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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 683140, 17 pages
http://dx.doi.org/10.1155/2011/683140
Research Article

𝛼-Well-Posedness for Mixed Quasi Variational-Like Inequality Problems

School of Mathematics, Chongqing Normal University, Chongqing 400047, China

Received 15 January 2011; Revised 18 March 2011; Accepted 14 April 2011

Academic Editor: V.Β Zeidan

Copyright Β© 2011 Jian-Wen Peng and Jing Tang. 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.

Abstract

The concepts of 𝛼-well-posedness, 𝛼-well-posedness in the generalized sense, L-𝛼-well-posedness and L-𝛼-well-posedness in the generalized sense for mixed quasi variational-like inequality problems are investigated. We present some metric characterizations for these well-posednesses.

1. Introduction

Well-posedness plays a crucial role in the stability theory for optimization problems, which guarantees that, for an approximating solution sequence, there exists a subsequence which converges to a solution. The study of well-posedness for scalar minimization problems started from Tykhonov [1] and Levitin and Polyak [2]. Since then, various notions of well-posedness for scalar minimization problems have been defined and studied in [3–8] and the references therein. It is worth noting that the recent study for various types of well-posedness has been generalized to variational inequality problems [9–13], generalized variational inequality problems [14, 15], quasi variational inequality problems [16], generalized quasi variational inequality problems [17], generalized vector variational inequality problems [18], vector quasi variational inequality problems [19], mixed quasi variational-like inequality problems [20], and many other problems.

In this paper, we are interested in investigating four classes of well-posednesses for a mixed quasi variational-like inequality problem. The paper is organized as follows. In Section 2, we introduce the definitions of 𝛼-well-posedness, 𝛼-well-posedness in the generalized sense, L-𝛼-well-posedness and L-𝛼-well-posedness in the generalized sense for a mixed quasi variational-like inequality problem. In Section 3, some characterizations of 𝛼-well-posedness, and L-𝛼-well-posedness for a mixed quasi variational-like inequality problem are obtained. In Section 4, some characterizations of 𝛼-well-posedness in the generalized sense and L-𝛼-well-posedness in the generalized sense for a mixed quasi variational-like inequality problem are presented.

2. Preliminaries

Throughout this paper, without other specification, let 𝐸 be a real Banach space with the dual πΈβˆ—, let 𝐾 be a nonempty closed convex subset of 𝐸, and let π‘†βˆΆπΎβ†’2𝐾 be a set-valued map. Let πΉβˆΆπΎβ†’2πΈβˆ— be a set-valued map with nonempty values, let πœ‚βˆΆπΎΓ—πΎβ†’πΈ be a single-valued map, and let π‘“βˆΆπΎβ†’π‘… be a real-valued function. Ceng et al. [20] introduced the following mixed quasi variational-like inequality problem, which is to find a point π‘₯0∈𝐾 such that, for some 𝑒0∈𝐹(π‘₯0), (MQVLI)π‘₯0ξ€·π‘₯βˆˆπ‘†0ξ€Έ,𝑒0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έξ€·π‘₯βˆ’π‘“(𝑦)≀0,βˆ€π‘¦βˆˆπ‘†0ξ€Έ.(2.1)

Denote by Ξ“ the solution set of (MQVLI). Let 𝛼>0; we introduce the notions of several classes of 𝛼-well-posednesses for (MQVLI).

Definition 2.1. A sequence (π‘₯𝑛)𝑛 in 𝐾 is an 𝛼-approximating sequence for (MQVLI) if(i)there exists a sequence (𝑒𝑛)𝑛 in πΈβˆ—, with π‘’π‘›βˆˆπΉ(π‘₯𝑛),forallπ‘›βˆˆπ‘;(ii)there exists a sequence (πœ€π‘›)𝑛,πœ€π‘›>0, πœ€π‘›β†’0 such that 𝑑π‘₯𝑛π‘₯,π‘†π‘›ξ€Έξ€Έβ‰€πœ€π‘›ξ«π‘’,βˆ€π‘›βˆˆπ‘,𝑛π‘₯,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2β‰€πœ€π‘›ξ€·π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ,βˆ€π‘›βˆˆπ‘.(2.2)

Definition 2.2. (MQVLI) is said to be 𝛼-well-posed (resp., 𝛼-well-posed in the generalized sense) if it has a unique solution π‘₯0 and every 𝛼-approximating sequence (π‘₯𝑛)𝑛 strongly converges to π‘₯0 (resp., if the solution set Ξ“ of (MQVLI) is nonempty and for every 𝛼-approximating sequence (π‘₯𝑛)𝑛 has a subsequence which strongly converges to a point of Ξ“).

Definition 2.3. A sequence (π‘₯𝑛)𝑛 is an L-𝛼-approximating sequence for (MQVLI) if there exists a real number sequence (πœ€π‘›)𝑛,πœ€π‘›>0, πœ€π‘›β†’0 such that 𝑑π‘₯𝑛π‘₯,π‘†π‘›ξ€Έξ€Έβ‰€πœ€π‘›ξ«ξ€·π‘₯,βˆ€π‘›βˆˆπ‘,𝑣,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2β‰€πœ€π‘›ξ€·π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ,π‘£βˆˆπΉ(𝑦),π‘›βˆˆπ‘.(2.3)

Definition 2.4. (MQVLI) is said to be L-𝛼-well-posed (resp., L-𝛼-well-posed in the generalized sense) if it has a unique solution π‘₯0 and every L-𝛼-approximating sequence (π‘₯𝑛)𝑛 strongly converges to π‘₯0 (resp., if the solution set Ξ“ of (MQVLI) is nonempty and for every L-𝛼-approximating sequence (π‘₯𝑛)𝑛 has a subsequence which strongly converges to a point of Ξ“).

It is worth noting that if 𝛼=0, then the definitions of 𝛼-well-posedness, 𝛼-well-posedness in the generalized sense, L-𝛼-well-posedness, and L-𝛼-well-posedness in the generalized sense for (MQVLI), respectively, reduce to those of the well-posedness, well-posedness in the generalized sense, L-well-posedness, and L-well-posedness in the generalized sense for (MQVLI) in [20]. We also note that Definition 2.2 generalizes and extends 𝛼-well-posedness and 𝛼-well-posedness in the generalized sense of variational inequalities in [10] which are related to the continuously differentiable gap function of variational inequalities introduced by Fukushima [21].

In order to investigate the 𝛼-well-posedness for (MQVLI), we need the following definitions.

We recall the notion of Mosco convergence [22]. A sequence (𝐻𝑛)𝑛 of subsets of 𝐸 Mosco converges to a set 𝐻 if 𝐻=liminf𝑛𝐻𝑛=π‘€βˆ’limsup𝑛𝐻𝑛,(2.4) where liminf𝑛𝐻𝑛 and π‘€βˆ’limsup𝑛𝐻𝑛 are, respectively, the PainlevΓ©-Kuratowski strong limit inferior and weak limit superior of a sequence (𝐻𝑛)𝑛, that is, liminf𝑛𝐻𝑛=ξ€½π‘¦βˆˆπΈβˆΆβˆƒπ‘¦π‘›βˆˆπ»π‘›,π‘›βˆˆπ‘,with𝑦𝑛,β†’π‘¦π‘€βˆ’limsup𝑛𝐻𝑛=ξ€½π‘¦βˆˆπΈβˆΆβˆƒπ‘›π‘˜β†‘+∞,π‘›π‘˜βˆˆπ‘,βˆƒπ‘¦π‘›π‘˜βˆˆπ»π‘›π‘˜,π‘˜βˆˆπ‘,withπ‘¦π‘›π‘˜ξ€Ύ,⇀𝑦(2.5) where β€œβ‡€β€ means weak convergence, and β€œβ†’β€ means strong convergence.

If 𝐻=liminf𝑛𝐻𝑛, we call the sequence (𝐻𝑛)𝑛 of subsets of 𝐸 Lower Semi-Mosco convergent to a set 𝐻.

It is easy to see that a sequence (𝐻𝑛)𝑛 of subsets of 𝐸 Mosco converges to a set 𝐻 implies that the sequence (𝐻𝑛)𝑛 also Lower Semi-Mosco converges to the set 𝐻, but the converse is not true in general.

