# On a System of Nonlinear Variational Inclusions withHh,η-Monotone Operators

Authors: Zeqing Liu; Jeong Sheok Ume; Shin Min Kang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is concerned mainly with the existence and iterative approximation of solutions for a system of nonlinear variational inclusions involving the stronglyHh,η-monotone operators in Hilbert spaces. The results presented in this paper extend, improve, and unify many known results in the literature.


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## Body

## 1. Introduction

Recently, a few authors introduced and studied several classes of systems of nonlinear variational inequalities and inclusions in Hilbert spaces and established the existence of solutions or the approximate solutions for these systems of nonlinear variational inequalities and inclusions [1–19]. Using projection methods, Liu et al. [12], Verma [14–17], Rhoades and Verma [18], and Wu et al. [19] suggested some iterative algorithms for approximating the solutions of several classes of systems of variational inequalities involving relaxed monotone operators, strongly monotone operators, relaxed cocoercive operators, and pseudocontractive operators, respectively. Applying the resolvent operator techniques, Liu et al. [11] and Nie et al. [13] discussed the existence and uniqueness of solutions and suggested iterative algorithms for a system of general quasivariational-like inequalities and a system of nonlinear variational inequalities, respectively, and gave the convergence analysis for the iterative algorithms. Utilizing the resolvent operator method associated with (H,η)-monotone operators, Fang et al. [5] investigated the existence and uniqueness of solutions for a class of system of variational inclusions in Hilbert spaces, constructed an iterative algorithm for approximating the solution of this system of variational inclusions, and discussed the convergence of iterative sequences generated by the algorithm.Motivated and inspired by the results [1–19], in this paper, we introduce two new classes of strictly Hh,η-monotone and strongly Hh,η-monotone operators and investigate the existence and Lipschitz continuity of the resolvent operators with respect to the strictly Hh,η-monotone and strongly Hh,η-monotone operators. We introduce also a class of system of nonlinear variational inclusions involving strongly Hh,η-monotone operators in Hilbert spaces and suggest two new iterative algorithms for approximating solutions of the system of nonlinear variational inclusions by using the resolvent operator technique associated with Hh,η-monotone operators. The convergence criteria of iterative sequences generated by the algorithms are established. The results presented in this paper extend, improve, and unify some known results in the literature.This paper is organized as follows. In Section2, we recall and introduce some definitions, notation, and a lemma. In Section 3, we study properties of the resolvent operators with respect to strictly Hh,η-monotone and strongly Hh,η-monotone operators, respectively, in Hilbert spaces. In Section 4, we introduce a new class of system of nonlinear variational inclusions in Hilbert spaces and use the resolvent operator technique with respect to strongly Hh,η-monotone operators to investigate the existence and uniqueness of solution and suggest two new iterative algorithms for the system of nonlinear variational inclusions. The convergence criteria of the sequences generated by the iterative algorithms are also given under certain conditions.


## 2. Preliminaries and Lemmas

In this section, we recall and introduce some notation, definitions, and a lemma, which will be used in this paper. LetH be a real Hilbert space endowed with an inner product 〈·,·〉 and norm ∥·∥, respectively. I stands for the identity mapping on H and ℝ=(-∞,+∞). Let T:H→2H be a set-valued operator. The graph of T, denoted by Graph(T), is defined as follows:
(2.1)Graph(T)={(x,y):x∈H,y∈T(x)}.Definition 2.1.
Letη:H×H→H and M:H→2H be operators. The operator M is said to be(1)
monotone if(2.2)〈u-v,x-y〉≥0,∀x,y∈H,u∈M(x),v∈M(y);(2)
η-monotone if
(2.3)〈u-v,η(x,y)〉≥0,∀x,y∈H,u∈M(x),v∈M(y);(3)
strictly monotone if(2.4)〈u-v,x-y〉>0,∀x,y∈H,x≠y,u∈M(x),v∈M(y);(4)
strictlyη-monotone if
(2.5)〈u-v,η(x,y)〉>0,∀x,y∈H,x≠y,u∈M(x),v∈M(y);(5)
strongly monotone if there exists a constantr>0 satisfying
(2.6)〈u-v,x-y〉≥r∥x-y∥2,∀x,y∈H,u∈M(x),v∈M(y);(6)
stronglyη-monotone if there exists a constant r>0 satisfying
(2.7)〈u-v,η(x,y)〉≥r∥x-y∥2,∀x,y∈H,u∈M(x),v∈M(y);(7)
maximal monotone (resp.,  maximally strictly monotone, maximally strongly monotone) ifM is monotone (resp., strictly monotone, strongly monotone) and (I+λM)(H)=H for any λ>0;(8)
maximalη-monotone (resp., maximally strictly η-monotone, maximally strongly η-monotone) if M is η-monotone (resp., strictly η-monotone, strongly η-monotone) and (I+λM)(H)=H for any λ>0;(9)
Hh-monotone (resp., strictly Hh-monotone, strongly Hh-monotone) if M is monotone (resp., strictly monotone, strongly monotone) and (h+λM)(H)=H for any λ>0;(10)
Hh,η-monotone (resp., strictly Hh,η-monotone, strongly Hh,η-monotone) if M is η-monotone (resp., strictly η-monotone, strongly η-monotone) and (h+λM)(H)=H for any λ>0.Definition 2.2.
