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

Volume 7, Issue 3, Article 114, 2006

GENERALIZED NONLINEAR MIXED QUASI-VARIATIONAL INEQUALITIES INVOLVING MAXIMALη-MONOTONE MAPPINGS

MAO-MING JIN

DEPARTMENT OFMATHEMATICS

FULINGTEACHERSCOLLEGE

FULING, CHONGQING408003 PEOPLESREPUBLIC OFCHINA

mmj1898@163.com

Received 27 July, 2004; accepted 09 May, 2006 Communicated by S.S. Dragomir

ABSTRACT. In this paper, we introduce and study a new class of generalized nonlinear mixed quasi-variational inequalities involving maximalη-monotone mapping. Using the resolvent op- erator technique for maximalη-monotone mapping, we prove the existence of solution for this kind of generalized nonlinear multi-valued mixed quasi-variational inequalities without com- pactness and the convergence of iterative sequences generated by the new algorithm. We also discuss the convergence and stability of the iterative sequence generated by the perturbed itera- tive algorithm for solving a class of generalized nonlinear mixed quasi-variational inequalities.

Key words and phrases: Mixed quasi-variational inequality, Maximal η-monotone mapping, Resolvent operator, Conver- gence, stability.

2000 Mathematics Subject Classification. 49J40; 47H10.

1. INTRODUCTION

In recent years, variational inequalities have been generalized and extended in many differ- ent directions using novel and innovative techniques. These have been used to study wider classes of unrelated problems arising in optimization and control, economics and finance, trans- portation and electrical networks, operations research and engineering sciences in a general and unified framework, see [1] – [15], [18] – [27] and the references therein. An important and use- ful generalization of variational inequality is called the variational inclusion. It is well known that one of the most important and interesting problems in the theory of variational inequalities is the development of an efficient and implementable algorithm for solving variational inequal- ities. For the past years, many numerical methods have been developed for solving various classes of variational inequalities, such as the projection method and its variant forms, linear approximation, descent, and Newton’s methods.

ISSN (electronic): 1443-5756 c

2006 Victoria University. All rights reserved.

This work was supported by the National Natural Science Foundation of China (69903012) and the Educational Science Foundation of Chongqing, Chongqing of China (021301).

144-04

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Recently, Huang and Fang [10] introduced a new class of maximal η-monotone mappings and proved the Lipschitz continuity of the resolvent operator for maximal η-monotone map- pings in Hilbert spaces. They also introduced and studied a new class of generalized variational inclusions involving maximalη-monotone mappings and constructed a new algorithm for solv- ing this class of generalized variational inclusions by using the resolvent operator technique for maximalη-monotone mappings.

The main purpose of this paper is to introduce and study a new class of generalized nonlinear mixed quasi-variational inequalities involving maximalη-monotone mappings. Using the resol- vent operator technique for maximalη-monotone mappings, we prove the existence of a solution for this kind of generalized nonlinear multivalued mixed quasi-variational inequalities without compactness and the convergence of iterative sequences generated by the new algorithm. We also discuss the convergence and stability of the iterative sequence generated by the perturbed iterative algorithm for solving a class of generalized nonlinear mixed quasi-variational inequal- ities. The results shown in this paper improve and extend the previously known results in this area.

2. PRELIMINARIES

LetH be a real Hilbert space endowed with a normk·kand an inner product h·,·i, respec- tively. Let 2H, CB(H), and H(·,·)denote the family of all the nonempty subsets of H, the family of all the nonempty closed bounded subsets ofH, and the Hausdorff metric onCB(H), respectively. Let η, N : H×H → H be two single-valued mappings with two variables and g : H → H be a single-valued mapping. Let S, T, G : H → CB(H) be three multivalued mappings andM :H×H → 2H be a multivalued mapping such that for eacht ∈H, M(·, t) is maximal η-monotone with Ran(g)T

DomM(·, t) 6= ∅. Now we consider the following problem:

Findu∈H, x∈Su, y ∈T u, andz ∈Gusuch thatg(u)∈Dom(M(·, z))and

(2.1) 0∈N(x, y) +M(g(u), z)).

Problem (2.1) is called a generalized nonlinear multivalued mixed quasi-variational inequality.

