The Tikhonov Regularization Method for Set-Valued Variational Inequalities

Abstract

This paper aims to establish the Tikhonov regularization theory for set-valued variational inequalities. For this purpose, we firstly prove a very general existence result for set-valued variational inequalities, provided that the mapping involved has the so-called variational inequality property and satisfies a rather weak coercivity condition. The result on the Tikhonov regularization improves some known results proved for single-valued mapping.

1. Introduction

This paper discusses the generalized variational inequality problem (in short, GVIP(𝐹,𝐾)) which is to find 𝑥∈𝐾 and 𝑥^∗∈𝐹(𝑥) such that𝑥^∗,𝑦−𝑥≥0,∀𝑦∈𝐾, (1.1) where 𝐾 is a nonempty closed convex set in ℝ^𝑛 and 𝐹∶𝐾→2^ℝ𝑛 is a set-valued mapping with nonempty values. We use GVIP(𝐹,𝐾) and SOL(𝐹,𝐾) to denote the problem (1.1) and its solution set, respectively.

Generalized variational inequality has been extensively studied in the literature; see [1–7] and the references therein.

The Tikhonov regularization method is an important method for the ill-posed variational inequalities; see Pages 307 and  1224 in [8]. To our best knowledge, the Tikhonov regularization method has been discussed only for the case where the mapping 𝐹 is single valued. This paper develops the Tikhonov regularization method for set-valued variational inequality GVIP(𝐹,𝐾).

As a preparation, we firstly give an existence result for GVIP(𝐹,𝐾). It is well known if 𝐹 is upper semicontinuous with nonempty compact convex values and the set 𝐾 is compact and convex, then GVIP(𝐹,𝐾) has a solution. If 𝐾 is noncompact, one usually requires some kind of coerciveness conditions for the existence of solutions to GVIP(𝐹,𝐾). Thus many researchers have attempted to search coerciveness condition as weak as possible; see [2, 3, 8, 9] and the references therein. In particular, [9] proved the following result.

Theorem 1.1. If 𝐹 is single valued and continuous, and if the following condition is satisfied:(H) There exists 𝑛∈ℕ such that for every 𝑥∈𝐾⧵𝐾_𝑛, there is 𝑦∈𝐾 with ‖𝑦‖<‖x‖ satisfying ⟨𝐹(𝑥),𝑥−𝑦⟩≥0,then the variational inequality has a solution.

Example  3.1 in [9] shows that the condition (H) is strictly weaker than many known coerciveness conditions. So far, it is not known whether Theorem 1.1 could be extended to the situation where 𝐹 is a set-valued mapping. An affirmative answer is given in Corollary 3.9 of this paper, which says that if 𝐹 is upper semicontinuous with nonempty compact convex values, then the coerciveness condition (A), which reduces to the condition (H) when 𝐹 is single-valued, implies that GVIP(𝐹,𝐾) has a solution. Actually, a more general existence result is verified in Theorem 3.8 which does not require that 𝐹 have any kind of continuity. Theorem 3.8 shows that if 𝐹 has the so-called variational inequality property, then the condition (A) implies that GVIP(𝐹,𝐾) has a solution. If the mapping 𝐹 is either upper semicontinuous (with nonempty compact convex values), or quasimonotone and upper hemicontinuous (with nonempty compact convex values), then it has the variational inequality property. Thus Theorem 3.8 unifies many known existence results for GVIP(𝐹,𝐾).

The Tikhonov regularization method has been much discussed in the literature. In particular, assuming that 𝐾 is a box and 𝐹 is a single-valued mapping, [10] proved that SOL(𝐹+𝜀𝐼,𝐾) is nonempty for any 𝜀>0 (here 𝐼 stands for the identity mapping), provided that 𝐹 is a continuous 𝑃₀-function and SOL(𝐹,𝐾) is nonempty and bounded. This result was extended by [11] to the situation where 𝐾 is a closed convex set, the mapping 𝐹 is single valued, and a coercivity condition is used to replace the assumption of SOL(𝐹,𝐾) being bounded. It should be noted that the coercivity condition assumed in [11] does not necessarily imply that SOL(𝐹,𝐾) is bounded. [12] further improves the result of [11] by assuming a weaker coercivity condition. All the above results on the Tikhonov regularization assume that the mapping 𝐹 is single valued. The last part of this paper aims to establish the Tikhonov regularization theory for set-valued variational inequality (1.1). Theorem 4.1 improves the main result of [11, 12] by assuming a weaker coercivity condition and by allowing 𝐹 to be a set-valued mapping (without monotonicity).

