On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxed 𝜂-𝛼 Monotone Mapping

Abstract

We introduce two new concepts of weakly relaxed 𝜂-𝛼 monotone mappings and weakly relaxed 𝜂-𝛼 semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed 𝜂-𝛼 monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed 𝜂-𝛼 semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.

1. Introduction

The variational inequality theory provides us with a simple, natural, unified, and elegant framework to study a wide class of linear and nonlinear problems arising in many fields, such as mechanics, engineering sciences, elasticity, optimization, control, programming, economics, transportation, oceanography, and regional. Because of their wide applicability, various extensions and generalizations of the classical variational inequality problem have been proposed and studied in recent years. Variational-like inequalities problems is one of cornerstone in this field. Some special case of generalized variational inequalities and variational-like inequalities have been studied by several authors including Bai et al. [1], Chang et al. [2], dos Santos et al. [3], Xiao and Huang [4], Zhao and Xia [5, 6], and references therein.

It is well known that the monotonicity and generalized monotonicity play an important role of the study in variational inequality theory. In recent years, a substantial number of papers on existence results for solving variational inequality problems and variational-like inequality problems based on different generalization of monotonicity such as pseudomonotonicity, quasimonotonicity, relaxed monotonicity, semimonotonicity, and 𝑝-monotonicity (see [7–15]) appeared. In [16], Fang and Huang introduced a new concept of relaxed 𝜂-𝛼 monotonicity and obtained the existence of solutions for variational-like inequalities with relaxed 𝜂-𝛼 monotone mappings in reflexive Banach spaces. Recently, Sintunavarat [17] established the existence of solution of mixed equilibrium problem with the weakly relaxed 𝛼-monotone bi-function in Banach spaces.

Inspired and motivated by the work of Fang and Huang [16] and Sintunavarat [17], in this paper we introduce the two new concepts of weakly relaxed 𝜂-𝛼 monotone mappings and weakly relaxed 𝜂-𝛼 semimonotone mappings as well as two classes of variational-like inequalities with weakly relaxed 𝜂-𝛼 monotone mappings and weakly relaxed 𝜂-𝛼 semimonotone mappings. By using the KKM technique, we study some existence of solutions for variational-like inequalities with weakly relaxed 𝜂-𝛼 monotone mappings in reflexive Banach spaces. We also obtain the solvability of variational-like inequalities with weakly relaxed 𝜂-𝛼 semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem. Our results in this paper extend and improve the results of Fang and Huang [16] and many results in the literature.

2. Preliminaries

In this paper, unless otherwise specified, 𝐾 is a nonempty closed convex subset of a real reflexive Banach space 𝐸 with dual space 𝐸^∗. The following basic knowledge will be useful in our work.

Definition 1 (see [18]). Let 𝑇∶𝐾→𝐸^∗ and 𝜂∶𝐾×𝐾→𝐸 be two mappings. A mapping 𝑇 is said to be 𝜂-hemicontinuous if, for any fixed 𝑥,𝑦∈𝐾, the mapping 𝑓∶[0,1]→ℝ defined by 𝑓(𝑡)=𝑇𝑡𝑦+(1−𝑡)𝑥,𝜂𝑦,𝑥 (1) is continuous at 0⁺.

Definition 2 (see [16]). Let 𝑇∶𝐾→𝐸^∗ and 𝜂∶𝐾×𝐾→𝐸 be two mappings. A mapping 𝑇 is said to be 𝜂-coercive with respect to a proper function 𝑓∶𝐾→ℝ∪{∞} if there exists 𝑥₀∈𝐾 such that 𝑇𝑥−𝑇𝑥₀,𝜂𝑥,𝑥₀+𝑓(𝑥)−𝑓𝑥₀𝜂𝑥,𝑥₀⟶∞, (2) whenever ‖𝑥‖ is large enough.

Remark 3. If 𝑓=𝛿_𝐾, where 𝛿_𝐾 is the indicator function of 𝐾, then Definition 2 coincides with the definition of 𝜂-coercivity in the sense of Yang and Chen [15].

Definition 4 (see [16]). A mapping 𝑇∶𝐾→𝐸^∗ is said to be relaxed 𝜂-𝛼 monotone if there exist a function 𝜂∶𝐾×𝐾→𝐸 and 𝛼∶𝐸→ℝ with 𝛼(𝑡𝑧)=𝑡^𝑝𝛼(𝑧) for all 𝑡>0 and 𝑧∈𝐸 such that 𝑇𝑥−𝑇𝑦,𝜂𝑥,𝑦≥𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐾, (3) where 𝑝>1 is a constant.

