Existence Results for Nonsmooth Vector Quasi-Variational-Like Inequalities

Abstract

We introduce nonsmooth vector quasi-variational-like inequalities (NVQVLI) by means of a bifunction. We establish some existence results for solutions of these inequalities by using Fan-KKM theorem and a maximal element theorem. By using the technique and methodology adopted in Al-Homidan et al. (2012), one can easily derive the relations among these inequalities and a vector quasi-optimization problem. Hence, the existence results for a solution of a vector quasi-optimization problem can be derived by using our results. The results of this paper extend several known results in the literature.

1. Introduction

The theory of quasi-variational inequalities (QVI) was started with a pioneerwork of A. Bensoussan and J. L. Lions in 1973, perhaps motivated by the stochastic control and impulse control problems. It was the paper of Bensoussan et al. [1] in which the term quasi-variational inequality was introduced. The quasi-variational inequality is an extension of a variational inequality [2] in which the underlying set depends on the solution itself. For further details on quasi-variational inequalities, we refer to [3–5] and the references therein. In 1980, Giannessi [6] initiated the theory of vector variational inequalities with applications to vector optimization. Since then, it has been growing up in different directions. One of such directions is the application to the theory of vector optimization. However, if the underlying objective function is not differentiable and not convex, then we need to define a nonsmooth vector variational-like inequality by means of Dini directional derivatives or Clarke directional derivatives. For studying such problems by using vector variational-like inequalities, Alshahrani et al. [7], Al-Homidan et al. [8], Ansari and Lee [9], Crespi et al. [10], and Lalitha and Mehta [11] considered a vector variational inequality, defined by means of Dini directional derivatives, called nonsmooth vector variational inequality. The nonsmooth vector optimization is studied in these references by using nonsmooth vector variational inequalities. Motivated by the extension of variational inequalities for vector-valued functions, several researchers started to study the QVI for vector-valued functions, known as vector quasi-variational inequalities (VQVI); see, for example, [12–16] and the references therein. An optimization problem in which the feasible set depends on the solution itself is called quasi-optimization problem [14]. Such problems can be solved by using the vector quasi-variational inequality technique. To the best of our knowledge, no study has been done in the literature to study nonsmooth quasi-variational inequalities which are defined by means of a bifunction, in particular by means of Dini or Clarke directional derivatives. This paper can be treated as the beginning of the study of nonsmooth (vector) quasi-variational inequalities and nonsmooth vector quasi-optimization problem.

In this paper, we consider the vector quasi-variational-like inequality problems defined by means of a bifunction and present some existence results for solutions of these problems by using Fan-KKM theorem and a maximal element theorem. By using the technique and methodology adopted in [8], one can easily derive the relations among these inequalities and a vector quasi-optimization problem. Hence, the existence results for a solution of a vector quasi-optimization problem can be derived by using our results. The results of this paper extend several known results in the literature.

2. Formulations

We adopt the following ordering relations. We consider the cones 𝐶=ℝ^ℓ₊ and 𝐶^∘∶=intℝ^ℓ₊, where ℝ^ℓ₊ is the nonnegative orthant of ℝ^ℓ, and 𝟎 is the origin of ℝ^ℓ; let 𝐷 be a set of ℝ^ℓ₊. Then for all 𝑥,𝑦∈𝐷, 𝑥≥_𝐶𝑦⟺𝑥−𝑦∈𝐶;𝑥≱_𝐶𝑦⟺𝑥−𝑦∉𝐶;𝑥≤_𝐶𝑦⟺𝑦−𝑥∈𝐶;𝑥≰_𝐶𝑦⟺𝑦−𝑥∉𝐶;𝑥≥^∘_𝐶𝑦⟺𝑥−𝑦∈𝐶^∘;𝑥≱^∘_𝐶𝑦⟺𝑥−𝑦∉𝐶^∘;𝑥≤^∘_𝐶𝑦⟺𝑦−𝑥∈𝐶^∘;𝑥≰^∘_𝐶𝑦⟺𝑦−𝑥∉𝐶^∘. (1)

Let 𝑔∶ℝ^𝑛→ℝ be a real-valued function. The upper Dini directional derivative of 𝑔 at 𝑥∈𝐾 in the direction 𝑑∈ℝ^𝑛 is defined as 𝑔^𝐷(𝑥;𝑑)=limsup_𝑡→0+𝑔(𝑥+𝑡𝑑)−𝑔(𝑥)𝑡. (2)

For further details on Dini directional derivatives, we refer to the recent book [2].

Let 𝐾 be a nonempty subset of ℝ^𝑛, 𝐴∶𝐾→2^𝐾 a set-valued map, and 𝜂∶𝐾×𝐾→ℝ^𝑛 a mapping. Let ℎ=(ℎ₁,…,ℎ_ℓ)∶𝐾×ℝ^𝑛→ℝ^ℓ be a vector-valued function such that, for each fixed 𝑥∈𝐾, ℎ(𝑥;𝑑) is positively homogeneous in 𝑑. In particular, we consider ℎ(𝑥;𝑑)=𝑓^𝐷(𝑥;𝑑) where 𝑓=(𝑓₁,…,𝑓_ℓ)∶ℝ^𝑛→ℝ^ℓ a vector-valued function and 𝑓^𝐷(𝑥;𝑑)=𝑓^𝐷₁(𝑥;𝑑),…,𝑓^𝐷_ℓ(𝑥;𝑑). (3)

The nonsmooth (Stampacchia or Minty type) vector quasi-variational-like inequality problems are defined as follows.