General variational inequalities have been extensively discussed; see [13–16]. It should be interesting to discuss the Tikhonov regularization method for general variational inequality in a similar way.

2. Preliminaries

Unless stated otherwise, we assume that 𝐾⊂ℝ^𝑛 is a nonempty closed convex set and 𝐹∶𝐾→2^ℝ𝑛 is a set-valued mapping with nonempty values. For 𝑟>0, 𝐾_𝑟∶={𝑥∈𝐾∶‖𝑥‖≤𝑟}.

Definition 2.1. Let 𝐹∶𝐾→2^ℝ𝑛 be a set-valued mapping. 𝐹 is said to be(i)monotone on 𝐾 if for each pair of points 𝑥,𝑦∈𝐾 and for all 𝑥^∗∈𝐹(𝑥) and 𝑦^∗∈𝐹(𝑦), ⟨𝑦^∗−𝑥^∗,𝑦−𝑥⟩≥0,(ii)maximal monotone on 𝐾 if, for any 𝑢∈𝐾, ⟨𝜉−𝑥^∗,𝑢−𝑥⟩≥0 for all 𝑥∈𝐾 and all 𝑥^∗∈𝐹(𝑥) implies 𝜉∈𝐹(𝑢),(iii)quasimonotone on 𝐾 if for each pair of points 𝑥,𝑦∈𝐾 and for all 𝑥^∗∈𝐹(𝑥) and 𝑦^∗∈𝐹(𝑦), ⟨𝑥^∗,𝑦−𝑥⟩>0 implies that ⟨𝑦^∗,𝑦−𝑥⟩≥0,(iv)𝐹 is said to be upper semicontinuous at 𝑥∈𝐾 if for every open set 𝑉 containing 𝐹(𝑥), there is an open set 𝑈 containing 𝑥 such that 𝐹(𝑦)⊂𝑉 for all 𝑦∈𝐾∩𝑈; if 𝐹 is upper semicontinuous at every 𝑥∈𝐾, we say 𝐹 is upper semicontinuous on 𝐾,(v)upper hemicontinuous on 𝐾 if the restriction of 𝐹 to every line segment of 𝐾 is upper semicontinuous.

 The following result is celebrated; see [17].

Lemma 2.2. Let 𝐾 be a nonempty convex subset of a Hausdorff topological vector space 𝐸, and let 𝐺∶𝐾→2^𝐸 be a set-valued mapping from 𝐾 into 𝐸 satisfying the following properties:(i)𝐺 is a KKM mapping: for every finite subset 𝐴 of 𝐾, co(𝐴)⊂⋃_𝑥∈𝐴𝐺(𝑥), where 𝑐𝑜 denotes the convex hull;(ii)𝐺(𝑥) is closed in 𝐸 for every 𝑥∈𝐾;(iii)𝐺(𝑥₀) is compact in 𝐸 for some 𝑥₀∈𝐾. Then ⋂_𝑥∈𝐾𝐺(𝑥)≠∅.

3. Existence of Solutions and Coercivity Conditions

Definition 3.1. 𝐹 is said to have variational inequality property on 𝐾 if for every nonempty bounded closed convex subset 𝐷 of 𝐾, GVIP(𝐹,𝐷) has a solution.

Proposition 3.2. The following classes of mappings have the variational inequality property:(i)every upper semicontinuous set-valued mapping with nonempty compact convex values,(ii)every upper hemicontinuous quasimonotone set-valued mapping with nonempty compact convex values;(iii)if 𝐹 is a single-valued continuous mapping and 𝑇 is upper hemicontinuous and monotone with nonempty compact convex values, then 𝐹+𝑇 has the variational inequality property.