We will use the usual abbreviations usc and lsc for β€œupper semicontinuous” and β€œlower semicontinuous”, respectively. For any π‘₯,π‘¦βˆˆπΈ, [π‘₯,𝑦] will denote the line segment {𝑑π‘₯+(1βˆ’π‘‘)π‘¦βˆΆπ‘‘βˆˆ[0,1]}, while [π‘₯,𝑦) and (π‘₯,𝑦) are defined analogously. We will frequently use 𝑠,𝑀, and π‘€βˆ— to denote, respectively, the norm topology on 𝐸, the weak topology on 𝐸, and the weak* topology on 𝐸. Given a convex set 𝐾, a multivalued map πΉβˆΆπΎβ†’2πΈβˆ— will be called upper hemicontinuous, if its restriction on any line segment [π‘₯,𝑦]βŠ†πΎ is usc with respect to the π‘€βˆ— topology on πΈβˆ—. πΉβˆΆπΎβ†’2πΈβˆ— will be called πœ‚-monotone if, for any π‘₯,π‘¦βˆˆπΎ, for all π‘’βˆˆπΉ(π‘₯),π‘£βˆˆπΉ(𝑦), βŸ¨π‘’βˆ’π‘£,πœ‚(π‘₯,𝑦)⟩β‰₯0. We refer the reader to [23, 24] for basic facts about multivalued maps.

Lemma 2.5 (see [25]). Let (𝐻𝑛)𝑛 be a sequence of nonempty subsets of a Banach space 𝐸 such that (i)𝐻𝑛 is convex for every π‘›βˆˆπ‘;(ii)𝐻0βŠ†liminf𝑛𝐻𝑛;(iii)there exists π‘šβˆˆπ‘ such that β‹‚int𝑛β‰₯π‘šπ»π‘›β‰ βˆ…. Then, for every 𝑒0∈int𝐻0, there exists a positive real number 𝛿 such that 𝑒int𝐡0ξ€Έ,π›ΏβŠ†π»π‘›,βˆ€π‘›β‰₯π‘š,(2.6) where 𝐡(𝑒0,𝛿) is a closed ball with a center 𝑒0 and radius 𝛿. If 𝐸 is a finite dimensional space, then assumption (iii) can be replaced by (iii)β€²int𝐻0β‰ βˆ….

The following lemmas play important role in this paper.

Lemma 2.6. Let 𝐸 be a real separable Banach space with the dual πΈβˆ—, let 𝑆0 be a nonempty convex subset of 𝐸, and let πΉβˆΆπ‘†0β†’2πΈβˆ— be a set-valued map with nonempty, weakly* compact convex valued, πœ‚-monotone, and upper hemicontinuous. Let πœ‚βˆΆπ‘†0×𝑆0→𝐸 be a single-valued map with πœ‚(π‘₯,π‘₯)=0,forallπ‘₯βˆˆπ‘†0, and let π‘“βˆΆπ‘†0→𝑅 be a convex lsc function. Assume that the map π‘¦β†¦βŸ¨π‘’,πœ‚(π‘₯,𝑦)⟩ is concave for each (𝑒,π‘₯)∈𝐹(𝑆0)×𝑆0 and usc. If 𝑆1 is a convex subset of 𝑆0 with the property that, for each π‘₯βˆˆπ‘†0 and each π‘¦βˆˆπ‘†1,(π‘₯,𝑦]βŠ†π‘†1, then for each π‘₯0βˆˆπ‘†0, the following conditions are equivalent: (i)Thereexists𝑒0∈𝐹(π‘₯0), such that forallπ‘¦βˆˆπ‘†0,βŸ¨π‘’0,πœ‚(π‘₯0,𝑦)⟩+𝑓(π‘₯0)βˆ’π‘“(𝑦)βˆ’(𝛼/2)β€–π‘₯0βˆ’π‘¦β€–2≀0,(ii)forallπ‘¦βˆˆπ‘†1,thereexistsπ‘£βˆˆπΉ(𝑦), such that βŸ¨π‘£,πœ‚(π‘₯0,𝑦)⟩+𝑓(π‘₯0)βˆ’π‘“(𝑦)βˆ’(𝛼/2)β€–π‘₯0βˆ’π‘¦β€–2≀0.

Proof. According to the πœ‚-monotonicity of 𝐹, (i) β‡’ (ii) is obvious.
Next prove (ii) β‡’ (i). Suppose that (ii) holds. Given any π‘¦βˆˆπ‘†1, let 𝑦𝑛=(1/𝑛)𝑦+(1βˆ’(1/𝑛))π‘₯0, for π‘›βˆˆπ‘. By the assumptions of 𝑆1,π‘¦π‘›βˆˆπ‘†1 for each π‘›βˆˆπ‘. It follows from the condition (ii) that for each π‘›βˆˆπ‘, there exists π‘£π‘›βˆˆπΉ(𝑦𝑛) such that 𝑣𝑛π‘₯,πœ‚0,𝑦𝑛π‘₯+𝑓0ξ€Έξ€·π‘¦βˆ’π‘“π‘›ξ€Έβˆ’π›Ό2β€–β€–π‘₯0βˆ’π‘¦π‘›β€–β€–2≀0.(2.7) Then 𝑣0β‰₯𝑛π‘₯,πœ‚0,𝑦𝑛π‘₯+𝑓0ξ€Έξ€·π‘¦βˆ’π‘“π‘›ξ€Έβˆ’π›Ό2β€–β€–π‘₯0βˆ’π‘¦π‘›β€–β€–2=𝑣𝑛π‘₯,πœ‚0,1𝑛1𝑦+1βˆ’π‘›ξ‚π‘₯0ξ€·π‘₯+𝑓0ξ€Έξ‚€1βˆ’π‘“π‘›ξ‚€1𝑦+1βˆ’π‘›ξ‚π‘₯0ξ‚βˆ’π›Ό2β€–β€–β€–π‘₯0βˆ’1𝑛1π‘¦βˆ’1βˆ’π‘›ξ‚π‘₯0β€–β€–β€–2β‰₯1𝑛𝑣𝑛π‘₯,πœ‚0+ξ‚€1,𝑦1βˆ’π‘›ξ‚ξ«π‘£π‘›ξ€·π‘₯,πœ‚0,π‘₯0ξ€·π‘₯+𝑓0ξ€Έβˆ’1𝑛𝑓1(𝑦)βˆ’1βˆ’π‘›ξ‚π‘“ξ€·π‘₯0ξ€Έβˆ’π›Ό21𝑛2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2=1𝑛𝑣𝑛π‘₯,πœ‚0+1,𝑦𝑛𝑓π‘₯0ξ€Έβˆ’1𝑛𝛼𝑓(𝑦)βˆ’2𝑛2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2,(2.8) which implies that 𝑣𝑛π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’β€–β€–π‘₯2𝑛0β€–β€–βˆ’π‘¦2≀0,βˆ€π‘›βˆˆπ‘.(2.9) It follows that for each π‘›βˆˆπ‘, βˆƒπ‘£π‘›ξ€·π‘¦βˆˆπΉπ‘›ξ€Έ,𝑣𝑛π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2≀0.(2.10) Since 𝐹 is a weak* compact valued and (𝑠,π‘€βˆ—)-usc on the line segment [π‘₯0,𝑦], 𝐹 is (𝑠,π‘€βˆ—)-closed, and (𝑠,π‘€βˆ—)-subcontinuous on [π‘₯0,𝑦], it follows from lim𝑛𝑦𝑛=π‘₯0 and π‘£π‘›βˆˆπΉ(𝑦𝑛) that {𝑣𝑛} has a subsequence weak* converging to some π‘£βˆˆπΉ(π‘₯0). By taking the limit of subsequence in (2.10) we get βˆ€π‘¦βˆˆπ‘†1ξ€·π‘₯,βˆƒπ‘£βˆˆπΉ0ξ€Έ,π‘₯𝑣,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2≀0.(2.11) Define the bifunction πœ™(𝑣,𝑦) on 𝐹(π‘₯0)×𝑆0 by πœ™ξ«ξ€·π‘₯(𝑣,𝑦)=𝑣,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2.(2.12) For each π‘¦βˆˆπ‘†0,πœ™(β‹…,𝑦) is weakly* lsc and quasiconvex on the weakly* compact convex set 𝐹(π‘₯0) while for each π‘£βˆˆπΉ(π‘₯0),πœ™(𝑣,β‹…) is usc and quasiconcave on the convex set S1. Hence, according to the Sion Minimax Theorem [26], supπ‘¦βˆˆπ‘†1minξ€·π‘₯π‘£βˆˆπΉ0ξ€Έπœ™(𝑣,𝑦)=minξ€·π‘₯π‘£βˆˆπΉ0ξ€Έsupπ‘¦βˆˆπ‘†1πœ™(𝑣,𝑦).(2.13) By (2.11), we have supπ‘¦βˆˆπ‘†1minπ‘£βˆˆπΉ(π‘₯0)πœ™(𝑣,𝑦)≀0; hence, minπ‘£βˆˆπΉ(π‘₯0)supπ‘¦βˆˆπ‘†1πœ™(𝑣,𝑦)≀0, which implies that there exists 𝑣0∈𝐹(π‘₯0), such that 𝑣0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2≀0,βˆ€π‘¦βˆˆπ‘†1.(2.14)
Finally, for each π‘¦βˆˆπ‘†0, choose π‘§βˆˆπ‘†1, and a sequence (𝑦𝑛)𝑛 in (𝑦,𝑧]βŠ†π‘†1 converging to 𝑦. The function πœ™(𝑣,β‹…) is usc and concave on 𝑆0; hence its restriction on any line segment is continuous [27, Theorem  2.35]. Accordingly, (2.14) implies thereexists𝑣0∈𝐹(π‘₯0),forallπ‘¦βˆˆπ‘†0, 𝑣0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2=lim𝑛𝑣0ξ€·π‘₯,πœ‚0,𝑦𝑛π‘₯+𝑓0ξ€Έξ€·π‘¦βˆ’π‘“π‘›ξ€Έβˆ’π›Ό2β€–β€–π‘₯0βˆ’π‘¦π‘›β€–β€–2≀0.(2.15) Hence, (i) holds.