Nonsmooth Stampacchia Vector Quasi-Variational-Like Inequality Problem (NSVQVLIP). Find 𝑥∈𝐾 such that 𝑥∈𝐴(𝑥) and ℎ𝑥;𝜂𝑦,𝑥=ℎ₁𝑥;𝜂𝑦,𝑥,…,ℎ_ℓ𝑥;𝜂𝑦,𝑥≰^∘_𝐶(𝟎,∀𝑦∈𝐴𝑥). (4)

Minty Vector Quasi-Variational-Like Inequality Problem (NMVQVLIP). Find 𝑥∈𝐾 such that 𝑥∈𝐴(𝑥) and ℎ𝑦;𝜂=ℎ𝑥,𝑦₁𝑦;𝜂𝑥,𝑦,…,ℎ_ℓ𝑦;𝜂𝑥,𝑦≱^∘_𝐶(𝟎,∀𝑦∈𝐴𝑥). (5)

When 𝐴(𝑥)=𝐾, for all 𝑥∈𝐾, then these problems were studied in [8, 9, 11] with applications to vector optimization. Furthermore, if we consider the previous Dini directional derivative as a bifunction ℎ(𝑥;𝑑), with 𝑥 referring to a point in ℝ^𝑛 and 𝑑 referring to a direction from ℝ^𝑛, that is, if ℎ(𝑥;⋅)=𝑓^𝐷(𝑥;⋅), then the previously mentioned problems are studied in [7, 8, 10] and the references therein.

The main motivation of this paper is to establish some existence results for solutions of NMVQVLIP and NSVQVLIP by using Fan-KKM theorem or a maximal element theorem. Of course, by using the technique of [8], we can easily establish some results on the relations among NMVQVLIP, NSVQVLIP, and vector quasi-optimization problems [14]. Since the results are straightforward, we are not including them here.

3. Preliminaries

Let 𝐾⊆ℝ^𝑛 be a nonempty set. We denote by 𝐾, int 𝐴, and co(𝐾) the closure of 𝐾, the interior of 𝐾, and the convex hull of 𝐾, respectively.

Definition 1. Let 𝐾⊆ℝ^𝑛 be a nonempty set and 𝜂∶𝐾×𝐾→𝑋 a mapping. The set 𝐾 is said to be invex with respect to 𝜂 if, for all 𝑥,𝑦∈𝐾 and all 𝑡∈[0,1], we have 𝑥+𝑡𝜂(𝑦,𝑥)∈𝐾.

We say that the map 𝜂 is skew if, for all 𝑥,𝑦∈𝐾, 𝜂𝑦,𝑥+𝜂𝑥,𝑦=𝟎. (6)

Condition C. Let 𝐾⊆ℝ^𝑛 be an invex set with respect to 𝜂∶𝐾×𝐾→ℝ^𝑛. Then, for all 𝑥,𝑦∈𝐾, 𝑡∈[0,1], we have 𝜂𝑥,𝑥+𝑡𝜂𝑦,𝑥=−𝑡𝜂𝑦,𝑥. (7)

We adopt the following definition of affineness.

A vector-valued function 𝑔∶ℝ^𝑛→ℝ^ℓ is called affine if, for all 𝑥₁,𝑥₂,…,𝑥_𝑚∈ℝ^𝑛 and 𝑡_𝑖≥0 for all 𝑖=1,2,…,𝑚 with ∑^𝑚_𝑖=1𝑡_𝑖=1, we have 𝑔^𝑚_𝑖=1𝑡_𝑖𝑥_𝑖=^𝑚_𝑖=1𝑡_𝑖𝑔𝑥_𝑖. (8)

The following lemma can be easily proved.

Lemma 2 (see [17]). Let 𝐾 be a nonempty convex subset of a vector space 𝑋 and 𝜂∶𝐾×𝐾→𝑋 a mapping. If 𝜂 is affine in the first argument and skew, then it is also affine in the second argument.

Definition 3 (see [18, 19]). Let 𝐾⊆ℝ^𝑛 be a nonempty set. A vector-valued function 𝑔∶𝐾→ℝ^ℓ is said to be 𝐶-lower semicontinuous (resp., 𝐶-upper semicontinuous) at 𝑥∈𝐾 if for any neighborhood 𝑉 of 𝑔(𝑥), there exists a neighborhood 𝑈 of 𝑥 such that 𝑔(𝑦)∈𝑉+𝐶 for all 𝑦∈𝑈∩𝐾 (resp., 𝑔(𝑦)∈𝑉−𝐶 for all 𝑦∈𝑈∩𝐾). 𝑔 is said to be 𝐶-lower semicontinuous (resp., 𝐶-upper semicontinuous) on 𝐾 if it is 𝐶-lower semicontinuous (resp., 𝐶-upper semicontinuous) at every point 𝑥∈𝐾.

It is shown in [18] that a function 𝑔∶𝐾→ℝ^ℓ is 𝐶-lower semicontinuous if and only if, for all 𝛼∈ℝ^ℓ, the set {𝑥∈𝐾∶𝑔(𝑥)≱^∘_𝐶𝛼} is closed in 𝐾.

Definition 4. Let 𝐾⊆ℝ^𝑛 be a nonempty convex set. A vector-valued function 𝑔∶𝐾→ℝ^ℓ is said to be 𝐶-convex if, for all 𝑥,𝑦∈𝐾 and all 𝑡∈[0,1], 𝑔𝑡𝑥+(1−𝑡)𝑦≤_𝐶𝑡𝑔(𝑥)+(1−𝑡)𝑔𝑦. (9)

Definition 5 (see [18, 19]). Let 𝐾⊆ℝ^𝑛 be a nonempty convex set. A vector-valued function 𝑔∶𝐾→ℝ^ℓ is said to be 𝐶-quasiconvex if, for all 𝛼∈ℝ^𝑛, the set {𝑥∈𝐾∶𝑔(𝑥)≤_𝐶𝛼} is convex.

