Further, we discuss the convergence and stability of the iterative sequence generated by the perturbed algorithm

(1)

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

Volume 7, Issue 3, Article 91, 2006

NEW PERTURBED ITERATIONS FOR A GENERALIZED CLASS OF STRONGLY NONLINEAR OPERATOR INCLUSION PROBLEMS IN BANACH SPACES

HENG-YOU LAN, HUANG-LIN ZENG, AND ZUO-AN LI DEPARTMENT OFMATHEMATICS

SICHUANUNIVERSITY OFSCIENCE ANDENGINEERING

ZIGONG, SICHUAN643000, P. R. CHINA

hengyoulan@163.com

Received 05 March, 2006; accepted 19 March, 2006 Communicated by R.U. Verma

ABSTRACT. The purpose of this paper is to introduce and study a new kind of generalized strongly nonlinear operator inclusion problems involving generalizedm-accretive mapping in Banach spaces. By using the resolvent operator technique for generalized m-accretive mapping due to Huang and Fang, we also prove the existence theorem of the solution for this kind of operator inclusion problems and construct a new class of perturbed iterative algorithm with mixed errors for solving this kind of generalized strongly nonlinear operator inclusion problems in Banach spaces. Further, we discuss the convergence and stability of the iterative sequence generated by the perturbed algorithm. Our results improve and generalize the corresponding results of [3, 6, 11, 12].

Key words and phrases: Generalizedm-accretive mapping; Generalized strongly nonlinear operator inclusion problems; Per- turbed iterative algorithm with errors; Existence; Convergence and stability.

2000 Mathematics Subject Classification. 68Q25, 49J40, 47H19, 47H12.

1. INTRODUCTION

LetXbe a real Banach space andT :X →2^X is a multi-valued operator, where2^X denotes the family of all the nonempty subsets of X. The following operator inclusion problem of findingx∈X such that

(1.1) 0∈T(u)

has been studied extensively because of its role in the modelization of unilateral problems, nonlinear dissipative systems, convex optimizations, equilibrium problems, knowledge engineering, etc. For details, we can refer to [1] – [6], [8] – [15] and the references therein. Concerning the development of iterative algorithms for the problem (1.1) in the literature, a very common

ISSN (electronic): 1443-5756 c

This work was supported by the Educational Science Foundation of Sichuan, Sichuan of China No. 2004C018, 2005A140.

Authors are thankful to Prof. Ram U. Verma and Y. J. Cho for their valuable suggestions.

093-06

(2)

assumption is thatT is a maximal monotone operator orm-accretive operator. WhenT is maximal monotone orm-accretive, many iterative algorithms have been constructed to approximate the solutions of the problem (1.1).

In many practical cases,T is split in the formT = F +M, whereF, M :X →2^X are two multi-valued operators. So the problem (1.1) reduces to the following: Findx∈Xsuch that

(1.2) 0∈F(x) +M(x),

which is called the variational inclusion problem. When bothF andM are maximal monotone or M is m-accretive, some approximate solutions for the problem (1.2) have been developed (see [10, 13] and the references therein). IfM =∂ϕ, where∂ϕis the subdifferential of a proper convex lower semi-continuous functionalϕ:X →R∪ {+∞}, then the problem (1.2) reduces to the variational inequality problem:

Findx∈X andu∈F(x)such that

(1.3) hu, y−xi+ϕ(y)−ϕ(x)≥0, y∈X.

Many iterative algorithms have been established to approximate the solution of the problem (1.3) whenF is strongly monotone. Recently, the problem (1.2) was studied by several authors whenF andM need not to be maximal monotone orm-accretive. Further, Ding [3], Huang [6], and Lan et al. [11] developed some iterative algorithms to solve the following quasi-variational inequality problem of findingx∈X andu∈F(x),v ∈V(x)such that

(1.4) hu, y−x)i+ϕ(y, v)−ϕ(x, v)≥0, ∀y∈X

by introducing the concept of subdifferential ∂ϕ(·, t) of a proper functionalϕ(·,·)fort ∈ X, which is defined by

∂ϕ(·, t) ={f ∈X :ϕ(y, t)−ϕ(x, t)≥ hf, y−x)i, y ∈X},

whereϕ(·, t) : X → R∪ {+∞}is a proper convex lower semi-continuous functional for all t∈X.