Lemma 2.7. Let 𝐸 be a real Banach space with the dual πΈβˆ—, let 𝐾 be a nonempty convex subset of 𝐸, and let 𝑆 be a convex-valued set-valued map from 𝐾 to 2𝐾. Let πΉβˆΆπΎβ†’2πΈβˆ— be a set-valued map with nonempty values, let πœ‚βˆΆπΎΓ—πΎβ†’πΈ be a single-valued map with πœ‚(π‘₯,π‘₯)=0,forallπ‘₯∈𝐾, and let π‘“βˆΆπΎβ†’π‘… be a convex function. Assume that the function π‘¦β†¦βŸ¨π‘’,πœ‚(π‘₯,𝑦)⟩ is concave, for each (𝑒,π‘₯)∈𝐹(𝐾)×𝐾. Then π‘₯0βˆˆΞ“ if and only if the following condition holds: βˆƒπ‘’0ξ€·π‘₯∈𝐹0ξ€Έ,π‘₯0ξ€·π‘₯βˆˆπ‘†0ξ€Έ,𝑒0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2ξ€·π‘₯≀0,βˆ€π‘¦βˆˆπ‘†0ξ€Έ.(2.16)

Proof. The necessity is easy to get; next we start to prove the sufficiency. Let forallπ‘¦βˆˆπ‘†(π‘₯0),forallπ‘‘βˆˆ(0,1),𝑦𝑑=𝑑𝑦+(1βˆ’π‘‘)π‘₯0. Since 𝑒0∈𝐹(π‘₯0),π‘₯0βˆˆπ‘†(π‘₯0), and 𝑆 is convex-valued, π‘¦π‘‘βˆˆπ‘†(π‘₯0), it follows that 𝑒0ξ€·π‘₯,πœ‚0,𝑦𝑑π‘₯+𝑓0ξ€Έξ€·π‘¦βˆ’π‘“π‘‘ξ€Έβˆ’π›Ό2β€–β€–π‘₯0βˆ’π‘¦π‘‘β€–β€–2≀0,βˆ€π‘‘βˆˆ(0,1).(2.17) Thus, 𝑒0β‰₯0ξ€·π‘₯,πœ‚0,𝑦𝑑π‘₯+𝑓0ξ€Έξ€·π‘¦βˆ’π‘“π‘‘ξ€Έβˆ’π›Ό2β€–β€–π‘₯0βˆ’π‘¦π‘‘β€–β€–2𝑒β‰₯𝑑0ξ€·π‘₯,πœ‚0𝑒,𝑦+(1βˆ’π‘‘)0ξ€·π‘₯,πœ‚0,π‘₯0ξ€·π‘₯+𝑓0ξ€Έξ€·π‘₯βˆ’π‘‘π‘“(𝑦)βˆ’(1βˆ’π‘‘)𝑓0ξ€Έβˆ’π›Ό2𝑑2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2𝑒=𝑑0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2𝑑‖‖π‘₯0β€–β€–βˆ’π‘¦2ξ‚„,(2.18) which implies that 𝑒0ξ€·π‘₯∈𝐹0ξ€Έ,π‘₯0ξ€·π‘₯βˆˆπ‘†0ξ€Έ,𝑒0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2𝑑‖‖π‘₯0β€–β€–βˆ’π‘¦2ξ€·π‘₯≀0βˆ€π‘¦βˆˆπ‘†0ξ€Έ,βˆ€π‘‘βˆˆ(0,1).(2.19) The above inequality implies, for 𝑑 converging to zero, that π‘₯0 is a solution of (MQVLI). This completes the proof.

3. The Characterizations of Well-Posedness for (MQVLI)

In this section, we investigate some metric characterizations of 𝛼-well-posedness and L-𝛼-well-posedness for (MQVLI).

For any πœ€>0, we consider the sets π‘„πœ€=𝛼π‘₯βˆˆπΎβˆΆπ‘‘(π‘₯,𝑆(π‘₯))β‰€πœ€,βˆƒπ‘’βˆˆπΉ(π‘₯)βˆΆβŸ¨π‘’,πœ‚(π‘₯,𝑦)⟩+𝑓(π‘₯)βˆ’π‘“(𝑦)βˆ’2β€–π‘₯βˆ’π‘¦β€–2,πΏβ‰€πœ€,βˆ€π‘¦βˆˆπ‘†(π‘₯)πœ€=𝛼π‘₯βˆˆπΎβˆΆπ‘‘(π‘₯,𝑆(π‘₯))β‰€πœ€,βŸ¨π‘£,πœ‚(π‘₯,𝑦)⟩+𝑓(π‘₯)βˆ’π‘“(𝑦)βˆ’2β€–π‘₯βˆ’π‘¦β€–2.β‰€πœ€,βˆ€π‘¦βˆˆπ‘†(π‘₯),βˆ€π‘£βˆˆπΉ(𝑦)(3.1)

Theorem 3.1. Let the same assumptions be as in Lemma 2.7. Then, one has the following. (a)(MQVLI) is 𝛼-well-posed if and only if the solution set Ξ“ of (MQVLI) is nonempty and limπœ€β†’0diamπ‘„πœ€=0.(b)Moreover, if 𝐹 is πœ‚-monotone, then (MQVLI) is L-𝛼-well-posed if and only if the solution set Ξ“ of (MQVLI) is nonempty and limπœ€β†’0diamπΏπœ€=0.

Proof. We only prove (a). The proof of (b) is similar and is omitted here. Suppose that (MQVLI) is 𝛼-well-posed; then Ξ“β‰ βˆ…. It follows from Lemma 2.7 that π‘„πœ€β‰ βˆ…. Suppose by contradiction that exists a real number 𝛽, such that limπœ€β†’0diamπ‘„πœ€>𝛽>0; then there exists πœ€π‘›>0, with πœ€π‘›β†˜0, and (𝑀𝑛)𝑛,(𝑧𝑛)π‘›βˆˆπ‘„πœ€π‘›, such that β€–π‘€π‘›βˆ’π‘§π‘›β€–>𝛽,forallπ‘›βˆˆπ‘. Since the sequences (𝑀𝑛)𝑛, and (𝑧𝑛)𝑛 are both 𝛼-approximating sequences for (MQVLI), (𝑀𝑛)𝑛 and (𝑧𝑛)𝑛 strongly converge to the unique solution 𝑒0, and this gives a contradiction. Therefore, limπœ€β†’0diamπ‘„πœ€=0.
Conversely, let (π‘₯𝑛)π‘›βŠ‚πΎ be an 𝛼-approximating sequence for (MQVLI). Then there exists a sequence (𝑒𝑛)𝑛 in πΈβˆ— with π‘’π‘›βˆˆπΉ(π‘₯𝑛) and a sequence (πœ€π‘›)𝑛 in 𝑅+ with πœ€π‘›β†’0, such that 𝑑π‘₯𝑛π‘₯,π‘†π‘›ξ€Έξ€Έβ‰€πœ€π‘›,𝑒𝑛π‘₯,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2β‰€πœ€π‘›ξ€·π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ,βˆ€π‘›βˆˆπ‘.(3.2) That is, π‘₯π‘›βˆˆπ‘„πœ€π‘›,forallπ‘›βˆˆπ‘. It is easy to see limπœ€β†’0diamπ‘„πœ€=0 and Ξ“β‰ βˆ… imply that Ξ“ is a singleton point set. Indeed, if there exist two different solutions 𝑧1,𝑧2, then from Lemma 2.7, we know that 𝑧1,𝑧2βˆˆπ‘„πœ€,forallπœ€>0. Thus, limπœ€β†’0diamπ‘„πœ€β‰₯‖𝑧1βˆ’π‘§2β€–β‰ 0, a contradiction. Let π‘₯0 be the unique solution of (MQVLI). It follows from Lemma 2.7 that π‘₯0βˆˆπ‘„πœ€π‘›. Thus, limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘₯0‖≀limπ‘›β†’βˆždiamπ‘„πœ€π‘›=0. So (π‘₯𝑛)𝑛 strongly converges to π‘₯0. Therefore, (MQVLI) is 𝛼-well-posed.