It is shown in [18] that if 𝑔 is 𝐶-quasiconvex, then the set {𝑥∈𝐾∶𝑔(𝑥)≤^∘_𝐶𝛼} is convex.

A vector-valued function 𝑔∶ℝ^𝑛→ℝ^ℓ is called positively homogeneous if for all 𝑥∈ℝ^𝑛 and all 𝑟>0, 𝑔(𝑟𝑥)=𝑟𝑔(𝑥).

Definition 6. Let 𝑈 be a nonempty subset of a topological vector space 𝐸. A set-valued map 𝑇∶𝑈→2^𝑈 is said to be a KKM map provided and co(𝑀)⊆𝑇(𝑀)=⋃_𝑥∈𝑀𝑇(𝑥) for each finite subset 𝑀 of 𝑈, where co(𝑀) denotes the convex hull of 𝑀.

The following Fan-KKM theorem [20] will be used in the sequel.

Theorem 7 (see [20]). Let 𝑈 be a nonempty subset of a Hausdorff topological vector space 𝐸. Assume that 𝑇∶𝑈→2^𝑈⧵{∅} is a KKM map satisfying the following conditions: (i)for each 𝑥∈𝑈, 𝑇(𝑥) is closed; (ii)for at least one 𝑥∈𝑈, 𝑇(𝑥) is compact.
Then, ⋂_𝑥∈𝑈𝑇(𝑥)≠∅.

We will use the following maximal element theorem to prove the existence of solutions of nonsmooth vector quasi-variational-like inequality problems.

Theorem 8 (see [21, Corollary 3.2]). Let 𝐾 be a nonempty convex subset of a Hausdorff topological vector space 𝑋 and 𝑆,𝑇∶𝐾→2^𝐾 two set-valued maps. Assume that the following conditions hold: (i)for all 𝑥∈𝐾, co(𝑆(𝑥))⊆𝑇(𝑥); (ii)for all 𝑥∈𝐾, 𝑥∉𝑇(𝑥) and 𝑆⁻¹(𝑦)={𝑥∈𝐾∶𝑦∈𝑆(𝑥)} is open in 𝐾; (iii)there exist a nonempty compact convex subset 𝐷⊆𝐾 and a nonempty compact subset 𝐵 of 𝐾 such that for each 𝑥∈𝐾⧵𝐵, there exists 𝑦∈𝐷 such that 𝑥∈int𝑆⁻¹(𝑦).
Then, there exists 𝑥∈𝐾 such that 𝑆(𝑥)=∅.

4. Existence Results

Definition 9 (see [8]). Let 𝐾⊆ℝ^𝑛 be a nonempty set and 𝜂∶𝐾×𝐾→ℝ^𝑛 a mapping. A vector-valued bifunction ℎ=(ℎ₁,…,ℎ_ℓ)∶𝐾×ℝ^𝑛→ℝ^ℓ is said to be(a)𝐶-pseudomonotone with respect to 𝜂 on 𝐾 if, for all 𝑥,𝑦∈𝐾, ℎ𝑥;𝜂𝑦,𝑥≰^∘_𝐶𝟎impliesℎ𝑦;𝜂𝑥,𝑦≱^∘_𝐶𝟎, (10)(b)𝐶-properly subodd if ℎ𝑥;𝑑₁+ℎ𝑥;𝑑₂+⋯+ℎ𝑥;𝑑_𝑚≥_𝐶𝟎, (11) for every 𝑑_𝑖∈ℝ^𝑛, 𝑖=1,2,…,𝑚 with ∑^𝑚_𝑖=1𝑑_𝑖=𝟎 and 𝑥∈𝐾.

The definition of proper suboddness is considered in [11]. Of course, if 𝑚=2, the definition of proper suboddness reduces to the definition of suboddness.

Definition 10. Let 𝐾 be a nonempty convex subset of ℝ^𝑛. A function 𝑔∶𝐾→ℝ^ℓ is said to be hemicontinuous if, for all 𝑥,𝑦∈𝐾, the mapping 𝑡↦𝑔(𝑦+𝑡(𝑥−𝑦)) is continuous. The upper and lower hemicontinuity can be defined analogously.

Definition 11. Let 𝐾⊆ℝ^𝑛 be an invex set with respect to 𝜂∶𝐾×𝐾→ℝ^𝑛. A function 𝑔∶𝐾→ℝ^ℓ is said to be 𝜂-hemicontinuous if, for all 𝑥,𝑦∈𝐾, the mapping 𝑡↦𝑔(𝑦+𝑡𝜂(𝑥,𝑦)) is continuous. The upper and lower 𝜂-hemicontinuity can be defined analogously.

The following concept of 𝜂-upper sign continuity for the bifunction ℎ is considered in [8].

Definition 12 (see [8]). Let 𝐾⊆ℝ^𝑛 be a nonempty invex set with respect to 𝜂∶𝐾×𝐾→ℝ^𝑛. A vector-valued bifunction ℎ=(ℎ₁,…,ℎ_ℓ)∶𝐾×ℝ^𝑛→ℝ^ℓ is said to be 𝜂-upper sign continuous if, for all 𝑥,𝑦∈𝐾 and 𝑡∈(0,1), ℎ𝑥+𝑡𝜂𝑦,𝑥;𝜂𝑦,𝑥≰^∘_𝐶𝟎impliesℎ𝑥;𝜂𝑦,𝑥≰^∘_𝐶𝟎. (12)

Remark 13. It can be easily seen that if 𝜂 is skew and ℎ is 𝜂-upper hemicontinuous in the first argument, then it is 𝜂-upper sign continuous, but the converse is not true in general.

