Volume 2009, Article ID 467512,12pages doi:10.1155/2009/467512
Research Article
On Solvability of a Generalized Nonlinear
Variational-Like Inequality
Zeqing Liu,
1Pingping Zheng,
1Jeong Sheok Ume,
2and Shin Min Kang
31Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea 3Department of Mathematics, The Research Institute of Natural Science, Gyeongsang National University,
Jinju 660-701, South Korea
Correspondence should be addressed to Jeong Sheok Ume,[email protected]
Received 12 July 2009; Accepted 28 September 2009
Recommended by Vy Khoi Le
A new generalized nonlinear variational-like inequality is introduced and studied. By applying the auxiliary principle technique and KKM theory, we construct a new iterative algorithm for solving the generalized nonlinear variational-like inequality. By means of the Banach fixed-point theorem, we establish the existence and uniqueness of solution for the generalized nonlinear variational-like inequality. The convergence of the sequence generated by the iterative algorithm is also discussed.
Copyrightq2009 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
It is well known that variational inequality theory has become a very effective and powerful tool for studying a wide range of problems arising in many diverse fields. One of the most interesting and important problems in the variational inequality theory is the development of an efficient iterative algorithm to compute approximate solutions. The researchers in1–
Motivated and inspired by the research work in 1–10, we introduce and study a generalized nonlinear variational-like inequality. By using the auxiliary principle technique and KKM theorem due to Zhang and Xiang 2, we suggest a new iterative scheme for solving the generalized nonlinear variational-like inequality. Utilizing the Banach fixed-point theorem, we prove the existence and uniqueness of solution for the generalized nonlinear variational-like inequality. Under certain conditions, we discuss the convergence of the iterative sequence generalized by the iterative algorithm.
2. Preliminaries
Throughout this paper, letHbe a real Hilbert space endowed with an inner product·,·and a norm · , respectively, andR −∞,∞ . LetKbe a nonempty closed convex subset ofH. Assume thata:K×K → Ris a coercive continuous bilinear form, that is, there exist positive constantsc, d >0 such that
a1 av, v ≥cv2, for allv∈K;
a2 |au, v | ≤duv, for allu, v∈K.
Remark 2.1. It follows froma1 anda2 thatc≤d.
Letb:K×K → Rbe nondifferentiable and satisfy the following conditions:
b1 bis linear in the first argument,
b2 bis convex in the second argument,
b3 bis bounded, that is, there exists a constantl >0 satisfying
|bu, v | ≤luv, ∀u, v∈K, 2.1
b4 bu, v −bu, w ≤bu, v−w , ∀u, v, w∈K.
Remark 2.2. It follows that
|bu, v −bu, w | ≤luv−w, ∀u, v, w∈K, 2.2
which implies thatbis continuous in the second argument.
LetA, B, C, D, S:K → H, N:H×H → H, η:K×K → Hbe mappings andf∈H. Now we consider the following generalized nonlinear variational-like inequality.
Findu∈Ksuch that
NAu, Bu −Su−f, ηv, u au, v−u ≥bu, u −bu, v , ∀v∈K. 2.3
Recall the following concepts and results.
Definition 2.3. LetA:K → H, N:H×H → H,andη:K×K → Hbe mappings.
