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volume 7, issue 2, article 66, 2006.

Received 14 October, 2005;

accepted 30 December, 2005.

Communicated by:R.U. Verma

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Journal of Inequalities in Pure and Applied Mathematics

GENERALIZED CO-COMPLEMENTARITY PROBLEMS IN p-UNIFORMLY SMOOTH BANACH SPACES

M. FIRDOSH KHAN AND SALAHUDDIN

Department of Mathematics Aligarh Muslim University Aligarh-202002, India EMail:khan_mfk@yahoo.com

c

2000Victoria University ISSN (electronic): 1443-5756 011-06

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Co-Complementarity Problems inp-Uniformly Smooth Banach

Spaces

M. Firdosh Khan and Salahuddin

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J. Ineq. Pure and Appl. Math. 7(2) Art. 66, 2006

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.

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

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

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. Con- sequently 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.

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 pa- rameters 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.

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2. Background of Problem Formulations

Throughout this article, we assume that E is a real Banach space whose norm is denoted byk · 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 onEinduced by the normk · k. As usual,h·,·iis the generalized duality pairing betweenEandE^?. 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 mapping J2 with gauge function φ(t) = t is called the normalized duality mapping. It is known that J_p(x) = kxk^p−2J₂(x) for all x 6= 0 and J_p is single valued if E^? is strictly convex. IfE = H is a Hilbert space, thenJ2 becomes the identity mapping onH.

Proposition 2.1 ([7]). Let E be a real Banach space. For 1 < p < ∞, the duality mappingJ_p :E →2^E^? has the following basic properties:

1. Jp(x)6=∅for allx∈E andD(Jp)(the domain ofJp)=E,

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2. J_p(x) = kxk^p−2J₂(x)for allx∈E,(x6= 0), 3. Jp(αx) =α^p−1Jp(x)for allα∈[0,∞), 4. J_p(−x) = −J_p(x),

5. J_p is bounded i.e., for any bounded subset A ⊂ E, J_p(A) is a bounded subset inE^?,

6. J_pcan 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 if Jp is single valued and uniformly continuous on any bounded subset ofE(Xu and Roach [24]).

Definition 2.1. Let E be 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 anyx, y ∈K,u∈T(x)andv ∈T(y)there existsj₂ ∈J₂(x−y)such that

hu−v, j₂i ≥0,

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

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2. T is said to be strongly accretive if for any x, 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 indepen- dently 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 →Eandm, g :E →E be the single-valued mappings and F, G, T : E → CB(E)the multivalued mappings. Let X 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)).

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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) If Gandg 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∈Esuch 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) IfAis zero mapping, then (2.4) is equivalent to a problem of findingx∈E such thatBx ∈E^?and

(2.5) hBx, xi= 0,

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

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3. The Characterization of Problem and Solutions

In this section, we briefly consider some basic concepts and results, which will be used throughout the paper. The real Banach spaceE is 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 ifJ_p is single-valued and uniformly continuous on any bounded subset of E and 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 be p-uniformly smooth) if there exists a constantc >0such that

ρ_E(τ)≤cτ^p for0< τ <∞.

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

ρ_E(τ)<









 1

p τ^p, 1< p <2

(E =Lp, lp orW_m^p) p−1

2 τ², p≤2

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

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

(i) retraction onΩifQ^p_Ω =QΩ;

(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 3.1 ([4,13]). Q_Ωis sunny nonexpansive retraction if and only if for all x, y ∈E,

hx−Q^?_Ω, J(Q_Ωx−y)i ≥0.

Lemma 3.2. Let E 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).

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Proof. From Proposition 2.1, it follows thatJ_p(x) = ∂ϕ(x)(subdifferential of ψ), whereψ(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, j_pi, for allj_p ∈J_p(x+y).

This completes the proof.

Theorem 3.3 ([9]). Let E be a Banach space, Ω a nonempty closed convex subset ofE, andm: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 Lemma3.1and the argument of [1, Theorem 8.1].

Theorem 3.4. Let E be a real p-uniformly smooth Banach space and X a closed convex cone in E. LetF, G, T : E → CB(E)be the multivalued map- pings,m, g:E →E the two single-valued mappings andN :E×E →Ethe nonlinear mapping. LetK :E → 2^E andK(z) =m(z) +X forx ∈E. 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), i.e.,g(x)∈K(z)and

N(u, v)∈(J(K(z)−g(x)))^?.

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(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 Theorem3.3and3.4, we have the following result.

Theorem 3.5. Let E be a p-uniformly smooth Banach space and X a closed convex cone in B. Let m, 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 3.6. LetE be a real Banach space and j_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 Theorem3.3, and hence will be omitted.

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4. Iterative Algorithms and Pertinent Concepts

We now propose the following iterative algorithm for computing the approxi- mate solution of (GCCP).

Algorithm 1. Let g, m: E → Ebe the two single-valued mappings,F, G, T : E → CB(E) the multivalued mappings and N : E ×E → E a 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.

Since u₀ ∈ F(x₀) ∈ CB(E), v₀ ∈ G(x₀) ∈ CB(E)and z₀ ∈ T(x₀) ∈ CB(E), by Nadler’s Theorem [20], there exists u₁ ∈ F(x₁), v₁ ∈ G(x₁) and z₁ ∈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)].

