Vector Variational-Like Inequalities with Generalized Semimonotone Mappings

Abstract

We introduce the concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity. We consider a class of generalized vector variationa-llike inequality problem involving generalized relaxed semimonotone mapping. By using Kakutani-Fan-Glicksberg’s fixed-point theorem, we prove the solvability for this class of vector variational-like inequality with relaxed monotonicity assumptions. The results presented in this paper generalize some known results for vector variational inequality in recent years.

1. Introduction

Vector variational inequalities were initially introduced and considered by Giannessi [1] in a finite-dimensional Euclidean space in 1980. Due to its wide application the theory of the vector variational inequality is generalized in different directions and many existence results and algorithms for vector variational inequality problems have been established under various conditions; see for examples [2–7] and references therein.

The concept monotonicity and the compactness operators are very useful in nonlinear functional analysis and its applications. In 1968, Browder [8] first combined the compactness and accretion of operators and posed the concept of a semiaccretive operator. Motivated by this idea, Chen [9] studied the concept of a semimonotone operator, which combines the compactness and monotonicity of an operator and posed it to the study of variational inequalities. Recently in 2003, Fang and Huang [10] introduced relaxed 𝜂-𝛼-semimonotone mapping, a generalized concept of semimonotonicity, and they established several existence results for the variational-like inequality problem.

In this paper, we pose two new concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity as well as two classes of generalized vector variational-like inequalities with generalized relaxed monotone mappings and generalized relaxed semimonotone mappings. We investigate the solvability of vector variational-like inequalities with generalized relaxed semimonotone mappings by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper generalize the results of Chen [9], Fang and Huang [10], Usman and Khan [11], and Zheng [7].

2. Preliminaries

Throughout the paper unless otherwise specified, let 𝑋 and 𝑌 be two real Banach spaces, 𝐾⊂𝑋 be a nonempty closed and convex subset of 𝑋. 𝑃 is said to be a closed convex and pointed cone with its apex at the origin, if the following conditions hold: (i)𝜆𝑃⊂𝑃, for all 𝜆>0, (ii)𝑃+𝑃⊂𝑃,(iii)𝑃∩(−𝑃)={0}.

The partial order ≤_𝑃 in 𝑌, induced by the pointed cone 𝑃, is defined by declaring 𝑥≤_𝑃𝑦 if and only if 𝑦−𝑥∈𝑃 for all 𝑥,𝑦 in 𝑌. An ordered Banach space is a pair (𝑌,𝑃) with the partial order induced by 𝑃. The weak order ≤_int𝑃 in an ordered Banach space (𝑌,𝑃) with int𝑃≠∅ is defined as 𝑥≤_int𝑃𝑦 if and only if 𝑦−𝑥∈int𝑃 for all 𝑥,𝑦 in 𝑌, where int𝑃 denotes the interior of 𝑃. Let 𝐿(𝑋,𝑌) denote the space of all continuous linear mappings from 𝑋 into 𝑌. A set-valued mapping 𝑃∶𝐾→2^𝑌 be such that for each 𝑥∈𝐾,𝑃(𝑥) is a proper, closed, convex cone with int𝑃(𝑥)≠∅ and let 𝑃₋=⋂_𝑥∈𝐾𝑃(𝑥).

First, we give the concept of generalized relaxed monotone mapping. In order to do so, the definition of a vector monotone mapping is needed, which was posed by Chen [9].

Definition 1 (see [9]). Let 𝑇∶𝐾→𝐿(𝑋,𝑌) be a mapping, 𝐾⊂𝑋 be a nonempty, closed and convex subset in 𝑋. Let 𝑃∶𝐾→2^𝑌 be such that for each 𝑥∈𝐾,𝑃(𝑥) is a proper, closed, convex cone with int𝑃(𝑥)≠∅.𝑇 is said to be 𝑃₋-monotone on 𝐾 if and only if it satisfies the following condition: 𝑇𝑦−𝑇(𝑥),𝑦−𝑥≥_𝑃−0,∀𝑥,𝑦∈𝐾, (1) where 𝑃₋=⋂_𝑥∈𝐾𝑃(𝑥).

We now give the concept of generalized relaxed monotone mapping.

Definition 2.  Let 𝑃∶𝐾→2^𝑌 be such that for each 𝑥∈𝐾,𝑃(𝑥) is a proper, closed, convex cone with int𝑃(𝑥)≠∅ and a mapping 𝜂∶𝐾×𝐾→𝑋. A mapping 𝑇∶𝐾→𝐿(𝑋,𝑌) is said to be generalized relaxed monotone, if 𝑇𝑦−𝑇𝑥,𝜂𝑦,𝑥−𝛼𝑥,𝑦≥_𝑃−0,∀𝑥,𝑦∈𝐾, (2) where 𝛼∶𝑋×𝑋→𝑌 is a mapping such that lim_𝑡→0⁺(𝛼(𝑥,𝑡𝑦+(1−𝑡)𝑥)/𝑡)=0.

