(1)http://jipam.vu.edu.au/ Volume 7, Issue 2, Article 66, 2006 GENERALIZED CO-COMPLEMENTARITY PROBLEMS INp-UNIFORMLY SMOOTH BANACH SPACES M

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http://jipam.vu.edu.au/

Volume 7, Issue 2, Article 66, 2006

GENERALIZED CO-COMPLEMENTARITY PROBLEMS INp-UNIFORMLY SMOOTH BANACH SPACES

M. FIRDOSH KHAN AND SALAHUDDIN DEPARTMENT OFMATHEMATICS

ALIGARHMUSLIMUNIVERSITY

ALIGARH-202002, INDIA

khan_mfk@yahoo.com

Received 14 October, 2005; accepted 30 December, 2005 Communicated by R.U. Verma

ABSTRACT. The objective of this paper is to study the iterative solutions of a class of generalized co-complementarity problems inp-uniformly smooth Banach spaces, with the devotion of sunny retraction mapping,p-strongly accretive,p-relaxed accretive and Lipschitzian (or more generally uniformly continuous) mappings. Our results are new and represents a significant improvement of previously known results. Some special cases are also discussed.

Key words and phrases: Generalized co-complementarity problems, Iterative algorithm, Sunny retraction, Sunny nonexpan- sive mapping,p-strongly accretive,p-relaxed accretive mapping, Lipschitzian mapping, Hausdorff metric,p-uniformly smooth Banach spaces.

2000 Mathematics Subject Classification. 49J40, 90C33, 47H10.

1. INTRODUCTION

The theory of complementarity problems initiated by Lemke [19] and Cottle and Dantzing [10] in the early sixties and later developed by other mathematicians see for example [6, 9, 11, 14, 17, 22] plays an important role and is fundamental in the study of a wide class of problems arising in optimization, game theory, economics and engineering sciences [3, 6, 8, 11, 15].

On the other hand, the accretive operators are of interest because several physically resolvent problems can be modeled by nonlinear evolution systems involving operators of the accretive type. Very closely related to the accretive operators is the class of dissipative operators, where an operatorT is said to be dissipative if and only if(−T)is accretive. The concepts of strictly strongly andm-(or sometimes hyper-) dissipativity are similarly defined.

These classes of operators have attracted a lot of interest because of their involvement in evolution systems modeling several real life problems. Consequently several authors have studied the existence, uniqueness and iterative approximations of solutions of nonlinear equations involving such operators, see [5, 12, 18] and the references cited therein.

ISSN (electronic): 1443-5756

011-06

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It is our purpose in this article to establish the strong convergence of the iterative algorithm to a solution of the generalized co-complimentarity problems in p−uniformly smooth Banach spaces when the operators are accretive, strictly accretive, strongly accretive, relaxed accretive and Lipschitzian. Our iteration processes are simple and independent of the geometry of E and iteration parameters can be chosen at the start of the iteration process. Consequently, most important results known in this connection will be special cases of our problem.

2. BACKGROUND OFPROBLEMFORMULATIONS

Throughout this article, we assume thatE is a real Banach space whose norm is denoted by k · k, E^? its topological dual space. CB(E) denotes the family of all nonempty closed and bounded subsets ofE. D(·,·)is the Hausdorff metric onCB(E)defined by

max

sup

x∈A

d(x, B), sup

y∈B

d(A, y)

=D(A, B), where

d(x, B) = inf

y∈Bd(x, y) and d(A, y) = inf

x∈Ad(x, y),

dis the metric onE induced by the normk · k. As usual,h·,·iis the generalized duality pairing betweenE andE^?. For1< p <∞, the mappingJ_p :E →2^E^? defined by

J_p(x) = {f^? ∈E^? :hx, f^?i=kfk · kxkandkfk=kxk^p−1} for allx∈E,

is called the duality mapping with gauge function φ(t) = t^p−1. In particular for p = 2, the duality mappingJ₂with gauge functionφ(t) =tis called the normalized duality mapping. It is known thatJ_p(x) =kxk^p−2J₂(x)for allx 6= 0andJ_p is single valued ifE^? is strictly convex.