Theorem 3.2. Let 𝐸 be a real separable Banach space with the dual πΈβˆ—, let 𝐾 be a nonempty closed convex subset of 𝐸, and let πœ‚βˆΆπΎΓ—πΎβ†’πΈ be a single-valued map with πœ‚(π‘₯,π‘₯)=0,forallπ‘₯∈𝐾, which is (𝑠,𝑀)-continuous in each of its variables separately. And let π‘“βˆΆπΎβ†’π‘… be a convex lsc function; let π‘†βˆΆπΎβ†’2𝐾 and πΉβˆΆπΎβ†’2πΈβˆ— be two set-valued maps. Assume the following conditions hold: (i)𝑆 is nonempty convex-valued and, for each sequence (π‘₯𝑛)𝑛 in 𝐾 converging to π‘₯0, the sequence (𝑆(π‘₯𝑛))𝑛 Lower Semi-Mosco converging to 𝑆(π‘₯0);(ii)for every converging sequence (𝑀𝑛)𝑛, there exists π‘šβˆˆπ‘, such that β‹‚int𝑛β‰₯π‘šπ‘†(𝑀𝑛)β‰ βˆ…;(iii)πΉβˆΆπΎβ†’2πΈβˆ— is nonempty, weak* compact convex valued, πœ‚-monotone, and upper hemicontinuous;(iv)the map π‘¦β†¦βŸ¨π‘’,πœ‚(π‘₯,𝑦)⟩ is concave for each (𝑒,π‘₯)∈𝐹(𝐾)×𝐾.Then, (MQVLI) is 𝛼-well-posed if and only if π‘„πœ€β‰ βˆ…,βˆ€πœ€β‰₯0,limπœ€β†’0diamπ‘„πœ€=0.(3.3)

The proof of the above theorem relies on the following lemma.

Lemma 3.3. Let the same assumptions be made as in Theorem 3.2. Let (π‘₯𝑛)𝑛 in 𝐾 be an 𝛼-approximating sequence. If (π‘₯𝑛)𝑛 converges to some π‘₯0∈𝐾, then π‘₯0 is a solution of (MQVLI).

Proof. Since (π‘₯𝑛)𝑛 is an 𝛼-approximating sequence for (MQVLI), there exists a sequence (𝑒𝑛)𝑛 in πΈβˆ— with π‘’π‘›βˆˆπΉ(π‘₯𝑛) and a sequence (πœ€π‘›)𝑛 in 𝑅+ with πœ€π‘›β†’0, such that 𝑑π‘₯𝑛π‘₯,π‘†π‘›ξ€Έξ€Έβ‰€πœ€π‘›,𝑒𝑛π‘₯,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2β‰€πœ€π‘›ξ€·π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ,βˆ€π‘›βˆˆπ‘.(3.4)
For each π‘›βˆˆπ‘, choose π‘₯ξ…žπ‘›βˆˆπ‘†(π‘₯𝑛), such that β€–π‘₯π‘›βˆ’π‘₯ξ…žπ‘›β€–<𝑑(π‘₯𝑛,𝑆(π‘₯𝑛))+πœ€π‘›β‰€2πœ€π‘›. It follows from π‘₯𝑛→π‘₯0 and πœ€π‘›β†’0 that π‘₯ξ…žπ‘›β†’π‘₯0. It follows from the assumption (i) that liminf𝑛𝑆(π‘₯𝑛)=𝑆(π‘₯0). Thus, π‘₯0βˆˆπ‘†(π‘₯0).
Assumption (ii) applied to the constant sequence 𝑀𝑛=π‘₯0,forallπ‘›βˆˆπ‘, implies that int𝑆(π‘₯0)β‰ βˆ…. For every π‘¦βˆˆint𝑆(π‘₯0), it follows from assumptions (i) and (ii) and Lemma 2.5 that there exist π‘šβˆˆπ‘ and 𝛿>0 such that int𝐡(𝑦,𝛿)βŠ†π‘†(π‘₯𝑛),forall𝑛>π‘š. Therefore, for 𝑛 sufficiently large, we have π‘¦βˆˆπ‘†(π‘₯𝑛). Notice that πœ‚(β‹…,𝑦) is (𝑠,𝑀)-continuous, 𝑓 is lsc, 𝐹 is πœ‚-monotone, and (π‘₯𝑛)𝑛 is an approximating sequence; we have, for every π‘£βˆˆπΉ(𝑦)π‘₯𝑣,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έβˆ’π‘“(𝑦)≀liminf𝑛π‘₯𝑣,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έξ€Ύβˆ’π‘“(𝑦)≀liminf𝑛𝑒𝑛π‘₯,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έξ€Ύβˆ’π‘“(𝑦)≀liminfπ‘›ξ‚ƒπœ€π‘›+𝛼2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2ξ‚„=𝛼2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2.(3.5)
Thus, for every π‘¦βˆˆint𝑆(π‘₯0) and every π‘£βˆˆπΉ(𝑦), we get βŸ¨π‘£,πœ‚(π‘₯0,𝑦)⟩+𝑓(π‘₯0)βˆ’π‘“(𝑦)βˆ’(𝛼/2)β€–π‘₯0βˆ’π‘¦β€–2≀0. Let 𝑆0=𝑆(π‘₯0) and 𝑆1=int𝑆(π‘₯0); it follows from Lemma 2.6 that there exists 𝑒0∈𝐹(π‘₯0) such that for all π‘¦βˆˆπ‘†(π‘₯0),βŸ¨π‘’0,πœ‚(π‘₯0,𝑦)⟩+𝑓(π‘₯0)βˆ’π‘“(𝑦)βˆ’(𝛼/2)β€–π‘₯0βˆ’π‘¦β€–2≀0. According to Lemma 2.7, π‘₯0 is a solution of (MQVLI).

Proof of Theorem 3.2. The necessity follows from Theorem 3.1 and Lemma 2.7. Now we prove the sufficiency. Suppose that (3.3) holds. Let us show that there exists at most one solution of (MQVLI). Indeed, if there exist two different solutions 𝑧1,𝑧2, then from Lemma 2.7, we know that 𝑧1,𝑧2βˆˆπ‘„πœ€,forallπœ€>0. Thus, limπœ€β†’0diamπ‘„πœ€β‰₯‖𝑧1βˆ’π‘§2β€–β‰ 0, a contradiction. Note also that there exist 𝛼-approximate sequences for (MQVLI); indeed, for any sequence (πœ€π‘›)𝑛 in 𝑅+ with πœ€π‘›β†’0, and any choice of π‘₯π‘›βˆˆπ‘„πœ€π‘› (which is nonempty by assumption), (π‘₯𝑛)𝑛 is an 𝛼-approximate sequence.
Let (π‘₯𝑛)𝑛 be an 𝛼-approximating sequence for (MQVLI); then π‘₯π‘›βˆˆπ‘„πœ€π‘›,forallπ‘›βˆˆπ‘. In light of (3.3), (π‘₯𝑛)𝑛 is a Cauchy sequence and strongly converging to a point π‘₯0∈𝐾. Applying Lemma 3.3, we get that π‘₯0 is a solution of (MQVLI) and so (MQVLI) is 𝛼-well-posed.

Now, we present a result in which assumption (ii) and the monotonicity of 𝐹 are dropped, while the continuity requirements are strengthened.