Remark 3. (i) If 𝛼(𝑥,𝑦)=𝛼(𝑥−𝑦), where 𝛼∶𝑋→ℝ with 𝛼(𝜆𝑧)=𝜆^𝑝𝛼(𝑧) for 𝜆>0,𝑝>1, then inclusion (2) 𝑇𝑦−𝑇𝑥,𝜂𝑦,𝑥−𝛼𝑥,𝑦≥_𝑃−0,∀𝑥,𝑦∈𝐾 (3) reduces to 𝑇𝑦−𝑇𝑥,𝜂𝑦,𝑥−𝛼𝑥−𝑦∈𝑃₋,∀𝑥,𝑦∈𝐾. (4) Also if 𝑃(𝑥)=ℝ₊, for all 𝑥∈𝐾, then 𝑃₋=⋂_𝑥∈𝐾𝑃(𝑥)=ℝ₊. As a result the above inclusion reduces to ⇒𝑇𝑦−𝑇𝑥,𝜂𝑦,𝑥−𝛼𝑥−𝑦≥0,∀𝑥,𝑦∈𝐾,𝑇𝑦−𝑇𝑥,𝜂𝑦,𝑥≥𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐾, (5) then 𝑇 is said to be relaxed 𝜂-𝛼-monotone, introduced and studied by Fang and Huang [10].
(ii) In the above inclusion (5), if we take 𝜂(𝑦,𝑥)=𝑦−𝑥, for all 𝑥,𝑦∈𝐾, then it reduces to 𝑇𝑦−𝑇𝑥,𝑦−𝑥≥𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐾, (6) then 𝑇 is said to be 𝛼-relaxed monotone.
(iii) In the above inclusion (6), if we take 𝛼(𝑧)=𝑘‖𝑧‖^𝑝, where 𝑘>0 is constant, then it reduces to 𝑇𝑦−𝑇𝑥,𝑦−𝑥≥𝑘𝑦−𝑥^𝑝,∀𝑥,𝑦∈𝐾. (7)
Then 𝑇 is called 𝑝-monotone, see for example [12–14].

We recall the following concepts and results which are needed in the sequel.

Definition 4. A mapping 𝑓∶𝐾→𝑌 is said to be (i)𝑃₋-convex, if 𝑓(𝑡𝑥+(1−𝑡)𝑦)≤_𝑃−𝑡𝑓(𝑥)+(1−𝑡)𝑓(𝑦), for all 𝑥,𝑦∈𝐾,𝑡∈[0,1];(ii)𝑃₋-concave, if −𝑓 is 𝑃₋-convex.

Definition 5. A mapping 𝜂∶𝐾×𝐾→𝑋 is said to be affine in the first argument if for any 𝑥_𝑖∈𝐾 and 𝜆_𝑖≥0,(1≤𝑖≤𝑛), with ∑^𝑛_𝑖=1𝜆_𝑖=1 and any 𝑦∈𝐾, 𝜂^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖,𝑦=^𝑛_𝑖=1𝜆_𝑖𝜂𝑥_𝑖,𝑦. (8)

Lemma 6 (see [3]). Let (𝑌,𝑃) be an ordered Banach space with a proper, closed, convex and solid cone 𝑃(𝑥) then for all 𝑥,𝑦,𝑧∈𝑌, one has (i)𝑧≰_{int𝑃(𝑥)}𝑥 and 𝑥≥_𝑃(𝑥)𝑦⇒𝑧≰_{int𝑃(𝑥)}𝑦,(ii)𝑧≱_{int𝑃(𝑥)}𝑥 and 𝑥≤_𝑃(𝑥)𝑦⇒𝑧≱_{int𝑃(𝑥)}𝑦.

Lemma 7 (see [15]). Let 𝐾 be a subset of a topological vector space 𝑋 and let 𝐹∶𝐾→2^𝑋 be a KKM mapping. If for each 𝑥∈𝐾, 𝐹(𝑥) is closed and for at least one 𝑥∈𝐾, 𝐹(𝑥) is compact, then _𝑥∈𝐾𝐹(𝑥)≠∅. (9)

Definition 8. A mapping 𝑇∶𝐾→𝐿(𝑋,𝑌) is said to be 𝜂-hemicontinuous, if for any 𝑥,𝑦∈𝐾, the mapping 𝑡→⟨𝑇𝑥+𝑡(𝑦−𝑥),𝜂(𝑦,𝑥)⟩ is continuous at 0⁺.

Now, we have the following Minty’s type Lemma.

