Moreover, we show that the strong well-posedness for a multi-valued variational inequality is equivalent to the existence and uniqueness of its solution

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Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 461–468 DOI: 10.18514/MMN.2018.2035

CHARACTERIZATIONS OF STRONG WELL-POSEDNESS FOR A CLASS OF MULTI-VALUED VARIATIONAL INEQUALITIES

M. OVEISIHA Received 03 June, 2016

Abstract. In this paper, by using the limiting subdifferential we consider the well-posedness for multi-valued variational inequalities and give some equivalence formulations for them. Moreover, we show that the strong well-posedness for a multi-valued variational inequality is equivalent to the existence and uniqueness of its solution.

2010Mathematics Subject Classification: 49K40; 49J52; 90C31

Keywords: limiting subdifferential, variational inequality, well-posedness, approximating sequence

1. INTRODUCTION

The classical concept of well-posedness for a minimization problem, which has been known as the Tykhonov well-posedness, was introduced by Tykhonov [7] in 1966. A minimization problem is Tykhonov well-posed if it has a unique solution and every minimizing sequence of the problem converges to the unique solution. In the last decades, various concepts of well-posedness such as˛-well-posedness, Hadam- ard well-posedness, Levitin-Polyak well-posedness and well-posedness by perturbations have been presented and studied for optimization problems, see [3,4,10,12]

and references therein. The concept of well-posedness for hemivariational inequality was first introduced by Goeleven and Mentagui [2] to provide some conditions guar- anteing the existence and uniqueness of a solution for a hemivariational inequality.

Later, Xiao and Huang [10] considered a concept of well-posedness for a variational- hemivariational inequality and obtained the equivalence of well-posedness between the variational-hemivariational inequality and the corresponding inclusion problem.

Very recently, Xiao et al. [9] established two kinds of conditions under which the strong and weak well-posedness for the hemivariational inequality are equivalent to the existence and uniqueness of its solutions, respectively.

In this paper, by using the limiting subdifferential we extend the concept of well- posedness to a class of multi-valued variational inequality which include as a special

c 2018 Miskolc University Press

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case the classical variational and hemivariational inequalities. Moreover, we establish some equivalence results for them. The paper is organized as follows: Section 2 prepares briefly some preliminary notions and results used in sequel. In Section 3, we show that the strong well-posedness for a multi-valued variational inequality is equivalent to the existence and uniqueness of its solution. Also, a metric characterization for the strong well-posedness of multi-valued variational inequality is obtained.

2. NOTATIONS AND PRELIMINARIES

LetX be a Banach space andXits topological dual space. The norm inX and Xwill be denoted byjj:jj:We denoteh:; :i; Œx; yandx; yŒthe dual pair betweenX andX, the line segment forx; y 2X, and the interior ofŒx; y, respectively. Now, we recall some concepts of subdifferentials that we need in the next section.

Definition 1([6]). LetX be a normed vector space,˝ be a nonempty subset of X,x2˝and"0. The set of"-normals to˝atxis

Nb".xI˝/WD fx2Xjlim sup

u!^˝x

hx; u xi jju xjj "g: Assume thatx2˝, the limiting normal cone to˝atxis

N.xNI˝/WD lim sup

x! Nx;"#0

Nb".xI˝/:

LetJ WX ! NRbe finite atxN 2X; the limiting subdifferential ofJ atxN is defined as follows

@_MJ.x/N WD fx2Xj.x; 1/2N..x; J.N x//N IepiJ /g:

Remark1 ([6]). The set-valued mappingx7!@MJ.x/has closed graph for locally Lipschitz functions.

Definition 2. LetT WX!2^X be a set-valued mapping. T is said to be relaxed invariant monotone with respect to if there exists a constant ˛ such that for any x; y2X and anyu2T .x/; v2T .y/;one has

hv; .x; y/i C hu; .y; x/i ˛.jj.x; y/jj²C jj.y; x/jj²/:

Remark2. (1) WhenT WX !Xis a single-valued operator, we obtain the definition of a relaxed invariant monotone operator.