Letg:H→H and η:H×H→H be operators. The operator g is called(1)
Lipschitz continuous if there exists a constantr>0 satisfying
(2.8)∥g(x)-g(y)∥≤r∥x-y∥,∀x,y∈H;(2)
monotone if(2.9)〈g(x)-g(y),x-y〉≥0,∀x,y∈H;(3)
η-monotone if
(2.10)〈g(x)-g(y),η(x,y)〉≥0,∀x,y∈H;(4)
strongly monotone if there exists a constantr>0 satisfying
(2.11)〈g(x)-g(y),x-y〉≥r∥x-y∥2,∀x,y∈H;(5)
stronglyη-monotone if there exists a constant r>0 satisfying
(2.12)〈g(x)-g(y),η(x,y)〉≥r∥x-y∥2,∀x,y∈H;(6)
relaxed Lipschitz if there exists a constantr>0 satisfying
(2.13)〈g(x)-g(y),x-y〉≤-r∥x-y∥2,∀x,y∈H.Definition 2.3.
LetN:H×H→H and a,g:H→H be operators. The operator N is said to be(1)
strongly monotone with respect toa and g in the first argument if there exists a constant r>0 satisfying
(2.14)〈N(a(x),z)-N(a(y),z),g(x)-g(y)〉≥r∥x-y∥2,∀x,y,z∈H;(2)
Lipschitz continuous in the first argument if there exists a constantr>0 satisfying
(2.15)∥N(x,w)-N(y,w)∥≤r∥x-y∥,∀x,y,w∈H.
Similarly, we could define the strong monotonicity ofN with respect to a and g in the second argument and the Lipschitz continuity of N in the second argument.Remark 2.4.
Forh=I, the definition of the Hh-monotone (resp., strictly Hh-monotone, strongly Hh-monotone) operator reduces to the definition of the maximal monotone (resp., maximally strictly monotone, maximally strongly monotone) operator.Remark 2.5.
Notice thatM is maximal monotone (resp., maximally strictly monotone, maximally strongly monotone) if and only if M is monotone (resp., strictly monotone, strongly monotone) and there is no other monotone (resp., strictly monotone, strongly monotone) operator whose graph contains strictly the graph Graph(M) of M.Lemma 2.6 (see [20]).
Let{αn}n≥0, {βn}n≥0 and {γn}n≥0 be nonnegative sequences satisfying
(2.16)αn+1≤(1-λn)αn+βnλn+γn,∀n≥0,
where {λn}n≥0⊂[0,1], ∑n=0∞‍λn=∞,limn→∞βn=0 and ∑n=0∞‍γn<∞. Then limn→∞αn=0.


## 3. The Properties of StrictlyHh,η-Monotone Operators and Strongly Hh,η-Monotone Operators

In this section, we discuss some properties of the set-valued strictlyHh,η-monotone operators and set-valued strongly Hh,η-monotone operators, respectively, dealing with a η-monotone operator h in Hilbert spaces.Theorem 3.1.
LetH be a real Hilbert space and let h:H→H,η:H×H→H and M:H→2H be operators such that h is η-monotone and M is strictly Hh,η-monotone. Then(i)
Mis maximally strictly η-monotone;(ii)
(h+λM)-1:H→H is a single-valued operator for each λ>0.Proof.
(i) SinceM is strictly Hh,η-monotone, it follows that M is strictly η-monotone. Suppose that there exists a strictly η-monotone set-valued operator A:H→2H satisfying Graph(A)⊃≠ Graph(M), that is, there exists (u0,x0)∈ Graph(A)∖Graph(M) such that
(3.1)〈x0-y,η(u0,v)〉>0,∀(v,y)∈Graph(M)withv≠u0.
Notice that h(u0)+λx0∈H=(h+λM)(H) for any λ>0. Thus there exists (v0,y0)∈ Graph(M) satisfying
(3.2)h(v0)+λy0=h(u0)+λx0,∀λ>0.
Suppose that u0=v0. Equation (3.2) means that x0=y0. Therefore,
(3.3)Graph(M)∋(v0,y0)=(u0,x0)∈Graph(A)∖Graph(M),
which is impossible. Suppose that u0≠v0. It follows from (3.1), (3.2), and the η-monotonicity of h and strict η-monotonicity of A that
(3.4)0<〈x0-y0,η(u0,v0)〉=-λ-1〈h(u0)-h(v0),η(u0,v0)〉≤0,
which is a contradiction. Hence M is maximally strictly η-monotone.
(ii) Suppose that there exists someu∈H such that (h+λM)-1(u) contains at least two different elements x and y. Since M is strictly Hh,η-monotone, h is η-monotone, and λ-1(u-h(x))∈M(x), λ-1(u-h(y))∈M(y), it follows that
(3.5)0<〈λ-1(u-h(x))-λ-1(u-h(y)),η(x,y)〉=-λ-1〈h(x)-h(y),η(x,y)〉≤0,
which is a contradiction. Consequently, the operator (h+λM)-1 is single valued. This completes the proof.Definition 3.2.
LetH be a real Hilbert space and let h:H→H,η:H×H→H and M:H→2H be operators such that h is η-monotone and M is strictly Hh,η-monotone. Then for each λ>0, the resolvent operator JM,λh:H→H is defined by
(3.6)JM,λh,η(x)=(h+λM)-1(x),∀x∈H.Theorem 3.3.
LetH be a real Hilbert space and let η:H×H→H be Lipschitz continuous with constant Lη. Assume that h:H→H is η-monotone and M:H→2H is strongly Hh,η-monotone with constant r. Then for every λ>0, the resolvent operator JM,λh,η:H→H is Lipschitz continuous with constant  Lη/λr.Proof.