The following result provides the relations between NSVQVLIP and NMVQVLIP when the set-valued map 𝐴∶𝐾→2^𝐾 is invex valued.

Proposition 14. Let 𝐾⊆ℝ^𝑛 be a nonempty invex set with respect to  𝜂∶𝐾×𝐾→ℝ^𝑛 such that Condition C holds. Let 𝐴∶𝐾→2^𝐾 be a set-valued map such that, for each 𝑥∈𝐾, 𝐴(𝑥) is a nonempty and invex set with respect to 𝜂. Let the vector-valued bifunction ℎ∶𝐾×ℝ^𝑛→ℝ^ℓ be 𝐶-properly subodd, 𝐶-pseudomonotone with respect to 𝜂, and 𝜂-upper sign continuous such that, for each fixed 𝑥∈𝐾, ℎ(𝑥;⋅) is positively homogeneous. Then, 𝑥∈𝐾 is a solution of NSVQVLIP if and only if it is a solution of NMVQVLIP.

Proof. It is similar to the proof of Proposition 7.7 in [8]. However, we include it for the sake of completeness of the paper.
The 𝐶-pseudomonotonicity of ℎ with respect to 𝜂 implies that every solution of NSVQVLIP is a solution of NMVQVLIP.
Conversely, let 𝑥∈𝐾 be a solution of NMVQVLIP. Then, 𝑥∈𝐴(𝑥), and ℎ𝑦;𝜂𝑥,𝑦≱^∘_𝐶𝟎,∀𝑦∈𝐴(𝑥). (13)
Since 𝐴(𝑥) is invex, we have 𝑦_𝑡=𝑥+𝑡𝜂(𝑦,𝑥)∈𝐴(𝑥) for all 𝑡∈(0,1), and therefore, (13) becomes ℎ𝑦_𝑡;𝜂𝑥,𝑦_𝑡≱^∘_𝐶𝟎. (14)
By Condition C, 𝜂(𝑥,𝑦_𝑡)=−𝑡𝜂(𝑦,𝑥), and thus, ℎ𝑦_𝑡;−𝑡𝜂𝑦,𝑥≱^∘_𝐶𝟎. (15)
By positive homogeneity and 𝐶-proper suboddness of ℎ, we have ℎ𝑦_𝑡;𝜂𝑦,𝑥≰^∘_𝐶𝟎. (16)
Thus, the 𝜂-upper sign continuity of ℎ yields 𝑥∈𝐾 is a solution of NSVQVLIP.

The following result gives the equivalence between NSVQVLIP and NMVQVLIP when the set-valued map 𝐴∶𝐾→2^𝐾 is convex valued.

Proposition 15. Let 𝐾⊆ℝ^𝑛 be a nonempty convex set, and let 𝜂∶𝐾×𝐾→ℝ^𝑛 be affine in the first argument and skew. Let 𝐴∶𝐾→2^𝐾 be a set-valued map such that, for each 𝑥∈𝐾, 𝐴(𝑥) is a nonempty and convex set. Let the vector-valued bifunction ℎ∶𝐾×ℝ^𝑛→ℝ^ℓ be 𝐶-properly subodd, 𝐶-pseudomonotone with respect to 𝜂, and 𝜂-upper sign continuous such that, for each fixed 𝑥∈𝐾, ℎ(𝑥;⋅) is positively homogeneous. Then, 𝑥∈𝐾 is a solution of NSVQVLIP if and only if it is a solution of NMVQVLIP.

Proof. It is similar to the proof of Proposition 7.8 in [8]. However, we include it for the sake of completeness of the paper.
The 𝐶-pseudomonotonicity of ℎ with respect to 𝜂 implies that every solution of NSVQVLIP is a solution of NMVQVLIP.
Conversely, let 𝑥∈𝐾 be a solution of NMVQVLIP. Then, 𝑥∈𝐴(𝑥), and ℎ𝑦;𝜂𝑥,𝑦≱^∘_𝐶𝟎,∀𝑦∈𝐴(𝑥). (17)
Since 𝐴(𝑥) is convex, we have 𝑦_𝑡=𝑥+𝑡(𝑦−𝑥)∈𝐴(𝑥) for all 𝑡∈(0,1), and therefore, (17) becomes ℎ𝑦_𝑡;𝜂𝑥,𝑦_𝑡≱^∘_𝐶𝟎. (18)
Since 𝜂 is affine in the first argument and skew, by Lemma 2, 𝜂 is also affine in the second argument. Since 𝜂(𝑥,𝑥)=𝟎 by skewness of 𝜂, we obtain ℎ𝑦_𝑡;𝜂𝑥,𝑦_𝑡=ℎ𝑦_𝑡;𝑡𝜂𝑥,𝑦+(1−𝑡)𝜂(𝑥,𝑥)=ℎ𝑦_𝑡;𝑡𝜂𝑥,𝑦≱^∘_𝐶𝟎. (19)
By positive homogeneity of ℎ in the second argument, we have ℎ𝑦_𝑡;𝜂𝑥,𝑦≱^∘_𝐶𝟎. (20)
Since 𝜂(𝑦,𝑥)+𝜂(𝑥,𝑦)=𝟎 by skewness of 𝜂, the 𝐶-proper suboddness of ℎ implies that ℎ𝑦_𝑡;𝜂𝑦,𝑥≰^∘_𝐶𝟎. (21)
The 𝜂-upper sign continuity of ℎ yields 𝑥∈𝐾 is a solution of NSVQVLIP.