1 Ais said to beLipschitz continuousif there exists a constantt >0 such that
Ax−Ay≤tx−y, ∀x, y∈K. 2.4
2 Ais said to beη-relaxed Lipschitzif there exists a constantλ >0 such that
Su−Sv, ηu, v ≤ −λu−v2, ∀u, v∈K. 2.5
3 Nis said to beη-strongly monotonewith respect toAin the first argument if there exists a constantβ >0 such that
NAu, y−NAv, y, ηu, v ≥βu−v2, ∀u, v∈K, ∀y∈H. 2.6
4 Nis said to beη-monotonewith respect toAin the second argument if
Ny, Au−Ny, Av, ηu, v ≥0, ∀u, v∈K, ∀y∈H. 2.7
5 Nis said to beη-relaxed cocoercivewith respect toAin the second argument if there exists a constants >0 such that
Ny, Au−Ny, Av, ηu, v ≥ −sNy, Au−Ny, Av2, ∀u, v∈K, ∀y∈H. 2.8
6 N is said to be Lipschitz continuous in the second argument if there exists a constantγ >0 such that
N
y, u−Ny, v≤γu−v, ∀u, v, y∈H. 2.9
7 ηis said to beLipschitz continuousif there exists a constantδ >0 such that
ηu, v ≤δu−v, ∀u, v∈K. 2.10
8 N and S are said to beη-hemicontinuous with respect toA andB inK if for anyx, y, u, v ∈ K, the mappingt → NAtx 1−t y , Btx 1−t y −Stx 1 −
Lemma 2.4see2 . LetX be a nonempty closed convex subset of a Hausdorfflinear topological spaceE, and letφ, ψ:X×X → Rbe mappings satisfying the following conditions:
a ψx, y ≤φx, y , for allx, y∈X,andψx, x ≥0, for allx∈X;
b for eachx∈X, φx,· is upper semicontinuous onX;
c for eachy∈X,the set{x∈X:ψx, y <0}is a convex set;
d there exists a nonempty compact setY ⊂Xandx0∈Y such thatψx0, y <0, for ally∈
X\Y.
Then there existsy∈Y such thatφx,y ≥0, for allx∈X.
Lemma 2.5see11 . Let{an}n≥0,{bn}n≥0,and{cn}n≥0be nonnegative sequences satisfying
an1≤1−λn anλnbncn, ∀n≥0, 2.11
where
{λn}∞n0⊂0,1,
∞
n0
λn ∞, ∞
n0
cn<∞, nlim→ ∞bn0. 2.12
Thenlimn→ ∞an0.
Assumption 2.6. Letη:K×K → Hsatisfy that
1 ηx, y −ηy, x , for allx, y∈K;
2 for anyx, y ∈ K,the mappingv → NAx, Bx −Sx−f, ηv, y is convex and lower semicontinuous inK.
3. Auxiliary Problem and Algorithm
Now we consider the following auxiliary problem with respect to the generalized nonlinear variational-like inequality2.3 . For eachu∈K,findw ∈Ksuch that
w, v −w ≥ u, v−w −ρNAw, B w −Sw−f, ηv,w
−ρaw, v −w −ρbu, v ρbu,w , ∀v∈K, 3.1
whereρ >0 is a constant.
with respect toAandBinK. Assume thatAssumption 2.6holds and there exists a positive constant ρsatisfying
ρ βσ2s2−α−λ−c<1. 3.2
Then for eachu∈K, the auxiliary problem3.1 has a unique solution inK.
Proof. Letube inK.Define two functionalsφandψ:K×K → Rby
φv, w v, v−w − u, v−wρNAv, Bv −Sv−f, ηv, w ρav, v−w −ρbu, w ρbu, v ,
ψv, w w, v−w − u, v−wρNAw, Bw −Sw−f, ηv, w ρaw, v−w −ρbu, w ρbu, v
3.3
for allv, w∈K.
Now we prove that the functionalsφandψ satisfy all the conditions ofLemma 2.4in the weak topology. It is easy to see for allv, w∈K,
φv, w −ψv, w v−w2ρNAv, Bv −NAw, Bv , ηv, w
ρNAw, Bv −NAw, Bw , ηv, w −ρSv−Sw, ηv, w ρav−w, v−w
≥1−ρ βσ2s2−α−λ−cv−w2
≥0, ψv, v 0,
3.4
which imply thatφandψsatisfy conditiona ofLemma 2.4. Sinceais a coercive continuous bilinear form,bis convex and continuous in the second argument, and for givenw∈K,the mappingv→ NAw, Bw −Sw−f, ηv, w is convex and lower semicontinuous inK, it follows that for eachv∈K, φv,· is weakly upper semicontinuous in the second argument and the set{v∈K:ψv, w <0}is convex for eachw∈K.That is, the conditionsb andc ofLemma 2.4hold. Letv∈K,
E1−ρ βσ2s2−α−λ−c−1
×v−uρdvρδNAv, Bv −MCv, Dv −fρlu1,
Y {w∈K:w−v ≤E}.