(2) If˛D0, then the Definition2reduces to the definition of an invariant monotone map.

Definition 3([11]). A mappingT WX !Xis said to be hemicontinuous if for any x1; x22X, the function t 7! hT .x1Ctx2/; x2ifrom Œ0; 1into 1;C1Œis continuous at0_C.

ConditionC ([5]). LetWXX!X. Then, for anyx; y2X; 2Œ0; 1

.y; yC.x; y//D .x; y/; .x; yC.x; y//D.1 /.x; y/:

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Remark3. By some computation, we can see that if ConditionC holds, then for anyx1; x22X and2Œ0; 1

.x1C.x2; x1/; x1/D.x2; x1/:

Now, suppose that J WX !R, WXX !X, AWX !X is a mapping and f 2Xis some given element. Consider the following multi-valued variational-like inequality associated with.A; f; J /:

M VLI.A; f; J /: FindxN2X such that for anyx2X, there exists2@MJ.x/;N that hAxN f C; .x;x/N i 0:

Definition 4. A sequencefxng X is said to be an approximating sequence for theM VLI.A; f; J /, if there existsfngwith n#0 such that for anyx2X there existsx_n2@MJ.xn/, that

hAx_n f Cx_n; .x; x_n/i _njj.x; x_n/jj:

Definition 5. The multi-valued variational-like inequalityM VLI.A; f; J /is said to be strongly well-posed if it has a unique solutionxN onX and for every approximating sequencefxng,.x; xN n/converges strongly to0.

3. MAIN RESULTS

In this section, we establish some conditions under which the well-posedness for the multi-valued variational-like inequality is equivalent to the existence and uniqueness of its solution. Theorems in this section extend theorems in [8,9] from hemivariational inequalities with Clarke’s subdifferential which is convex to multi-valued variational-like inequalities with limiting subdifferential which is not necessarily convex.

Theorem 1. Assume that operator AWX !X is hemicontinuous and relaxed invariant monotone with constantcandJ is locally Lipschitz such that@MJ satisfies relaxed invariant monotonicity condition with constant ˛. Consider the following assertions:

(i) xN is a solution of theM VLI.A; f; J /.

(ii) xN is a solution of the following associated multi-valued variational-like inequality:

AM VLI.A; f; J /: Find xN 2X such that for any x 2X there exists x2

@MJ.x/such thathAx f Cx; .x;x/N i 0:

IfcC˛0andis skew, then.i /).i i /. Ifsatisfies Condition C, then.i i /).i /.

Proof. .i /).i i /. Let xN 2X be a solution ofM VLI.A; f; J /. Hence, for any x2X there exists2@MJ.x/N such that

hAxN f C; .x;x/N i 0: (3.1)

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By the relaxed invariant monotonicity of@MJ, for anyx2X andx2@MJ.x/, one has

hx; .x; x/N i C h; .x;x/N i ˛.jj.x;x/N jj²C jj.x; x/N jj²/: (3.2) It follows from relaxed invariant monotonicity of the operatorA, (3.1) and (3.2), that

hAx f Cx;.x;x/N i hAxCx; .x;x/N i hAxNC; .x;x/N i

D ŒhAx; .x; x/N i C hAx; .x;N x/N i C hx; .x; x/N i C h; .x;x/N i .cC˛/.jj.x;x/N jj²C jj.x; x/N jj²/;

which shows thatxN is a solution ofAM VLI.A; f; J /.

.i i /).i /. Conversely, letxN be a solution to theAM VLI.A; f; J /. Hence, for any x2X there existsx2@_MJ.x/such that

hAx f Cx; .x;x/N i 0: (3.3) For any´2X andt2Œ0; 1, setx.t /D NxCt .´;x/N in inequality (3.3), we have

hA.xNCt .´;x//N f Cx_t; .xNCt .´;x/;N x/N i 0;

thatx_t 2@MJ.x.t //. It follows Condition C, that

hA.xNCt .´;x//N f Cx_t; .´;x/N i 0:

SinceJ is locally Lipschitz, we deduce that@MJ is locally bounded (Corollary 1.81 in [6]). Hence, there exists a neighborhood ofxN and a constant` > 0such that for each´in this neighborhood andx2@MJ.´/, we havejjxjj `:Sincex.t /! Nx when t !0 fort to be sufficiently small jjx_tjj `, without loss of generality we may assume thatx_t !xin weak-topology. Now, hemicontinuity of the operator AonX implies that

hAxN f Cx; .´;x/N i 0:

By the arbitrariness of´2X, we conclude that xN is a solution ofM VLI.A; f; J /.