SinceM is strongly Hh,η-monotone with constant r, it follows that M is strictly Hh,η-monotone. Let x,y be in H. In view of λ-1(x-h(JM,λh,η(x)))∈M(JM,λh,η(x)) and λ-1(y-h(JM,λh,η(y)))∈M(JM,λh,η(y)) and the strong monotonicity of M, we deduce that
(3.7)λ-1〈x-y,η(JM,λh,η(x),JM,λh,η(y))〉-λ-1〈h(JM,λh,η(x))-h(JM,λh,η(y)),η(JM,λh,η(x),JM,λh,η(y))〉=〈λ-1(x-h(JM,λh,η(x)))-λ-1(y-h(JM,λh,η(y))),η(JM,λh,η(x),JM,λh,η(y))〉≥r∥JM,λh,η(x)-JM,λh,η(y)∥2.
This leads to
(3.8)Lη∥x-y∥∥JM,λh,η(x)-JM,λh,η(y)∥≥〈x-y,η(JM,λh,η(x),JM,λh,η(y))〉≥〈h(JM,λh,η(x))-h(JM,λh,η(y)),η(JM,λh,η(x),JM,λh,η(y))〉+λr∥JM,λh,η(x)-JM,λh,η(y)∥2≥λr∥JM,λh,η(x)-JM,λh,η(y)∥2,
which yields that
(3.9)∥JM,λh,η(x)-JM,λh,η(y)∥≤Lηλr∥x-y∥.
This completes the proof.Theorem 3.4.
LetH be a real Hilbert space and let η:H×H→H be Lipschitz continuous with constant Lη. Assume that h:H→H is strongly η-monotone with constant s and M:H→2H is strongly Hh,η-monotone with constant r. Then for every λ>0, the resolvent operator JM,λh,η:H→H is Lipschitz continuous with constant Lη/(s+λr).Proof.
As in the proof of Theorem3.3, by the strong monotonicity of h and M, we infer that for any x,y∈H(3.10)Lη∥x-y∥∥JM,λh,η(x)-JM,λh,η(y)∥≥∥x-y∥∥η(JM,λh,η(x)-JM,λh,η(y))∥≥〈x-y,η(JM,λh,η(x)-JM,λh,η(y))〉≥〈h(JM,λh,η(x))-h(JM,λh,η(y)),η(JM,λh,η(x)-JM,λh,η(y))〉+λr∥JM,λh,η(x)-JM,λh,η(y)∥2≥(s+λr)∥JM,λh,η(x)-JM,λh,η(y)∥2,
which means that
(3.11)∥JM,λh,η(x)-JM,λh,η(y)∥≤Lηs+λr∥x-y∥.
This completes the proof.The following example shows that Theorem3.3 is different from Lemma 2.2 in [5].Example 3.5.
LetH=ℝ with the usual norm. Define single-valued and set-valued operators h:H→H,η:H×H→H and M:H→2H by
(3.12)h(x)=3x3-2,∀x∈H,η(x,y)=2(x-y),∀x,y∈H,M(x)={{3x+1}forx<0,[1,5]forx=0,{5+4x}forx>0.
It is clear that η is Lipschitz continuous with constant 2 and
(3.13)〈h(x)-h(y),η(x,y)〉=6(x-y)2(x2+xy+y2)≥0,∀x,y∈H,
which yields that h is η-monotone. On the other hand, for any r>0, there exist (xr,yr)=(r/10,r/20)∈H×H satisfying
(3.14)〈h(xr)-h(yr),η(xr,yr)〉=21r280000<r2400=r(xr-yr)2,
that is, h is not strongly η-monotone. Now we claim that M is strongly Hh,η-monotone. Let x,y be in H with y>x. We have to consider the following five cases.Case 1.
Suppose thatx<y<0. It follows that
(3.15)〈M(x)-M(y),η(x,y)〉=6(x-y)2≥2(x-y)2.Case 2.
Suppose thatx<0=y. For any z∈[1,5], we get that
(3.16)〈M(x)-z,η(x,y)〉=2(3x+1-z)(y-x)=2(y-x)2+2x(1+2x-z)=2(x-y)2.Case 3.
Suppose thatx<0<y. Clearly, we have
(3.17)〈M(x)-M(y),η(x,y)〉=2(3x-4-4y)(x-y)=8(x-y)2+2(4+x)(y-x)≥2(x-y)2.Case 4.
Suppose thatx=0<y. For each z∈[0,1], we deduce that
(3.18)〈z-M(y),η(x,y)〉=2(z-5-4y)(x-y)≥8(x-y)2+2(5-z-x)≥2(x-y)2.Case 5.
Suppose that0<x<y. It is easy to verify that
(3.19)〈M(x)-M(y),η(x,y)〉=8(x-y)2≥2(x-y)2.
Hence M is strongly η-monotone with constant 2.Letλ>0. In order to show that (h+λM)(H)=H, we need only to verify that (h+λM)(H)⊇H. Assume that t∈[λ-2,5λ-2]⊆H. We know that t∈(h+λM)(0)=[λ-2,5λ-2]. Assume that t∈(-∞,λ-2)⊆H. Define a function f:ℝ→ℝ by
(3.20)f(x)=3x3+3λx+λ-2-t,∀x∈ℝ.
Notice that the function f is continuous and
(3.21)f(0)f(-λ-2-tλ)=(λ-2-t)[-3(λ-2-tλ)3-2(λ-2-t)2]<0.
It follows that there exists x0∈(-(λ-2-t)/λ,0) such that f(x0)=0, that is,
(3.22)t=3x03+3λx0+λ-2=(h+λM)(x0).