Throughout the rest of the paper, unless otherwise specified, we assume that 𝐴∶𝐾→2^𝐾 is a set-valued map such that 𝐴(𝑥) is nonempty convex for all 𝑥∈𝐾, 𝐴⁻¹(𝑦) is open for all 𝑦∈𝐾, and the set ℱ={𝑥∈𝐾∶𝑥∈𝐴(𝑥)} is closed.

We present some existence results for the solutions of NSVQVLIP and NMVQVLIP without boundedness assumption on the underlying set 𝐾.

Theorem 16. Let 𝐾⊆ℝ^𝑛 be a nonempty convex set, and 𝜂∶𝐾×𝐾→ℝ^𝑛 be skew, affine, and lower semicontinuous in the first argument. Let ℎ=(ℎ₁,…,ℎ_ℓ)∶𝐾×ℝ^𝑛→ℝ^ℓ be 𝐶-properly subodd, positively homogeneous in the second argument, and 𝐶-pseudomonotone with respect to 𝜂 such that 𝑥↦ℎ(𝑦;𝜂(𝑥,𝑦)) is continuous. Assume that there exist a nonempty compact convex subset 𝐷 of 𝐾 and 𝑦∈𝐷 such that, for all 𝑥∈𝐾⧵𝐷, 𝑦∈𝐴(𝑥) and ℎ(𝑥;𝜂(𝑦,𝑥))≤^∘_𝐶𝟎 Then, there exists a solution 𝑥∈𝐾 of MVQVLIP.
Furthermore, if  ℎ is 𝜂-upper sign continuous, then 𝑥∈𝐾 is a solution of SVQVLIP.

