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volume 7, issue 3, article 91, 2006.

Received 05 March, 2006;

accepted 19 March, 2006.

Communicated by:R.U. Verma

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Journal of Inequalities in Pure and Applied Mathematics

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 of Mathematics

Sichuan University of Science and Engineering Zigong, Sichuan 643000, P.R. China.

EMail:hengyoulan@163.com

2000c Victoria University ISSN (electronic): 1443-5756 093-06

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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

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Abstract

The purpose of this paper is to introduce and study a new kind of generalized strongly nonlinear operator inclusion problems involving generalized m- accretive mapping in Banach spaces. By using the resolvent operator technique for generalizedm-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].

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

Key words: Generalizedm-accretive mapping; Generalized strongly nonlinear operator inclusion problems; Perturbed iterative algorithm with errors; Exis- tence; Convergence and stability.

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

The authors thank Prof. Ram U. Verma and Y. J. Cho for their valuable suggestions.

1. Introduction

Let X be a real Banach space and T : X → 2^X is a multi-valued operator, where2^X denotes the family of all the nonempty subsets ofX. The following operator inclusion problem of findingx∈Xsuch that

(1.1) 0∈T(u)

has been studied extensively because of its role in the modelization of unilat- eral 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 assumption is that T is a maximal monotone operator orm-accretive operator. WhenT is maximal monotone or m-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 both F and M 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).

If M = ∂ϕ, where ∂ϕ is the subdifferential of a proper convex lower semi- continuous functional ϕ :X → R∪ {+∞}, then the problem (1.2) reduces to the variational inequality problem:

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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) when F is strongly monotone. Recently, the problem (1.2) was studied by several authors when F and M need not to be maximal monotone or m-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∈Xandu∈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 functional ϕ(·,·)fort ∈X, which is defined by

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

where ϕ(·, t) : X → R∪ {+∞} is a proper convex lower semi-continuous functional for allt ∈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 generalized m-accretive mapping, which is a generalization of anm-accretive mapping, and gave the definition and properties of the resolvent operator for the generalized

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m-accretive mapping in a Banach space. Later, by using the resolvent operator technique, which is a very important method for finding 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)),

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 and M : X ×X → 2^X is a generalized m-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 generalized m-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

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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.

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2. Generalized m-Accretive Mapping

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

J_q(x) ={x^∗ ∈X^∗ : hx, x^∗i=kxk^q, kx^∗k=kxk^q−1}, ∀x∈X, where q > 1 is a constant. In particular, J2 is the usual normalized duality mapping. It is well known that, in general,J_q(x) = kxk^q−2J₂(x)for allx6= 0 and J_q is single-valued if X^∗ is strictly convex (see, for example, [16]). If X =His a Hilbert space, thenJ2 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 →X is 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. Let h, g : X → X be two single-valued mappings. The map- pingN : X×X×X →X is said to be

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1. σ-strongly accretive with respect tohin the first argument, if for anyx, y ∈ X, there existsjq(x−y)∈Jq(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 to g in the second argument, if for any x, 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;

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

kN(x,·,·)−N(y,·,·)k ≤kx−yk, ∀x, y ∈X.

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

Definition 2.3 ([7]). Let η : X×X → X^∗ be a single-valued mapping and A : 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);

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

Remark 1. Huang and Fang gave one example of the generalizedm-accretive mapping in [7]. If X = X^∗ = H is a Hilbert space, then (1), (2) of Defini- tion2.3reduce to the definition ofη-monotonicity and maximalη-monotonicity respectively; if X is uniformly smooth and η(x, y) = J₂(x−y), then (1) and (2) of Definition2.3reduce 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 of X is 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 if limt→0ρX(t)

t = 0 and X is calledq-uniformly smooth if there exists a constantc > 0such that ρX ≤ct^q, whereq >1is a real number.

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It is well known that Hilbert spaces, L_p (or l_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 2.1. 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 constant c_q > 0such that for all x, y ∈ X, j_q(x) ∈ J_q(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)⁻¹ is single-valued, if η : X ×X → X^∗ is strict monotone andA : X → 2^X is a generalizedm-accretive mapping, whereI is the identity mapping. Based on this fact, Huang and Fang [7] gave the following definition:

Definition 2.5. LetA: 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 2.2 ([7, 8]). Let η : X ×X → X^∗ be τ-Lipschitz continuous and δ- strongly monotone. LetA : X → 2^X be a generalized m-accretive mapping.

Then the resolvent operatorJ_A^ρ forAis Lipschitz continuous with constant ^τ_δ, i.e.,

kJ_A^ρ(x)−J_A^ρ(y)k ≤ τ

δkx−yk, ∀x, y ∈X.

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3. Existence Theorems

In this section, we shall give the existence theorem of problem (1.6). Firstly, from the definition of the resolvent operator for a generalizedm-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(·,g(x))^ρ = (I+ρM(·, g(x)))⁻¹ andρ >0is a constant.