Assume that t∈(5λ-2,+∞)⊆H. Define g:ℝ→ℝ by
(3.23)g(x)=3x3+4λx+5λ-2-t,∀x∈ℝ.
Obviously, g is continuous and
(3.24)g(0)g(t-5λ+2λ)=3(5λ-2-t)[(t-5λ+2λ)3+t-5λ+2]<0,
which implies that there exists x0∈(0,(t-5λ+2)/λ) with g(x0)=0. Consequently,
(3.25)t=3x03+4λx0+5λ-2=(h+λM)(x0).
Therefore, H⊆(h+λM)(H). Thus M is strongly Hh,η-monotone. It follows from Theorem 3.3 that the resolvent operator JM,λh,η is Lipschitz continuous with constant 1/λ. However, we cannot invoke Lemma 2.2 in [5] to prove the Lipschitz continuity of the resolvent operator JM,λh,η because h is not strongly η-monotone.


## 4. A System of Nonlinear Variational Inclusions

In this section, we investigate a new class of system of nonlinear variational inclusions involving stronglyHh,η-monotone operators in Hilbert spaces and suggest two new iterative algorithms for approximating solutions of the system of nonlinear variational inclusions by using the resolvent operator technique.Letai,bi,fi,gi,hi:H→H,ηi,Ni:H×H→H be single-valued operators and Mi:H→2H set-valued operators and pi∈H for i∈{1,2}. We now consider the following system of nonlinear variational inclusions:find(x,y)∈H×H such that
(4.1)p1∈N1(a1(x),b1(y))+M1(f1(x)-g1(x)),p2∈N2(a2(x),b2(y))+M2(f2(y)-g2(y)).It is easy to see that the system of nonlinear variational inclusions (4.1) includes a lot of variational inequalities, quasivariational inequalities, variational-like inequalities, variational inclusions, and systems of variational inequalities, quasivariational inequalities, variational-like inequalities, and variational inclusions in [1–16] and the references therein as special cases.Now we use the resolvent operator technique to establish the equivalence between the existence of solutions for the system of nonlinear variational inclusions (4.1) and the existence of fixed points for the single-valued operator Q defined by (4.3) below.Lemma 4.1.
LetH be a real Hilbert space. Let ti∈(0,1) and ρi>0 be constants, ai,bi,fi,gi,hi:H→H,ηi,Ni:H×H→H and Mi:H→2H be single-valued and set-valued operators, respectively, such that hi is ηi-monotone and Mi is strictly Hhi,ηi-monotone for i∈{1,2}. Then the following statements are pairwise equivalent.(a)
The system of nonlinear variational inclusions (4.1) has a solution (x,y)∈H×H.(b)
There exists(x,y)∈H×H satisfying
(4.2)f1(x)-g1(x)=JM1,ρ1h1,η1(h1(f1(x)-g1(x))+ρ1p1-ρ1N1(a1(x),b1(y))),f2(y)-g2(y)=JM2,ρ2h2,η2(h2(f2(y)-g2(y))+ρ2p2-ρ2N2(a2(x),b2(y))).(c)
The single-valued operatorQ:H×H→H×H defined by
(4.3)Q(u,v)=(G1(u,v),G2(u,v)),∀(u,v)∈H×H,where
(4.4)G1(u,v)=(1-t1)u+t1[{JM1,ρ1h1,η1}u-(f1(u)-g1(u))+t1+JM1,ρ1h1,η1(h1(f1(u)-g1(u))+ρ1p1-ρ1N1(a1(u),b1(v)))],∀(u,v)∈H×H,G2(u,v)=(1-t2)v+t2[{JM1,ρ1h1,η1}v-(f2(v)-g2(v))+t1+JM2,ρ2h2,η2(h2(f2(v)-g2(v))+ρ2p2-ρ2N2(a2(u),b2(v)))],∀(u,v)∈H×H
has a fixed point (x,y)∈H×H.Proof.
It follows from Theorem3.1, (4.3), and (4.4) that (x,y)∈H×H is a solution of the system of nonlinear variational inclusions (4.1) if and only if
(4.5)p1∈N1(a1(x),b1(y))+M1(f1(x)-g1(x)),p2∈N2(a2(x),b2(y))+M2(f2(y)-g2(y))⟺h1(f1(x)-g1(x))+ρ1p1-ρ1N1(a1(x),b1(y))∈(h1+ρ1M1)(f1(x)-g1(x)),⟺h2(f2(y)-g2(y))+ρ2p2-ρ2N2(a2(x),b2(y))∈(h2+ρ2M2)(f2(y)-g2(y)),⟺f1(x)-g1(x)=JM1,ρ1h1,η1(h1(f1(x)-g1(x))+ρ1p1-ρ1N1(a1(x),b1(y))),⟺f2(y)-g2(y)=JM2,ρ2h2,η2(h2(f2(y)-g2(y))+ρ2p2-ρ2N2(a2(x),b2(y))),⟺x=(1-t1)x⟺x=+t1[{JM2,ρ2h2,η2}x-(f1(x)-g1(x))⟺x=+t1+JM1,ρ1h1,η1(h1(f1(x)-g1(x))+ρ1p1-ρ1N1(a1(x),b1(y)))]⟺x=G1(x,y),xx⟺y=(1-t2)y⟺y=+t2[{JM2,ρ2h2,η2}y-(f2(y)-g2(y))⟺x=+t1+JM2,ρ2h2,η2(h2(f2(y)-g2(y))+ρ2p2-ρ2N2(a2(x),b2(y)))]⟺x=G2(x,y),⟺Q(x,y)=(G1(x,y),G2(x,y))=(x,y).