Proof. (i) is well known in the literature. (ii) is verified in [18]. (iii) is a consequence of the Debrunner-Flor lemma [19] and [20, Theorem 41.1]. Indeed, let 𝐷 be a bounded closed convex subset of 𝐾, and let 𝑁_𝐷(𝑥)∶=𝜉∈ℝ^𝑛∶sup_𝑦∈𝐷𝜉,𝑦−𝑥≤0, (3.1) the normal cone of 𝐷 at 𝑥∈𝐷. By [20, Theorem 41.1], 𝑀(𝑥)∶=𝑇(𝑥)+𝑁_𝐷(𝑥) is a maximal monotone mapping. By the Debrunner-Flor lemma [19], there is 𝑢∈𝐷 such that 𝜉+𝐹(𝑢),𝑥−𝑢≥0,∀𝑥∈𝐷,∀𝜉∈𝑀(𝑥). (3.2) Since 𝑀 is maximal monotone, −𝐹(𝑢)∈𝑀(𝑢)≡𝑇(𝑢)+𝑁_𝐷(𝑢). By the definition of 𝑁_𝐷, 𝑢∈SOL(𝐹+𝑇,𝐷).

Proposition 3.3 below shows that Proposition 3.2(iii) can be extended to the case where 𝐹 is a set-valued mapping.

Proposition 3.3. If 𝐹∶𝐾→2^ℝ𝑛 is upper semicontinuous with nonempty compact convex values and 𝑇∶𝐾→2^ℝ𝑛 is monotone and upper hemicontinuous on 𝐾 with nonempty compact convex values, then 𝐹+𝑇 has the variational inequality property on 𝐾.

Proof. Let 𝐷 be a bounded closed convex subset of 𝐾. Define 𝐺,𝐻∶𝐷→2^𝐷 by G𝑦∶=𝑥∈𝐷∶sup_{(𝑥)𝜉∈𝑇}𝜉,𝑦−𝑥+sup_{(𝑥)𝜁∈𝐹},𝐻𝑦𝜁,𝑦−𝑥≥0∶=𝑥∈𝐷∶inf_{𝜉∈𝑇(𝑦)}𝜉,𝑦−𝑥+sup_{𝜁∈𝐹(𝑥)}𝜁,𝑦−𝑥≥0. (3.3) Since 𝑇 is monotone, 𝐺(𝑦)⊂𝐻(𝑦). Since 𝐹 is upper semicontinuous, 𝐻(𝑦) is closed and hence compact in 𝐷 for every 𝑦∈𝐷. Now we prove that 𝐺 is a KKM map. If not, there is {𝑦₁,…,𝑦_𝑛}⊂𝐷 and 𝑥₀=∑𝜆_𝑖𝑦_𝑖 such that 𝑥₀∉⋃^𝑛_𝑖=1𝐺(𝑦_𝑖)0=sup_{𝜉∈𝑇(𝑥0)}𝜉,𝜆_𝑖𝑦_𝑖−𝑥₀+sup_{𝜁∈𝐹(𝑥0)}𝜁,𝜆_𝑖𝑦_𝑖−𝑥₀≤𝜆_𝑖sup_{𝑥𝜉∈𝑇0}𝜉,𝑦_𝑖−𝑥₀+sup_{𝑥𝜁∈𝐹0}𝜁,𝑦_𝑖−𝑥₀<0. (3.4) This contradiction shows that 𝐺 is a KKM map, so is 𝐻. By Lemma 2.2, there is 𝑥∈∩_𝑦∈𝐷𝐻(𝑦). Fix any 𝑦∈𝐷. Let 𝑦_𝑡∶=𝑥+𝑡(𝑦−𝑥). Then for small 𝑡∈(0,1), 𝑦_𝑡∈𝐷 and hence 0≤inf_{𝜉∈𝑇(𝑦𝑡)}𝜉,𝑦_𝑡−𝑥+sup_{𝜁∈𝐹(𝑥)}𝜁,𝑦_𝑡−𝑥=𝑡inf_{𝜉∈𝑇(𝑦𝑡)}𝜉,𝑦−𝑥+𝑡sup_{𝜁∈𝐹(𝑥)}𝜁,𝑦−𝑥. (3.5) Dividing 𝑡>0 on both sides, we have sup_{𝜉∈𝑇(𝑦𝑡)}𝜉,𝑦−𝑥+sup_{𝜁∈𝐹(𝑥)}𝜁,𝑦−𝑥≥0. (3.6) Letting 𝑡→0+ yields that 𝑥∈𝐺(𝑦), as 𝑇 is upper hemicontinuous. Since 𝑦∈𝐷 is arbitrary, 𝑥∈⋂_𝑦∈𝐷𝐺(𝑦): sup_{𝜉∈𝑇(𝑥)}sup_{𝜁∈𝐹(𝑥)}𝜉+𝜁,𝑦−𝑥≥0,∀𝑦∈𝐷. (3.7) Since 𝐹(𝑥) and 𝑇(𝑥) are compact and convex, the Sion minimax theorem implies the existence of 𝑢∈𝐹(𝑥) and 𝑣∈𝑇(𝑥) such that 𝑢+𝑣,𝑦−𝑥≥0,∀𝑦∈𝐷. (3.8) Thus 𝑥 solves GVIP(𝐹+𝑇,𝐷).