Some special cases of the problem (2.1):

(I) Ifη(x, y) = x−yfor allx, y inH andGis the identity mapping, then problem (2.1) reduces to findingu∈H, x∈Su, y ∈T usuch thatg(u)∈Dom(M(·, u))and

(2.2) 0∈N(x, y) +M(g(u), u).

Problem (2.2) is called the multivalued quasi-variational inclusion, which was studied by Noor [18] – [22].

(II) IfS, T are single-valued mappings andGis the identity mapping, then problem (2.1) is equivalent to findingu∈H such thatg(u)∈Dom(M(·, u))and

(2.3) 0∈N(Su, T u) +M(g(u), u)).

Problem (2.3) is called a generalized nonlinear mixed quasi-variational inequality.

(III) IfM(·, t) = ∂ϕ(·, t), whereϕ :H×H →RS{+∞}is a functional such that for each fixedtinH,ϕ(·, t) :H→RS{+∞}is lower semicontinuous andη-subdifferentiable on H, and∂ϕ(·, t)denotes the η-subdifferential ofϕ(·, t), then problem (2.1) reduces to the following problem:

Findu∈H, x∈Suandy∈T usuch that

(2.4) hN(x, y), η(v, g(u))i ≥ϕ(g(u), z)−ϕ(v, z)

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for allv inH, which which appears to be a new one. Furthermore, ifN(x, y) = x−y for all x, y inH, S, T are single-valued mappings andGis the identity mapping, then problem (2.4) reduces to the general quasi-variational-like inclusion considered by Ding and Luo [3].

(IV) IfS, T :H →Hare single-valued mappings,gis an identity mapping,N(x, y) =x−y for allx, yinH, andM(·, t) = ∂ϕfor alltinH, where∂ϕdenotes theη-subdifferential of a proper convex lower semicontinuous functionϕ :H → RS

{+∞}, then problem (2.1) reduces to the following problem:

Findu∈H such that

(2.5) hSu−T u, η(v, u)i ≥ϕ(u)−ϕ(v)

for allvinH, which is called the strongly nonlinear variational-like inclusion problem considered by Lee et al. [15].

(V) IfGis an identity mapping,η(x, y) = x−yandM(·, t) = ∂ϕfor eacht ∈ H, where ϕ : H → RS

{+∞} is a proper convex lower semicontinuous function on H and g(H)T

Dom(∂ϕ(·, t)) 6= ∅for eacht ∈ H and∂ϕ(·, t)denotes the subdifferential of functionϕ(·, t), then problem (2.1) reduces to findingu∈H, x∈Suandy∈T usuch thatg(u)∈Dom(∂ϕ(·, t))and

(2.6) hN(x, y), v−g(u)i ≥ϕ(g(u))−ϕ(v)

for all v inH. Furthermore, if N(x, y) = x−y for allx, y in H, andg is an identity mapping, then the problem (2.6) is equivalent to the set-valued nonlinear generalized variational inclusion considered by Huang [6] and, in turn, includes the variational in- clusions studied by Hassouni and Moudafi [5] and Kazmi [14] as special cases.

For a suitable choice ofN, η, M, S, T, G, g, and for the space H, one can obtain a number of known and new classes of variational inclusions, variational inequalities, and corresponding optimization problems from the general set-valued variational inclusion problem (2.1). Further- more, these types of variational inclusions enable us to study many important problems arising in the mathematical, physical, and engineering sciences in a general and unified framework.

Definition 2.1. Let T be a selfmap ofH, x0 ∈ H and letxn+1 = f(T, xn)define an iteration procedure which yields a sequence of points{xn}n=0inH. Suppose that{x∈H :T x=x} 6=

∅and{xn}n=0converges to a fixed pointxofT. Let{yn} ⊂Hand letn=||yn+1−f(T, yn)||.

If lim

n→∞n = 0 implies that lim

n→∞yn = x, then the iteration procedure defined by xn+1 = f(T, xn)is said to beT-stable or stable with respect toT.

Lemma 2.1 ([16]). Let{an},{bn}, and{cn}be three sequences of nonnegative numbers satis- fying the following conditions: there existsn0 such that

an+1 ≤(1−tn)an+bntn+cn, for all n ≥ n0, where tn ∈ [0,1], P

n=0tn = ∞, lim

n→∞bn = 0 and P

n=0cn < ∞. Then

n→∞lim an = 0.