Proof. For all 𝑥∈𝐾, we define two set-valued maps 𝑆₁,𝑆₂∶𝐾→2^𝐾 by 𝑆₁(𝑥)=𝑦∈𝐾∶ℎ𝑥;𝜂𝑦,𝑥≤^∘_𝐶𝟎,𝑆₂(𝑥)=𝑦∈𝐾∶ℎ𝑦;𝜂𝑥,𝑦≥^∘_𝐶𝟎. (22)
For all 𝑥,𝑦∈𝐾 and for each 𝑖=1,2, we also define other two set-valued maps 𝑃_𝑖∶𝐾→2^𝐾 and 𝑄_𝑖∶𝐾→2^𝐾 by 𝑃_𝑖(𝑥)=𝐴(𝑥)∩𝑆_𝑖(𝑥)𝐴(𝑥)𝑄,if𝑥∈ℱ,,if𝑥∈𝐾⧵ℱ,_𝑖𝑦=𝐾⧵𝑃_𝑖⁻¹𝑦. (23)
For each 𝑖=1,2 and for all 𝑦∈𝐾, we have (see, e.g., [22]) 𝑃_𝑖⁻¹𝑦=(𝐾⧵ℱ)∪𝑆_𝑖⁻¹𝑦∩𝐴⁻¹𝑦, (24) and therefore, 𝑄_𝑖𝑦=ℱ∩𝐾⧵𝑆_𝑖⁻¹𝑦∪𝐾⧵𝐴⁻¹𝑦. (25)
The rest of the proof is divided into the following four steps.
(a) We claim that 𝑄₁ is a KKM map on 𝐾.
Assume the contrary that 𝑄₁ is not a KKM map. Then, there exist a finite set {𝑦₁,…,𝑦_𝑚} in 𝐾 and 𝑡₁,…,𝑡_𝑚≥0 with ∑^𝑚_𝑖=1𝑡_𝑖=1 such that 𝑥=∑^𝑚_𝑖=1𝑡_𝑖𝑦_𝑖∉𝑄₁(𝑦_𝑖) for all 𝑖=1,…,𝑚; that is, 𝑥∈𝑃₁⁻¹𝑦_𝑖⟺𝑦_𝑖∈𝑃₁(𝑥)∀𝑖=1,…,𝑚. (26)
If 𝑥∈ℱ, then 𝑃₁(𝑥)=𝐴(𝑥)∩𝑆₁(𝑥), and therefore, 𝑦_𝑖∈𝑆₁(𝑥),𝑦_𝑖∈𝐴(𝑥)∀𝑖=1,…,𝑚. (27)
Hence, ℎ𝑥;𝜂𝑦_𝑖,𝑥≤^∘_𝐶𝟎,𝑦_𝑖∈𝐴(𝑥)∀𝑖=1,…,𝑚. (28)
Since 𝐶^∘ is a convex cone and 𝑡_𝑖≥0 with ∑^𝑚_𝑖=1𝑡_𝑖=1, we have ^𝑚_𝑖=1𝑡_𝑖ℎ𝑥;𝜂𝑦_𝑖,𝑥≤^∘_𝐶𝟎. (29)
Since 𝜂 is skew, we have 𝜂(𝑥,𝑥)=𝟎. By the affineness of 𝜂 in the first argument, we have ^𝑚_𝑖=1𝑡_𝑖𝜂𝑦_𝑖,𝑥=𝜂^𝑚_𝑖=1𝑡_𝑖𝑦_𝑖,𝑥=𝜂(𝑥,𝑥)=𝟎. (30)
Since ℎ is 𝐶-proper subodd, we have ^𝑚_𝑖=1ℎ𝑥;𝑡_𝑖𝜂𝑦_𝑖,𝑥≥_𝐶𝟎. (31)
By positive homogeneity of ℎ, we obtain ^𝑚_𝑖=1𝑡_𝑖ℎ𝑥;𝜂𝑦_𝑖,𝑥≥_𝐶𝟎, (32) a contradiction of (29).
If 𝑥∈𝐾⧵ℱ, then 𝑥∉𝐴(𝑥). By the definition of 𝑄₁, we have 𝑃₁(𝑥)=𝐴(𝑥), and therefore, 𝑦_𝑖∈𝑃₁(𝑥)=𝐴(𝑥) for all 𝑖=1,…,𝑚. Since 𝐴(𝑥) is convex, we obtain 𝑥∈𝐴(𝑥), again a contradiction. Hence, 𝑄₁ is a KKM map.
(b) We show that 𝑄₁(𝑦)=𝐾⧵𝑃₁⁻¹(𝑦)⊆𝐷, where 𝑦 and 𝐷 are the same as in the hypothesis.
Indeed, if 𝑥∈𝑄₁(𝑦)⧵𝐷, then 𝑥∈[ℱ∩(𝐾⧵𝑆₁⁻¹(𝑦))]∪[𝐾⧵𝐴⁻¹(𝑦)]; that is, either 𝑥∈ℱ∩(𝐾⧵𝑆₁⁻¹(𝑦)) or 𝑥∈𝐾⧵𝐴⁻¹(𝑦).
If 𝑥∈ℱ∩(𝐾⧵𝑆₁⁻¹(𝑦)), then 𝑥∈ℱ and 𝑥∈𝐾⧵𝑆₁⁻¹(𝑦); that is, 𝑥∈𝐴(𝑥) and ℎ(𝑥;𝜂(𝑦,𝑥))≰^∘_𝐶𝟎, a contradiction to our assumption that ℎ(𝑥;𝜂(𝑦,𝑥))≤^∘_𝐶𝟎.
If 𝑥∈𝐾⧵𝐴⁻¹(𝑦), then 𝑥∉𝐴⁻¹(𝑦) if and only if 𝑦∉𝐴(𝑥), again a contradiction to our assumption that 𝑦∈𝐴(𝑥). Hence, 𝑄₁(𝑦)⊆𝐷.
(c) We show that ⋂_𝑦∈𝐾𝑄₁(𝑦)≠∅.
Since 𝐷 is compact, 𝑄₁(𝑦) is also compact. Moreover, since 𝑄₁ is a KKM map, co𝑦₁,…,𝑦_𝑚⊆^𝑚_𝑖=1𝑄₁𝑦_𝑖⊆^𝑚_𝑖=1𝑄₁𝑦_𝑖, (33) for each finite subset {𝑦₁,…,𝑦_𝑚} of 𝐾. Then by Theorem 7, we get ⋂_𝑦∈𝐾𝑄₁(𝑦)≠∅.
(d) Next, we claim that ⋂_𝑦∈𝐾𝑄₁(𝑦)⊆⋂_𝑦∈𝐾𝑄₂(𝑦).
Let 𝑧∈⋂_𝑦∈𝐾𝑄₁(𝑦); then 𝑧∈𝑄₁(𝑦) for each 𝑦∈𝐾. For an arbitrary element 𝑦∈𝐾, we have to show that 𝑧∈𝑄₂(𝑦).
Since 𝑧∈𝑄₁(𝑦), there exists a sequence {𝑧_𝑚}⊆𝑄₁(𝑦) such that {𝑧_𝑚} converges to 𝑧. Since {𝑧_𝑚}⊆𝑄₁(𝑦), we have 𝑧_𝑚⊆𝐾⧵𝑃₁⁻¹𝑦=ℱ∩𝐾⧵𝑆₁⁻¹𝑦∪𝐾⧵𝐴⁻¹𝑦. (34)
Then, either {𝑧_𝑚}⊆ℱ∩(𝐾⧵𝑆₁⁻¹(𝑦)) or {𝑧_𝑚}⊆𝐾⧵𝐴⁻¹(𝑦).
If {𝑧_𝑚}⊆ℱ∩(𝐾⧵𝑆₁⁻¹(𝑦)), then {𝑧_𝑚}⊆ℱ and {𝑧_𝑚}⊆𝐾⧵𝑆₁⁻¹(𝑦). It follows that {𝑧_𝑚}⊆ℱ and ℎ(𝑧_𝑚;𝜂(𝑦,𝑧_𝑚))≰^∘_𝐶𝟎. Since ℱ is closed and 𝑧_𝑚→𝑧, we have 𝑧∈ℱ; that is, 𝑧∈𝐴(𝑧). By 𝐶-pseudomonotonicity of ℎ, we obtain 𝑧∈𝐴(𝑧),ℎ𝑦;𝜂𝑧_𝑛,𝑦≱𝟎. (35)
By the continuity of 𝑥↦ℎ(𝑦;𝜂(𝑥,𝑦))ℎ, we get ℎ(𝑦,𝜂(𝑧,𝑦))≱^∘_𝐶0. This implies that 𝑧∈𝐴(𝑧) and 𝑦∉𝑆₂(𝑧); that is, 𝑧∉(𝐾⧵ℱ)∪𝑆₂⁻¹(𝑦), and hence, 𝑧∉𝑃₂⁻¹(𝑦). Therefore, 𝑧∈𝐴(𝑧), and 𝑧∈𝐾⧵𝑃₂⁻¹(𝑦)=𝑄₂(𝑦).
Let {𝑧_𝑚}⊆𝐾⧵𝐴⁻¹(𝑦). Since 𝐴⁻¹(𝑦) is open in 𝐾, for all 𝑦∈𝐾, 𝐾⧵𝐴⁻¹(𝑦) is closed in 𝐾. Since 𝑧_𝑚→𝑧, we have 𝑧∈𝐾⧵𝐴⁻¹(𝑦). Hence, 𝑧∉𝐴⁻¹(𝑦)⇔𝑦∉𝐴(𝑧), which implies that 𝑦∉𝑃₂(𝑧)⟺𝑧∉𝑃₂⁻¹𝑦⟺𝑧∈𝐾⧵𝑃₂⁻¹𝑦⟺𝑧∈𝑄₂𝑦. (36)
From (c), we get ⋂_𝑦∈𝐾𝑄₂(𝑦)≠∅. Hence, there exists 𝑥∈𝐾 such that 𝑥∈_𝑦∈𝐾𝐾⧵𝑃₂⁻¹𝑦=𝐾⧵_𝑦∈𝐾𝑃₂⁻¹𝑦. (37)
This implies that 𝑃₂(𝑥)=∅.
If 𝑥∈𝐾⧵ℱ, then 𝑃₂(𝑥)=𝐴(𝑥)=∅, a contradiction. Otherwise, 𝑥∈ℱ; then ∅=𝑃₂(𝑥)=𝐴(𝑥)∩𝑆₂(𝑥). Therefore, 𝑥∈𝐴(𝑥) such that ℎ(𝑦;𝜂(𝑥,𝑦))≱_𝐶𝟎𝟎 for all 𝑦∈𝐴(𝑥). From Proposition 15, 𝑥∈𝐾 is a solution of SVQVLIP.