It is easy to see that the problem (1.4) is equivalent to the following:

Findx∈X such that

(1.5) 0∈F(x) +∂ϕ(x, V(x)).

Recently, Huang and Fang [7] first introduced the concept of a generalizedm-accretive mapping, which is a generalization of anm-accretive mapping, and gave the definition and prop- erties of the resolvent operator for the generalized m-accretive mapping in a Banach space.

Later, by using the resolvent operator technique, which is a very important method for find- ing solutions of variational inequality and variational inclusion problems, a number of nonlinear variational inclusions and many systems of variational inequalities, variational inclusions, complementarity problems and equilibrium problems. Bi, Huang, Jin and other authors introduced and studied some new classes of nonlinear variational inclusions involving generalized m-accretive mappings in Banach spaces, they also obtained some new corresponding existence and convergence results (see, [2, 5, 8] and the references therein). On the other hand, Huang, Lan, Zeng, Wang et al. discussed the stability of the iterative sequence generated by the algorithm for solving what they studied (see [6, 11, 15, 19]).

Motivated and inspired by the above works, in this paper, we introduce and study the following new class of generalized strongly nonlinear operator inclusion problems involving generalizedm-accretive mappings:

Findx∈X such that(p(x), g(x))∈DomM and

(1.6) f ∈N(S(x), T(x), U(x)) +M(p(x), g(x)),

(3)

where f is an any given element on X, a real Banach space, S, T, U, p, g : X → X and N : X×X×X→X are single-valued mappings andM : X×X→2^X is a generalizedm- accretive mapping with respect to the first argument,2^X denotes the family of all the nonempty subsets ofX. By using the resolvent operator technique for generalizedm-accretive mappings due to Huang and Fang [7, 8], we prove the existence theorems of the solution for these types of operator inclusion problems in Banach spaces, and discuss the convergence and stability of a new perturbed iterative algorithm for solving this class of nonlinear operator inclusion problems in Banach spaces. Our results improve and generalize the corresponding results of [3, 6, 11, 12].

We remark that for a suitable choice off, the mappingsN, η, S, T, U, M, p, g and the space X, a number of known or new classes of variational inequalities, variational inclusions and corresponding optimization problems can be obtained as special cases of the nonlinear quasi- variational inclusion problem (1.6). Moreover, these classes of variational inclusions provide us with a general and unified framework for studying a wide range of interesting and important problems arising in mechanics, optimization and control, equilibrium theory of transportation and economics, management sciences, and other branches of mathematical and engineering sciences, etc. See for more details [1, 3, 4, 6, 9, 11, 15, 17, 18] and the references therein.

2. GENERALIZEDm-ACCRETIVE MAPPING

Throughout this paper, letX be a real Banach space with dual space X^∗, h·,·ithe dual pair betweenXandX^∗, and2^X denote the family of all the nonempty subsets ofX. The generalized duality mappingJq :X →2^X^∗ is defined by

J_q(x) = {x^∗ ∈X^∗ : hx, x^∗i=kxk^q,kx^∗k=kxk^q−1}, ∀x∈X,

whereq >1is a constant. In particular,J₂ is the usual normalized duality mapping. It is well known that, in general, Jq(x) = kxk^q−2J2(x) for all x 6= 0 and Jq is single-valued if X^∗ is strictly convex (see, for example, [16]). If X = H is a Hilbert space, then J2 becomes the identity mapping ofH. In what follows we shall denote the single-valued generalized duality mapping byj_q.