Theorem 3.4. Let 𝐸 be a real separable Banach space with the dual πΈβˆ—, let 𝐾 be a nonempty closed convex subset of 𝐸, and let πœ‚βˆΆπΎΓ—πΎβ†’πΈ be a single-valued map with πœ‚(π‘₯,π‘₯)=0,forallπ‘₯∈𝐾, which is (𝑠,𝑠)-continuous. And let π‘“βˆΆπΎβ†’π‘… be a convex and continuous function, let π‘†βˆΆπΎβ†’2𝐾 and πΉβˆΆπΎβ†’2πΈβˆ— be two set-valued maps. Assume the following assumptions hold: (i)the multifunction 𝑆 is nonempty convex-valued and for each sequence (π‘₯𝑛)𝑛 in 𝐾 converging to π‘₯0, the sequence (𝑆(π‘₯𝑛))𝑛 Lower Semi-Mosco converging to 𝑆(π‘₯0);(ii)πΉβˆΆπΎβ†’2πΈβˆ— is nonempty, weak* compact, and convex valued, (𝑠,π‘€βˆ—)-usc;(iii)the map π‘¦β†¦βŸ¨π‘’,πœ‚(π‘₯,𝑦)⟩ is concave for each (𝑒,π‘₯)∈𝐹(𝐾)×𝐾.Then, (MQVLI) is 𝛼-well-posed if and only if (3.3) holds.

The proof of the above theorem relies on the following lemma.

Lemma 3.5. Let the assumptions be as in Theorem 3.4. Let (π‘₯𝑛)𝑛 in 𝐾 be an 𝛼-approximating sequence. If (π‘₯𝑛)𝑛 converges to some π‘₯0∈𝐾, then π‘₯0 is a solution of (MQVLI).

Proof. Since (π‘₯𝑛)𝑛 is an 𝛼-approximating sequence for (MQVLI), there exist a sequence (𝑒𝑛)𝑛 in πΈβˆ— with π‘’π‘›βˆˆπΉ(π‘₯𝑛) and a sequence (πœ€π‘›)𝑛 in 𝑅+,πœ€π‘›β†’0, such that 𝑑π‘₯𝑛π‘₯,π‘†π‘›ξ€Έξ€Έβ‰€πœ€π‘›,𝑒𝑛π‘₯,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2β‰€πœ€π‘›ξ€·π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ,βˆ€π‘›βˆˆπ‘.(3.6)
As in Lemma 3.3, we infer π‘₯0βˆˆπ‘†(π‘₯0). Since 𝑆(π‘₯𝑛) Lower Semi-Mosco converges to 𝑆(π‘₯0), for every π‘¦βˆˆπ‘†(π‘₯0), there exists a sequence π‘¦π‘›βˆˆπ‘†(π‘₯𝑛),forallπ‘›βˆˆπ‘, such that lim𝑛𝑦𝑛=𝑦 in the strongly topology. Since πœ‚ is (𝑠,𝑠)-continuous, the sequence (πœ‚(π‘₯𝑛,𝑦𝑛))𝑛 converges strongly to πœ‚(π‘₯0,𝑦). It follows from (ii) and Proposition  2.19 in [24] that there exists a subsequence (𝑒𝑛𝑗)𝑗 of (𝑒𝑛)𝑛 weak* converging to some 𝑒0βˆˆπΈβˆ—. It follows from (ii) and Proposition  2.17 in [24] that 𝐹 is (𝑠,π‘€βˆ—)-closed, and so 𝑒0∈𝐹(π‘₯0). Thus, we have |||𝑒𝑛𝑗π‘₯,πœ‚π‘›π‘—,π‘¦π‘›π‘—βˆ’ξ«π‘’ξ‚ξ‚­0ξ€·π‘₯,πœ‚0|||≀|||𝑒,𝑦𝑛𝑗π‘₯,πœ‚π‘›π‘—,𝑦𝑛𝑗π‘₯βˆ’πœ‚0ξ€Έξ‚­|||+|||𝑒,𝑦0βˆ’π‘’π‘›π‘—ξ€·π‘₯,πœ‚0ξ€Έξ‚­|||≀‖‖𝑒,π‘¦π‘›π‘—β€–β€–β€–β€–πœ‚ξ‚€π‘₯𝑛𝑗,𝑦𝑛𝑗π‘₯βˆ’πœ‚0ξ€Έβ€–β€–βˆ’|||𝑒,𝑦0βˆ’π‘’π‘›π‘—ξ€·π‘₯,πœ‚0ξ€Έξ‚­|||,π‘¦βŸΆ0.(3.7) Hence, βŸ¨π‘’π‘›π‘—,πœ‚(π‘₯𝑛𝑗,𝑦𝑛𝑗)βŸ©β†’βŸ¨π‘’0,πœ‚(π‘₯0,𝑦)⟩ and so 𝑒0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2=lim𝑗𝑒𝑛𝑗π‘₯,πœ‚π‘›π‘—,𝑦𝑛𝑗π‘₯+π‘“π‘›π‘—ξ‚ξ‚€π‘¦βˆ’π‘“π‘›π‘—ξ‚βˆ’π›Ό2β€–β€–π‘₯π‘›π‘—βˆ’π‘¦π‘›π‘—β€–β€–2≀limπ‘—πœ€π‘›π‘—=0.(3.8) Applying Lemma 2.7, π‘₯0 is a solution of (MQVLI).

Proof of Theorem 3.4. The necessity follows from Theorem 3.1 and Lemma 2.7. Now we prove the sufficiency. Suppose that (3.3) holds. It follows from the proof of Theorem 3.2 that there exists at most one solution of (MQVLI) and there exist 𝛼-approximate sequences for (MQVLI). Let (π‘₯𝑛)𝑛 be an 𝛼-approximating sequence for (MQVLI); then π‘₯π‘›βˆˆπ‘„πœ€π‘›,forallπ‘›βˆˆπ‘. In light of (3.3), (π‘₯𝑛)𝑛 is a Cauchy sequence and strongly converging to a point π‘₯0∈𝐾. Applying Lemma 3.5, we get that π‘₯0 is a solution of (MQVLI) and so (MQVLI) is 𝛼-well-posed.

We have analogous results for L-𝛼-well-posedness.

Theorem 3.6. Let 𝐸 be a real separable Banach space with the dual πΈβˆ—, let 𝐾 be a nonempty closed convex subset of 𝐸, and let πœ‚βˆΆπΎΓ—πΎβ†’πΈ be a single-valued map with πœ‚(π‘₯,π‘₯)=0,forallπ‘₯∈𝐾, which is (𝑠,𝑀)-continuous in each of its variables separately. And let π‘“βˆΆπΎβ†’π‘… be a convex lsc function; let π‘†βˆΆπΎβ†’2𝐾 and πΉβˆΆπΎβ†’2πΈβˆ— be two set-valued maps. Assume the following assumptions hold: (i)the multifunction 𝑆 is nonempty convex-valued and for each sequence (π‘₯𝑛)𝑛 in 𝐾 converging to π‘₯0, the sequence (𝑆(π‘₯𝑛))𝑛 Lower Semi-Mosco converging to 𝑆(π‘₯0);(ii)for every converging sequence (𝑀𝑛)𝑛, there exists π‘šβˆˆπ‘, such that β‹‚int𝑛β‰₯π‘šπ‘†(𝑀𝑛)β‰ βˆ…;(iii)πΉβˆΆπΎβ†’2πΈβˆ— is a set-valued map with nonempty, weak* compact convex valued, πœ‚-monotone and upper hemicontinuous;(iv)the map π‘¦β†¦βŸ¨π‘’,πœ‚(π‘₯,𝑦)⟩ is concave for each (𝑒,π‘₯)∈𝐹(𝐾)×𝐾.Then (MQVLI) is L-𝛼-well-posed if and only if πΏπœ€β‰ βˆ…,βˆ€πœ€β‰₯0,limπœ€β†’0diamπΏπœ€=0.(3.9)

Lemma 3.7. Let the same assumptions be as in Theorem 3.6. Let (π‘₯𝑛)𝑛 in 𝐾 be an L-𝛼-approximating sequence. If (π‘₯𝑛)𝑛 converges to some π‘₯0∈𝐾, then π‘₯0 is a solution of (MQVLI).