Remark 17. (a) Theorem 16 extends and generalizes [8, Theorem 7.34], [9, Theorem 5.1], and [11, Theorem 2.2].
(b) If 𝐴 is a closed map, then the set ℱ={𝑥∈𝐾∶𝑥∈𝐴(𝑥)} is closed.

By using maximal element Theorem 8, we present the following existence result for solutions of NSVQVLIP and NMVQVLIP without boundedness assumption on the underlying set.

Theorem 18. Let 𝐾⊆ℝ^𝑛 be a nonempty convex set, and let 𝜂∶𝐾×𝐾→ℝ be skew. Let ℎ∶𝐾×ℝ^𝑛→ℝ^ℓ be 𝐶-pseudomonotone with respect to 𝜂 such that ℎ(𝑥;𝟎)=𝟎 for all 𝑥∈𝐾; the set 𝑀={𝑦∈𝐾∶ℎ(𝑥,𝜂(𝑦,𝑥))≤^∘_𝐶𝟎} is convex, and the set 𝐿={𝑥∈𝐾∶ℎ(𝑦,𝜂(𝑥,𝑦))≱^∘_𝐶𝟎} is closed in 𝐾. Assume that there exist a nonempty compact convex subset 𝐷⊆𝐾 and a nonempty compact subset 𝐵 of 𝐾 such that, for each 𝑥∈𝐾⧵𝐵, there exists 𝑦∈𝐷 such that 𝑦∈𝐴(𝑥) and ℎ(𝑦;𝜂(𝑥,𝑦))≥^∘_𝐶𝟎. Then, there exists a solution 𝑥∈𝐾 of NMVQVLIP.
Furthermore, if ℎ is 𝐶-properly subodd, 𝜂-upper sign continuous, and for each fixed 𝑥∈𝐾, ℎ(𝑥;⋅) is positively homogeneous, then 𝑥∈𝐾 is a solution of NSVQVLIP.