Definition 2.1. The mappingg : X →Xis said to be

(1) α-strongly accretive, if for anyx, y ∈X, there existsj_q(x−y)∈J_q(x−y)such that hg(x)−g(y), j_q(x−y)i ≥αkx−yk^q,

whereα >0is a constant;

(2) β-Lipschitz continuous, if there exists a constantβ >0such that kg(x)−g(y)k ≤βkx−yk, ∀x, y ∈X.

Definition 2.2. Leth, g : X →Xbe two single-valued mappings. The mappingN : X×X× X →Xis said to be

(1) σ-strongly accretive with respect to hin the first argument, if for any x, y ∈ X, there existsj_q(x−y)∈J_q(x−y)such that

hN(h(x),·,·)−N(h(y),·,·), j_q(x−y)i ≥σkx−yk^q, whereσ >0is a constant;

(2) ς-relaxed accretive with respect tog in the second argument, if for anyx, y ∈X, there existsj_q(x−y)∈J_q(x−y)such that

hN(·, g(x),·)−N(·, g(y),·), jq(x−y)i ≥ −ςkx−yk^q whereς >0is a constant;

(4)

(3) -Lipschitz continuous with respect to the first argument, if there exists a constant >0 such that

kN(x,·,·)−N(y,·,·)k ≤kx−yk, ∀x, y ∈X.

Similarly, we can define theξ, γ-Lipschitz continuity in the second and third argument ofN(·,·,·), respectively.

Definition 2.3 ([7]). Letη: X×X →X^∗ be a single-valued mapping andA: X →2^X be a multi-valued mapping. ThenAis said to be

(1) η-accretive if

hu−v, η(x, y)i ≥0, ∀x, y ∈X, u∈A(x), v ∈A(y);

(2) generalizedm-accretive ifAisη-accretive and(I+λA)(X) =X for all (equivalently, for some)λ >0.

Remark 2.1. Huang and Fang gave one example of the generalized m-accretive mapping in [7]. IfX = X^∗ = His a Hilbert space, then (1), (2) of Definition 2.3 reduce to the definition of η-monotonicity and maximal η-monotonicity respectively; if X is uniformly smooth and η(x, y) = J₂(x−y), then (1) and (2) of Definition 2.3 reduce to the definitions of accretivity andm-accretivity in uniformly smooth Banach spaces, respectively (see [7, 8]).

Definition 2.4. The mappingη: X×X →X^∗ is said to be

(1) δ-strongly monotone, if there exists a constantδ >0such that hx−y, η(x, y)i ≥δkx−yk², ∀x, y ∈X;

(2) τ-Lipschitz continuous, if there exists a constantτ > 0such that kη(x, y)k ≤τkx−yk, ∀x, y ∈X.

The modules of smoothness ofXis the functionρ_X : [0,∞)→[0,∞)defined by ρ_X(t) = sup

1

2kx+yk+kx−yk −1 : kxk ≤1, kyk ≤t

. A Banach space X is called uniformly smooth iflimt→0 ρ_X(t)

t = 0 andX is called q- uniformly smooth if there exists a constant c >0such thatρ_X ≤ct^q, whereq >1is a real number.

It is well known that Hilbert spaces,L_p (orl_p) spaces, 1 < p < ∞, and the Sobolev spaces W^m,p, 1 < p < ∞, are all q-uniformly smooth. In the study of characteristic inequalities in q-uniformly smooth Banach spaces, Xu [16] proved the following result:

Lemma 2.2. Let q > 1 be a given real number and X be a real uniformly smooth Banach space. ThenX isq-uniformly smooth if and only if there exists a constantcq >0such that for allx, y ∈X,jq(x)∈Jq(x), there holds the following inequality

kx+yk^q ≤ kxk^q+qhy, j_q(x)i+c_qkyk^q.