IfE =H is a Hilbert space, thenJ₂becomes the identity mapping onH.

Proposition 2.1 ([7]). LetE be a real Banach space. For1 < p < ∞, the duality mapping J_p :E →2^E^? has the following basic properties:

(1) J_p(x)6=∅for allx∈EandD(J_p)(the domain ofJ_p)=E, (2) J_p(x) =kxk^p−2J₂(x)for allx∈E,(x6= 0),

(3) Jp(αx) = α^p−1Jp(x)for allα∈[0,∞), (4) Jp(−x) =−Jp(x),

(5) J_p is bounded i.e., for any bounded subsetA⊂E,J_p(A)is a bounded subset inE^?, (6) J_p can be equivalently defined as the subdifferential of the functionalϕ(x) = p⁻¹kxk^p

(Asplund [2]), i.e.,

J_p(x) =∂ϕ(x) ={f ∈E^? : ϕ(y)−ϕ(x)≥(f, y−x), for ally∈E}

(7) E is a uniformly smooth Banach space (equivalently,E^? is a uniformly convex Banach space) if and only ifJpis single valued and uniformly continuous on any bounded subset ofE(Xu and Roach [24]).

Definition 2.1. LetEbe a real Banach space andK a nonempty subset ofE. LetT :K →2^E be a multivalued mapping

(1) T is said to be accretive if for any x, y ∈ K, u ∈ T(x) and v ∈ T(y) there exists j₂ ∈J₂(x−y)such that

hu−v, j₂i ≥0,

or equivalently, there existsj_p ∈J_p(x−y),1< p <∞, such that

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(2) T is said to be strongly accretive if for anyx, y ∈ K, u ∈ T(x) and v ∈ T(y)there existsj₂ ∈J₂(x−y)such that

hu−v, j₂i ≥kkx−yk²,

or equivalently, there existsj_p ∈J_p(x−y),1< p <∞such that hu−v, j_pi ≥kkx−yk^p,

for some constantk > 0.

The concept of a single-valued accretive mapping was introduced independently by Browder [5] and Kato [18] in 1967. An early fundamental result in the theory of accretive mappings which is due to Browder states that the following initial value problem,

du(t)

dt +T u(t) = 0, u(0) =u0, is solvable ifT is locally Lipschitzian and accretive onE.

More precisely, letN : E×E → E andm, g : E →E be the single-valued mappings and F, G, T : E → CB(E)the multivalued mappings. LetX be a fixed closed convex cone ofE.

DefineK :E →2^E by

K(z) = m(z) +X for allx∈E, z∈T(x).

We shall study the following generalized co-complementarity problem (GCCP):

Findx∈E,u∈F(x),v ∈G(x),z ∈T(x)such thatg(x)∈K(z)and

(2.1) N(u, v)∈(J(K(z)−g(x)))^?,

where(J(K(z)−g(x)))^? is the dual cone of the setJ(K(z)−g(x)).

2.1. Special Cases.

(i) IfEis a Hilbert space,F, T are identity mappings andN(u, v) = Bx+Av, whereB, A are single-valued mappings, then Problem (2.1) reduces to a problem of findingx∈E, v ∈G(x)such thatg(x)∈K(z)and

(2.2) B(x) +A(v)∈(K(x)−g(x))^?

considered by Jou and Yao [16].

(ii) IfGandg are identity mappings, then (2.2) reduces to findingx∈K(x)such that

(2.3) B(x) +A(y)∈(K(x)−x)^?

which is called a strongly nonlinear quasi complementarity problem, studied by Noor [22].

(iii) Ifmis a zero mapping, then (2.3) is equivalent to findingx∈E such that

(2.4) Bx+Ax∈E^? and hBx+Ax, xi= 0,

which is known as the mildly nonlinear complementarity problem, studied by Noor [21].

