5] S.S. Chang, Y.J. Cho, H.Y. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Sci., New York, 00.

16 46 Thai J. Math. (03)/ R. Ahmad et al. [6] J.Y. Chen, N.C. Wong, J.C. Yao, Algorithm for generalized cocomplementarity problems in Banach spaces, Comput. Math. Appl. 43 () (00) [7] X.P. Ding, C.L. Lou, Perturbed proximal point algorithms for general quasi-variational like inclusions, J. Comput. Appl. Math. 0 (000) [8] H.Y. Lan, Y.J. Cho, R.U. Verma, Nonlinear relaxed cocoercive variational inclusions involving (A, η)-accretive mappings in Banach spaces, Comput. Math. Appl. 5 (006) [9] H.Y. Lan, J.H. Kim, Y.J. Cho, On a new system of nonlinear A- monotone multi-valued variational inclusions, J. Math. Anal. Appl. 37 (007) [0] H.Y. Lan, J.I. Kang, Y.J. Cho, Nonlinear (A, η)-monotone operator inclusion systems involving non-monotone set-valued mappings, Taiwanese J. Math. (007) [] B.S. Lee, M.K. Kang, S.J. Lee, K.H. Yang, Variational inequalities for L-pseudo- monotone maps, Nonlinear Anal. Forum 6 (00) [] M.A. Noor, Generalized set-valued variational inclusions and resolvent equations, J. Math. Anal. Appl. 8 (998) [3] Y.Z. Zou, N.J. Huang, H(, )-accretive operator with an application for solving variational inclusions in Banach spaces, Appl. Math. Comput. 04 (008) [4] R. Ahmad, M. Dilshad, M.M. Wong, J.C. Yao, H(, )-cocoercive operator and an application for solving generalized variational inclusions, Abs. Appl. Anal., Vol. 0 (0), Article ID 6534, pages. [5] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 6 () (99) [6] L.C. Zeng, S.M. Guu, J.C. Yao, Characterization of H-monotone operators with applications to variational inclusions in Banach spaces, Comput. Math. Appl. 50 (005)

17 H(, )-Co-Accretive Mapping with an Application for Solving [7] K.R. Kazmi, F.A. Khan, M. Shahzad, A system of generalized variational inclusions involving generalized H(, )-accretive mapping in real q-uniformly smooth Banach spaces, Appl. Math. Comput. 7 (0) [8] Y.P. Fang, N.J. Huang, H-monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput. 45 (003) [9] Y.P. Fang, N.J. Huang, Iterative algorithm for a system of variational inclusions involving H-accretive operators in Banach spaces, Acta Math. Hungar. 08 (3) (005) [0] X.P. Luo, N.J. Huang, A new class of variational inclusions with B- monotone operators in Banach spaces, J. Comput. Appl. Math. 33 (00) (Accepted 4 July 0) Thai J. Math.