In [7], Huang and Fang show that for anyρ >0, inverse mapping(I+ρA)⁻¹is single-valued, ifη: X×X →X^∗is strict monotone andA: X →2^X is a generalizedm-accretive mapping, where I is the identity mapping. Based on this fact, Huang and Fang [7] gave the following definition:

(5)

Definition 2.5. Let A : X → 2^X be a generalizedm-accretive mapping. Then the resolvent operatorJ_A^ρ forAis defined as follows:

J_A^ρ(z) = (I+ρA)⁻¹(z), ∀z ∈X,

whereρ >0is a constant andη: X×X →X^∗is a strictly monotone mapping.

Lemma 2.3 ([7, 8]). Letη:X×X →X^∗beτ-Lipschitz continuous andδ-strongly monotone.

LetA : X → 2^X be a generalized m-accretive mapping. Then the resolvent operatorJ_A^ρ for Ais Lipschitz continuous with constant ^τ_δ, i.e.,

kJ_A^ρ(x)−J_A^ρ(y)k ≤ τ

δkx−yk, ∀x, y ∈X.

3. EXISTENCETHEOREMS

In this section, we shall give the existence theorem of problem (1.6). Firstly, from the definition of the resolvent operator for a generalized m-accretive mapping, we have the following lemma:

Lemma 3.1. xis the solution of problem (1.6) if and only if

(3.1) p(x) = J_M^ρ_(·,g(x))[p(x)−ρ(N(S(x), T(x), U(x))−f)], whereJ_M^ρ_(·,g(x)) = (I +ρM(·, g(x)))⁻¹ andρ >0is a constant.

Theorem 3.2. LetX be aq-uniformly smooth Banach space,η : X×X →X^∗ beτ-Lipschitz continuous andδ-strongly monotone, M : X ×X → 2^X be a generalizedm-accretive map- ping with respect to the first argument, and mappingsS, T, U : X → X beκ, µ, ν-Lipschitz continuous, respectively. Letp: X → X beα-strongly accretive andβ-Lipschitz continuous, g : X → X beι-Lipschitz continuous,N : X ×X×X → X beσ-strongly accretive with respect toS in the first argument and ς-relaxed accretive with respect to T in the second ar- gument, and, ξ, γ-Lipschitz continuous in the first, second and third argument, respectively.

Suppose that there exist constantsρ >0andζ >0such that for eachx, y, z ∈X, (3.2)

J_M^ρ_(·,x)(z)−J_M(·,x)^ρ (z)

≤ζkx−yk and

(3.3)







h =ζι+ 1 + ^τ_δ

(1−qα+cqβ^q)¹^q <1, τh

(1−qρ(σ−ς) +c_qρ^q(κ+ξµ)^q)¹^q +ργνi

< δ(1−h), wherec_q is the same as in Lemma 2.2, then problem (1.6) has a unique solutionx^∗.

Proof. From Lemma 3.1, problem (1.6) is equivalent to the fixed problem (3.1), equation (3.1) can be rewritten as follows:

x=x−p(x)−J_M^ρ_(·,g(x))[p(x)−ρ(N(S(x), T(x), U(x))−f)].

For everyx∈X, take

(3.4) Q(x) =x−p(x)−J_M(·,g(x))^ρ [p(x)−ρ(N(S(x), T(x), U(x))−f)].

(6)

Thenx^∗ is the unique solution of problem (1.6) if and only ifx^∗is the unique fixed point ofQ.

In fact, it follows from (3.2), (3.4) and Lemma 2.3 that kQ(x)−Q(y)k

≤ kx−y−(p(x)−p(y))k+

J_M^ρ_(·,g(x))[p(x)−ρ(N(S(x), T(x), U(x))−f)]

− J_M^ρ_(·,g(y))[p(y)−ρ(N(S(y), T(y), U(y))−f)]

≤ kx−y−(p(x)−p(y))k+

J_M^ρ_(·,g(x))[p(x)−ρ(N(S(x), T(x), U(x))−f)]