Proof. For each 𝑥∈𝐾, define two set-valued maps 𝑃,𝑄∶𝐾→2^𝐾 by 𝑃(𝑥)=𝑦∈𝐾∶ℎ𝑦;𝜂𝑥,𝑦≥^∘_𝐶𝟎,𝑄(𝑥)=𝑦∈𝐾∶ℎ𝑥;𝜂𝑦,𝑥≤^∘_𝐶𝟎. (38)
Then, 𝑥∉𝑄(𝑥) for all 𝑥∈𝐾. Indeed, by skewness of 𝜂, 𝜂(𝑥,𝑥)=𝟎 for all 𝑥∈𝐾. By assumption, 𝟎=ℎ(𝑥;𝜂(𝑥,𝑥))=ℎ(𝑥;𝟎)=𝟎≰^∘_𝐶𝟎. Thus, 𝑥∉𝑄(𝑥).
By hypothesis, the complement of 𝑃⁻¹(𝑦) in 𝐾, 𝑃⁻¹𝑦^𝑐=𝑥∈𝐾∶ℎ𝑦,𝜂𝑥,𝑦≱^∘_𝐶𝟎, (39) is closed in 𝐾 for each 𝑦∈𝐾. Therefore, 𝑃⁻¹(𝑦) is open in 𝐾 for all 𝑦∈𝐾.
Define other two set-valued maps 𝑆,𝑇∶𝐾→2^𝐾 by 𝑆(𝑥)=𝐴(𝑥)(𝑥)𝐴(𝑥)𝑇(𝑥)=𝐴(𝑥)(𝑥)𝐴(𝑥)∩𝑃,if𝑥∈ℱ,,if𝑥∈𝐾⧵ℱ,∩𝑄,if𝑥∈ℱ,,if𝑥∈𝐾⧵ℱ. (40)
Since, for all 𝑥∈𝐾, 𝑥∉𝑄(𝑥), we have 𝑥∉𝑇(𝑥).
By 𝐶-pseudomonotonicity of ℎ, we have 𝑃(𝑥)⊆𝑄(𝑥) for all 𝑥∈𝐾. Since 𝐴(𝑥) and 𝑄(𝑥) are convex, for all 𝑥∈𝐾, we have 𝑆(𝑥)⊆𝑇(𝑥)⟹co(𝑆(𝑥))⊆co(𝑇(𝑥))=𝑇(𝑥). (41)
Since, for each 𝑦∈𝐾, 𝐴⁻¹(𝑦) and 𝑃⁻¹(𝑦) are open in 𝐾, (𝐴∩𝑃)⁻¹(𝑦)=𝐴⁻¹(𝑦)∩𝑃⁻¹(𝑦) is open in 𝐾. Also, since, for each 𝑦∈𝐾, 𝑆⁻¹𝑦=𝐴⁻¹𝑦∩𝑃⁻¹𝑦∪(𝐾⧵ℱ)∩𝐴⁻¹𝑦 (42) (see, e.g., the proof of [23, Lemma 2.3]) and 𝐾⧵ℱ is open in 𝐾, we have that 𝑆⁻¹(𝑦) is open in 𝐾. Therefore, by Theorem 8, there exists 𝑥∈𝐾 such that 𝑆(𝑥)=∅. If𝑥∈𝐾⧵ℱ, then 𝐴(𝑥)=∅, a contradiction to our assumption. So, 𝑥∈ℱ, and thus, 𝐴(𝑥)∩𝑃(𝑥)=∅. Therefore, 𝑥∈𝐴(𝑥),ℎ𝑦;𝜂𝑥,𝑦≱^∘_𝐶𝟎,∀𝑦∈𝐴(𝑥). (43)
Thus, 𝑥 is a solution of NMVQVLIP.

By Proposition 15, 𝑥∈𝐾 is a solution of NSVQVLIP.

Remark 19. If, for each fixed 𝑥∈𝐾, the vector-valued function 𝑦↦ℎ(𝑥,𝜂(𝑦,𝑥)) is 𝐶-quasiconvex, then the set 𝑀={𝑦∈𝐾∶ℎ(𝑥,𝜂(𝑦,𝑥))≤^∘_𝐶𝟎} is convex.

Remark 20. For all 𝑥∈𝐾, the set 𝑀={𝑦∈𝐾∶ℎ(𝑥,𝜂(𝑦,𝑥))≤^∘_𝐶𝟎} is convex, if 𝜂 is affine in the first argument and ℎ is 𝐶-convex in the second argument.

Indeed, let 𝑦₁,𝑦₂∈𝑀. Since 𝐶^∘ is a convex cone, for all 𝑡∈(0,1), we have 𝑡ℎ𝑥;𝜂𝑦₁,𝑥≤^∘_𝐶𝟎,(1−𝑡)ℎ𝑥;𝜂𝑦₂,𝑥≤^∘_𝐶𝟎. (44)

By adding these relations, we get 𝑡ℎ𝑥;𝜂𝑦₁,𝑥+(1−𝑡)ℎ𝑥;𝜂𝑦₂,𝑥≤^∘_𝐶𝟎. (45)

Since ℎ is 𝐶-convex in the second argument, we have ℎ𝑥;𝑡𝜂𝑦₁,𝑥+(1−𝑡)𝜂𝑦₂≤,𝑥_𝐶𝑡ℎ𝑥;𝜂𝑦₁,𝑥+(1−𝑡)ℎ𝑥;𝜂𝑦₂,𝑥. (46)

By combining relations (45) and (46), we obtain ℎ𝑥;𝑡𝜂𝑦₁,𝑥+(1−𝑡)𝜂𝑦₂,𝑥≤^∘_𝐶𝟎. (47)

Since 𝜂 is affine in the first argument, we get ℎ𝑥;𝜂𝑡𝑦₁+(1−𝑡)𝑦₂,𝑥≤^∘_𝐶𝟎, (48) and hence, 𝑡𝑦₁+(1−𝑡)𝑦₂∈𝑀. Thus, for all 𝑥∈𝐾, 𝑀 is a convex set.

Remark 21. The set 𝐿={𝑥∈𝐾∶ℎ(𝑦,𝜂(𝑥,𝑦))≱^∘_𝐶𝟎} is closed in 𝐾 if the vector-valued function 𝑥↦ℎ(𝑦,𝜂(𝑥,𝑦)) is 𝐶-lower semicontinuous for each fixed 𝑦∈𝐾.

5. Conclusions

In this paper, we defined vector quasi-variational-like inequality problems by means of a bifunction and established two existence results for solutions of these problems. One can treat upper Dini directional derivative as a bifunction, and hence, the bifunction can be replaced by upper Dini directional derivative. Then, we get the so-called nonsmooth vector quasi-variational-like inequality problem. By using the technique and methodology given in [8], one can easily derive some relations between vector quasi-variational-like inequality problems and a vector quasi-optimization problem. Since the results are straightforward, we have not included them in this paper. Of course, the results of this paper extend several known results in the literature, namely, [9, Theorem 5.1], [11, Theorem 3.1], and [8, Theorem 7.34] from vector variational (-like) inequality problems to vector quasi-variational (-like) inequality problems. This paper can be treated as the first in this direction.

Acknowledgments

This research was done during the visit of the second author to King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, and it was supported by KFUPM Funded Research Project no. IN121035.