Clearly,Y is a weakly compact subset ofK. For eachw∈K\Y,we infer that
ψv, w w, v−w − u, v−wρNAw, Bw −Sw−f, ηv, w ρaw, v−w −ρbu, w ρbu, v
−v−w, v−wv−u, v−w −ρNAv, Bw −NAw, Bw , ηv, w
−ρNAv, Bv −NAv, Bw , ηv, w ρSv−Sw, ηv, w ρNAv, Bv −Sv−f, ηv, w −ρav−w, v−w
ρav, v−w −ρbu, w ρbu, v
≤ −v−w1−ρ βσ2s2−α−λ−cv−w − v−u
−ρδNAv, Bv −MCv, Dv −f−ρdv −ρlu
<0,
3.6
which means that the conditiond ofLemma 2.4holds. ThusLemma 2.4ensures that there existsw ∈Y ⊆Ksuch thatφv,w ≥0 for allv∈K,that is,
v, v−w ≥ u, v−w −ρNAv, Bv −Sv−f, ηv,w
−ρav, v−w −ρbu, v ρbu,w , ∀v∈K. 3.7
Putvttv 1−t wfort∈0,1andv∈K.Replacingvbyvtin3.7 , we obtain that
vt, vt−w ≥ u, vt−w −ρNAvt, Bvt −Svt−f, ηvt,w −ρavt, vt−w −ρbu, vt ρbu,w , ∀v∈K.
3.8
Notice thatbis convex in the second argument. It follows fromAssumption 2.6and3.8 that
tvt, v−w ≥t
u, v−w −ρNAvt, Bvt −Svt−f, ηv,w −ρavt, v−w −ρbu, v ρbu,w
, ∀v∈K,
3.9
which implies that
vt, v−w ≥ u, v−w −ρNAvt, Bvt −Svt−f, ηv,w −ρavt, v−w −ρbu, v ρbu,w , ∀v∈K.
Lettingt → 0in3.10 , we conclude that
w, v −w ≥ u, v−w −ρNAw, B w −Sw−f, ηv,w
−ρaw, v −w −ρbu, v ρbu,w , ∀v∈K. 3.11
That is,w ∈Kis a solution of the auxiliary problem3.1 .
Now we prove the uniqueness of solution for the auxiliary problem3.1. Suppose that
w1, w2∈Kare two solutions of the auxiliary problem3.1 with respect tou. It follows that
w1, v−w1 ≥ u, v−w1 −ρNAw1, Bw1 −Sw1−f, ηv, w1
−ρaw1, v−w1 −ρbu, v ρbu, w1 , ∀v∈K,
3.12
w2, v−w2 ≥ u, v−w2 −ρNAw2, Bw2 −Sw2−f, ηv, w2
−ρaw2, v−w2 −ρbu, v ρbu, w2 , ∀v∈K.
3.13
Takingvw2in3.12 andvw1in3.13 , we get that
w1, w2−w1 ≥ u, w2−w1 −ρNAw1, Bw1 −Sw1−f, ηw2, w1
−ρaw1, w2−w1 −ρbu, w2 ρbu, w1 ,
3.14
w2, w1−w2 ≥ u, w1−w2 −ρNAw2, Bw2 −Sw2−f, ηw1, w2
−ρaw2, w1−w2 −ρbu, w1 ρbu, w2 .
3.15
Adding3.14 and3.15 , we deduce that
w1−w22 ≤ −ρNAw1, Bw1 −NAw2, Bw1 , ηw1, w2
−ρNAw2, Bw1 −NAw2, Bw2 , ηw1, w2
ρSw1−Sw2, ηw1, w2 −ρaw1−w2, w1−w2
≤ρ βσ2s2−α−λ−cw
1−w22,
3.16
which yields thatw1w2by3.2 . That is,w is the unique solution of the auxiliary problem
3.1 . This completes the proof.