− J_M^ρ_(·,g(x))[p(y)−ρ(N(S(y), T(y), U(y))−f)]

+

J_M^ρ_(·,g(x))[p(y)−ρ(N(S(y), T(y), U(y))−f)]

− J_M^ρ_(·,g(y))[p(y)−ρ(N(S(y), T(y), U(y))−f)]

≤ 1 + τ

δ

kx−y−(p(x)−p(y))k +τ

δ{kx−y−ρ[(N(S(x), T(x), U(x))−N(S(y), T(x), U(x))) + (N(S(y), T(x), U(x))−N(S(y), T(y), U(x)))]k

+ρkN(S(y), T(y), U(x))−N(S(y), T(y), U(y))k}

(3.5)

+ζkg(x)−g(y))k.

By the hypothesis ofg, p, S, T, U, N and Lemma 2.2, now we know there exists c_q > 0such that

kg(x)−g(y)k ≤ιkx−yk, (3.6)

kx−y−(p(x)−p(y))k^q ≤(1−qα+c_qβ^q)kx−yk^q, (3.7)

kN(S(y), T(y), U(x))−N(S(y), T(y), U(y))k ≤γνkx−yk, (3.8)

kx−y−ρ[(N(S(x), T(x), U(x))−N(S(y), T(x), U(x))) + (N(S(y), T(x), U(x))−N(S(y), T(y), U(x)))]k^q

≤ kx−yk^q−qρh(N(S(x), T(x), U(x))−N(S(y), T(x), U(x))) + (N(S(y), T(x), U(x))−N(S(y), T(y), U(x))), j_q(x−y)i +c_qρ^qk(N(S(x), T(x), U(x))−N(S(y), T(x), U(x))) + (N(S(y), T(x), U(x))−N(S(y), T(y), U(x)))k^q

≤ kx−yk^q−qρ[hN(S(x), T(x), U(x))−N(S(y), T(x), U(x)), j_q(x−y)i +hN(S(y), T(x), U(x))−N(S(y), T(y), U(x)), j_q(x−y)i]

+c_qρ^q[kN(S(x), T(x), U(x))−N(S(y), T(x), U(x))k +kN(S(y), T(x), U(x))−N(S(y), T(y), U(x))k]^q

≤[1−qρ(σ−ς) +cqρ^q(κ+ξµ)^q]kx−yk^q. (3.9)

Combining (3.5) – (3.9), we get

(3.10) kQ(x)−Q(y)k ≤θkx−yk,

(7)

where

θ =h+τ δ

h

(1−qρ(σ−ς) +c_qρ^q(κ+ξµ)^q)¹^q +ργνi , (3.11)

h=ζι+

1 + τ δ

(1−qα+cqβ^q)¹^q.

It follows from (3.3) that0< θ <1and soQ:X → Xis a contractive mapping, i.e.,Qhas a

unique fixed point inX. This completes the proof.

Remark 3.3. IfXis a 2-uniformly smooth Banach space and there existsρ >0such that











h=ζι+ 1 + ^τ_δ p

1−2α+c₂β² <1, 0< ρ < ^δ(1−h)_{τ γν} , γν <√

c₂(κ+ξµ), τ(σ−ς)> δγν(1−h) +p

[c₂(κ+ξµ)²−γ²ν²][τ²−δ²(1−h)²],

ρ− τ(σ−ς)+δγν(h−1) τ[c2(κ+ξµ)²−γ²ν²]

< [τ(σ−ς)−δγν(1−h)]²−[c₂(κ+ξµ)²−γ²ν²][τ²−δ²(1−h)²] τ[c2(κ+ξµ)²−γ²ν²] ,

then (3.3) holds. We note that the Hilbert space and L_p (or l_p) (2 ≤ p < ∞) spaces are 2-uniformly Banach spaces.

4. PERTURBED ALGORITHM ANDSTABILITY

In this section, by using the following definition and lemma, we construct a new perturbed iterative algorithm with mixed errors for solving problem (1.6) and prove the convergence and stability of the iterative sequence generated by the algorithm.