Lemma 9.  Let 𝐾⊂𝑋 be a nonempty, closed, and convex subset of 𝑋. Let 𝑃∶𝐾→2^𝑌 be such that for each 𝑥∈𝐾,𝑃(𝑥) is a proper, closed, convex cone with int𝑃(𝑥)≠∅. Let 𝑓∶𝐾×𝐾→𝑌 be 𝑃₋-convex in the first argument with the condition 𝑓(𝑥,𝑥)=0, for all 𝑥∈𝐾. Suppose the following conditions hold: (i)𝜂∶𝐾×𝐾→𝑋 is an affine in the first argument with the condition 𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0, for all 𝑥,𝑦∈𝐾; (ii)the set-valued mapping 𝑊∶𝐾→2^𝑌 defined as 𝑊(𝑥)=𝑌⧵{−int𝑃(𝑥)} for all 𝑥∈𝐾 is weakly upper semicontinuous; (iii)𝑇∶𝐾→𝐿(𝑋,𝑌) is 𝜂-hemicontinuous and generalized relaxed monotone mapping.
Then the following two problems are equivalent: (A) 𝑥∈𝐾,⟨𝑇𝑥,𝜂(𝑦,𝑥)⟩+𝑓(𝑦,𝑥)≰_{int𝑃(𝑥)}0, for  all 𝑦∈𝐾,(B) 𝑥∈𝐾,⟨𝑇𝑦,𝜂(𝑦,𝑥)⟩+𝑓(𝑦,𝑥)−𝛼(𝑥,𝑦)≰_{int𝑃(𝑥)}0, for  all 𝑦∈𝐾.

Proof. Following the lines of proof given by Chen [9], one can easily prove.

Theorem 10. Let 𝑋 be real reflexive Banach space, and 𝑌 be a Banach space. Let 𝐾⊂𝑋 be a nonempty, bounded, closed, and convex subset of 𝑋. Let 𝑃∶𝐾→2^𝑌 be such that for each 𝑥∈𝐾,𝑃(𝑥) is a proper, closed, convex cone with int𝑃(𝑥)≠∅. Let 𝑓∶𝐾×𝐾→𝑌 be 𝑃₋-convex and upper semicontinuous in the first and second arguments, respectively, with the condition 𝑓(𝑥,𝑥)=0 for all 𝑥∈𝐾. Let conditions (i)–(iii) of Lemma 9 be satisfied and suppose the following conditions hold: (i)𝜂∶𝐾×𝐾→𝑋 is lower semicontinuous in the second argument; (ii)𝛼∶𝑋×𝑋→𝑌 is weakly lower semicontinuous and 𝑃₋-convex in the first and second arguments, respectively. Then there exist 𝑥∈𝐾, such that 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦,𝑥≰_{int𝑃(𝑥)}0,∀𝑦∈𝐾. (10)