(iv) IfA is zero mapping, then (2.4) is equivalent to a problem of findingx ∈ E such that Bx∈E^? and

(2.5) hBx, xi= 0,

considered by Habetler [14] and Karamardian [17].

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3. THECHARACTERIZATION OFPROBLEM ANDSOLUTIONS

In this section, we briefly consider some basic concepts and results, which will be used throughout the paper. The real Banach spaceEis said to be uniformly smooth if its modulus of smoothnessρ_E(τ)defined by

ρ_E(τ) = sup

kx+yk+kx−yk

2 −1 : kxk= 1,kyk ≤τ

satisfies ^ρ^E_τ^(τ) → 0 as τ → 0. It follows that E is uniformly smooth if and only if J_p is single-valued and uniformly continuous on any bounded subset ofEand there exists a complete duality between uniform convexity and uniform smoothness. E is uniformly convex (smooth) if and only if E^? is uniformly smooth (convex). Recall that E is said to have the modulus of smoothness of power type p > 1 (and E is said to bep-uniformly smooth) if there exists a constantc >0such that

ρ_E(τ)≤cτ^p for0< τ <∞.

Remark 3.1. It is known that all Hilbert spaces and Banach spaces,L_p, l_pandW_m^p (1< p < ∞) are uniformly smooth and

ρ_E(τ)<









 1

p τ^p, 1< p <2

(E =L_p, l_porW_m^p) p−1

2 τ², p≤2

thereforeE is ap-uniformly smooth Banach space with modulus of smoothness of power type p <1andJ_pwill always represent the single-valued duality mapping.

Definition 3.1 ([4, 13]). Let E be a p-uniformly smooth Banach space and let Ωa nonempty closed convex subset ofE. A mappingQ_Ω :E →Ωis said to be

(i) retraction onΩifQ^p_Ω =Q_Ω;

(ii) nonexpansive retraction if it satisfies the inequality

kQ_Ω(x)−Q_Ω(y)k ≤ kx−yk, for allx, y ∈E; (iii) sunny retraction if for allx∈Eand for all−∞< t <∞

Q_Ω(Q_Ω(x) +t(x−Q_Ω(x)) =Q_Ω(x).

The following characterization of a sunny nonexpansive retraction mapping can be found in [4, 13].

Lemma 3.2 ([4, 13]). Q_Ωis sunny nonexpansive retraction if and only if for allx, y ∈E, hx−Q^?_Ω, J(QΩ x−y)i ≥0.

Lemma 3.3. LetE be a real Banach space andJ_p :E →2^E^?,1< p < ∞a duality mapping.

Then, for any givenx, y ∈E, we have

kx+yk^p ≤ kxk^p+phy, j_pi, for allj_p ∈J_p(x+y).

Proof. From Proposition 2.1, it follows that J_p(x) = ∂ϕ(x) (subdifferential of ψ), where ψ(x) =p⁻¹kxk^p. Also, it follows from the definition of the subdifferential ofψ that

ψ(x)−ψ(x+y)≥ hx−(x+y), j_pi, for allj_p ∈J_p(x+y). Substitutingψ(x)byp⁻¹kxk^p, we have

kx+yk^p ≤ kxk^p+phy, jpi, for alljp ∈Jp(x+y).

This completes the proof.

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Theorem 3.4 ([9]). Let E be a Banach space, Ωa nonempty closed convex subset of E, and m:E →E. Then for allx, y ∈E, we have

Q_Ω+m(z)(x) =m(z) +Q_Ω(x−m(z)).

We mention the following characterization theorem for the solution of a generalized co- complementarity problem which can be easily proved by using Lemma 3.2 and the argument of [1, Theorem 8.1].

Theorem 3.5. LetE be a realp-uniformly smooth Banach space andX a closed convex cone in E. Let F, G, T : E → CB(E) be the multivalued mappings, m, g : E → E the two single-valued mappings andN : E×E → E the nonlinear mapping. Let K : E → 2^E and K(z) =m(z) +X forx∈E. Then the following statements are equivalent:

(i) x ∈ E, u ∈ F(x), v ∈ G(x) and z ∈ T(x)are solutions of the Problem (2.1), i.e., g(x)∈K(z)and

N(u, v)∈(J(K(z)−g(x)))^?. (ii) x∈E,u∈F(x),v ∈G(x)andz ∈T(x)andτ >0

g(x) =Q_K(z)[g(x)−τ N(u, v)].