Definition 2.2. A mappingg :H →His said to be

(i) α-strongly monotone if there exists a constantα >0such that hg(u1)−g(u2), u1−u2i ≥αku1 −u2k2, for allui ∈H, i= 1,2;

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(ii) β-Lipschitz continuous if there exists a constantβ >0such that kg(u1)−g(u2)k ≤βku1−u2k,

for allui ∈H, i= 1,2.

Definition 2.3. A multivalued mappingS:H →CB(H)is said to be (i) H-Lipschitz continuous if there exists a constantγ >0such that

H(Su1, Su2)≤γku1−u2k, for allui ∈H, i= 1,2;

(ii) strongly monotone with respect to the first argument ofN(·,·) : H×H → H, if there exists a constantµ >0such that

hN(x1,·)−N(x2,·), u1−u2i ≥µku1−u2k2, for allxi ∈Sui, ui ∈H, i = 1,2.

Definition 2.4. A mappingN(·,·) :H×H →His said to be Lipschitz continuous with respect to the first argument if there exists a constantν >0such that

kN(u1,·)−N(u2,·)k ≤νku1−u2k, for allui ∈H, i= 1,2.

In a similar way, we can define Lipschitz continuity of N(·,·) with respect to the second argument.

Definition 2.5. Let η : H ×H → H be a single-valued mapping. A multivalued mapping M :H →2H is said to be

(i) η-monotone if

hx−y, η(u, v)i ≥0 for allu, v ∈H, x∈M u, y ∈M v;

(ii) strictlyη-monotone if

hx−y, η(u, v)i ≥0 for allu, v ∈H, x∈M u, y ∈M v and equality holds if and only ifu=v;

(iii) stronglyη-monotone if there exists a constantr >0such that

hx−y, η(u, v)i ≥rku−vk2 for allu, v ∈H, x∈M u, y ∈M v;

(iv) maximalη-monotone ifM isη-monotone and(I+λM)(H) = Hfor anyλ >0.

Remark 2.2.

(1) Ifη(u, v) = u−v for allu, v inH, then (i)-(iv) of Definition 2.5 reduce to the classi- cal definitions of monotonicity, strict monotonicity, strong monotonicity, and maximal monotonicity, respectively.

(2) Huang and Fang gave one example of maximalη-monotone mapping in [10].

Lemma 2.3 ([10]). Letη :H×H →Hbe strictly monotone andM :H →2H be a maximal η-monotone mapping. Then the following conclusions hold:

(1) hx−y, η(u, v)i ≥0for all(v, y)∈Graph(M)implies that(u, x)∈Graph(M), where Graph(M) = {(u, x)∈H×H :x∈M u};

(2) the inverse mapping(I+λM)−1is single-valued for anyλ >0.

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Based on Lemma 2.3, we can define the resolvent operator for a maximalη-monotone map- pingM as follows:

(2.7) JρM(z) = (I+ρM)−1(z) for allz∈H,

whereρ >0is a constant andη:H×H →His a strictly monotone mapping.

Lemma 2.4 ([10]). Letη : H×H → H be strongly monotone and Lipschtiz continuous with constantsδ >0andτ >0, respectively. LetM :H →2H be a maximalη-monotone mapping.

Then the resolvent operatorJρM forM is Lipschitz continuous with constantτ /δ, i.e., kJρM(u)−JρM(v)k ≤ τ

δku−vk for allu, v ∈H.

3. ITERATIVEALGORITHMS

We first transfer problem (2.1) into a fixed point problem.

Lemma 3.1. For givenu ∈ H, x ∈ Su, y ∈ T u, andz ∈ Gu,(u, x, y, z)is a solution of the problem (2.1) if and only if

(3.1) g(u) =JρM(·,z)(g(u)−ρN(x, y)), whereJρM(·,z) = (I+ρM(·, z))−1andρ >0is a constant.

Proof. This directly follows from the definition ofJρM(·,u). Remark 3.2. Equality (3.1) can be written as

u= (1−λ)u+λ[u−g(u) +JρM(·,z)(g(u)−ρN(x, y))],

where0< λ ≤1is a parameter andρ > 0is a constant. This fixed point formulation enables us to suggest the following iterative algorithm for problem (2.1) as follows:

Algorithm 3.1. Letη, N :H×H →H, g :H →H be single-valued mappings andS, T, G: H → CB(H) be multivalued mappings. Let M : H ×H → 2H be such that for each fixed t∈H,M(·, t) :H→2H is a maximalη-monotone mapping satisfyingg(u)∈Dom(M(·, z)).