This completes the proof.Based on Lemma4.1, we suggest the following new iterative algorithms for the system of nonlinear variational inclusions (4.1).Algorithm 4.2 (The Mann iteration method with errors).
For anyx0,y0∈H, compute {xn}n≥0,{yn}n≥0⊂H by
(4.6)xn+1=(1-σn)xn+σn[JM1,ρ1h1,η1(h1(f1(xn)-g1(xn))ρ1p1-ρ1N1(a1(xn),b1(yn)))xn-(f1(xn)-g1(xn))+JM1,ρ1h1,η1(h1(f1(xn)-g1(xn))ρ1p1-ρ1N1(a1(xn),b1(yn)))]+ξn,∀n≥0,yn+1=(1-δn)yn+δn[JM2,ρ2h2,η2(h2(f2(yn)-g2(yn))ρ2p2-ρ2N2(a2(xn),b2(yn)))yn-(f2(yn)-g2(yn))+JM2,ρ2h2,η2(h2(f2(yn)-g2(yn))ρ2p2-ρ2N2(a2(xn),b2(yn)))]+ξn,∀n≥0,
where {σn}n≥0 and {δn}n≥0 are sequences in (0,1), {ξn}n≥0 and {ζn}n≥0 are sequences in H introduced to take into account possible in inexact computation with ∑n=0∞‍(∥ξn∥+∥ζn∥)<+∞.Algorithm 4.3 (The implicit Mann iteration method with errors).
For anyx0,y0∈H, compute {xn}n≥0,{yn}n≥0⊂H by
(4.7)xn+1=(1-σn)xn+σn[{JM2,ρ2h2,η2)}xn+1-(f1(xn+1)-g1(xn+1))+JM1,ρ1h1,η1(h1(f1(xn+1)-g1(xn+1))+ρ1p1-ρ1N1(a1(xn+1),b1(yn+1)))]+ξn,∀n≥0,yn+1=(1-δn)yn+δn[{JM2,ρ2h2,η2)}yn+1-(f2(yn+1)-g2(yn+1))+JM2,ρ2h2,η2(h2(f2(yn+1)-g2(yn+1))+ρ2p2-ρ2N2(a2(xn+1),b2(yn+1)))]+ζn,∀n≥0,
where {σn}n≥0 and {δn}n≥0 are sequences in (0,1), and {ξn}n≥0 and {ζn}n≥0 are sequences in H introduced to take into account possible in inexact computation with ∑n=0∞‍(∥ξn∥+∥ζn∥)<+∞.
Now we investigate those conditions under which the approximate solutions generated by Algorithms4.2 and 4.3 converge strongly to the exact solutions of the system of nonlinear variational inclusions (4.1).Theorem 4.4.
LetH be a real Hilbert space and let ai,bi,fi,gi,hi:H→H,ηi:H×H→H be Lipschitz continuous with constants Lai,Lbi,Lfi,Lgi,Lhi,Lηi, respectively, hiηi-monotone, fi strongly monotone with constant αi, gi relaxed Lipschitz with constant βi for i∈{1,2}. Let Mi:H→2H be strongly Hhi,ηi-monotone with constant ri for i∈{1,2}. Assume that Ni:H×H→H is Lipschitz continuous with constants LNi1 and LNi2 in the first and second arguments, respectively, Ni is strongly monotone with constant γi with respect to ai and fi-gi in the first argument for i∈{1,2}. Let
(4.8)K1=1-2(α1-β1)+(Lf1+Lg1)2+Lη1ρ1r1[(Lf1+Lg1)(1+Lh1)+(Lf1+Lg1)2-2ρ1γ1+LN112La12],K2=1-2(α2-β2)+(Lf2+Lg2)2+Lη2ρ2r2[(Lf2+Lg2)(1+Lh2)+(Lf2+Lg2)2-2ρ2γ2+LN222Lb22],V1=1r1Lη1LN12Lb1,V2=1r2Lη2LN21La2.
If the following condition is fulfilled:
(4.9)V1V2<(1-K1)(1-K2),max{K1,K2}<1,
then the system of nonlinear variational inclusions (4.1) has a unique solution (x,y)∈H×H. Moreover, if
(4.10)sup{1-(1-K1)σn+V2δn,1-(1-K2)δn+V1σn:n≥0}<1,
then for each x0,y0∈H, the sequences {xn}n≥0 and {yn}n≥0 generated by Algorithm 4.2 converge strongly to x and y, respectively.Proof.
First of all we prove that the operatorQ defined by (4.3) is a contraction on the Banach space H×H with norm ∥(u,v)∥1=∥u∥+∥v∥ for (u,v)∈H×H. It follows from (4.9) that
(4.11)V1V2<(1-K1)(1-K2)⟺V21-K1<1-K2V1⟺thereexistst≠1withV21-K1<t<1-K2V1⟺thereexistt1,t2∈(0,1)witht1={t(1-t)fort∈(0,1),t1+tfort>1,t2={1-tfort∈(0,1),11+tfort>1
such that
(4.12)V2t2<(1-K1)t1,V1t1<(1-K2)t2⟺thereexistt1,t2∈(0,1)satisfying(4.11)and(4.13)max{1-(1-K1)t1+t2V2,1-(1-K2)t2+t1V1}<1.