Proof. Define two set-valued mappings 𝐹,𝐺∶𝐾→2^𝐾 as follows: 𝐹𝑦=≰𝑥∈𝐾∶𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦,𝑥_{int𝑃(𝑥)}0𝐺𝑦=≰,∀𝑦∈𝐾,𝑥∈𝐾∶𝑇𝑦,𝜂𝑦,𝑥+𝑓𝑦,𝑥−𝛼𝑥,𝑦_{int𝑃(𝑥)}0,∀𝑦∈𝐾. (11) Then 𝐹(𝑦) and 𝐺(𝑦) are nonempty since 𝑦∈𝐺(𝑦)∩𝐹(𝑦). We claim that 𝐹 is a KKM mapping. If this is not true, then there exist a finite set {𝑦₁,…,𝑦_𝑛}⊂𝐾 and 𝑡_𝑖≥0,𝑖=1,…,𝑛 with ∑^𝑛_𝑖=1𝑡_𝑖=1 such that 𝑦=∑^𝑛_𝑖=1𝑡_𝑖𝑦_𝑖∉⋃^𝑛_𝑖=1𝐹(𝑦_𝑖). Now by the definition of 𝐹, we have 𝑇𝑦,𝜂𝑦_𝑖,𝑦+𝑓𝑦_𝑖,𝑦≤_{int𝑃(𝑦)}0,𝑖=1,…,𝑛. (12) Now we have =0=𝑇𝑦,𝜂𝑦,𝑦+𝑓𝑦,𝑦𝑇𝑦,𝜂^𝑛_𝑖=1𝑡_𝑖𝑦_𝑖,𝑦+𝑓^𝑛_𝑖=1𝑡_𝑖𝑦_𝑖=,𝑦_𝑛𝑖=1𝑡_𝑖𝑇𝑦,𝜂𝑦_𝑖,𝑦+𝑓𝑦_𝑖≤,𝑦_{int𝑃(𝑦)}0, (13) which leads to a contradiction since 𝑃 is a proper cone. Thus our claim is verified. So 𝐹 is a KKM mapping.
Now we prove 𝐹(𝑦)⊂𝐺(𝑦) for every 𝑦∈𝐾. Indeed let 𝑥∈𝐹(𝑦). Then, we have 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦,𝑥≰_{int𝑃(𝑥)}0. (14) Since 𝑇 is generalized relaxed monotone, we have ≥𝑇𝑦,𝜂𝑦,𝑥+𝑓𝑦,𝑥_𝑃−𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦,𝑥+𝛼𝑥,𝑦. (15) From Lemma 6, we have 𝑇𝑦,𝜂𝑦,𝑥+𝑓𝑦,𝑥−𝛼𝑥,𝑦≰_{int𝑃(𝑥)}0, (16) that is, 𝑥∈𝐺(𝑦), which follows 𝐹(𝑦)⊂𝐺(𝑦), for each 𝑦∈𝐾 and so 𝐺 is also a KKM mapping. Now we claim that for each 𝑦∈𝐾,𝐺(𝑦)⊂𝐾 is closed in the weak topology of 𝑋.
Indeed suppose 𝑥∈𝐺(𝑦)^𝑤, the weak closure of 𝐺(𝑦). Since 𝑋 is reflexive, there is a sequence {𝑥_𝑛} in 𝐺(𝑦) such that {𝑥_𝑛} converges weakly to 𝑥∈𝐾. Then 𝑇𝑦,𝜂𝑦,𝑥_𝑛+𝑓𝑦,𝑥_𝑛−𝛼𝑥_𝑛,𝑦≰_{int𝑃(𝑥𝑛)}0, (17) which implies that 𝑇𝑦,𝜂𝑦,𝑥_𝑛+𝑓𝑦,𝑥_𝑛−𝛼𝑥_𝑛,𝑦∈𝑌⧵−int𝑃𝑥_𝑛. (18) Since 𝜂(𝑦,⋅) and 𝑓(𝑦,⋅) are lower and upper semicontinuous and 𝛼(⋅,𝑦) and 𝑌⧵{−int𝑃(𝑥_𝑛)} are weakly lower semicontinuous, therefore 𝑇𝑦,𝜂𝑦,𝑥+𝑓𝑦,𝑥−𝛼𝑥,𝑦∈𝑌⧵{−int𝑃(𝑥)}. (19) Thus we get 𝑇𝑦,𝜂𝑦,𝑥+𝑓𝑦,𝑥−𝛼𝑥,𝑦≰_{int𝑃(𝑥)}0 (20) and so 𝑥∈𝐺(𝑦). This shows that 𝐺(𝑦) is weakly closed for each 𝑦∈𝐾. Our claim is then verified. Since 𝑋 is reflexive and 𝐾⊂𝑋 is nonempty, bounded, closed, and convex, 𝐾 is a weakly compact subset of 𝑋 and so 𝐺(𝑦) is also weakly compact. According to Lemma 7, _𝑦∈𝐾𝐺𝑦≠∅. (21) This implies that there exists 𝑥∈𝐾 such that 𝑇𝑦,𝜂𝑦,𝑥+𝑓𝑦,𝑥−𝛼𝑥,𝑦≰_{int𝑃(𝑥)}0,∀𝑦∈𝐾. (22) Therefore by Minty’s type Lemma 9, we conclude that there exist 𝑥∈𝐾 such that 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦,𝑥≰_{int𝑃(𝑥)}0,∀𝑦∈𝐾. (23) This completes the proof.

3. Main Results

Throughout this section, let 𝑋 be real reflexive Banach space, 𝑌 a Banach space. Let 𝐾⊂𝑋 be a nonempty, bounded, closed, and convex subset of 𝑋. Let 𝑃∶𝐾→2^𝑌 be such that for each 𝑥∈𝐾,𝑃(𝑥) is a proper, closed, convex cone with int𝑃(𝑥)≠∅ and let 𝑃₋=⋂_𝑥∈𝐾𝑃(𝑥).

Some nonlinear mapping consisting with two variables, may be monotone with respect to the first variable and compact with respect to the second one. However, we cannot always expect for them to be monotone or compact with respect to the two variables simultaneously. Keeping this complexity in mind we are interested in the so called semimonotone mapping. We now give the concept of a generalized relaxed semimonotone mapping.

Definition 11. Let 𝐾⊂𝑋 be a nonempty, closed, and convex subset of 𝑋. Let 𝜂∶𝐾×𝐾→𝑋 be a mapping and let 𝛼∶𝑋×𝑋→𝑌 be a bifunction such that lim_𝑡→0⁺(𝛼(𝑥,𝑡𝑦+(1−𝑡)𝑥)/𝑡)=0 and 𝑓∶𝐾×𝐾→𝑌. A mapping 𝐴∶𝐾×𝐾→𝐿(𝑋,𝑌) is said to be generalized relaxed semimonotone mapping, if the following conditions hold: (i)for every 𝑢∈𝐾,𝐴(𝑢,⋅) is a generalized relaxed monotone mapping, that is, ⟨𝐴𝑢,𝑦−𝐴(𝑢,𝑥),𝜂𝑦,𝑥⟩−𝛼𝑥,𝑦≥_𝑃−0,∀𝑥,𝑦∈𝐾; (24)(ii)for every 𝑦∈𝐾,𝐴(⋅,𝑦) is completely continuous, that is, when 𝑢_𝑛→^𝑤𝑢,𝐴(𝑢_𝑛,𝑦)→𝐴(𝑢,𝑦) (by the norm of operators), where →^𝑤 denotes the weak convergence.