Proof. Since (π‘₯𝑛)𝑛 is an L-𝛼-approximating sequence for (MQVLI), there exists a sequence (πœ€π‘›)𝑛 in 𝑅+, πœ€π‘›β†’0, such that 𝑑(π‘₯𝑛,𝑆(π‘₯𝑛))β‰€πœ€π‘›, and π‘₯𝑣,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2β‰€πœ€π‘›ξ€·π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ,π‘£βˆˆπΉ(𝑦),π‘›βˆˆπ‘.(3.10) From the proof of Lemma 3.3, (i) and (ii), we can obtain π‘₯0βˆˆπ‘†(π‘₯0), int𝑆(π‘₯0)β‰ βˆ…, and for each π‘¦βˆˆint𝑆(π‘₯0), one has π‘¦βˆˆπ‘†(π‘₯𝑛) for 𝑛 sufficiently large. It follows from (iii) that for every π‘¦βˆˆint𝑆(π‘₯0) and every π‘£βˆˆπΉ(𝑦), we have π‘₯𝑣,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2≀liminf𝑛π‘₯𝑣,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2≀liminfπ‘›πœ€π‘›=0.(3.11)
Let 𝑆0=𝑆(π‘₯0) and 𝑆1=int𝑆(π‘₯0); it follows from Lemma 2.6 that there exists 𝑒0∈𝐹(π‘₯0) such that for all π‘¦βˆˆπ‘†(π‘₯0),βŸ¨π‘’0,πœ‚(π‘₯0,𝑦)⟩+𝑓(π‘₯0)βˆ’π‘“(𝑦)βˆ’(𝛼/2)β€–π‘₯0βˆ’π‘¦β€–2≀0. According to Lemma 2.7, π‘₯0 is a solution of (MQVLI).

Proof of Theorem 3.6. Assume that (3.9) holds. Let (π‘₯𝑛)𝑛 in 𝐾 be an L-𝛼-approximating sequence for (MQVLI); then there exists a sequence (πœ€π‘›)𝑛 in 𝑅+, such that π‘₯π‘›βˆˆπΏπœ€π‘›. It is easy to see that limπœ€β†’0diamπΏπœ€=0 and Ξ“β‰ βˆ… imply that Ξ“ is a singleton point set. Indeed, if there exist two different solutions 𝑧1,𝑧2, then from Lemma 2.7 and the πœ‚-monotonicity of 𝐹, we know that 𝑧1,𝑧2βˆˆπΏπœ€,forallπœ€>0. Thus, limπœ€β†’0diamπΏπœ€β‰₯‖𝑧1βˆ’π‘§2β€–β‰ 0, a contradiction. Let π‘₯0 be the unique solution of (MQVLI). It follows from Lemma 2.7 and the πœ‚-monotonicity of 𝐹 that π‘₯0βˆˆπΏπœ€π‘›. Thus, limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘₯0‖≀limπ‘›β†’βˆždiamπΏπœ€π‘›=0. So (π‘₯𝑛)𝑛 strongly converge to π‘₯0. It follows from Lemma 3.7 that π‘₯0βˆˆΞ“. Therefore, (MQVLI) is L-𝛼-well-posed.
Conversely, assume that the problem is L-𝛼-well-posed, It follows from the πœ‚-monotonicity of 𝐹 that βˆ…β‰ Ξ“βŠ‚πΏπœ€,βˆ€πœ€>0. Suppose by contradiction that a real number 𝛽 exists, such that limπœ€β†’0diamπΏπœ€>𝛽>0; then there exists πœ€π‘›>0, with πœ€π‘›β†’0, and (𝑀𝑛)𝑛,(𝑧𝑛)π‘›βˆˆπΏπœ€π‘›, such that β€–π‘€π‘›βˆ’π‘§π‘›β€–>𝛽,βˆ€π‘›βˆˆπ‘. Since the sequences (𝑀𝑛)𝑛 and (𝑧𝑛)𝑛 are both L-𝛼-approximating sequences for (MQVLI), (𝑀𝑛)𝑛 and (𝑧𝑛)𝑛 strongly converge to the unique solution 𝑒0, and this gives a contradiction. Therefore, limπœ€β†’0diamπΏπœ€=0.

Theorem 3.8. Let 𝐸 be a real separable Banach space, let 𝐾 be a nonempty closed convex subset of 𝐸, and let πœ‚βˆΆπΎΓ—πΎβ†’πΈ be a single-valued map with πœ‚(π‘₯,π‘₯)=0,forallπ‘₯∈𝐾, which is (𝑠,𝑠)-continuous. And let π‘“βˆΆπΎβ†’π‘… be a convex continuous function; 𝑆 be a set-valued map from 𝐾 to 2𝐾. Assume the following assumptions hold: (i)the multifunction 𝑆 is nonempty convex-valued and for each sequence (π‘₯𝑛)𝑛 in 𝐾 converging to π‘₯0, the sequence (𝑆(π‘₯𝑛))𝑛 Lower Semi-Mosco converging to 𝑆(π‘₯0);(ii)πΉβˆΆπΎβ†’2πΈβˆ— is a set-valued map with nonempty, weak* compact convex-valued, (𝑠,π‘€βˆ—)-usc, and πœ‚-monotone;(iii)the map π‘¦β†¦βŸ¨π‘’,πœ‚(π‘₯,𝑦)⟩ is concave for each (𝑒,π‘₯)∈𝐹(𝐾)×𝐾.
Then (MQVLI) is L-𝛼-well-posed if and only if (3.9) holds.

Lemma 3.9. Let the same assumptions be as in Theorem 3.8. Let (π‘₯𝑛)𝑛 in 𝐾 be an L-𝛼-approximating sequence. If (π‘₯𝑛)𝑛 converges to some π‘₯0∈𝐾, then π‘₯0 is a solution of (MQVLI).

Proof. Since (π‘₯𝑛)𝑛 is an L-𝛼-approximating sequence for (MQVLI), there exists a sequence (πœ€π‘›)𝑛 in 𝑅+, πœ€π‘›β†’0, such that 𝑑(π‘₯𝑛,𝑆(π‘₯𝑛))β‰€πœ€π‘›, and π‘₯𝑣,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2β‰€πœ€π‘›ξ€·π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ,βˆ€π‘£βˆˆπΉ(𝑦),π‘›βˆˆπ‘.(3.12)
It follows from the Lower Semi-Mosco convergence of 𝑆 and the proof of Lemma 3.3 that π‘₯0βˆˆπ‘†(π‘₯0). Since 𝑆(π‘₯𝑛) Lower Semi-Mosco converges to 𝑆(π‘₯0), for every π‘¦βˆˆπ‘†(π‘₯0), there exists a sequence π‘¦π‘›βˆˆπ‘†(π‘₯𝑛),forallπ‘›βˆˆπ‘, strongly converging to 𝑦. For each π‘›βˆˆπ‘ select π‘£π‘›βˆˆπΉ(𝑦𝑛). It follows from (ii) and Proposition  2.19 in [24] that there exists a subsequence (𝑣𝑛𝑗)𝑗 of (𝑣𝑛)𝑛 weak* converging to some π‘£βˆˆπΈβˆ—. It follows from (ii) and Proposition  2.17 in [24] that 𝐹 is (𝑠,π‘€βˆ—)-closed, and so π‘£βˆˆπΉ(𝑦). By the continuity of πœ‚ and similar argument with the proof of Lemma 3.5, we know that 𝑣𝑛𝑗π‘₯,πœ‚π‘›π‘—,π‘¦π‘›π‘—βŸΆξ«ξ€·π‘₯𝑣,πœ‚0.,𝑦(3.13) It follows from (3.12) that 𝑣𝑛𝑗π‘₯,πœ‚π‘›π‘—,𝑦𝑛𝑗π‘₯+π‘“π‘›π‘—ξ‚ξ‚€π‘¦βˆ’π‘“π‘›π‘—ξ‚βˆ’π›Ό2β€–β€–π‘₯π‘›π‘—βˆ’π‘¦π‘›π‘—β€–β€–2β‰€πœ€π‘›π‘—.(3.14) We deduce from the above inequality that ξ€·π‘₯βˆ€π‘¦βˆˆπ‘†0π‘₯,βˆƒπ‘£βˆˆπΉ(𝑦),𝑣,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2≀0.(3.15) Let 𝑆0=𝑆1=𝑆(π‘₯0); by Lemma 2.6 we know that there exists 𝑒0∈𝐹(π‘₯0), such that ξ€·π‘₯βˆ€π‘¦βˆˆπ‘†0ξ€Έ,𝑒0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2≀0.(3.16) Then using Lemma 2.7, π‘₯0 is a solution of (MQVLI).