Before making further discussion, we need to state some coercivity conditions. The relationships of these coercivity conditions are well known in the literature; however, we provide the proof for completeness.

Consider the following coercivity conditions.(A)There exists 𝑟>0 such that for every 𝑥∈𝐾⧵𝐾_𝑟, there is 𝑦∈𝐾 with ‖𝑦‖<‖𝑥‖ satisfying inf_{𝑥∗∈𝐹(𝑥)}⟨𝑥^∗,𝑥−𝑦⟩≥0.(B)There exists 𝑟>0 such that for every 𝑥∈𝐾⧵𝐾_𝑟, there is 𝑦∈𝐾_𝑟 satisfying inf_{𝑥∗∈𝐹(𝑥)}⟨𝑥^∗,𝑥−𝑦⟩≥0.(C)There exists 𝑟>0 such that for every 𝑥∈𝐾⧵𝐾_𝑟 and every 𝑥^∗∈𝐹(𝑥), there exists some 𝑦∈𝐾_𝑟 such that ⟨𝑥^∗,𝑥−𝑦⟩>0.(D)There exists 𝑟>0 such that for every 𝑥∈𝐾⧵𝐾_𝑟, there exists some 𝑦∈𝐾_𝑟 such that sup_{𝑦∗∈𝐹(𝑦)}⟨𝑦^∗,𝑥−𝑦⟩>0.(E)There exists 𝑦₀∈𝐾 such that the set𝐿𝑦₀∶=𝑥∈𝐾∶inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗,𝑥−𝑦₀<0 (3.9)

 is bounded, if nonempty.

Proposition 3.4. The following statements hold.(i)(C)⇒(B) if 𝐹 is of convex values.(ii)(D)⇒(B) if 𝐹 is quasimonotone.(iii)(E)⇒(B)⇒(A).

Proof. (i) By (C), for every 𝑥∈𝐾⧵𝐾_𝑟, inf_{𝑥∗∈𝐹(𝑥)}sup_{𝑦∈𝐾𝑟}⟨𝑥^∗,𝑥−𝑦⟩≥0. Since 𝐹(𝑥) is convex and 𝐾_𝑟 is compact convex, the Kneser minimax theorem implies that sup_{𝑦∈𝐾𝑟}inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗,𝑥−𝑦=inf_{𝑥∗∈𝐹(𝑥)}sup_{𝑦∈𝐾𝑟}𝑥^∗,𝑥−𝑦≥0. (3.10) Since 𝑦↦inf_{𝑥∗∈𝐹(𝑥)}⟨𝑥^∗,𝑥−𝑦⟩ is upper semicontinuous and since 𝐾_𝑟 is compact, there is 𝑦(𝑥)∈𝐾_𝑟 such that inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗,𝑥−𝑦(𝑥)=sup_{𝑦∈𝐾𝑟}inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗,𝑥−𝑦≥0. (3.11) This verifies (B). (ii) The implication of (D) ⇒ (B) is an immediate consequence of 𝐹 being quasimonotone. (iii) (E) ⇒ (B). If 𝐿(𝑦₀)=∅, then for any 𝑥∈𝐾, inf_{𝑥∗∈𝐹(𝑥)}⟨𝑥^∗,𝑥−𝑦₀⟩≥0; thus (B) holds. If 𝐿(𝑦₀)≠∅, then by (E), there is 𝑟>0 such that 𝐿(𝑦₀)∪{𝑦₀}⊂𝐾_𝑟. Thus (B) is still true. (B) ⇒ (A). Let 𝑟>0 be such that (B) holds. Then 𝑥∈𝐾⧵𝐾_𝑟+1, 𝑥∉𝐾_𝑟. By (B), there is 𝑦∈𝐾_𝑟 such that inf_{𝑥∗∈𝐹(𝑥)}⟨𝑥^∗,𝑥−𝑦⟩≥0. Obviously, ‖𝑦‖≤𝑟<𝑟+1<‖𝑥‖. Thus (A) is verified with 𝑟 replaced by 𝑟+1.