Proposition 1. LetCXbe nonempty, closed, convex and bounded,'WX ! Rbe proper, convex and lower semi-continuous andy2X be arbitrary. Assume that is continuous and affine with respect to the first argument and for eachx2X, there existsx.x/2Csuch that

hx.x/; .x; y/i '.y/ '.yC.x; y//:

Then, there existsy2Csuch that

hy; .x; y/i '.y/ '.yC.x; y//; 8x2X:

Proof. With some minor modification in the proof of Proposition 3.3 in [1], we

can deduce the proof.

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Theorem 2. LetAWX !Xbe relaxed invariant monotone with constantcand J a l.s.c. function that its limiting subdifferential satisfies relaxed invariant monotonicity condition with constant˛. If˛Cc > 0andis continuous and affine with respect to the second argument and skew, thenM VLI.A; f; J /is strongly well-posed if and only if it has a unique solution onX.

Proof. The necessity is obvious. For the sufficiency, suppose thatM VLI.A; f; J / has a unique solutionx. SinceN xN is the unique solution ofM VLI.A; f; J /, for any x2X, there existsx2@MJ.x/N such that

hAxN f Cx; .x;x/N i 0: (3.4) Suppose thatfxngis an approximating sequence forM VLI.A; f; J /. It follows that there existsn#0such that for anyx2X, there existsn.x/2@MJ.xn/that

hAxn f Cn.x/; .x; xn/i njj.x; xn/jj:

Now, consider the nonempty, convex and bounded set cofAxn fCnjn2@MJ.xn/g. Hence, it follows from Proposition1 with '.x/Dnjjx xnjj that there exists n

which is independent onx, such that

hAxn f Cn; .x; xn/i njj.x; xn/jj; 8x2X:

By choosing n, we can set nDPm

iD1i_nⁱ thatm2N, Pm

iD1i D1 and _nⁱ 2

@MJ.xn/. Hence,

m

X

iD1

ihAxn f C_nⁱ; .x; xn/i njj.x; xn/jj; 8x2X:

Now, setxD Nxin above inequality, yields

m

X

iD1

ihAxn f C_nⁱ; .x; xN n/i njj.x; xN n/jj:

Hence, it follows from relaxed invariant monotonicity of the operator A, relaxed invariant monotonicity of the@MJ, the skewness ofand above inequality that

njj.x; xN n/jj

m

X

iD1

ihAxn f C_nⁱ; .x; xN n/i

m

X

iD1

iŒhAxnC_nⁱ; .x; xN n/i hAxNC_nⁱ; .x; xN n/i

D

m

X

iD1

iŒ hAxN AxnC_nⁱ _nⁱ; .x; xN n/i

m

X

iD1

2_i.cC˛/jj.x; xN _n/jj²D 2.cC˛/jj.x; xN _n/jj²;

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where_nⁱ 2@MJ.x/N is obtained from (3.4) by setting xDxn. Since cC˛ > 0, it follows that

jj.x; xN n/jj n

2.cC˛/:

Taking limit at both sides of the above inequality, implies that.x; xN n/ converges

strongly to0.

Example1. LetX DR; f D0,Abe the identity map andJ be defined as J.x/D

x²C2x ifx > 0;

x² x ifx0:

The limiting subdifferential ofJ is

@MJ.x/D 8

<

:

2xC2 ifx > 0;

[-1,2] ifxD0;

2x 1 ifx < 0:

Letbe defined as.x; y/WDk.x y/, such that0 < k1. Then by some computation we can see that@MJ; Aare relaxed invariant monotone with constants˛D1, cD¹₂;respectively. Hence, all assumptions of Theorem2are fulfilled andxN D0is a unique solution of.M VLI /and therefore it is strongly well-posed.