Definition 4.1. LetS be a selfmap ofX,x₀ ∈ X, and let x_n+1 = h(S, x_n)define an iteration procedure which yields a sequence of points{x_n}^∞_n=0inX. Suppose that{x∈X :Sx=x} 6=

∅and{x_n}^∞_n=0converges to a fixed pointx^∗ofS. Let{u_n} ⊂Xand let_n =ku_n+1−h(S, u_n)k.

Iflim_n = 0implies thatu_n →x^∗, then the iteration procedure defined byx_n+1 =h(S, x_n)is said to beS-stable or stable with respect toS.

Lemma 4.1 ([12]). Let {a_n},{b_n},{c_n} be three nonnegative real sequences satisfying the following condition:

there exists a natural numbern0such that

a_n+1 ≤(1−t_n)a_n+b_nt_n+c_n, ∀n≥n₀, wheret_n∈[0,1],P∞

n=0t_n =∞,limn→∞b_n = 0,P∞

n=0c_n <∞. Thena_n→0(n→ ∞).

The relation (3.1) allows us to construct the following perturbed iterative algorithm with mixed errors.

Algorithm 4.1. Step 1. Choosex₀ ∈X.

Step 2. Let

(4.1)











x_n+1 = (1−α_n)x_n+α_n[y_n−p(y_n) +J_M^ρ_(·,g(y

n))(p(y_n)−ρ(N(S(y_n), T(y_n), U(y_n))−f))] +α_nu_n+ω_n, yn = (1−βn)xn+βn[xn−p(xn)

+J_M^ρ_(·,g(x

n))(p(x_n)−ρ(N(S(x_n), T(x_n), U(x_n))−f))] +v_n,

Step 3. Choose sequences{α_n},{β_n},{u_n},{v_n}and{ω_n}such that forn≥0,{α_n},{β_n} are two sequences in[0,1], {u_n},{v_n},{ω_n}are sequences inX satisfying the following conditions:

(8)

(i) u_n =u⁰_n+u⁰⁰_n;

(ii) limn→∞ku⁰_nk= limn→∞kv_nk= 0;

(iii) P∞

n=0ku⁰⁰_nk<∞, P∞

n=0kω_nk<∞,

Step 4. If x_n+1, y_n, α_n, β_n, u_n, v_n and ω_n satisfy (4.1) to sufficient accuracy, go to Step 5;

otherwise, setn :=n+ 1and return to Step 2.

Step 5. Let{z_n}be any sequence inXand define{ε_n}by

(4.2)











ε_n =kz_n+1− {(1−α_n)z_n+α_n[t_n−p(t_n) +J_M^ρ_(·,g(t

n))(p(t_n)−ρ(N(S(t_n), T(t_n), U(t_n))−f))] +α_nu_n+ω_n}k, t_n= (1−β_n)z_n+β_n[z_n−p(z_n)

+J_M^ρ_(·,g(z

n))(p(z_n)−ρ(N(S(z_n), T(z_n), U(z_n))−f))] +v_n.

Step 6. If ε_n, z_n+1, t_n, α_n, β_n, u_n, v_n and ω_n satisfy (4.2) to sufficient accuracy, stop;

otherwise, setn :=n+ 1and return to Step 3.

Theorem 4.2. Suppose that X, S, T, U, p, g, N, η and M are the same as in Theorem 3.2, θ is defined by (3.11). If P∞

n=0α_n = ∞ and conditions (3.2), (3.3) hold, then the perturbed iterative sequence{x_n}defined by (4.1) converges strongly to the unique solution of problem (1.6). Moreover, if there existsa ∈ (0, α_n]for alln ≥ 0, thenlimn→∞z_n = x^∗ if and only if limn→∞ε_n = 0, whereε_nis defined by (4.2).