Remark 3.5. The coercivity condition (A) is actually (C’) in [3] where it is shown that if 𝐹 is quasimonotone and upper hemicontinuous with nonempty compact convex values, then (A) implies that GVIP(𝐹,𝐾) has a solution. However, it seems unknown whether this assertion still holds if one replaces “quasimonotone and upper hemicontinuous” by “upper semicontinuous.” An affirmative answer is given by Corollary 3.9.

Remark 3.6. The coercivity condition (B) is the condition (C) in [3]. The condition (D) appears in [21]. The condition (E) appears essentially in Corollary  3.1 in [22]; see also Proposition  2.2.3 in [8]. If 𝐹 is single valued, then (E) reduces to Proposition  2.2.3(a) in [8].

Remark 3.7. Example  3.1 in [9] shows that (A) does not necessarily imply (B), even if 𝐹 is single-valued and continuous.

From the above discussion, (A) is the weakest coercivity condition among them. [9] proved if 𝐹 is single valued and continuous, then (A) implies that GVIP(𝐹,𝐾) has a solution. Corollary 3.9 shows that this assertion still holds even if 𝐹 is a set-valued mapping.

Theorem 3.8. Let 𝐾⊂ℝ^𝑛 be a nonempty closed convex set, and let 𝐹∶𝐾→2^ℝ𝑛 be a mapping with nonempty compact convex values. Suppose that (A) holds. If 𝐹 has the variational inequality property on 𝐾, then GVIP(𝐹,𝐾) has a solution.

Proof. Let 𝑚>𝑟. Since 𝐾_𝑚 is bounded closed convex and 𝐹 has the variational inequality property, there is 𝑥_𝑚∈𝐾_𝑚 such that sup_{𝑥∗∈𝐹(𝑥𝑚)}⟨𝑥^∗,𝑦−𝑥_𝑚⟩≥0,∀𝑦∈𝐾_𝑚. (3.12)  (i) If ‖𝑥_𝑚‖=𝑚, then ‖𝑥_𝑚‖>𝑟, and by assumption, there is 𝑦₀∈𝐾 with ‖𝑦₀‖<‖𝑥_𝑚‖ such that sup_{𝑥∗∈𝐹(𝑥𝑚)}⟨𝑥^∗,𝑦₀−𝑥_𝑚⟩≤0. (3.13) Fix any 𝑦∈𝐾. Since ‖𝑦₀‖<‖𝑥_𝑚‖≤𝑚, there is 𝑡∈(0,1) such that 𝑧_𝑡∶=𝑦₀+𝑡(𝑦−𝑦₀)∈𝐾_𝑚. It follows that 0≤sup_{𝑥∗∈𝐹(𝑥𝑚)}⟨𝑥^∗,𝑧_𝑡−𝑥_𝑚⟩≤𝑡sup_{𝑥∗∈𝐹(𝑥𝑚)}⟨𝑥^∗,𝑦−𝑥_𝑚⟩+(1−𝑡)sup_{𝑥∗∈𝐹(𝑥𝑚)}⟨𝑥^∗,𝑦₀−𝑥_𝑚⟩≤𝑡sup_{𝑥∗∈𝐹(𝑥𝑚)}⟨𝑥^∗,𝑦−𝑥_𝑚⟩. (3.14) Therefore, sup_{𝑥∗∈𝐹(𝑥𝑚)}⟨𝑥^∗,𝑦−𝑥_𝑚⟩≥0. Since 𝑦∈𝐾 is arbitrary, the conclusion is verified. (ii) If ‖𝑥_𝑚‖<𝑚, then for any 𝑦∈𝐾, there is 𝑡∈(0,1) such that 𝑧_𝑡∶=𝑥_𝑚+𝑡𝑦−𝑥_𝑚∈𝐾_𝑚. (3.15) It follows that 0≤sup_{𝑥∗∈𝐹(𝑥𝑚)}⟨𝑥^∗,𝑧_𝑡−𝑥_𝑚⟩=𝑡sup_{𝑥∗∈𝐹(𝑥𝑚)}⟨𝑥^∗,𝑦−𝑥_𝑚⟩. (3.16) Since 𝑦∈𝐾 is arbitrary, 𝑥_𝑚 solves GVIP(𝐹,𝐾).