Put (u1,v1),(u2,v2)∈H×H. By the Lipschitz continuity and various monotonicity of ai,bi,fi,gi,hi,Mi,Ni for i∈{1,2}, (4.4) and (4.9), we deduce that
(4.14)∥G1(u1,v1)-G1(u2,v2)∥≤(1-t1)∥u1-u2∥+t1∥u1-u2-[f1(u1)-g1(u1)-(f1(u2)-g1(u2))]∥+t1∥JM1,ρ1h1,η1(h1(f1(u1)-g1(u1))+ρ1p1-ρ1N1(a1(u1),b1(v1)))-JM1,ρ1h1,η1(h1(f1(u2)-g1(u2))+ρ1p1-ρ1N1(a1(u2),b1(v2)))∥≤[1-t1(1-1-2(α1-β1)+(Lf1+Lg1)2)]∥u1-u2∥+t1Lη1ρ1r1[∥h1(f1(u1)-g1(u1))-h1(f1(u2)-g1(u2))∥+t1Lη1ρ1r1+∥f1(u1)-g1(u1)-(f1(u2)-g1(u2))∥+t1Lη1ρ1r1+∥f1(u1)-g1(u1)-(f1(u2)-g1(u2))-ρ1(N1(a1(u1),b1(v1))-N1(a1(u2),b1(v1)))∥+t1Lη1ρ1r1+ρ1∥N1(a1(u2),b1(v1))-N1(a1(u2),b1(v2))∥]≤[1-t1(1-1-2(α1-β1)+(Lf1+Lg1)2)]∥u1-u2∥+t1Lη1ρ1r1[{(Lf1+Lg1)2-2ρ1γ1+ρ12LN112La12}(1+Lh1)(Lf1+Lg1)+(Lf1+Lg1)2-2ρ1γ1+ρ12LN112La12]∥u1-u2∥+t1Lη1r1LN12Lb1∥v1-v2∥=[1-(1-K1)t1]∥u1-u2∥+t1V1∥v1-v2∥,∥G2(u1,v1)-G2(u2,v2)∥≤(1-t2)∥v1-v2∥+t2∥v1-v2-[f2(v1)-g2(v1)-(f2(v2)-g2(v2))]∥+t2∥JM2,ρ2h2,η2(h2(f2(v1)-g2(v1))+ρ2p2-ρ2N2(a2(u1),b2(v1)))-JM2,ρ2h2,η2(h2(f2(v2)-g2(v2))+ρ2p2-ρ2N2(a2(u2),b2(v2)))∥≤[1-t2(1-1-2(α2-β2)+(Lf2+Lg2)2)]∥v1-v2∥+t2Lη2ρ2r2[∥h2(f2(v1)-g2(v1))-h2(f2(v2)-g2(v2))∥+t2Lη2ρ2r2+∥f2(v1)-g2(v1)-(f2(v2)-g2(v2))∥+t2Lη2ρ2r2+∥{(N2(a2(u1),b2(v1))-N2(a2(u1),b2(v2)))}f2(v1)-g2(v1)-(f2(v2)-g2(v2))-ρ2(N2(a2(u1),b2(v1))-N2(a2(u1),b2(v2)))∥+t2Lη2ρ2r2+ρ2∥N2(a2(u1),b2(v2))-N2(a2(u2),b2(v2))∥]≤[1-t2(1-1-2(α2-β2)+(Lf2+Lg2)2)]∥v1-v2∥+t2Lη2ρ2r2[{(Lf2+Lg2)2-2ρ2γ2+ρ22LN222Lb22]}(1+Lh2)(Lf2+Lg2)+(Lf2+Lg2)2-2ρ2γ2+ρ22LN222Lb22]∥v1-v2∥+t2Lη2r2LN21La2∥u1-u2∥=[1-(1-K2)t2]∥v1-v2∥+t2V2∥u1-u2∥.
In light of (4.3) and (4.8)–(4.14), we infer that
(4.15)∥Q(u1,v1)-Q(u2,v2)∥1=∥G1(u1,v1)-G1(u2,v2)∥+∥G2(u1,v1)-G2(u2,v2)∥≤[1-(1-K1)t1+t2V2]∥u1-u2∥+[1-(1-K2)t2+t1V1]∥v1-v2∥≤max{1-(1-K1)t1+t2V2,1-(1-K2)t2+t1V1}×[∥u1-u2∥+∥v1-v2∥]=K∥(u1,v1)-(u2,v2)∥1,
where
(4.16)K=max{1-(1-K1)t1+t2V2,1-(1-K2)t2+t1V1}<1,
that is, Q is a contraction on (H×H,∥·∥1). Therefore, the operator Q has a unique fixed point (x,y)∈H×H, which is the unique solution of the system of nonlinear variational inclusions (4.1) by Lemma 4.1. It follows that
(4.17)x=(1-σn)x+σn[{-ρ1N1(a1(x),b1(y)))}{JM1,ρ1h1,η1}x-(f1(x)-g1(x))+JM1,ρ1h1,η1(h1(f1(x)-g1(x))+ρ1p1-ρ1N1(a1(x),b1(y))){{JM1,ρ1h1,η1}}],∀n≥0,y=(1-δn)y+δn[{-ρ1N1(a1(x),b1(y)))}{JM1,ρ1h1,η1}y-(f2(y)-g2(y))+JM2,ρ2h2,η2(h2(f2(y)-g2(y))+ρ2p2-ρ2N2(a2(x),b2(y))){{JM1,ρ1h1,η1}}],∀n≥0.