Remark 12. When 𝑌=ℝ,𝑃(𝑥)=ℝ⁺, and 𝜂(𝑦,𝑥)=𝑦−𝑥, for  all 𝑥,𝑦∈𝐾 and 𝛼≡0, then this is exactly the concept which was introduced by Chen [9].

Example 13. Let 𝑋=𝑌=ℝ²,𝐾=[0,1]×[0,1]. Let 𝑃∶𝐾→2^𝑌 be defined by 𝑃(𝑥)=𝑦₁,𝑦₂∈𝑌∣𝑦₁≥0,𝑦₂≥0,𝑥∈𝐾. (25) So 𝑃₋=_𝑥∈𝐾𝑃(𝑥)=𝑦₁,𝑦₂∈𝑌∣𝑦₁≥0,𝑦₂≥0. (26) Let 𝑇∶𝐾→𝐿(𝑋,𝑌) be defined by 𝑇(𝑥)=𝑥₁𝑥₂𝑥₁+1𝑥₂𝑥+1,∀𝑥=₁,𝑥₂∈𝐾. (27) Also 𝜂∶𝐾×𝐾→𝑋, 𝛼∶𝑋×𝑋→𝑌 are define by 𝜂(𝑦,𝑥)=𝑦−𝑥, for  all 𝑥,𝑦∈𝐾 and 𝛼(𝑥,𝑦)=(𝑥−𝑦)², for  all 𝑥,𝑦∈𝑋, respectively.
Let us suppose 𝐴∶𝐾×𝐾→𝐿(𝑋,𝑌) is defined by 𝐴=𝑥𝑥,𝑦¹+𝑦₁𝑥₂+𝑦₂𝑦₁𝑦₂,∀𝑥=𝑥₁,𝑥₂,𝑦=𝑦₁,𝑦₂∈𝐾. (28) First, we show that 𝐴 is a generalized relaxed monotone mapping. Indeed, for each 𝑢=(𝑢₁,𝑢₂),𝑣=(𝑣₁,𝑣₂)∈𝐾, it follows that 𝐴(()=𝑢𝑢,𝑦−𝐴𝑣,𝑦,𝜂𝑢−𝑣)−𝛼𝑣,𝑢₁−𝑣₁𝑢₂−𝑣₂𝑢₁−𝑣₁𝑢₂−𝑣₂𝑢₁−𝑣₁𝑢₂−𝑣₂−𝑣₁−𝑢₁²𝑣₂−𝑢₂²=𝑢₂−𝑣₂²𝑢₁−𝑣₁²≥_𝑃−0. (29)
The norm of 𝐴 is defined as ‖𝐴‖=|𝑥+𝑦|+|𝑦|, for  all 𝑥=(𝑥₁,𝑥₂), and 𝑦=(𝑦₁,𝑦₂)∈𝐾.
Now, for fixed 𝑣∈𝐾, if 𝑢_𝑛∈𝐾,𝑢∈𝐾, and 𝑢_𝑛→^𝑤𝑢, it is easy to prove that 𝐴𝑢_𝑛,𝑣−𝐴(𝑢,𝑣)→0. (30) Hence, for every 𝑣∈𝐾,𝐴(⋅,𝑣) is completely continuous.
Therefore, the mapping defined as above is a generalized relaxed semimonotone mapping.

Now we will pose the main problem of our study. In this paper, we investigate the following generalized vector variational-like inequality problem (for short, GVVLIP) is to find a vector 𝑢∈𝐾 satisfying 𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝑓(𝑣,𝑢)≰_{int𝑃(𝑢)}0,∀𝑣∈𝐾, (31) where 𝐴∶𝐾×𝐾→𝐿(𝑋,𝑌) is a nonlinear mapping and 𝜂∶𝐾×𝐾→𝑋, 𝑓∶𝐾×𝐾→𝑌 are the two vector-valued bimappings.

The GVVLIP (31) includes many variational inequality problems as special cases.

Some special cases of GVVLIP (31) are as follows. (I)If 𝜂(𝑣,𝑢)=𝑣−𝑢 and 𝑓≡0, then the GVVLIP (31) reduces to the following variational inequality problem of finding 𝑢∈𝐾 such that ⟨𝐴(𝑢,𝑢),𝑣−𝑢⟩≰_{int𝑃(𝑢)}0,∀𝑣∈𝐾, (32) which was introduced and studied by Zheng [7] (II)If 𝑌=ℝ,𝑃(𝑢)=[0,+∞), for  all 𝑢∈𝐾. Also if 𝜂(𝑣,𝑢)=𝑣−𝑢,𝑓≡0 and 𝐿(𝑋,𝑌) is the dual space 𝑋^∗ of 𝑋, then the GVVLIP (31) reduces to the following variational inequality problem of finding 𝑢∈𝐾 such that ⟨𝐴(𝑢,𝑢),𝑣−𝑢⟩≥0,∀𝑣∈𝐾, (33) which was introduced and investigated by Chen [9]. He obtained some existence results and discussed their applications in partial differential equations of divergence form.