For any > 0, consider the following two sets:

˝./D f NxW 8x2X; 9x2@MJ.x/N s.t. hAxN f Cx; .x;x/N i jj.x;x/N jjg; ./D f NxW 8x2X; 9x2@MJ.x/N s.t. h Ax; .x; x/N i C h f Cx; .x;x/N i

jj.x;x/N jjg: Lemma 1. Suppose thatAWX!Xis invariant monotone and hemicontinuous.

Then˝./D ./for all > 0.

Proof. Taking into account the invariant monotonicity of mappingA, it is easy to obtain that˝./ ./. For the other side suppose thatxN 2 ./. Then, for any x2X, there existsx2@MJ.x/N such that

h Ax; .x; x/N i C h f Cx; .x;x/N i jj.x;x/N jj: (3.5) SetxD NxCt .´;x/N in (3.5), that´2X andt2Œ0; 1, yields

h A.xNCt .´;x//; .N x;N xNCt .´;x//N i C h f Cx; .xNCt .´;x/;N x/N i jj.xNCt .´;x/;N x/N jj:

By using Condition C, we obtain

h A.xNCt .´;x//; .´;N x/N i C h f Cx; .´;x/N i jj.´;x/N jj: Now, it follows from the hemicontinuity of mappingAthat

hAxN f Cx; .´;x/N i jj.´;x/N jj;

which shows thatxN 2˝./. This completes the proof.

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Lemma 2. Suppose thatAWX!Xis a hemicontinuous mapping. IfJ is locally Lipschitz and is continuous with respect to the second argument, then ˝./ is closed inX for all > 0.

Proof. Letfxng ˝./be a sequence such thatxn! NxinX. Then for anyx2X, there existsx_n2@MJ.xn/such that

hAx_n f Cx_n; .x; x_n/i jj.x; x_n/jj: (3.6) Since J is locally Lipschitz, there exists a subsequence of x_nthat convergent to a x2@MJ.x/N in weak-topology. Consider (3.6) with this subsequence, taking limit at both sides of it and using this fact that Ais hemicontinuous andis continuous with respect to the second argument, we obtain

hAxN f Cx; .x;x/N i jj.x;x/N jj;

which implies thatxN2˝./. This completes the proof.

Theorem 3. Suppose thatAWX!Xis hemicontinuous and invariant monotone with respect tothatis continuous with respect to the second argument, satisfies ConditionC and skew. ThenM VLI.A; f; J /is strongly well-posed if and only if

˝./¤¿;8 > 0 and diam.˝.//!0 as !0:

Proof. “Necessity” follows similarly from the first part of Theorem 3.1 in [8].

Hence, we prove the “Sufficiency”. Suppose that fx_ng X is an approximating sequence forM VLI.A; f; J /. Then there exist a nonnegative sequencen!0such that for anyx2X, there existsn2@_MJ.xn/such that

hAxn f Cn; .x; xn/i njj.x; xn/jj; (3.7) it means thatxn2˝.n/. It follows from diam.˝.//!0, thatfxngis a cauchy sequence and so converges strongly to some pointxN2X. SinceJ is locally lipschitz, there exists a subsequence ofn.e.g. fnig/that is convergent to a2@MJ.x/N in weak-topology. Taking limit at the both side of (3.7) and using this fact thatAis monotone and is skew and continuous with respect to the second argument, we obtain

hAx f C; .x;x/N i D lim

i!1hAx f Cni; .x; xni/i lim

i!1hAxn_i f Cn_i; .x; xn_i/i lim

i!1 nijj.x; xni/jj D0:

Now, by using Theorem1,xN is a solution ofM VLI.A; f; J /. Since, diam.˝.//! 0when!0,M VLI.A; f; J /has a unique solution. This completes the proof.

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Author’s address

M. Oveisiha

Department of Mathematics, Faculty of Science, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin, Iran

E-mail address:oveisiha@sci.ikiu.ac.ir