In view of Algorithm 4.2 and (4.17), we deduce that for any n≥0(4.18)∥xn+1-x∥≤(1-σn)∥xn-x∥+σn∥xn-x-[f1(xn)-g1(xn)-(f1(x)-g1(x))]∥+σn∥{(ρ1N1(a1(x),b1(y)))}JM1,ρ1h1,η1(h1(f1(xn)-g1(xn))+ρ1p1-ρ1N1(a1(xn),b1(yn)))-JM1,ρ1h1,η1(h1(f1(x)-g1(x))+ρ1p1-ρ1N1(a1(x),b1(y)))∥+∥ξn∥≤[1-σn(1-1-2(α1-β1)+(Lf1+Lg1)2)]∥xn-x∥+σnLη1ρ1r1[∥h1(f1(xn)-g1(xn))-h1(f1(x)-g1(x))∥+σnLη1ρ1r1+∥f1(xn)-g1(xn)-(f1(x)-g1(x))∥+σnLη1ρ1r1+∥{(h1(f1(x)-g1(x))+ρ1p1-ρ1N1(a1(x),b1(y)))}f1(xn)-g1(xn)-(f1(x)-g1(x))-ρ1(N1(a1(xn),b1(yn))-N1(a1(x),b1(yn)))∥+σnLη1ρ1r1+ρ1∥N1(a1(x),b1(yn))-N1(a1(x),b1(y))∥]+∥ξn∥≤[1-σn(1-1-2(α1-β1)+(Lf1+Lg1)2)]∥xn-x∥+σnLη1ρ1r1[{(Lf1+Lg1)2-2ρ1γ1+ρ12LN112La12}(1+Lh1)(Lf1+Lg1)+(Lf1+Lg1)2-2ρ1γ1+ρ12LN112La12]∥xn-x∥+σnLη1r1LN12Lb1∥yn-y∥+∥ξn∥=[1-(1-K1)σn]∥xn-x∥+σnV1∥yn-y∥+∥ξn∥,∥yn+1-y∥≤(1-δn)∥yn-y∥+δn∥yn-y-[f2(yn)-g2(yn)-(f2(y)-g2(y))]∥+δn∥{(h2(f2(x)-g2(x))+ρ2p2-ρ2N2(a2(x),b2(y)))}JM2,ρ2h2,η2(h2(f2(yn)-g2(yn))+ρ2p2-ρ2N2(a2(xn),b2(yn)))-JM2,ρ2h2,η2(h2(f2(x)-g2(x))+ρ2p2-ρ2N2(a2(x),b2(y)))∥+∥ζn∥≤[1-(1-K2)δn]∥yn-y∥+δnV2∥xn-x∥+∥ζn∥.
By virtue of (4.18), we have
(4.19)∥xn+1-x∥+∥yn+1-y∥≤(1-(1-K1)σn)∥xn-x∥+σnV1∥yn-y∥+(1-(1-K2)δn)∥yn-y∥+δnV2∥xn-x∥+∥ξn∥+∥ζn∥≤max{1-(1-K1)σn+V2δn,1-(1-K2)δn+V1σn}×[∥xn-x∥+∥yn-y∥]+∥ξn∥+∥ζn∥≤K*(∥xn-x∥+∥yn-y∥)+∥ξn∥+∥ζn∥,∀n≥0,
where
(4.20)K*=sup{1-(1-K1)σn+V2δn,1-(1-K2)δn+V1σn:n≥0}<1
by (4.10). Lemma 2.6 and (4.19) ensure that the sequences {xn}n≥0 and {xn}n≥0 generated by Algorithm 4.2 converge strongly to x and y, respectively. This completes the proof.Theorem 4.5.
LetH,ai,bi,fi,gi,hi,ηi,Mi,Ni,Ki,Vi for i∈{1,2} be as in Theorem 4.4. If (4.9) is satisfied, then the system of nonlinear variational inclusions (4.1) has a unique solution (x,y)∈H×H. Moreover, if
(4.21)sup{K1σn+V2δn,V1σn+K2δn:n≥0}<1,(4.22)sup{1-σn1-K1σn,1-δn1-K2δn:n≥0}+sup{V1σn1-K1σn,V2δn1-K2δn:n≥0}<1
hold, then for each x0,y0∈H, the sequences {xn}n≥0 and {xn}n≥0 generated by Algorithm 4.3 converge strongly to x and y, respectively.Proof.
It follows from Theorem4.4 that the system of nonlinear variational inclusions (4.1) has a unique solution (x,y)∈H×H. Now we prove that Algorithm 4.3 is well defined. Let n≥0 be a fixed integer. For each (xn,yn)∈H×H, define S1,S2:H×H→H and S:H×H→H×H by
(4.23)S1(u,v)=(1-σn)xn+σn[JM1,ρ1h1,η1(h1(f1(u)-g1(u))+ρ1p1-ρ1N1(a1(u),b1(v)))u-(f1(u)-g1(u))+JM1,ρ1h1,η1(h1(f1(u)-g1(u))+ρ1p1-ρ1N1(a1(u),b1(v)))]+ξn,∀u,v∈H,S2(u,v)=(1-δn)yn+δn[JM2,ρ2h2,η2u-(f2(u)-g2(u))+JM2,ρ2h2,η2(h2(f2(u)-g2(u))+ρ2p2-ρ2N2(a2(u),b2(v))){JM1,ρ1h1,η1}]+ζn,∀u,v∈HS(u,v)=(S1(u,v),S2(u,v)),∀u,v∈H.