Corollary 3.9. Let 𝐾⊂ℝ^𝑛 be a nonempty closed convex set, and let 𝐹∶𝐾→2^ℝ𝑛 be upper semicontinuous set-valued mapping with nonempty compact convex values. Suppose that (A) holds. Then GVIP(𝐹,𝐾) has a solution.

Proof. By Proposition 3.2, 𝐹 has the variational inequality property. The conclusion follows immediately from Theorem 3.8.

Remark 3.10. Assuming that the mapping 𝐹 is single valued and continuous, Proposition  2.2.3 in [8] and Theorem  3.2 in [9] show that SOL(𝐹,𝐾) is nonempty if the conditions (E) and (A) hold, respectively. Therefore, Corollary 3.9 improves Proposition  2.2.3 in [8] and Theorem  3.2 in [9]: the mapping 𝐹 is set valued instead of single valued.

4. The Tikhonov Regularization

Theorem 4.1. Let 𝐾 be a nonempty closed convex set in ℝ^𝑛, and let 𝐹∶𝐾→2^ℝ𝑛 be upper semicontinuous with nonempty compact convex values. If assumption (A) holds, then for any 𝜀>0,(i)GVIP(𝐹+𝜀𝐼,𝐾) has a solution;(ii)the set {SOL(𝐹+𝑡𝐼,𝐾)∶𝑡∈(0,𝜀]} is bounded.

Proof. (i) Let 𝑟 be as in assumption (A). We claim that for every 𝑥∈𝐾⧵𝐾_𝑟, there is 𝑦∈𝐾 with ‖𝑦‖<‖𝑥‖ satisfying inf_{𝑥∗∈𝐹(𝑥)}⟨𝑥^∗+𝜀𝑥,𝑥−𝑦⟩≥0. (4.1) Granting this, we obtain that assumption (A) is satisfied with the mapping 𝐹 replaced by 𝐹+𝜀𝐼. Since 𝐹 is upper semicontinuous with nonempty compact convex values and 𝐼 is continuous and monotone, Theorem 3.8 and Proposition 3.3 imply that GVIP(𝐹+𝜀𝐼,𝐾) has a solution. Now we prove the claim. Since for every 𝑥∈𝐾⧵𝐾_𝑟, there is 𝑦∈𝐾 with ‖𝑦‖<‖𝑥‖ satisfying inf_{𝑥∗∈𝐹(𝑥)}⟨𝑥^∗,𝑥−𝑦⟩≥0, we obtain inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗+𝜀𝑥,𝑥−𝑦=inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗,𝑥−𝑦+𝜀𝑥,𝑥−𝑦=inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗,𝑥−𝑦+𝜀‖𝑥‖²−𝜀𝑥,𝑦≥𝜀‖𝑥‖‖𝑥‖−‖𝑦‖≥0. (4.2)  (ii) Let 𝑡∈(0,𝜀] and 𝑥(𝑡)∈SOL(𝐹+𝑡𝐼,𝐾). Then 𝑥(𝑡)∈𝐾_𝑛. If not, by assumption (A), there is 𝑦(𝑡)∈𝐾 with ‖𝑦(𝑡)‖<‖𝑥(𝑡)‖ such that inf_{𝑥∗∈𝐹(𝑥(𝑡))}𝑥^∗,𝑥(𝑡)−𝑦(𝑡)≥0. (4.3) Since 𝑥(𝑡)∈SOL(𝐹+𝑡𝐼,𝐾) and 𝑦(𝑡)∈𝐾, 0≥inf_{𝑥∗∈𝐹(𝑥(𝑡))}𝑥^∗+𝑡𝑥(𝑡),𝑥(𝑡)−𝑦(𝑡)=inf_{𝑥∗∈𝐹(𝑥(𝑡))}𝑥^∗,𝑥(𝑡)−𝑦(𝑡)+𝑡‖𝑥(𝑡)‖²𝑥(𝑡)(𝑡(𝑡)‖−𝑡,𝑦)≥𝑡‖𝑥²−𝑡‖𝑥(𝑡)‖‖𝑦(𝑡)‖. (4.4) Therefore ‖𝑥(𝑡)‖≤‖𝑦(𝑡)‖, a contradiction.

Remark 4.2. Theorem 4.1 improves in two ways: the mapping 𝐹 is set valued instead of single valued; Theorem  1 in [12] used the coercivity condition (B) while our Theorem 4.1 uses the weaker coercivity condition (A).