The proof of the below result is similar to that ofTheorem 3.1and is omitted.
continuous with constantξ,andNandSareη-hemicontinuous with respect toAandBinK. Assume thatAssumption 2.6holds and there exists a positive constantρsatisfying
ρξδ−α−c <1. 3.17
Then for eachu∈K, the auxiliary problem3.1 has a unique solution inK.
Based on Theorems3.1 and 3.2, we suggest the following iterative algorithm with errors for solving the generalized nonlinear variational-like inequality2.3.
Algorithm 3.3. For givenu0 ∈ K,compute sequence{un}n≥0 ⊂ Kby the following iterative
scheme:
un1, v−un1
≥ un, v−un1 −ρNAun1, Bun1 −Sun1−f, ηv, un1
−ρaun1, v−un1 −ρbun, v ρbun, un1 en, ηv, un1 , ∀v∈K, n≥0,
3.18
where ρ > 0 is a constant and{en}n≥0 is a sequence inK introduced to take into account
possible inexact computation and satisfies that
lim
n→ ∞en0. 3.19
4. Existence and Convergence
In this section, we prove the existence of solution for the generalized nonlinear variational-like inequality2.3 and discuss the convergence of the sequence generated byAlgorithm 3.3.
Theorem 4.1. LetKbe a nonempty closed convex subset of a real Hilbert spaceH,f ∈H,andη:K× K → HLipschitz continuous with constantδ. Assume thata:K×K → Ris a coercive continuous bilinear form satisfying (a1) and (a2),b:K×K → Rsatisfies (b1)–(b4). LetA, B, C, D, S:K → H andN : H×H → Hbe mappings such thatBis Lipschitz continuous with constants,Nis η-strongly monotone with respect toAin the first argument with constantα,η-relaxed cocoercive with respect toBand Lipschitz continuous in the second argument with constantsβandσ, respectively,S isη-relaxed Lipschitz with constantλ, andNandSareη-hemicontinuous with respect toAandBin K. Assume thatAssumption 2.6holds and
βσ2s2l < αλc. 4.1
Then the generalized nonlinear variational-like inequality2.3 possesses a unique solutionu ∈ K and the iterative sequence{un}n≥0generated byAlgorithm 3.3converges strongly tou.
auxiliary problem3.1. Next we show thatFis a contraction mapping inK. Letu1andu2be
arbitrary elements inKandρ >0 a constant. Using3.1, we deduce that
Fu1, v−Fu1 ≥ u1, v−Fu1 −ρNAFu1, BFu1 −SFu1−f, ηv, Fu1
−ρaFu1, v−Fu1 −ρbu1, v ρbu1, Fu1 , ∀v∈K,
4.2
Fu2, v−Fu2 ≥ u2, v−Fu2 −ρ
NAFu2, BFu2 −SFu2−f, ηv, Fu2
−ρaFu2, v−Fu2 −ρbu2, v ρbu2, Fu2 , ∀v∈K.