Combining Theorem 3.4 and 3.5, we have the following result.

Theorem 3.6. LetE be ap-uniformly smooth Banach space andX a closed convex cone inB.

Letm, g:E →E be the two single-valued mappings,F, G, T :E → CB(E)the multivalued mappings and N : E × E → E a nonlinear mapping. Then the following statements are equivalent:

(i) x∈E,u∈F(x),v ∈G(x)andz ∈T(x)are solutions of the Problem (2.1), (ii) x=x−g(x) +m(z) +Q_X[g(x)−τ N(u, v)−m(z)],for someτ > 0.

The following inequality will be used in our main results.

Lemma 3.7. LetEbe a real Banach space andj_p : E →2^E^?,1< p < ∞a duality mapping.

Then, for any givenx, y ∈E, we have

hx−y, j_p(x)−j_p(y)i ≤2d^pρ_E

4kx−yk d

, where

d^p =

kxk²+kyk² 2

.

Proof. The proof of the above inequalities are the generalized form of the proof of Theorem

3.4, and hence will be omitted.

4. ITERATIVEALGORITHMS AND PERTINENTCONCEPTS

We now propose the following iterative algorithm for computing the approximate solution of (GCCP).

Algorithm 4.1. Letg, m:E →E be the two single-valued mappings,F, G, T :E →CB(E) the multivalued mappings andN :E×E →Ea nonlinear mapping.

For any givenx₀ ∈E,u₀ ∈F(x₀),v₀ ∈G(x₀)andz₀ ∈T(x₀), let

x₁ =x₀−g(x₀) +m(z₀) +Q_X[g(x₀)−τ N(u₀, v₀)−m(z₀)]

whereτ > 0is a constant.

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Sinceu₀ ∈F(x₀)∈ CB(E),v₀ ∈G(x₀)∈CB(E)andz₀ ∈T(x₀)∈CB(E), by Nadler’s Theorem [20], there existsu₁ ∈F(x₁),v₁ ∈G(x₁)andz₁ ∈T(x₁)such that

ku₀−u₁k ≤(1 + 1)D(F(x₀), F(x₁)), kv₀ −v₁k ≤(1 + 1)D(G(x₀), G(x₁)), kz₀−z₁k ≤(1 + 1)D(T(x₀), T(x₁)), whereDis a Hausdorff metric onCB(E).

Let

x₂ =x₁−g(x₁) +m(z₁) +Q_X[g(x₁)−τ N(u₁, v₁)−m(z₁)].

Sinceu₁ ∈ F(x₁) ∈ CB(E), v₁ ∈ G(x₁) ∈ CB(E)andz₁ ∈ T(x₁) ∈ CB(E), there exists u₂ ∈F(x₂),v₂ ∈G(x₂)andz₂ ∈T(x₂)such that

ku₁ −u₂k ≤(1 + 2⁻¹)D(F(x₁), F(x₂)), kv₁−v₂k ≤(1 + 2⁻¹)D(G(x₁), G(x₂)), kz1−z2k ≤(1 + 2⁻¹)D(T(x1), T(x2)).

By induction, we can obtain{x_n},{u_n},{v_n}and{z_n}as

(4.1) x_n+1 =x_n−g(x_n) +m(z_n) +Q_X[g(x_n)−τ N(u_n, v_n)−m(z_n)].

un ∈F(xn);ku_n−un+1k ≤(1 + (1 +n)⁻¹)D(F(xn), F(xn+1)), v_n∈G(x_n);kv_n−v_n+1k ≤(1 + (1 +n)⁻¹)D(G(x_n), G(x_n+1)), z_n∈T(x_n);kz_n−z_n+1k ≤(1 + (1 +n)⁻¹)D(T(x_n), T(x_n+1)), n≥0, whereτ > 0is a constant.