Theorem 4.3. Let 𝐾 be a nonempty closed convex set in ℝ^𝑛, and let 𝐹∶𝐾→2^ℝ𝑛 be upper semicontinuous with nonempty compact convex values. Assume that there is a nonempty bounded closed convex set 𝐷⊂𝐾 such that 𝑥∈𝐾∶sup_𝑦∈𝐷inf_{𝑥∗(𝑥)∈𝐹}𝑥^∗,𝑥−𝑦<0isbounded,ifnonempty. (4.5) Then for any 𝜀>0, GVIP(𝐹+𝜀𝐼,𝐾) has a solution and the set {SOL(𝐹+𝑡𝐽,𝐾)∶𝑡∈(0,𝜀]} is bounded.

Proof. Let 𝑟>0 be such that 𝐷𝑥∈𝐾∶sup_𝑦∈𝐷inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗,𝑥−𝑦<0⊂𝐾_𝑟. (4.6) Then for every 𝑥∈𝐾⧵𝐾_𝑟, sup_𝑦∈𝐷inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗,𝑥−𝑦≥0. (4.7) Since 𝐷 is bounded closed convex and since 𝑦↦inf_{𝑥∗∈𝐹(𝑥)}⟨𝑥^∗,𝑥−𝑦⟩ is upper semicontinuous, there is 𝑦₀∈𝐷 such that inf_{𝑥∗∈𝐹(𝑥)}𝑥^∗,𝑥−𝑦₀≥0. (4.8) Since 𝑦₀∈𝐷⊂𝐾_𝑟, ‖𝑦₀‖<𝑟≤‖𝑥‖. Therefore, assumption (A) is satisfied. Then the conclusion follows from Theorem 4.1.

Remark 4.4. If 𝐹 is a single-valued mapping, and 𝐷={𝑥₀} for some 𝑥₀∈𝐾, then Theorem 4.3 reduces to Theorem  2.1 in [11].

For every 𝜀>0, let 𝐴_𝜀⊂ℝ^𝑛, and we definelimsup_𝜀→0+𝐴_𝜀∶=𝑥∈ℝ^𝑛∶∃𝜀_𝑛⟶0+and𝑥_𝑛∈𝐴_𝜀𝑛suchthat𝑥_𝑛⟶𝑥. (4.9)

Theorem 4.5. Let 𝐾 be a nonempty closed convex set in ℝ^𝑛, and let 𝐹∶𝐾→2^ℝ𝑛 be upper semicontinuous with nonempty compact convex values. If assumption (A) holds, then ∅≠limsup_𝜀→0+SOL(𝐹+𝜀𝐼,𝐾)⊂SOL(𝐹,𝐾). (4.10)

Proof. The set limsup_𝜀→0+SOL(𝐹+𝜀𝐼,𝐾) being nonempty follows from Theorem 4.1. Let 𝑥∈limsup_𝜀→0+SOL(𝐹+𝜀𝐼,𝐾). Then there are a sequence 𝜀_𝑛→0+ and 𝑥_𝑛∈SOL(𝐹+𝜀_𝑛𝐼,𝐾) such that 𝑥_𝑛→𝑥. This means that for some 𝑥^∗_𝑛∈𝐹(𝑥_𝑛), 𝑥^∗_𝑛+𝜀_𝑛𝑥_𝑛,𝑦−𝑥_𝑛≥0,∀𝑦∈𝐾. (4.11) Since 𝐹 is upper semicontinuous with nonempty compact values, {𝑥^∗_𝑛} is compact. Without loss of generality, assume lim_𝑛→∞𝑥^∗_𝑛=𝑥^∗ for some 𝑥^∗∈𝐹(𝑥). Thus for every 𝑦∈𝐾, 𝑥^∗_𝑛+𝜀_𝑛𝑥_𝑛,𝑦−𝑥_𝑛=𝑥^∗_𝑛,𝑦−𝑥_𝑛+𝜀_𝑛𝑥_𝑛,𝑦−𝜀_𝑛‖𝑥_𝑛‖²⟶𝑥^∗,𝑦−𝑥as𝑛⟶∞. (4.12) It follows from (4.11) that 𝑥∈SOL(𝐹,𝐾).

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (no. 10701059) and by the Sichuan Youth Science and Technology Foundation.