For givenλ∈[0,1],u0 ∈H, x0 ∈Su0, y0 ∈T u0 andz0 ∈Gu0, let u1 = (1−λ)u0

u0−g(u0) +JρM(·,z0)(g(u0)−ρN(x0, y0)) . By Nadler [17], there existx1 ∈Su1, y1 ∈T u1andz1 ∈Gu1such that

kx0−x1k ≤(1 + 1)H(Su0, Su1), ky0−y1k ≤(1 + 1)H(T u0, T u1), kz0−z1k ≤(1 + 1)H(Gu0, Gu1).

Let

u2 = (1−λ)u1

u1−g(u1) +JρM(·,z1)(g(u1)−ρN(x1, y1)) . By induction, we can obtain sequences{un},{xn},{yn}and{zn}satisfying

(3.2)









un+1 = (1−λ)un+λh

un−g(un) +JρM(·,zn)(g(un)−ρN(xn, yn))i , xn∈Sun, kxn−xn+1k ≤(1 + (1 +n)−1)H(Sun, Sun+1),

yn∈T un, kyn−yn+1k ≤(1 + (1 +n)−1)H(T un, T un+1), zn ∈gun, kzn−zn+1k ≤(1 + (1 +n)−1)H(Gun, Gun+1), forn = 1,2,3, . . . ,where0< λ≤1andρ >0are both constants.

Now we construct a new pertured iterative algorithm for solving the generalized nonlinear mixed quasi-variational inequality (2.3) as follows:

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Algorithm 3.2. Let η, N : H×H → H andS, T : H → H be single-valued mappings. Let M : H ×H → 2H be such that for each fixed t ∈ H, M(·, t) : H → 2H is a maximalη- monotone mapping satisfyingg(u)∈Dom(M(·, u)). For givenu0 ∈H, the perturbed iterative sequence{un}is defined by

(3.3)

un+1= (1−αn)unn[vn−g(vn) +JρM(·,vn)(g(vn)−ρN(Svn, T vn))] +αnen, vn = (1−βn)unn[un−g(un) +JρM(·,un)(g(un)−ρN(Sun, T un))] +βnfn, forn = 0,1,2, . . . ,where {en}and {fn}are two sequences of the elements ofH introduced to take into account possible inexact computation and the sequences {αn}, {βn} satisfy the following conditions:

0≤αn, βn≤1(n≥0), and

X

n=0

αn=∞.

Let{yn}be any sequence inHand define{n}by

(3.4)











 n =

yn+1−n

(1−αn)ynnh

xn−g(xn) +JρM(·,xn)(g(xn)−ρN(Sxn, T xn))i

neno , xn= (1−βn)ynnh

yn−g(yn) +JρM(·,yn)(g(yn)−ρN(Syn, T yn))i

nfn, forn = 0,1,2, . . . .

4. MAINRESULTS

In this section, we first prove the existence of a solution of problem (2.1) and the convergence of an iterative sequence generated by Algorithm 3.1.

Theorem 4.1. Let η : H ×H → H be strongly monotone and Lipschitz continuous with constants δ andτ, respectively. Let S, T, G : H → CB(H) beH-Lipschitz continuous with constantsα, β, γ, respectively,g :H →Hbeµ-Lipschitz continuous andν-strongly monotone.

LetN :H×H →Hbe Lipschitz continuous with respect to the first and second arguments with constantsξandζ, respectively, andS :H →CB(H)be strongly monotone with respect to the first argument ofN(·,·)with constantr. LetM :H×H →2H be a multivalued mapping such that for each fixed t ∈ H, M(·, t)is maximalη-monotone. Suppose that there exist constants ρ >0andκ >0such that for eachx, y, z ∈H,

(4.1)

JρM(·,x)(z)−JρM(·,y)(z)

≤κkx−yk, and

(4.2)













ρ− τ r−δ(1−h)ζβ τ(ξ2α2−ζ2β2)

<

[τ r−δ(1−h)ζβ]2−(ξ2α2−ζ2β2)(τ2−δ2(1−h)2) τ(ξ2α2−ζ2β2) , τ r > δ(1−h)ζβ+p

2α2−ζ2β2)(τ2−δ2(1−h)2), ξα > ζβ, h= (1 +δτ−1)p

1−2ν+µ2+κγ, ρτ ζβ < δ(1−h), h <1.