These iteration processes have been extensively investigated by various authors for approx- imating either the fixed point of nonlinear mappings or solutions of nonlinear equations in Banach spaces or variational inequalities, variational inclusions, or complementarity problems in Hilbert spaces.

Definition 4.1. A single valued mappingg :E →E is said to be

(i) p-strongly accretive if for allx, y ∈E there existsjp ∈Jp(x−y)such that hg(x)−g(y), j_p(x−y)i ≥κkx−yk^p

for some real constantκ∈(0,1)and1< p <∞.

(ii) Lipschitz continuous if for anyx, y ∈E, there exists constantβ >0, such that kg(x)−g(y)k ≤βkx−yk.

Definition 4.2. A multivalued mappingF :E →CB(E)is said to beD-Lipschitz continuous if for anyx, y ∈E,

D(F(x), F(y))≤µkx−yk forµ > 0andD(·,·)is Hausdorff metric defined onCB(E).

Definition 4.3. Let F : E → CB(E) be a multivalued mapping. A nonlinear mappingN : E×E →Eis said to be relaxed accretive with respect to the first argument of mapF, if there exists a constantα >0such that

hN(u_n,·)−N(u_n−1,·), j_p(x_n−x_n−1)i ≥ −αkx_n−x_n−1k^p; andN is Lipschitz continuous with respect to the first argument if

kN(u,·)−N(y,·)k ≤σkx−yk, forx, y ∈E,

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whereσ > 0is a constant.

Similarly, we define the Lipschitz continuity ofN with respect to second argument.

5. MAINRESULTS

In this section, we show that if E is a p-uniformly smooth Banach space, then the iterative process converges strongly to the given problem (2.1).

Theorem 5.1. LetE be a p-uniformly smooth real Banach space withρ_E(τ) ≤ cτ^p for some c > 0, 0< τ < ∞and1 < p < ∞. LetX be a closed convex cone ofE.Letm, g :E → E be the two single-valued mappings, F, G, T : E → CB(E) the multivalued mappings. Let K :E →2^E such thatK(z) =m(z) +Xfor allx∈E,z ∈T(x)and the following conditions hold.

(i) g andmare Lipschitz continuous;

(ii) g is strongly accretive;

(iii) F, GandT areD-Lipschitz continuous;

(iv) N is Lipschitz continuous with respect to the first as well as the second argument;

(v) N is p-relaxed accretive with respect to the first argument with mapping F : E → CB(E);

(vi)

q+ (1 +pατ+pc2^2p+1τ^pσ^pη^p)^1/p+τ δξ <1 and

(5.1) q = 2(1−pκ+c2^2p+1pβ^p)^1/p+ 2µρ.

Then for anyx₀ ∈E,u₀ ∈F(x₀),v₀ ∈G(x₀)andz₀ ∈T(x₀)the sequencesx_n, u_n, v_n and z_n generated by Algorithm 4.1, converge strongly to some x ∈ E, u ∈ F(x), v ∈G(x)andz ∈T(x), which solve the problem (2.1).

Proof. By the iterative schemes (4.1) and Definition 3.1, we have kx_n+1−x_nk

=kx_n−g(x_n) +m(z_n) +Q_X[g(x_n)−τ N(u_n, v_n)−m(z_n)]

− x_n−1+g(x_n−1)−m(z_n−1)−Q_X[g(x_n−1)−τ N(u_n−1, v_n−1)−m(z_n−1)]k

≤ kxn−xn−1−(g(xn)−g(xn−1))k+km(zn)−m(zn−1k +kQ_X[g(x_n)−τ N(u_n, v_n)−m(z_n)]

−Q_X[g(xn−1)−τ N(un−1, vn−1)−m(zn−1)]k

≤2kx_n−xn−1 −(g(x_n)−g(xn−1))k+ 2km(z_n)−m(zn−1)k +kx_n−x_n−1−τ(N(u_n, v_n)−N(u_n−1, v_n−1))k

≤2kxn−xn−1 −(g(xn)−g(xn−1))k+ 2km(zn)−m(zn−1)k +kx_n−xn−1−τ(N(u_n, v_n)−N(un−1, v_n))k

(5.2)

+τkN(un−1, v_n)−N(un−1, vn−1)k.