Proof of Theorem 3.8. Assume that (3.9) holds. If (π‘₯𝑛)𝑛 in 𝐾 is an L-𝛼-approximating sequence, then from the proof of Theorem 3.6, we know that (π‘₯𝑛)𝑛 converges to some π‘₯0∈𝐾. By Lemma 3.9, π‘₯0 is a solution of (MQVLI) and so (MQVLI) is L-𝛼-well-posed. The converse is exactly same as that in the proof of Theorem 3.6.

4. The Characterizations of 𝛼-Well-Posed in the Generalized Sense for (MQVLI)

In this section, we investigate some metric characterizations of 𝛼-well-posedness in the generalized sense for (MQVLI).

Definition 4.1 (see [8]). Let 𝐴 be a nonempty subset of 𝑋. The measure of noncompactness πœ‡ of the set 𝐴 is defined by ξƒ―πœ‡(𝐴)=infπœ€>0βˆΆπ΄βŠ†π‘›ξšπ‘–=1𝐴𝑖,diam𝐴𝑖.<πœ€,𝑖=1,2,…,𝑛(4.1)

Definition 4.2 (see [8]). Let (𝑋,𝑑) be a metric space and let 𝐴 and 𝐡 be nonempty subsets of 𝑋. The Hausdorff distance 𝐻(β‹…,β‹…) between 𝐴 and 𝐡 is defined by 𝐻(𝐴,𝐡)=max{𝑒(𝐴,𝐡),𝑒(𝐡,𝐴)},(4.2) where 𝑒(𝐴,𝐡)=supπ‘Žβˆˆπ΄π‘‘(π‘Ž,𝐡) with 𝑑(π‘Ž,𝐡)=infπ‘βˆˆπ΅β€–π‘Žβˆ’π‘β€–.

Theorem 4.3. Let the same assumptions be as in Lemma 2.7. Then, one has the following. (a)(MQVLI) is 𝛼-well-posed in the generalized sense if and only if the solution set Ξ“ of (MQVLI) is nonempty compact and 𝑒(π‘„πœ€,Ξ“)β†’0,π‘Žπ‘ πœ€β†’0.(b)Moreover, if 𝐹 is πœ‚-monotone, then (MQVLI) is 𝐿-𝛼-well-posed in the generalized sense if and only if the solution set Ξ“ of (MQVLI) is nonempty compact and 𝑒(πΏπœ€,Ξ“)β†’0,π‘Žπ‘ πœ€β†’0.

Proof. We only prove (a). The proof of (b) is similar and is omitted here. Assume that (MQVLI) is 𝛼-well-posed in the generalized sense; then Ξ“ is nonempty and compact. It follows from Lemma 2.7 that π‘„πœ€β‰ βˆ…. Now we show that 𝑒(π‘„πœ€,Ξ“)β†’0,π‘Žπ‘ πœ€β†’0. Suppose by contradiction that there exists 𝛽>0,πœ€π‘›β†’0 and π‘€π‘›βˆˆπ‘„πœ€π‘›, such that 𝑑(𝑀𝑛,Ξ“)>𝛽. It follows from π‘€π‘›βˆˆπ‘„πœ€π‘› that (𝑀𝑛)𝑛 is an 𝛼-approximating sequence for (MQVLI). Since (MQVLI) is 𝛼-well-posedness in the generalized sense, there exists a subsequence (π‘€π‘›π‘˜)π‘˜ of (𝑀𝑛)𝑛 strongly converging to a point of Ξ“. This contradicts 𝑑(𝑀𝑛,Ξ“)>𝛽. Thus 𝑒(π‘„πœ€,Ξ“)β†’0,π‘Žπ‘ πœ€β†’0.
For the converse, let (π‘₯𝑛)𝑛 be an 𝛼-approximating sequence for (MQVLI); then π‘₯π‘›βˆˆπ‘„πœ€π‘›. It follows from 𝑒(π‘„πœ€π‘›,Ξ“)β†’0 that there exists a sequence π‘§π‘›βŠ‚Ξ“, such that 𝑑(π‘₯𝑛,𝑧𝑛)β†’0. Since Ξ“ is compact, there exists a subsequence (π‘§π‘›π‘˜)π‘˜ of (𝑧𝑛)𝑛 strongly converging to π‘₯0βˆˆΞ“. Thus the corresponding subsequence (π‘₯π‘›π‘˜)π‘˜ of (π‘₯𝑛)𝑛 is strongly converging to π‘₯0. Therefore, (MQVLI) is 𝛼-well-posed in the generalized sense.

Theorem 4.4. Let the same assumptions be as in Theorem 3.4. Then (MQVLI) is 𝛼-well-posed in the generalized sense if and only if π‘„πœ€β‰ βˆ…,βˆ€πœ€>0,limπœ€β†’βˆžπœ‡ξ€·π‘„πœ€ξ€Έ=0.(4.3)

Proof. Assume that (MQVLI) is 𝛼-well-posed in the generalized sense; so π‘„πœ€β‰ βˆ…,forallπœ€>0. By Theorem 4.3(a), Ξ“ is nonempty compact and limπœ€β†’0𝑒(π‘„πœ€,Ξ“)β†’0. For any πœ€>0, we have π»ξ€·π‘„πœ€ξ€Έξ€½π‘’ξ€·π‘„,Ξ“=maxπœ€ξ€Έξ€·,Ξ“,𝑒Γ,π‘„πœ€ξ€·π‘„ξ€Έξ€Ύ=π‘’πœ€ξ€Έ,,Ξ“(4.4) since Ξ“ is compact, πœ‡(Ξ“)=0. For every πœ€>0, the following relation holds (see, e.g., [13]) πœ‡ξ€·π‘„πœ€ξ€Έξ€·π‘„β‰€2π»πœ€ξ€Έξ€·π‘„,Ξ“+πœ‡(Ξ“)=2π»πœ€ξ€Έξ€·π‘„,Ξ“=2π‘’πœ€ξ€Έ.,Ξ“(4.5) It follows from limπœ€β†’0𝑒(π‘„πœ€,Ξ“)β†’0 that limπœ€β†’0πœ‡(π‘„πœ€)=0.
Conversely, assume that (4.3) holds. Then, for any πœ€>0, cl(π‘„πœ€) is nonempty closed and increasing with πœ€>0. By (4.3), limπœ€β†’0πœ‡(cl(π‘„πœ€))=limπœ€β†’0πœ‡(π‘„πœ€)=0, where cl(π‘„πœ€) is the closure of π‘„πœ€. By the generalized Cantor theorem [23, page 412], we know that limπœ€β†’0𝐻𝑄clπœ€ξ€Έξ€Έ,Ξ”=0,asπœ€βŸΆ0,(4.6) where β‹‚Ξ”=πœ€>0cl(π‘„πœ€) is nonempty compact.
Now we show that Ξ“=Ξ”.(4.7) It follows from Lemma 2.7 that Ξ“βŠ†Ξ”. So we need to prove that Ξ”βŠ†Ξ“. Indeed, let π‘₯0βˆˆΞ”. Then 𝑑(π‘₯0,π‘„πœ€)=0 for every πœ€>0. Given πœ€π‘›>0, πœ€π‘›β†’0, for every 𝑛 there exists π‘₯π‘›βˆˆπ‘„πœ€π‘› such that 𝑑(π‘₯0,π‘₯𝑛)<πœ€π‘›. Hence, π‘₯𝑛→π‘₯0 and 𝑑π‘₯𝑛π‘₯,π‘†π‘›ξ€Έξ€Έβ‰€πœ€π‘›,(4.8)βˆƒπ‘’π‘›ξ€·π‘₯βˆˆπΉπ‘›ξ€Έ,𝑒𝑛π‘₯,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έβˆ’π‘“(𝑦)β‰€πœ€π‘›+𝛼2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2ξ€·π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ.(4.9)
It follows from (4.8), π‘₯𝑛→π‘₯0, and the proof of Lemma 3.3 that π‘₯0βˆˆπ‘†(π‘₯0).
Since 𝑆(π‘₯𝑛) Lower Semi-Mosco converges to 𝑆(π‘₯0), for every π‘¦βˆˆπ‘†(π‘₯0), there exists a sequence π‘¦π‘›βˆˆπ‘†(π‘₯𝑛),forallπ‘›βˆˆπ‘, such that lim𝑛𝑦𝑛=𝑦 in the strong topology.
Since πœ‚ is (𝑠,𝑠)-continuous, the sequence (πœ‚(π‘₯𝑛,𝑦𝑛))𝑛 converges strongly to πœ‚(π‘₯0,𝑦). It follows from (ii) and Proposition  2.19 in [24] that there exists a subsequence (𝑒𝑛𝑗)𝑗 of (𝑒𝑛)𝑛 weak* converging to some 𝑒0βˆˆπΈβˆ—. It follows from (ii) and Proposition  2.17 in [24] that 𝐹 is (𝑠,π‘€βˆ—)-closed, and so 𝑒0∈𝐹(π‘₯0). It follows from the proof of Lemma 3.5 that 𝑒𝑛𝑗π‘₯,πœ‚π‘›π‘—,π‘¦π‘›π‘—βŸΆξ«π‘’ξ‚ξ‚­0ξ€·π‘₯,πœ‚0.,𝑦(4.10) Hence, 𝑒0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2=lim𝑗𝑒𝑛𝑗π‘₯,πœ‚π‘›π‘—,𝑦𝑛𝑗π‘₯+π‘“π‘›π‘—ξ‚ξ‚€π‘¦βˆ’π‘“π‘›π‘—ξ‚βˆ’π›Ό2β€–β€–π‘₯π‘›π‘—βˆ’π‘¦π‘›π‘—β€–β€–2≀limπ‘—πœ€π‘›π‘—=0,(4.11) that is, βˆƒπ‘’0ξ€·π‘₯∈𝐹0ξ€Έ,𝑒0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2ξ€·π‘₯≀0,βˆ€π‘¦βˆˆπ‘†0ξ€Έ.(4.12) By Lemma 2.7, we know that π‘₯0βˆˆΞ“. Thus, Ξ”βŠ†Ξ“. It follows from (4.6) and (4.7) that limπœ€β†’0𝑒(π‘„πœ€,Ξ“)=0. It follows from the compactness of Ξ“ and Theorem 4.3(a) that (MQVLI) is 𝛼-well-posed in the generalized sense. The proof is completed.