Then the iterative sequences{un},{xn},{yn}and{zn} generated by Algorithm 3.1 converge strongly tou, x, y andz, respectively and(u, x, y, z)is a solution of problem (2.1).

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Proof. It follows from (3.2), (4.1) and Lemma 2.4 that kun+1−unk

=

(1−λ)(un−un−1) +λ

[un−un−1−(g(un)−g(un−1))

+JρM(·,zn)(g(un)−ρN(xn, yn))−JρM(·,zn−1)(g(un−1)−ρN(xn−1, yn−1))

≤(1−λ)kun−un−1k+λkun−un−1−(g(un+1)−g(un))k +λ

JρM(·,zn)(g(un)−ρN(xn, yn))−JρM(·,zn)(g(un−1)−ρN(xn−1, yn−1)) +λ

JρM(·,zn)(g(un−1)−ρN(xn−1, yn−1))

− JρM(·,zn−1)(g(un−1)−ρN(xn−1, yn−1))

≤(1−λ)kun−un−1k+λkun−un−1−(g(un)−g(un−1))k +λτ

δkg(un)−g(un−1)−ρ(N(xn, yn)−N(xn−1, yn−1))k+λκkzn−zn−1k

≤(1−λ)kun−un−1k+λ 1 + τ

δ

kun−un−1−(g(un)−g(un−1))k +λτ

δkun−un−1 −ρ(N(xn, yn)−N(xn−1, yn))k +λρτ

δkN(xn−1, yn)−N(xn−1, yn−1)k+λκkzn−zn−1k.

(4.3)

Sinceg is strongly monotone and Lipschitz continuous, we obtain kun−un−1−(g(un)−g(un−1))k2

=kun−un−1k2−2hun−un−1, g(un)−g(un−1)i+kg(un)−g(un−1)k2

≤(1−2ν+µ2)kun−un−1k2. (4.4)

SinceS isH-Lipschitz continuous and strongly monotone with respect to the first argument of N(·,·)andN is Lipschitz continuous with respect to the first argument, we have

kun−un−1−ρ(N(xn, yn)−N(xn−1, yn))k2

=kun−un−1k2−2ρhun−un−1, N(xn, yn)−N(xn−1, yn)i +ρ2kN(xn, yn)−N(xn−1, yn)k2

≤(1−2ρr+ρ2ξ2(1 +n−1)2α2)kun−un−1k2. (4.5)

Further, sinceT, G areH-Lipschitz continuous and N is Lipschitz continuous with respect to the second argument, we get

kN(xn−1, yn)−N(xn−1, yn−1)k ≤ζkyn−yn−1k ≤ζβ(1 +n−1)kun−un−1k, (4.6)

kzn−zn−1k ≤γ(1 +n−1)kun−un−1k.

(4.7)

By (4.3) – (4.7), we obtain

kun−un−1k ≤(1−λ+λ(1 +τ δ−1)p

1−2ν+µ2 +λτ δ−1p

1−2ρr+ρ2ξ2(1 +n−1)2α2 +λρτ δ−1ζβ(1 +n−1) +λκγ(1 +n−1)

= (1−λ+λhn+λtn(ρ))kun−un−1k

nkun−un−1k, (4.8)

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where

hn= (1 +τ δ−1)p

1−2ν+µ2+κγ(1 +n−1), tn(ρ) =τ δ−1p

1−2ρr+ρ2ξ2(1 +n−1)2α2+ρτ δ−1ζβ(1 +n−1) and θn= 1−λ+λhn+λtn(ρ).