By Lemmas 3.3, 3.7, p-strongly accretive, Lipschitz continuity ofg and j_p ∈ J_p(x+y), we have

kx_n−x_n−1−(g(x_n)−g(x_n−1))k^p

≤ kxn−xn−1k^p+ph−(g(xn)−g(xn−1)), jpi

≤ kx_n−xn−1k^p−phg(x_n)−g(xn−1), j_p(x_n−xn−1−(g(x_n)−g(xn−1))i

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≤ kx_n−xn−1k^p−phg(x_n)−g(xn−1), j_p(x_n−xn−1)i

−phg(x_n)−g(x_n−1), j_p(x_n−x_n−1−(g(x_n)−g(x_n−1)))−j_p(x_n−x_n−1)i

≤ kx_n−xn−1k^p−p^kkx_n−xn−1k^p+ 2pd^pρ_E

4kg(x_n)−g(xn−1)k d

≤ kxn−xn−1k^p−p^kkxn−xn−1k^p+ 2pd^pc4^pkg(x_n)−g(x_n−1)k^p d^p

≤ kx_n−xn−1k^p−pkkx_n−xn−1k^p+ 2^2p+1cpβ^pkx_n−xn−1k^p

≤(1−pk+c2^2p+1pβ^p)kxn−xn−1k^p. (5.3)

By the Lipschitz continuity ofmandD-Lipschitz continuity ofT, we have km(z_n)−m(zn−1)k ≤µkz_n−zn−1k

≤µ(1 +n⁻¹)D(T(x_n), T(xn−1))

≤µ(1 +n⁻¹)ρkxn−xn−1k.

(5.4)

SinceF is η-Lipschitz continuous, Gis ξ-Lipschitz continuous and N is Lipschitz continuous with respect to the first and second arguments with positive constants σ and δ respec- tively. Using a similar argument to that of Xiaolin He [23], we have for everyun, u⁰_n ∈F(xn), N(un, v) = N(u⁰_n, v). On the other handun−1 ∈F(xn−1), and from the definition of Hausdorff metric and compactness ofF(x_n), there is au⁰_n ∈F(x_n)such that

ku⁰_n−un−1k ≤D(F(xn), F(xn−1)).

Hence

kN(u_n, v_n)−N(un−1, v_n)k=kN(u⁰_n, v_n)−N(un−1, v_n)k

≤σku⁰_n−un−1k

≤σ(1 +n⁻¹)D(F(xn), F(xn−1))

≤σ(1 +n⁻¹)ηkx_n−x_n−1k.

(5.5)

Similarly, we get

(5.6) kN(un−1, v_n)−N(un−1, vn−1)k ≤δξ(1 +n⁻¹)kx_n−xn−1k.

By using Lemma 3.7, thep-relaxed accretive mapping and (5.5), we have kx_n−xn−1−τ(N(u_n, v_n)−N(un−1, v_n))k^p

≤ kx_n−xn−1k^p

−pτhN(u_n, v_n)−N(u_n−1, v_n), j_p(x_n−x_n−1−τ(N(u_n, v_n)−N(u_n−1, v_n)))i

≤ kx_n−xn−1k^p−pτhN(u_n, v_n)−N(un−1, v_n), j_p(x_n−xn−1)i

−phτ(N(u_n, v_n)−N(un−1, v_n)),

j_p(x_n−xn−1−τ(N(u_n, v_n)−N(un−1, v_n)))−j_p(x_n−xn−1)i

≤ kx_n−xn−1k^p+pτ αkx_n−xn−1k^p+ 2pd^pρ_E

τ4kN(u_n, v_n)−N(un−1, v_n)k d

≤ kx_n−xn−1k^p+pατkx_n−xn−1k^p+ 2pτ^pc4^pkN(u_n, v_n)−N(un−1, v_n)k^p

≤ kx_n−xn−1k^p+pατkx_n−xn−1k^p+pτ^pc2^2p+1(1 +n⁻¹)^pσ^pη^pkx_n−xn−1k^p

≤(1 +pατ +pc2^2p+1(1 +n⁻¹)^pτ^pσ^pη^p)kx_n−xn−1k^p. (5.7)