As in the proof of Theorem 4.4, we infer that for any (u1,v1),(u2,v2)∈H×H(4.24)∥S1(u1,v1)-S1(u2,v2)∥≤σn{∥{JM1,ρ1h1,η1}u1-u2-[f1(u1)-g1(u1)-(f1(u2)-g1(u2))]+JM1,ρ1h1,η1(h1(f1(u1)-g1(u1))+ρ1p1-ρ1N1(a1(u1),b1(v1)))-JM1,ρ1h1,η1(h1(f1(u2)-g1(u2))+ρ1p1-ρ1N1(a1(u2),b1(v2)))∥}≤K1σn∥u1-u2∥+V1σn∥v1-v2∥,∥S2(u1,v1)-S2(u2,v2)∥≤K2δn∥v1-v2∥+V2δn∥u1-u2∥,
which yield that
(4.25)∥S(u1,v1)-S(u2,v2)∥1=∥S1(u1,v1)-S1(u2,v2)∥+∥S2(u1,v1)-S2(u2,v2)∥≤(K1σn+V2δn)∥u1-u2∥+(V1σn+K2δn)∥v1-v2∥≤max{K1σn+V2δn,V1σn+K2δn}∥(u1,v1)-(u2,v2)∥1.
(4.21) and (4.25) ensure that S is a contraction on H×H. Thus S has a unique fixed point (xn+1,yn+1)∈H×H. Consequently, for each (xn,yn)∈H×H, there exists a unique element (xn+1,yn+1)∈H×H satisfying
(4.26)xn+1=(1-σn)xn+σn[JM1,ρ1h1,η1(h1(f1(xn+1)-g1(xn+1))+ρ1p1-ρ1N1(a1(xn+1),b1(yn+1)))xn+1-(f1(xn+1)-g1(xn+1))+JM1,ρ1h1,η1(h1(f1(xn+1)-g1(xn+1))+ρ1p1-ρ1N1(a1(xn+1),b1(yn+1)))]+ξn,yn+1=(1-δn)yn+δn[{-ρ1N1(a1(u),b1(v)))}JM2,ρ2h2,η2(h2(f2(yn+1)-g2(yn+1))+ρ2p2-ρ2N2(a2(xn+1),b2(yn+1)))yn+1-(f2(yn+1)-g2(yn+1))+JM2,ρ2h2,η2(h2(f2(yn+1)-g2(yn+1))+ρ2p2-ρ2N2(a2(xn+1),b2(yn+1))){JM1,ρ1h1,η1}]+ζn.
That is, Algorithm 4.3 is well defined. Next we show the convergence of the sequences {xn}n≥0 and {xn}n≥0 generated by Algorithm 4.3. In light of (4.17) and (4.26), we conclude that for any n≥0(4.27)∥xn+1-x∥≤(1-σn)∥xn-x∥+σn∥xn+1-x-[f1(xn+1)-g1(xn+1)-(f1(x)-g1(x))]∥+σn∥JM1,ρ1h1,η1({-ρ1N1(a1(u),b1(v)))}h1(f1(xn+1)-g1(xn+1))+ρ1p1-ρ1N1(a1(xn+1),b1(yn+1)))-JM1,ρ1h1,η1(h1(f1(x)-g1(x))+ρ1p1-ρ1N1(a1(x),b1(y)))∥+∥ξn∥≤(1-σn)∥xn-x∥+K1σn∥xn+1-x∥+V1σn∥yn+1-y∥+∥ξn∥,
which means that
(4.28)∥xn+1-x∥≤1-σn1-K1σn∥xn-x∥+V1σn1-K1σn∥yn+1-y∥+11-K1σn∥ξn∥,∀n≥0.
Similarly, we have
(4.29)∥yn+1-y∥≤1-δn1-K2δn∥yn-y∥+V2δn1-K2δn∥xn+1-x∥+11-K2δn∥ζn∥,∀n≥0.
In view of (4.28) and (4.29), we deduce that
(4.30)∥xn+1-x∥+∥yn+1-y∥≤max{1-σn1-K1σn,1-δn1-K2δn}[∥xn-x∥+∥yn-y∥]+max{V1σn1-K1σn,V2δn1-K2δn}[∥xn+1-x∥+∥yn+1-y∥]+max{11-K1σn,11-K2δn}[∥ξn∥+∥ζn∥],∀n≥0,
which implies that
(4.31)∥xn+1-x∥+∥yn+1-y∥≤max{(1-σn)/(1-K1σn),(1-δn)/(1-K2δn)}1-max{(V1σn)/(1-K1σn),(V2δn)/(1-K2δn)}[∥xn-x∥+∥yn-y∥]+max{1/(1-K1σn),1/(1-K2δn)}1-max{V1σn/1-K1σn,V2δn/1-K2δn}[∥ξn∥+∥ζn∥]≤a1-b[∥xn-x∥+∥yn-y∥]+max{1/(1-K1),1/(1-K2)}1-b[∥ξn∥+∥ζn∥],∀n≥0,
where
(4.32)a=sup{1-σn1-K1σn,1-δn1-K2δn:n≥0},b=sup{V1σn1-K1σn,V2δn1-K2δn:n≥0}.
Note that (4.22) yields that a/(1-b)<1. Consequently, Lemma 2.6 and (4.31) yield that the sequences {xn}n≥0 and {xn}n≥0 generated by Algorithm 4.3 converge strongly to x and y, respectively. This completes the proof.Remark 4.6.
Lemma4.1 and Theorems 4.4 and 4.5 extend, improve, and unify the corresponding results in [2–5, 11–19].



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*Source: 643828-2012-11-22.xml*