We recall the following fixed-point theorem, by Zeidle [16], which will play an important role in establishing our existing results for GVVLIP (31).

Theorem 14 (see [16]). The set-valued mapping 𝐹∶𝑀→2^𝑀 has a fixed point, if the following conditions are satisfied: (1)𝑀 is compact, convex, and nonempty set in locally convex space; (2)𝐹(𝑥) is convex, closed, and nonempty for every 𝑥∈𝑀; (3)𝐹 is upper semicontinuous on 𝑀.

Now, we have the following existence results for GVVLIP (31) involving a generalized relaxed semimonotone mapping in reflexive Banach spaces.

Theorem 15. Let 𝑋 be real reflexive Banach space, and let 𝑌 be a Banach space. Let 𝐾⊂𝑋 be a nonempty, bounded, closed, and convex subset of 𝑋. Let 𝑃∶𝐾→2^𝑌 be such that for each 𝑥∈𝐾,𝑃(𝑥) is a proper, closed, convex cone with int𝑃(𝑥)≠∅. Let 𝐴∶𝐾×𝐾→𝐿(𝑋,𝑌) be a generalized relaxed semimonotone mapping. Suppose the following conditions hold: (i)𝑓∶𝐾×𝐾→𝑌 is 𝑃₋-convex with 𝑓(𝑥,𝑥)=0, for  all 𝑥∈𝐾 and upper semicontinuous in the second argument; (ii)𝜂∶𝐾×𝐾→𝑋 is an affine mapping with 𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0, for  all 𝑥,𝑦∈𝐾 and lower semicontinuous in the second argument; (iii)The set-valued mapping 𝑊∶𝐾→2^𝑌 defined as 𝑊(𝑥)=𝑌⧵{−int𝑃(𝑥)}, for  all 𝑥∈𝐾, is weakly upper semicontinuous and concave; (iv)𝛼∶𝑋×𝑋→𝑌 is 𝑃₋-concave and weakly lower semicontinuous in the second and first argument, respectively; (v)For each fixed 𝑣∈𝐾, 𝐴(𝑣,⋅)∶𝐾×𝐾→𝐿(𝑋,𝑌) is continuous on each finite dimensional subspace of 𝑋. Then there exists a 𝑢₀∈𝐾 such that 𝐴𝑢₀,𝑢₀,𝜂𝑣,𝑢₀+𝑓𝑣,𝑢₀≰_{int𝑃(𝑢0)}0,∀𝑣∈𝐾. (34)