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Now from (5.2) – (5.7), we get

kx_n−xn−1k ≤2(1−pk+c2^2p+1pβ^p)^1/pkx_n−xn−1k+ 2µρ(1 +n⁻¹)kx_n−xn−1k + (1 +pατ+pc2^2p+1(1 +n⁻¹)^pτ^pσ^pη^p)^1/pkxn−xn−1k

+τ δξ(1 +n⁻¹)kx_n−x_n−1k

≤[2(1−pk+c2^2p+1pβ^p)^1/p+ 2µρ(1 +n⁻¹)

+ (1 +pατ+pc2^2p+1(1 +n⁻¹)^pτ^pσ^pη^p)^1/p+τ δξ(1 +n⁻¹)]

≤[q_n+ (1 +pατ +pc2^2p+1(1 +n⁻¹)^pτ^pσ^pη^p)^1/p +τ δξ(1 +n⁻¹)]kx_n−xn−1k

≤θ_nkx_n−xn−1k, (5.8)

where

θ_n=q_n+ (1 +pατ+pc2^2p+1(1 +n⁻¹)^pτ^pσ^pη^p)^1/p+τ δξ(1 +n⁻¹) and

(5.9) qn = 2(1−pk+c2^2p+1pβ^p)^1/p+ 2µρ 1 +n⁻¹ . Letting

(5.10) θ =q+ (1 +pατ +pc2^2p+1τ^pσ^pη^p)^1/p+τ δξ and

q= 2(1−pk+c2^2p+1pβ^p)^1/p+ 2µρ.

We know thatθn→θasn→ ∞. From condition (5.1), it follows thatθ <1. Henceθn<1, forn sufficiently large. Consequently{x_n}is a Cauchy sequence and this converges to some x∈E. By Algorithm 4.1 and theD-Lipschitz continuity ofF, GandT, it follows that

ku_n−un−1k ≤(1 +n⁻¹)D(F(x_n), F(xn−1))

≤(1 +n⁻¹)ηkx_n−xn−1k, kv_n−vn−1k ≤(1 +n⁻¹)D(G(x_n), G(xn−1))

≤(1 +n⁻¹)ξkx_n−xn−1k, kz_n−zn−1k ≤(1 +n⁻¹)D(T(x_n), T(xn−1))

≤(1 +n⁻¹)ρkxn−xn−1k,

which means that{u_n}, {v_n} and{z_n}are all Cauchy sequences inE. Therefore there exist u ∈ E, v ∈ E and z ∈ E such that u_n → u, v_n → v and z_n → z as n → ∞. Since g, m, F, G, T, N andQ_X are all continuous, we have

x=x−g(x) +Q_X[g(x)−τ N(u, v)−m(z)].

Finally, we prove thatu∈F(x). In fact, sinceu_n∈F(x_n)and d(un, F(x))≤max

(

d(un, F(x)), sup

u∈F(x)

d(T(xn), u) )

≤max (

sup

y∈F(xn)

d(y, F(x)), sup

u∈F(x)

d(F(x_n), u) )

=D(F(x_n), F(x)),

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We have

d(u, F(x))≤ ku−unk+d(un, F(x))

≤ ku−u_nk+D(F(x_n), F(x))

≤ ku−u_nk+ηkx_n−xk →0 asn→ ∞,

which implies thatd(u, F(x)) = 0. SinceF(x)∈CB(E), it follows thatu∈F(x). Similarly, we can prove thatG(x)∈CB(E)i.e.,v ∈G(x)andT(x)∈CB(E)i.e.,z ∈T(x). Hence by

Theorem 3.6, we get the conclusion.