Lettingθ = 1−λ+λh+λt(ρ),where h= (1 +τ δ−1)p

1−2ν+µ2 +κγ and t(ρ) = τ δ−1p

1−2ρr+ρ2ξ2α2 +ρτ δ−1ζβ, we have thathn →h, tn(ρ)→t(ρ)andθn →θasn→ ∞. It follows from condition (4.2) that θ < 1. Henceθn < 1forn sufficiently large. Therefore, (4.8) implies that{un}is a Cauchy sequence inHand so we can assume thatun→u ∈Hasn→ ∞. By the Lipschitz continuity ofS, T andGwe obtain

kxn−xn−1k ≤(1 + (1 +n)−1)H(Sun, Sun−1)≤α(1 + (1 +n)−1)kun−un−1k, kyn−yn−1k ≤(1 + (1 +n)−1)H(T un, T un−1)≤β(1 + (1 +n)−1)kun−un−1k, kzn−zn−1k ≤(1 + (1 +n)−1)H(Gun, Gun−1)≤γ(1 + (1 +n)−1)kun−un−1k.

It follows that {xn},{yn} and {zn} are also Cauchy sequences in H. We can assume that xn→x, yn →yandzn→z, respectively. Note that forxn ∈Sun, we have

d(x, Su)≤ kx−xnk+d(xn, Su)

≤ kx−xnk+H(Sun, Su)

≤ kx−xnk+αkun−uk →0,

as n → ∞. Hence we must have x ∈ Su. Similarly, we can show that y ∈ T u and z ∈Gu. From

un+1 = (1−λ)un

un−g(un) +JρM(·,zn)(g(un)−ρN(xn, yn)) , it follows that

g(u) =JρM(·,z)(g(u)−ρN(x, y)).

By Lemma 3.1,(u, x, y, z)is a solution of problem (2.1). This completes the proof.

Remark 4.2. For an appropriate and suitable choice of the mappings η, N, S, T, G, g, M and the spaceH, we can obtain several known results in [1], [3], [5] – [8], [14], [18] – [22], [24] – [26] as special cases of Theorem 4.1.

Now we prove the convergence and stability of the iterative sequence generated by the Algo- rithm 3.2.

Theorem 4.3. Let η : H ×H → H be strongly monotone and Lipschitz continuous with constants δ and τ, respectively. Let S, T : H → H be Lipschitz continuous with constants α, β, respectively, g : H → H beµ-Lipschitz continuous andν-strongly monotone. LetN : H ×H → H be Lipschitz continuous with respect to the first and second arguments with constantsξandζ, respectively, andS :H →H be strongly monotone with respect to the first argument ofN(·,·)with constantr. LetM :H×H →2H be a multivalued mapping such that for each fixedt ∈H,M(·, t)is maximalη-monotone. Suppose that there exist constantsρ > 0 andκ >0such that for eachx, y, z ∈H,

(4.9)

JρM(·,x)(z)−JρM(·,y)(z)

≤κkx−yk,

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and

(4.10)













ρ− τ r−δ(1−h)ζβ τ(ξ2α2−ζ2β2)

<

[τ r−δ(1−h)ζβ]2−(ξ2α2−ζ2β2)(τ2−δ2(1−h)2) τ(ξ2α2−ζ2β2) , τ r > δ(1−h)ζβ+p

2α2−ζ2β2)(τ2−δ2(1−h)2), ξα > ζβ, h= (1 +δτ−1)p

1−2ν+µ2+κ, ρτ ζβ < δ(1−h), h <1.

If lim

n→∞kenk= 0, lim

n→∞kfnk= 0, then

(I) The sequence{un}defined by Algorithm 3.2 converges strongly to the unique solution u of problem (2.3).

(II) IfP

n=0n <∞, then lim

n→∞yn =u. (III) If lim

n→∞yn =u, then lim

n→∞n= 0.

Proof. (I) It follows from Theorem 4.1 that there existsu ∈H which is a solution of problem (2.3) and so

(4.11) g(u) = JρM(·,u)(g(u)−ρN(Su, T u)).