Proof. From Theorem 3.2, we know that problem (1.6) has a unique solutionx^∗ ∈X. It follows from (4.1), (3.11) and the proof of (3.10) in Theorem 3.2 that

kx_n+1−x^∗k

≤(1−αn)kxn−x^∗k+αnθkyn−x^∗k+αn(ku⁰_nk+ku⁰⁰_nk) +kωnk

≤(1−α_n)kx_n−x^∗k+α_nθky_n−x^∗k+α_nku⁰_nk+ (ku⁰⁰_nk+kω_nk).

(4.3)

Similarly, we have

(4.4) ky_n−x^∗k ≤(1−β_n+β_nθ)kx_n−x^∗k+kv_nk.

Combining (4.3) – (4.4), we obtain

(4.5) kx_n+1−x^∗k ≤[1−α_n(1−θ(1−β_n+β_nθ))]kx_n−x^∗k

+αn(ku⁰_nk+θkvnk) + (ku⁰⁰_nk+kωnk).

Sinceθ < 1,0< β_n ≤1 (n≥0), we have1−β_n+β_nθ <1and1−θ(1−β_n+β_nθ)>1−θ >0.

Therefore, (4.5) implies

(4.6) kx_n+1−x^∗k ≤[1−α_n(1−θ)]kx_n−x^∗k +α_n(1−θ)· 1

1−θ(ku⁰_nk+θkv_nk) + (ku⁰⁰_nk+kω_nk).

SinceP∞

n=0α_n =∞, it follows from Lemma 4.1 and (4.6) thatkx_n−x^∗k →0(n → ∞), i.e., {xn}converges strongly to the unique solutionx^∗ of the problem (1.6).

Now we prove the second conclusion. By (4.2), we know (4.7) kzn+1−x^∗k ≤ k(1−αn)zn+αn[tn−p(tn)

+J_M(·,g(t^ρ

n))(p(t_n)−ρ(N(S(t_n), T(t_n), U(t_n))−f)))

+α_nu_n+ω_n−x^∗k+ε_n.

(9)

As the proof of inequality (4.6), we have (4.8) k(1−α_n)z_n+α_n[t_n−p(t_n)

+J_M(·,g(t^ρ

n))(p(t_n)−ρ(N(S(t_n), T(t_n), U(t_n))−f))) +α_nu_n+ω_n−x^∗k

≤[1−α_n(1−θ)]kz_n−x^∗k +α_n(1−θ)· 1

1−θ(ku⁰_nk+θkv_nk) + (ku⁰⁰_nk+kω_nk).

Since0< a≤α_n, it follows from (4.7) and (4.8) that kz_n+1−x^∗k

≤[1−α_n(1−θ)]kz_n−x^∗k+α_n(1−θ)· 1

1−θ(ku⁰_nk+θkv_nk) + (ku⁰⁰_nk+kω_nk) +ε_n

≤[1−α_n(1−θ)]kz_n−x^∗k+α_n(1−θ)· 1 1−θ

ku⁰_nk+θkv_nk+ εn

a

+ (ku⁰⁰_nk+kω_nk).

Suppose thatlimε_n = 0. Then fromP∞

n=0α_n=∞and Lemma 4.1, we havelimz_n=x^∗. Conversely, iflimzn =x^∗, then we get

ε_n=kz_n+1− {(1−α_n)z_n+α_n[t_n−p(t_n) + J_M^ρ_(·,g(t

n))(p(t_n)−ρ(N(S(t_n), T(t_n), U(t_n))−f))] +α_nu_n+ω_n}

≤ kzn+1−x^∗k+k(1−αn)zn+αn[tn−p(tn) + J_M^ρ_(·,g(t

n))(p(t_n)−ρ(N(S(t_n), T(t_n), U(t_n))−f))) +α_nu_n+ω_n−x^∗

≤ kz_n+1−x^∗k+ [1−α_n(1−θ)]kz_n−x^∗k

+αn(ku⁰_nk+θkvnk) + (ku⁰⁰_nk+kωnk)→0 (n→ ∞).