4.3
LettingvFu2in4.2 andvFu1in4.3, and adding these inequalities, we arrive at
Fu1−Fu22Fu1−Fu2, Fu1−Fu2
≤ u1−u2, Fu1−Fu2
−ρNAFu1, BFu1 −SFu1−NAFu2, BFu2 SFu2, ηFu1, Fu2
−ρaFu1−Fu2, Fu1−Fu2 ρbu1−u2, Fu2−Fu1
≤ u1−u2Fu1−Fu2
−ρNAFu1, BFu1 −NAFu2, BFu1 , ηFu1, Fu2
−ρNAFu2, BFu1 −NAFu2, BFu2 , ηFu1, Fu2
ρSFu1−SFu2, ηFu1, Fu2
−ρaFu1−Fu2, Fu1−Fu2 ρbu1−u2, Fu2−Fu1
≤1ρlu1−u2Fu1−Fu2ρ βσ2s2−α−λ−c
Fu1−Fu22,
4.4
that is,
Fu1−Fu2 ≤θu1−u2, 4.5
where
θ 1ρl
1−ρβσ2s2−α−λ−c <1 4.6
by4.1 . Therefore,F :K → Kis a contraction mapping. It follows from the Banach fixed-point theorem thatFhas a unique fixed pointu∈K.In light of3.1 , we get that
u, v−u ≥ u, v−u −ρNAu, Bu −Su−f, ηv, u
which implies that
NAu, Bu −Su−f, ηv, u au, v−u ≥bu, u −bu, v , ∀v∈K, 4.8
that is,u∈Kis a solution of the generalized nonlinear variational-like inequality2.3 . Now we prove the uniqueness. Suppose that the generalized nonlinear variational-like inequality2.3 has two solutionsu, u 0∈K. It follows that
NAu, Bu −Su−f, ηv,u au, v −u ≥bu,u −bu, v , 4.9
NAu0, Bu0 −Su0−f, ηv, u0 au0, v−u0 ≥bu0, u0 −bu0, v 4.10
for allv∈K.Takingvu0in4.9 andvuin4.10 , we obtain that
NAu, Bu −Su−f, ηu0,u au, u 0−u ≥bu,u −bu, u0 ,
NAu0, Bu0 −Su0−f, ηu, u0 au0,u−u0 ≥bu0, u0 −bu0,u .
4.11
Adding4.11 , we deduce that
αλc−βσ2s2−lu−u 02
≤ NAu, Bu −NAu0, Bu0 , ηu, u0 − Su−Su0, ηu, u0
au−u0,u−u0 −bu−u0,u−u0
≤0,
4.12
which together with4.1 implies thatuu0.That is, the generalized nonlinear
variational-like inequality2.3 has a unique solution inK.
Next we discuss the convergence of the iterative sequence generated byAlgorithm 3.3. Takingvun1in4.7 andvuin3.18 , and adding these inequalities, we infer that
un1−u2≤ un−u, un1−u −ρNAun1, Bun1 −NAu, Bun1 , ηun1, u
−ρNAu, Bun1 −NAu, Bu , ηun1, u
ρSun1−Su, ηun1, u −ρaun1−u, un1−u
ρbun−u, u−un1 en, ηun1, u
≤1ρlun−uun1−u −ρ βσ2s2−α−λ−c
un1−u2
δenun1−u, ∀n≥0.
That is,
un1−u ≤θun−u δ
1−ρβσ2s2−α−λ−cen, ∀n≥0, 4.14
where θ is defined by 4.6 . It follows from 3.19 , 4.1, 4.14 , and Lemma 2.5 that the iterative sequence {un}n≥0 generated by Algorithm 3.3 converges strongly to u. This
completes the proof.
As in the proof ofTheorem 4.1, we have the following theorem.
Theorem 4.2. LetKbe a nonempty closed convex subset of a real Hilbert spaceH,f ∈H,andη:K× K → HLipschitz continuous with constantδ. Assume thata:K×K → Ris a coercive continuous bilinear form satisfying (a1) and (a2),b:K×K → Rsatisfies (b1)–(b4). LetA, B, C, D, S:K → H andN : H×H → Hbe mappings such thatN isη-strongly monotone with respect toAin the first argument with constantα,η-monotone with respect toBin the second argument,Sis Lipschitz continuous with constantξ,andNandSareη-hemicontinuous with respect toAandBinK. Assume thatAssumption 2.6holds and
ξδl < αc. 4.15
Then the generalized nonlinear variational-like inequality2.3 possesses a unique solutionu ∈ K and the iterative sequence{un}n≥0generated byAlgorithm 3.3converges strongly tou.
Remark 4.3. The conditions of Theorems3.1,3.2,4.1, and4.2are different from the conditions of the results in 1–10. In particular, the mappings B and N with respect to the second argument in Theorems3.2and4.2are Lipschitz continuous, but other mappings in Theorems
3.2and4.2are not Lipschitz continuous.
Acknowledgments
This work was supported by the Science Research Foundation of Educational Department of Liaoning Province2009A419 and the Korea Research FoundationKRF Grant funded by the Korea governmentMEST 2009-0073655 .
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