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Since u₁ ∈ F(x₁) ∈ CB(E), v₁ ∈ G(x₁) ∈ CB(E) and z₁ ∈ T(x₁) ∈ CB(E), there existsu2 ∈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);kun−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 approximating 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 exists j_p ∈ J_p(x−y) such that

hg(x)−g(y), j_p(x−y)i ≥κkx−yk^p for some real constantκ ∈(0,1)and1< p <∞.

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(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 mapping F : E → CB(E) is said to be D- 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. LetF : 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(un−1,·), j_p(x_n−xn−1)i ≥ −αkx_n−xn−1k^p; andN is Lipschitz continuous with respect to the first argument if

kN(u,·)−N(y,·)k ≤σkx−yk, forx, y ∈E, whereσ >0is a constant.

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

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5. Main Results

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

Theorem 5.1. LetEbe ap-uniformly smooth real Banach space withρ_E(τ)≤ cτ^p for some c > 0, 0 < τ < ∞ and 1 < p < ∞. Let X be a closed convex cone of E. Let m, g : E → E be the two single-valued mappings, F, G, T : E → CB(E)the multivalued mappings. LetK : E →2^E such that K(z) = m(z) +X for allx∈E,z ∈T(x)and the following conditions hold.

(i) gandmare Lipschitz continuous;

(ii) gis 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.

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Then for any x₀ ∈ E, u₀ ∈ F(x₀), v₀ ∈ G(x₀) and z₀ ∈ T(x₀) the sequencesxn, un, vnand zngenerated by Algorithm 1, converge strongly to some x ∈ E, u ∈ F(x), v ∈ G(x) and z ∈ T(x), which solve the problem (2.1).

Proof. By the iterative schemes (4.1) and Definition3.1, 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)]

−xn−1+g(xn−1)−m(zn−1)

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

≤ kx_n−x_n−1−(g(x_n)−g(x_n−1))k+km(z_n)−m(z_n−1k +kQX[g(xn)−τ N(un, vn)−m(zn)]

−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−xn−1−τ(N(u_n, v_n)−N(un−1, vn−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))k

(5.2)

+τkN(un−1, vn)−N(un−1, vn−1)k.

By Lemmas 3.2, 3.6, p-strongly accretive, Lipschitz continuity of g and jp ∈ 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

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

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

−phg(xn)−g(xn−1), jp(xn−xn−1

−(g(x_n)−g(xn−1)))−j_p(x_n−xn−1)i

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

4kg(x_n)−g(xn−1)k d

≤ kx_n−xn−1k^p−p^kkx_n−xn−1k^p + 2pd^pc4^pkg(x_n)−g(xn−1)k^p

d^p

≤ kxn−xn−1k^p−pkkxn−xn−1k^p+ 2^2p+1cpβ^pkxn−xn−1k^p

≤(1−pk+c2^2p+1pβ^p)kx_n−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)ρkx_n−xn−1k.

(5.4)

Since F is η-Lipschitz continuous, G is ξ-Lipschitz continuous and N is Lipschitz continuous with respect to the first and second arguments with posi- tive constantsσandδrespectively. Using a similar argument to that of Xiaolin He [23], we have for every u_n, u⁰_n ∈ F(x_n), N(u_n, v) = N(u⁰_n, v). On the

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other hand un−1 ∈ F(xn−1), and from the definition of Hausdorff metric and compactness ofF(xn), there is au⁰_n∈F(xn)such that

ku⁰_n−un−1k ≤D(F(x_n), F(xn−1)).

Hence

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

≤σku⁰_n−un−1k

≤σ(1 +n⁻¹)D(F(x_n), F(xn−1))

≤σ(1 +n⁻¹)ηkx_n−xn−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 Lemma3.6, 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(un−1, v_n),

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

jp(xn−xn−1−τ(N(un, vn)−N(un−1, vn)))−jp(xn−xn−1)i

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

τ4kN(u_n, v_n)−N(u_n−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−x_n−1k^p. (5.7)

Now from (5.2) – (5.7), we get kxn−xn−1k

≤2(1−pk+c2^2p+1pβ^p)^1/pkx_n−x_n−1k + 2µρ(1 +n⁻¹)kxn−xn−1k

+ (1 +pατ +pc2^2p+1(1 +n⁻¹)^pτ^pσ^pη^p)^1/pkx_n−x_n−1k +τ δξ(1 +n⁻¹)kxn−xn−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−x_n−1k, (5.8)

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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) q_n = 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 that θn → θ as n → ∞. From condition (5.1), it follows that θ < 1. Henceθ_n < 1, fornsufficiently large. Consequently{x_n}is a Cauchy sequence and this converges to some x ∈ E. By Algorithm 1 and the D- 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)ρkx_n−xn−1k,

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which means that {u_n}, {v_n}and{z_n}are all Cauchy sequences inE. There- fore there existu∈E,v ∈Eandz ∈Esuch thatun →u,vn→vandzn →z asn→ ∞. Sinceg, 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(u_n, F(x))≤max

(

d(u_n, F(x)), sup

u∈F(x)

d(T(x_n), u) )

≤max (

sup

y∈F(x_n)

d(y, F(x)), sup

u∈F(x)

d(F(x_n), u) )

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

d(u, F(x))≤ ku−u_nk+d(u_n, F(x))

≤ ku−u_nk+D(F(x_n), F(x))

≤ ku−unk+ηkxn−xk →0 asn→ ∞,

which implies thatd(u, F(x)) = 0. Since F(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 Theorem3.5, we get the conclusion.

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