This completes the proof.

Remark 4.3. Ifu_n =v_n=ω_n= 0 (n≥0)in Algorithm 4.1, then the conclusions of Theorem 4.2 also hold. The results of Theorems 3.2 and 4.2 improve and generalize the corresponding results of [3, 6, 11, 12].

REFERENCES

[1] C. BAIOCCHIANDA. CAOPELO, Variational and Quasivariational Inequalities, Application to Free Boundary Problems, Wiley, New York, 1984.

[2] Z.S. BI, Z. HANANDY.P. FANG, Sensitivity analysis for nonlinear variational inclusions involving generalizedm-accretive mappings, J. Sichuan Univ., 40(2) (2003), 240–243.

[3] X.P. DING, Existence and algorithm of solutions for generalized mixed implicit quasi-variational inequalities, Appl. Math. Comput. 113 (2000), 67–80.

[4] F. GIANNESSIANDA. MAUGERI, Variational Inequalities and Network Equilibrium Problems, Plenum, New York, 1995.

[5] N.J. HUANG, Nonlinear implicit quasi-variational inclusions involving generalized m-accretive mappings, Arch. Inequal. Appl., 2(4) (2004), 413–425.

[6] N.J. HUANG, M.R. BAI, Y.J. CHO AND S.M. KANG, Generalized nonlinear mixed quasi- variational inequalities, Comput. Math. Appl., 40 (2000), 205–216.

[7] N.J. HUANG, Y.P. FANG, Generalizedm-accretive mappings in Banach spaces, J. Sichuan Univ., 38(4) (2001), 591–592.

(10)

[8] N.J. HUANG, Y.P. FANG AND C.X. DENG, Nonlinear variational inclusions involving generalized m-accretive mappings, Proceedings of the Bellman Continum: International Workshop on Uncertain Systems and Soft Computing, Beijing, China, July 24-27, 2002, pp. 323–327.

[9] M.M. JIN, Sensitivity analysis for strongly nonlinear quasi-variational inclusions involving generalizedm-accretive mappings, Nonlinear Anal. Forum, 8(1) (2003), 93–99.

[10] J.S. JUNGANDC.H. MORALES, The Mann process for perturbedm-accretive opertators in Ba- nach spaces, Nonlinear Anal., 46(2) (2001), 231–243.

[11] H.Y. LAN, J.K. KIMANDN.J. HUANG, On the generalized nonlinear quasi-variational inclusions involving non-monotone set-valued mappings, Nonlinear Funct. Anal. & Appl., 9(3) (2004), 451–

465.

[12] L.S. LIU, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive map- pings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114–135.

[13] A. MOUDAFI, On the regularization of the sum of two maximal monotone operators, Nonlin. Anal.

TMA, 42 (2000), 1203–1208.

[14] W.L. WANG, Z.Q. LIU, C. FENG AND S.H. KANG, Three-step iterative algorithm with errors for generalized strongly nonlinear quasi-variational inequalities, Adv. Nonlinear Var. Inequal., 7(2) (2004), 27–34.

[15] M.V. SOLODOV AND B.F. SVAITER, A hybrid projection-proximal point algorithm, J. Convex Anal., 6(1) (1999), 59–70.

[16] H.K. XU, Inequalities in Banach spaces with applications, Nonlinear Anal., 16(12) (1991), 1127–

1138.

[17] G.X.Z. YUAN, KKM Theory and Applications, Marcel Dekker, 1999.

[18] H.L. ZENG, A fast learning algorithm for solving systems of linear equations and related problems, J. Adv. Modelling Anal., 29 (1995), 67–71.

[19] H.L. ZENG, Stability analysis on continuous neural networks with slowly synapse-varying struc- tures, Proceedings of IEEE: International Sympostium on Circuits and Systems, United States of America, 1992, pp. 443–447.