REFERENCES

[1] YA ALBER, Metric and generalized projection operators in Banach spaces, properties and ap- plications, in Theory and Applications of Nonlinear Operators of Monotone and Accretive type, Kartsatos (Ed.), Marcel Dekker, New York (1996), 15–50.

[2] E. ASPLUND, Positivity of duality mappings, Bull. Amer. Math. Soc., 73 (1967), 200–203.

[3] C. BAIOCCHI AND A. CAPELO, Variational and Quasivariational Inequalities, Application to Free Boundary Problems, J. Wiley, New York, 1984.

[4] Y. BANYAMINIANDJ. LINDENSTRAUSS, Geometric nonlinear functional analysis, I, American Mathematical Society Colloqum Publication, 48 (2000), American Mathematical Society, Provi- dence, R.I.

[5] F.E. BROWDER, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull.

Amer. Math. Soc., 73 (1967), 875–885.

[6] A. CARBONE AND G. ISAC, The generalized order complementarity problem: Application to economics and an existence result, Nonlinear Studies, 5(2) (1998), 139–151.

[7] S.S. CHANG, Y.J. CHO, B.S. LEE, J.S. JUNGANDS.M. KANG, Iterative approximation of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl., 224 (1998), 149–165.

[8] S.S. CHANG, J.K. KIMANDK.H. KIM, On the existence and iterative approximation problems of solutions for set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl., 268 (2002), 89–108.

[9] J.Y. CHEN, N.C. WONGANDJ.C. YAO, Algorithm for generalized co-complementarity problems in Banach spaces, Comput. Math. Appl., 43 (2002), 49–54.

[10] R.W. COTTLEANDG.B. DANTZING, Complementarity pivot theory of mathematical program- ming, Linear Algebra Appl., 1 (1968), 163–185.

[11] R.W. COTTLE J.S. PANGANDR.E. STONE, The Linear Complementarity Problems, Academic Press, New York, 1992.

[12] L. DENGAND X.P. DING, Iterative approximation of Lipschitz strictly pseudo contractive map- pings in uniformly smooth Banach spaces, Nonlinear Analysis, TMA, 24(7) (1995), 981–987.

[13] K. GOEBELANDS. REICH, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Map- pings, Marcel Dekker, New York, 1984.

[14] S.J. HABETLER ANDA.L. PRICE, Existence theory for generalized nonlinear complementarity problems, J Optim. Theory Appl., 7 (1971), 223–239.

[15] D.H. HYERS, G. ISACAND TH. M. RASSIAS, Topics in Nonlinear Analysis and Applications, World Scientific, Singapore, New Jersey, London, Hong Kong, 1997.

(11)

[16] C.R. JOUANDJ.C. YAO, Algorithm for generalized multivalued variational inequalities in Hilbert spaces, Computers Math. Appl., 25(9) (1993), 7–13.

[17] S. KARAMARDIAN, Generalized complementarity problems, J. Optim. Theo. Appl., 8 (1971), 161–168.

[18] T. KATO, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 18/19 (1967), 508–

520.

[19] C.E. LEMKE, Bimatrix equilibrium point and mathematical programming, Management Science, 11 (1965), 681–689.

[20] S.B. NADLER Jr., Multivalued contraction mappings, Pacific J. Math., 30 (1969), 475–488.

[21] M.A. NOOR, On the nonlinear complementarity problem, J. Math. Anal. Appl., 13 (1987), 455–

460.

[22] M.A. NOOR, Nonlinear quasi-complementarity problems, Appl. Math. Lett., 2(3)(1980), 251–254.

[23] XIAOLIN HE, On Φ-strongly accretive mappings and some set-valued variational problems, J.

Math. Anal. Appl., 277(2003), 504–511.

[24] Z.B. XUANDG.F. ROACH, Characteristic inequalities in uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl., 157 (1991), 189–210.