Theorem 3.2. Let X be a q-uniformly smooth Banach space, η : X ×X → X^∗ be τ-Lipschitz continuous and δ-strongly monotone, M : X ×X → 2^X be a generalized m-accretive mapping with respect to the first argument, and mappingsS, T, U : X → X beκ,µ,ν-Lipschitz continuous, respectively. Let p : X → X beα-strongly accretive andβ-Lipschitz continuous,g : X → X beι-Lipschitz continuous,N : X×X×X → Xbeσ-strongly accretive with respect toSin the first argument andς-relaxed accretive with respect toT in the second argument, and, ξ,γ-Lipschitz continuous in the first, second and third argument, respectively. Suppose that there exist constants ρ > 0 and ζ > 0 such 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+c_qβ^q)¹^q <1, τh

(1−qρ(σ−ς) +c_qρ^q(κ+ξµ)^q)¹^q +ργνi

< δ(1−h),

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wherec_q is the same as in Lemma2.1, then problem (1.6) has a unique solution x^∗.

Proof. From Lemma3.1, 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(·,g(x))^ρ [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)].

Then x^∗ 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 Lemma2.2that

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

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≤ 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}

+ζkg(x)−g(y))k.

(3.5)

By the hypothesis ofg,p,S, T, U,Nand Lemma2.1, 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)), jq(x−y)i]

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+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) +c_qρ^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,

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+c_qβ^q)¹^q.

It follows from (3.3) that 0 < θ < 1 and so Q : X → X is a contractive mapping, i.e.,Qhas a unique fixed point inX. This completes the proof.

Remark 2. IfX is a 2-uniformly smooth Banach space and there existsρ > 0 such 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.

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4. Perturbed Algorithm and Stability

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. Let S be a selfmap of X, x0 ∈ X, and let xn+1 = h(S, xn) define an iteration procedure which yields a sequence of points {x_n}^∞_n=0 inX.

Suppose that {x ∈ X : Sx = x} 6= ∅and{x_n}^∞_n=0 converges to a fixed point x^∗ ofS. Let{un} ⊂X and letn =kun+1−h(S, un)k. Iflimn = 0 implies 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 {an},{bn},{cn} 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, where tn ∈ [0,1], P∞

n=0tn = ∞, limn→∞bn = 0, P∞

n=0cn < ∞. Then a_n→0(n→ ∞).

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

Algorithm 1. Step 1. Choosex₀ ∈X.

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Step 2. Let

(4.1)











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

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(·,g(x^ρ

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 for n ≥0,{α_n},{β_n}are two sequences in[0,1],{u_n},{v_n},{ω_n}are sequences inXsatisfying the following conditions:

(i) u_n=u⁰_n+u⁰⁰_n;

(ii) limn→∞ku⁰_nk= limn→∞kvnk= 0;

(iii) P∞

n=0ku⁰⁰_nk<∞, P∞

n=0kω_nk<∞,

Step 4. Ifx_n+1,y_n,α_n,β_n,u_n,v_nandω_nsatisfy (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 inX and 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(·,g(t^ρ

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(·,g(z^ρ

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

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Step 6. Ifε_n, z_n+1, t_n,α_n,β_n, u_n, v_n andω_n satisfy (4.2) to sufficient accu- racy, 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 Theorem3.2,θis defined by (3.11). IfP∞

n=0α_n=∞and conditions (3.2), (3.3) hold, then the perturbed iterative sequence {xn} 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 iflimn→∞ε_n= 0, where εnis defined by (4.2).

Proof. From Theorem 3.2, we know that problem (1.6) has a unique solution x^∗ ∈ X. It follows from (4.1), (3.11) and the proof of (3.10) in Theorem3.2 that

kxn+1−x^∗k

≤(1−α_n)kx_n−x^∗k+α_nθky_n−x^∗k+α_n(ku⁰_nk+ku⁰⁰_nk) +kω_nk

≤(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

+α_n(ku⁰_nk+θkv_nk) + (ku⁰⁰_nk+kω_nk).

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Sinceθ < 1, 0 < β_n ≤ 1 (n ≥ 0), we have1−β_n+β_nθ < 1and1−θ(1− βn+βnθ)>1−θ >0. Therefore, (4.5) implies

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

1−θ(ku⁰_nk+θkv_nk) + (ku⁰⁰_nk+kω_nk).

SinceP∞

n=0α_n = ∞, it follows from Lemma4.1and (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) kz_n+1−x^∗k ≤ k(1−α_n)z_n+α_n[t_n−p(t_n)

+J_M(·,g(t^ρ

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

+α_nu_n+ω_n−x^∗k+ε_n. 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).

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Since0< a≤α_n, it follows from (4.7) and (4.8) that kzn+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+ ε_n a

+ (ku⁰⁰_nk+kω_nk).

Suppose thatlimε_n = 0. Then fromP∞

n=0α_n =∞and Lemma4.1, we have limz_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}

≤ kz_n+1−x^∗k+k(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−x^∗

≤ kz_n+1−x^∗k+ [1−α_n(1−θ)]kz_n−x^∗k

+α_n(ku⁰_nk+θkv_nk) + (ku⁰⁰_nk+kω_nk)→0 (n→ ∞).

This completes the proof.

Remark 3. Ifu_n =v_n =ω_n = 0 (n≥0)in Algorithm1, 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].

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