From (4.9), (4.11) and Algorithm 3.2, it follows that kun+1−uk

=

(1−αn)(un−u)−αnh

vn−u−(g(vn)−g(u)) +JρM(·,vn)(g(vn)−ρN(Svn, T vn))

−JρM(·,u)(g(u)−ρN(Su, T u))i

nen

≤(1−αn)kun−uk+αnkvn−u−(g(vn)−g(u))k+αnkenk +αn

JρM(·,vn)(g(vn)−ρN(Svn, T vn))−JρM(·,vn)(g(u)−ρN(Su, T u)) +αn

JρM(·,vn)(g(u)−ρN(Su, T u))−JρM(·,u)(g(u)−ρN(Su, T u))

≤(1−αn)kun−uk+αnkvn−u−(g(vn)−g(u))k+αnkenk +αnτ

δkg(vn)−g(u)−ρ(N(Svn, T vn)−N(Su, T u))k+αnκkvn−uk

≤(1−αn)kun−uk+αn 1 + τ

δ

kun−u−(g(vn)−g(u))k+αnkenk +αnτ

δkvn−u−ρ(N(Svn, T vn)−N(Su, T vn))k +αnρτ

δkN(Su, T vn)−N(Su, T u)k+αnκkvn−uk.

(4.12)

By the Lipschitz continuity ofN, S, T, gand the strong monotonicity ofSandg, we obtain kvn−u−(g(vn)−g(u))k2 ≤(1−2ν+µ2)kvn−uk2,

(4.13)

kvn−u−ρ(N(Svn, T vn)−N(Su, T vn))k2 ≤(1−2ρr+ρ2ξ2α2)kvn−uk2, (4.14)

kN(Su, T vn)−N(Su, T u))k ≤ζβkvn−uk.

(4.15)

It follows from (4.12) – (4.15) that

(4.16) kun+1−uk ≤(1−αn)kun−uk+θαnkvn−uk+αnkenk,

(10)

where

θ =κ+ (1 +τ δ−1)p

1−2ν+µ2+τ δ−1p

1−2ρr+ρ2ξ2α2+ρτ δ−1ζβ.

Similarly, we have

(4.17) kvn−uk ≤(1−βn)kun−uk+θβnkun−uk+βnkfnk.

From (4.16) and (4.17), we have

kun+1−uk ≤[1−αn(1−θ)(1 +θβn)]kun−uk+αnβnθkfnk+αnkenk Condition (4.10) implies that0< θ <1, and so

(4.18) kun+1−uk ≤[1−αn(1−θ)]kun−uk+αn(1−θ)dn,

wheredn = (βnθkfnk+kenk)(1−θ)−1 → 0, as n → ∞. It follows from (4.18) and Lemma 2.1 thatun →uasn→ ∞.

Now we prove thatu is a unique solution of problem (2.3). In fact, ifuis also a solution of problem (2.3), then

g(u) =JρM(·,u)(g(u)−ρN(Su, T u)), and, as the proof of (4.16), we have

ku −uk ≤θku−uk,

where0< θ <1and sou =u. This completes the proof of Conclusion (I).

Next we prove Conclusion (II). Using (3.4) we obtain kyn+1−uk

yn+1

(1−αn)ynn

xn−g(xn) + JρM(·,xn)(g(xn)−ρN(Sxn, T xn))

nen +

(1−αn)ynn

xn−g(xn) + JρM(·,xn)(g(xn)−ρN(Sxn, T xn))

nen−u

=

(1−αn)ynn

xn−g(xn) (4.19)

+ JρM(·,xn)(g(xn)−ρN(Sxn, T xn))

nen−u +n. As the proof of inequality (4.18), we have

(4.20)

(1−αn)ynn

xn−g(xn) +JρM(·,xn)(g(xn)−ρN(Sxn, T xn))

nen−u

≤(1−αn(1−θ))kyn−uk+αn(1−θ)dn. It follows from (4.19) and (4.20) that

(4.21) kyn+1−uk ≤(1−αn(1−θ))kyn−uk+αn(1−θ)dn+n. SinceP

n=0n < ∞, dn → 0, andP

n=0αn <∞. It follows that (4.21) and Lemma 2.1 that

n→∞lim yn=u.

Now we prove Conclusion (III). Suppose that lim

n→∞yn =u. Then we have n =

yn+1−(1−αn)ynn xn

−g(xn) + JρM(·,xn)(g(xn)−ρN(Sxn, T xn))

nen

≤ kyn+1−uk+

(1−αn)ynn xn

−g(xn) + JρM(·,xn)(g(xn)−ρN(Sxn, T xn))

nen−u

≤ kyn+1−uk+ (1−αn(1−θ))kyn−uk+αn(1−θ)dn→0

(11)

asn → ∞. This completes the proof.

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