Proof. Let 𝐹 be a finite dimensional subspace of 𝑋 and 𝐾_𝐹=𝐾⋂𝐹≠∅. For each 𝑣∈𝐾_𝐹, we consider the following generalized vector variational-like inequality problem. Find 𝑢₀∈𝐾_𝐹 such that 𝐴𝑣,𝑢₀,𝜂𝑢,𝑢₀+𝑓𝑢,𝑢₀≰_{int𝑃(𝑢0)}0,∀𝑢∈𝐾_𝐹. (35) Since 𝐾_𝐹⊂𝐹 is bounded, closed, and convex, 𝐴(𝑣,⋅) is continuous on 𝐾_𝐹 and generalized relaxed monotone for each fixed 𝑣∈𝐾. From Theorem 10, we know our problem has solution 𝑢₀∈𝐾_𝐹.
Define a set-valued mapping 𝑇∶𝐾_𝐹→2^𝐾𝐹 as follows: 𝐹(𝑣)=𝑤∈𝐾_𝐹∶𝐴()(()≰𝑣,𝑤,𝜂𝑢,𝑤)+𝑓𝑢,𝑤_{int𝑃(𝑤)}0,∀𝑢∈𝐾_𝐹. (36) It follows from Minty’s type Lemma 9, that for each fixed 𝑣∈𝐾_𝐹𝑤∈𝐾_𝐹∶𝐴()(()≰𝑣,𝑤,𝜂𝑢,𝑤)+𝑓𝑢,𝑤_{int𝑃(𝑤)}0,∀𝑢∈𝐾_𝐹=𝑤∈𝐾_𝐹∶𝐴()(()()≰𝑣,𝑢,𝜂𝑢,𝑤)+𝑓𝑢,𝑤−𝛼𝑤,𝑢_{int𝑃(𝑤)}0,∀𝑢∈𝐾_𝐹. (37) Now we will use the fixed-point theorem to verify the existence of the solution of the problem in a finite dimensional. Since 𝐹 is of finite dimensional, hence 𝐾_𝐹 is compact. First, we claim that 𝐹(𝑣)=𝑤∈𝐾_𝐹∶𝐴()(()(){(𝑤(𝑤𝑣,𝑢,𝜂𝑢,𝑤)+𝑓𝑢,𝑤−𝛼𝑤,𝑢∈𝑌⧵−int𝑃)}=𝑊) (38) is convex. Indeed, let 𝑤₁,𝑤₂∈𝐹(𝑣) and 𝑚,𝑛≥0, such that 𝑚+𝑛=1𝑚𝐴(𝑣,𝑢),𝜂𝑢,𝑤₁+𝑓𝑢,𝑤₁−𝛼𝑤₁,𝑢=𝑚𝑊𝑤₁,𝑛𝐴(𝑣,𝑢),𝜂𝑢,𝑤₂+𝑓𝑢,𝑤₂−𝛼𝑤₂,𝑢=𝑚𝑊𝑤₂. (39) Since 𝜂(𝑢,⋅) is affine and 𝑓(𝑢,⋅),𝛼(⋅,𝑢) are 𝑃₋-convex, then from preceding two inclusions we have 𝐴(𝑣,𝑢),𝜂𝑢,𝑚𝑤₁+𝑛𝑤₂+𝑓𝑢,𝑚𝑤₁+𝑛𝑤₂−𝛼𝑚𝑤₁+𝑛𝑤₂,𝑢∈𝑚𝑊𝑤₁+𝑛𝑊𝑤₂. (40) Since 𝑊 is concave, we have 𝑚𝑤₁+𝑛𝑤₂∈𝐹(𝑣), that is, 𝐹(𝑣) is convex and our claim is then verified. Now, we claim that 𝐹(𝑣) is closed. Let 𝑤_𝑗∈𝐹(𝑣) such that 𝑤_𝑗→𝑤, then 𝐴(𝑣,𝑢),𝜂𝑢,𝑤_𝑗+𝑓𝑢,𝑤_𝑗−𝛼𝑤_𝑗𝑤,𝑢∈𝑌⧵−int𝑃_𝑗,𝐴(𝑣,𝑢),𝜂𝑢,𝑤_𝑗+𝑓𝑢,𝑤_𝑗−𝛼𝑤_𝑗,𝑢∈𝑊𝑤_𝑗. (41) Since 𝑓(𝑦,⋅)+𝛼(⋅,𝑦) is upper semicontinuous, also 𝐴(𝑣,𝑢)∈𝐿(𝑋,𝑌) and 𝑊 is upper semicontinuous; therefore 𝐴(𝑣,𝑢),𝜂𝑢,𝑤_𝑗+𝑓𝑢,𝑤_𝑗−𝛼𝑤_𝑗→𝐴()(()()(𝑤).,𝑢𝑣,𝑢,𝜂𝑢,𝑤)+𝑓𝑢,𝑤−𝛼𝑤,𝑢∈𝑊 (42) This implies 𝑤∈𝐹(𝑣); hence 𝐹(𝑣) is closed. Next, we claim that 𝐹 is upper semicontinuous. Let 𝑣_𝑗→𝑣,𝑤_𝑗→𝐹(𝑣_𝑗), and 𝑤_𝑗→𝑤, then we have 𝐴𝑣_𝑗,𝑢,𝜂𝑢,𝑤_𝑗+𝑓𝑢,𝑤_𝑗−𝛼𝑤_𝑗,𝑢∈𝑊𝑤_𝑗. (43) From the complete continuity of 𝐴(⋅,𝑢) and the lower semicontinuity of 𝜂(𝑢,⋅), we have 𝐴𝑣_𝑗,𝑢,𝜂𝑢,𝑤_𝑗→𝐴(𝑣,𝑢),𝜂(𝑢,𝑤) (44) which implies that 𝐴(𝑣,𝑢),𝜂(𝑢,𝑤)+𝑓(𝑢,𝑤)−𝛼(𝑤,𝑢)∈𝑊(𝑤), (45) that is, 𝑤∈𝐹(𝑣); thus our claim is then verified. Hence 𝐹 is upper semicontinuous. By Fan-Glicksberg fixed-point theorem, there exists a 𝑣₀∈𝐹(𝑣₀) that is, there exists a 𝑣₀∈𝐾_𝐹 such that 𝐴𝑣₀,𝑣₀,𝜂𝑢,𝑣₀+𝑓𝑢,𝑣₀≰_{int𝑃(𝑣0)}0,∀𝑢∈𝐾_𝐹. (46) Now we generalize this result to whole space.
Let Ω≡{𝐹⊂𝑋∶𝐹befinitedimensional,𝐹⋂𝐾≠∅} and let Γ_𝐹≡{𝑤∈𝐾∶⟨𝐴(𝑤,𝑢),𝜂(𝑢,𝑤)⟩+𝑓(𝑢,𝑤)−𝛼(𝑤,𝑢)≰_{int𝑃(𝑤)}0,forall𝑢∈𝐾_𝐹},𝐹∈Ω.
From the above we know that forall𝐹∈Ω,Γ_𝐹≠∅. Let Γ_𝐹^𝑤 denote the weak closure of Γ_𝐹. For any 𝐹_𝑖∈Ω,𝑖=1,…,𝑛, we know that Γ_⋃^𝑛_{𝑖=1𝐹𝑖}⊆⋂^𝑛_𝑖=1Γ_𝐹𝑖⊆⋂^𝑛_𝑖=1Γ_𝐹𝑖^𝑤. Therefore, ⋂^𝑛_𝑖=1Γ_𝐹𝑖^𝑤≠∅. Since 𝐾 is weakly compact, from the finite intersection property, we have ⋂_𝑥∈ΩΓ_𝐹^𝑤≠∅. Let 𝑤₀∈⋂_𝑥∈ΩΓ_𝐹^𝑤, we see that 𝐴𝑤₀,𝑤₀,𝜂𝑢,𝑤₀+𝑓𝑢,𝑤₀≰_{int𝑃(𝑤0)}0. (47) Indeed, for each 𝑢∈𝐾, let 𝐹∈Ω, such that 𝑢∈𝐾_𝐹, 𝑤₀∈𝐾_𝐹. From 𝑤₀∈Γ_𝐹^𝑤, there exists 𝑤_𝑗∈Γ_𝐹, that is, 𝐴𝑤_𝑗,𝑢,𝜂𝑢,𝑤_𝑗+𝑓𝑢,𝑤_𝑗−𝛼𝑤_𝑗𝑤,𝑢∈𝑌⧵−int𝑃_𝑗∈𝑊𝑤_𝑗, (48) such that 𝑤_𝑗→^𝑤𝑤₀, from the complete continuity of 𝐴(⋅,𝑣), upper semicontinuity of 𝑓(𝑢,⋅), and lower semicontinuity of 𝛼(⋅,𝑢), we have 𝐴𝑤₀,𝑢,𝜂𝑢,𝑤₀+𝑓𝑢,𝑤₀−𝛼𝑤₀,𝑢∈𝑊𝑤₀. (49) From Lemma 9, we have 𝐴𝑤₀,𝑤₀,𝜂𝑢,𝑤₀+𝑓𝑢,𝑤₀≰_{int𝑃(𝑤0)}0,∀𝑢∈𝐾. (50) This completes the proof.