Theorem 4.5. Let the same assumptions be as in Theorem 3.8. Then (MQVLI) is L-𝛼-well-posed in the generalized sense if and only if πΏπœ€β‰ βˆ…,βˆ€πœ€>0,limπœ€β†’0πœ‡ξ€·πΏπœ€ξ€Έ=0.(4.13)

Proof. Assume that (MQVLI) is L-𝛼-well-posed in the generalized sense. It follows from Lemma 2.7 and the πœ‚-monotonicity of 𝐹 that Ξ“βŠ‚πΏπœ€,forallπœ€>0. And so πΏπœ€β‰ βˆ…, for each πœ€>0. By similar argument with that in the proof of Theorem 4.3(a), we can get 𝑒(πΏπœ€,Ξ“)β†’0 as πœ€β†’0. From the proof of Theorem 4.4, we also obtain πœ‡ξ€·πΏπœ€ξ€Έξ€·πΏβ‰€2π»πœ€ξ€Έξ€·πΏ,Ξ“+πœ‡(Ξ“)=2π»πœ€ξ€Έξ€·πΏ,Ξ“=2π‘’πœ€ξ€Έ.,Ξ“(4.14) Thus, limπœ€β†’0πœ‡(πΏπœ€)=0.
Conversely, assume that (4.13) holds. Then, for any πœ€>0, cl(πΏπœ€) is nonempty closed and increasing with πœ€>0. By (4.13), limπœ€β†’0πœ‡(cl(πΏπœ€))=limπœ€β†’0πœ‡(πΏπœ€)=0, where cl(πΏπœ€) is the closure of πΏπœ€. By the generalized Cantor theorem [23, Page 412], we know that limπœ€β†’0𝐻𝐿clπœ€ξ€Έξ€Έ,Ξ”=0,asπœ€βŸΆ0,(4.15) where β‹‚Ξ”=πœ€>0cl(πΏπœ€) is nonempty compact.
Now we show that Ξ“=Ξ”.(4.16) It follows from Lemma 2.7 and the monotonicity of 𝐹 that Ξ“βŠ†Ξ”. So we need to prove that Ξ”βŠ†Ξ“. Indeed, let π‘₯0βˆˆΞ”. Then 𝑑(π‘₯0,πΏπœ€)=0 for every πœ€>0. Given πœ€π‘›>0, πœ€π‘›β†’0, for every 𝑛 there exists π‘₯π‘›βˆˆπΏπœ€π‘› such that 𝑑(π‘₯0,π‘₯𝑛)<πœ€π‘›. Hence, π‘₯𝑛→π‘₯0 and 𝑑π‘₯𝑛π‘₯,π‘†π‘›ξ€Έξ€Έβ‰€πœ€π‘›,(4.17)π‘₯𝑣,πœ‚π‘›ξ€·π‘₯,𝑦+π‘“π‘›ξ€Έβˆ’π‘“(𝑦)β‰€πœ€π‘›+𝛼2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦2ξ€·π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ,βˆ€π‘£βˆˆπΉ(𝑦).(4.18)
It follows from (4.17), π‘₯𝑛→π‘₯0, and the proof of Lemma 3.3 that π‘₯0βˆˆπ‘†(π‘₯0).
Since 𝑆(π‘₯𝑛) Lower Semi-Mosco converges to 𝑆(π‘₯0), for every π‘¦βˆˆπ‘†(π‘₯0), there exists a sequence π‘¦π‘›βˆˆπ‘†(π‘₯𝑛),forallπ‘›βˆˆπ‘, such that lim𝑛𝑦𝑛=𝑦 in the strong topology.
For each π‘›βˆˆπ‘ select π‘£π‘›βˆˆπΉ(𝑦𝑛). Since 𝐹 is (𝑠,π‘€βˆ—)-usc with weak* compact convex values, we can find a subsequence (𝑣𝑛𝑗)𝑗 of (𝑣𝑛)𝑛 weak* converging to some π‘£βˆˆπΉ(𝑦). By the continuity of πœ‚ and similar argument with the proof of Lemma 3.5, we know that 𝑣𝑛𝑗π‘₯,πœ‚π‘›π‘—,π‘¦π‘›π‘—βŸΆξ«ξ€·π‘₯𝑣,πœ‚0.,𝑦(4.19) Hence, 𝑣𝑛𝑗π‘₯,πœ‚π‘›π‘—,𝑦𝑛𝑗π‘₯+π‘“π‘›π‘—ξ‚ξ‚€π‘¦βˆ’π‘“π‘›π‘—ξ‚βˆ’π›Ό2β€–β€–π‘₯π‘›π‘—βˆ’π‘¦π‘›π‘—β€–β€–2β‰€πœ€π‘›π‘—.(4.20) We deduce from the above inequality that ξ€·π‘₯βˆ€π‘¦βˆˆπ‘†0π‘₯,βˆƒπ‘£βˆˆπΉ(𝑦),𝑣,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2≀0.(4.21) By Lemma 2.6 we know that there exist 𝑒0∈𝐹(π‘₯0), such that ξ€·π‘₯βˆ€π‘¦βˆˆπ‘†0ξ€Έ,𝑒0ξ€·π‘₯,πœ‚0ξ€·π‘₯,𝑦+𝑓0ξ€Έπ›Όβˆ’π‘“(𝑦)βˆ’2β€–β€–π‘₯0β€–β€–βˆ’π‘¦2≀0.(4.22) It follows from Lemma 2.7 that π‘₯0βˆˆΞ“. Thus, Ξ”βŠ†Ξ“. It follows from (4.15) and (4.16) that limπœ€β†’0𝑒(πΏπœ€,Ξ“)=0. It follows from the compactness of Ξ“ and Theorem 4.3(b) that (MQVLI) is L-𝛼-well-posed in the generalized sense. The problem is completed.

Remark 4.6. (i) It is easy to see that if 𝛼=0, then by the main results in our paper, we can recover the corresponding results in [20] with the weaker condition 𝑆(π‘₯𝑛) Lower Semi-Mosco converging to 𝑆(π‘₯0) instead of the condition 𝑆(π‘₯𝑛) Mosco converging to 𝑆(π‘₯0).
(ii) The proof methods of Theorems 4.4 and 4.5 are different from those ofTheorems  4.1 and  4.2in [20].

Acknowledgments

The authors would like to express their thanks to the referee for helpful suggestions. This research was supported by the National Natural Science Foundation of China (Grants 10771228 and 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), and the Research Project of Chongqing Normal University (Grant 08XLZ05).

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