If the boundedness of 𝐾 is dropped off, then we have the following theorem under certain coercivity condition.

Theorem 16. Let 𝑋 be real reflexive Banach space and 𝑌 be a Banach space. Let 0∈𝐾⊂𝑋 be a nonempty, closed, and convex subset of 𝑋. Let 𝑃∶𝐾→2^𝑌 be such that for each 𝑥∈𝐾,𝑃(𝑥) is a proper, closed, convex cone with int𝑃(𝑥)≠∅. Let 𝐴∶𝐾×𝐾→𝐿(𝑋,𝑌) be a generalized relaxed semimonotone mapping. Let conditions (i)–(v) of Theorem 15 be satisfied and lim_{‖𝑢‖→∞}𝐴(𝑢,𝑢),𝜂(𝑢,0)+𝑓(0,𝑢)≥_{int𝑃(𝑢)}0 (51) then there exist 𝑢∈𝐾 such that 𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝑓(𝑣,𝑢)≰_{int𝑃(𝑢)}0,∀𝑣∈𝐾. (52)

Proof. For each 𝑟>0, let 𝐵[0,𝑟] denote the closed ball in the Banach space 𝑋 with center 0 and radius 𝑟. By Theorem 15, for each 𝑟∈ℕ, there exists 𝑢_𝑟∈𝐵[0,𝑟]∩𝐾 such that 𝐴𝑢_𝑟,𝑢_𝑟,𝜂𝑣,𝑢_𝑟+𝑓𝑣,𝑢_𝑟≰_{int𝑃(𝑢𝑟)}[]0,∀𝑣∈𝐵0,𝑟∩𝐾. (53) Since 0∈𝐾, we have 𝐴𝑢_𝑟,𝑢_𝑟,𝜂0,𝑢_𝑟+𝑓0,𝑢_𝑟≰_{int𝑃(𝑢𝑟)}0. (54) From condition (vi), we know that {𝑢_𝑟} is bounded. So, we may suppose that 𝑢_𝑟 converges to 𝑢 as 𝑟→∞. It follows from Lemma 9, that 𝐴𝑢_𝑟,𝑣,𝜂𝑣,𝑢_𝑟+𝑓𝑣,𝑢_𝑟−𝛼𝑢_𝑟,𝑣≰_{int𝑃(𝑢𝑟)}0,∀𝑣∈𝐾∩𝐵_𝑟. (55) Since 𝐴 is completely continuous, 𝜂, 𝛼 are lower semicontinuous, and 𝑓,𝑊 are upper semicontinuous. Therefore, it follows that 𝐴(𝑢,𝑣),𝜂(𝑣,𝑢)+𝑓(𝑣,𝑢)−𝛼(𝑢,𝑣)≰_{int𝑃(𝑢)}0∀𝑣∈𝐾∩𝐵_𝑟. (56) Again from Lemma 9, we get 𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝑓(𝑣,𝑢)≰_{int𝑃(𝑢)}0,∀𝑣∈𝐾. (57) This completes the proof.

Remark 17. Theorems 15 and 16 improve and generalize Theorems 3.1 and 3.2 of Fang and Huang [10] and Theorem 2.1 to Theorem 2.6 of Chen [9].