Remark 5. (1) If 𝜂(𝑥,𝑦)=𝑥−𝑦 for all 𝑥,𝑦∈𝐾, then (3) reduces to 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐾, (4) and 𝑇 is said to be 𝛼-monotone.
(2) If 𝜂(𝑥,𝑦)=𝑥−𝑦 for all 𝑥,𝑦∈𝐾 and 𝛼(𝑧)=𝑘‖𝑧‖^𝑝, where 𝑘>0 and 𝑝>1, then (3) becomes 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥𝑘𝑥−𝑦^𝑝,∀𝑥,𝑦∈𝐾, (5) and 𝑇 is said to be 𝑝-monotone (see in [14, 19, 20]).
(3) Every monotone mapping is relaxed 𝜂-𝛼 monotone with 𝜂(𝑥,𝑦)=𝑥−𝑦 for all 𝑥,𝑦∈𝐾 and 𝛼≡0.

Definition 6. Let 𝐹∶𝐾→2^𝐸 be a set-valued mapping. Then, 𝐹 is said to be KKM mapping if, for any finite subset {𝑦₁,𝑦₂,…,𝑦_𝑛} of 𝐾, we have co{𝑦₁,𝑦₂,…,𝑦_𝑛}⊂⋃^𝑛_𝑖=1𝐹(𝑦_𝑖), where co{𝑦₁,𝑦₂,…,𝑦_𝑛} denotes the convex hull of {𝑦₁,𝑦₂,…,𝑦_𝑛}.

Remark 7. Let 𝐹,𝐺∶𝐾→2^𝐸. If 𝐹 is KKM mapping and 𝐹(𝑦)⊂𝐺(𝑦) for all 𝑦∈𝐾, then 𝐺 is also KKM mapping.

Lemma 8 (see [21]). Let 𝑀 be a nonempty subset of a Hausdorff topological vector space 𝑋, and let 𝐹∶𝑀→2^𝑋 be a KKM mapping. If 𝐹(𝑦) is closed in 𝑋 for all 𝑦∈𝑀 and compact for some 𝑦∈𝑀, then ⋂_𝑦∈𝑀𝐹(𝑦)≠∅.

3. Variational-Like Inequalities Problems with Weakly Relaxed 𝜂-𝛼 Monotone Mapping

In this section, we introduce the new class of mapping which generalizes several classes. Using KKM technique, we study and prove the existence of solutions for variational-like inequalities problems with mapping in such class in Banach spaces.

Definition 9. A mapping 𝑇∶𝐾→𝐸^∗ is said to be weakly relaxed 𝜂-𝛼 monotone if there exist a function 𝜂∶𝐾×𝐾→𝐸 and 𝛼∶𝐸→ℝ with lim_𝑡→0+𝛼(𝑡𝑥)=0,lim_𝑡→0+𝑑𝛼()𝑑𝑡𝑡𝑥=0, (6)(7) for all 𝑡>0 and 𝑥∈𝐸 such that 𝑇𝑥−𝑇𝑦,𝜂𝑥,𝑦≥𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐾. (8)

Remark 10. We obtain that the relaxed 𝜂-𝛼 monotonicity implies weakly relaxed 𝜂-𝛼 monotonicity. So, class of relaxed 𝜂-𝛼 monotone mapping is subclass of weakly relaxed 𝜂-𝛼 monotone mapping class. Also, we get that classes of relaxed 𝛼 monotone mapping, 𝑝-monotone mapping, and monotone mapping are subclass of weakly relaxed 𝜂-𝛼 monotone mapping class.

Theorem 11. Let 𝑇∶𝐾→𝐸^∗ be an 𝜂-hemicontinuous and weakly relaxed 𝜂-𝛼 monotone and let 𝑓∶𝐾→ℝ∪{∞} be a proper convex function. Suppose that the following conditions hold: (a)𝜂(𝑥,𝑥)=0 for all 𝑥∈𝐾; (b)for any fixed 𝑦,𝑧∈𝐾, the mapping 𝑥↦⟨𝑇𝑧,𝜂(𝑥,𝑦)⟩ is convex.
Then, the following problems (9) and (10) are equivalent: 𝑦(𝑥)𝑓𝑖𝑛𝑑𝑥∈𝐾𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑇𝑥,𝜂𝑦,𝑥+𝑓−𝑓≥0,∀𝑦∈𝐾,𝑦(𝑥),𝑓𝑖𝑛𝑑𝑥∈𝐾𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑇𝑦,𝜂𝑦,𝑥+𝑓−𝑓≥𝛼𝑦−𝑥∀𝑦∈𝐾. (9)(10)

Proof. Suppose that (9) has a solution. So, there exists 𝑥∈𝐾 such that 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥0,∀𝑦∈𝐾. (11) Since 𝑇 is weakly relaxed 𝜂-𝛼 monotone, we have 𝑇𝑦−𝑇𝑥,𝜂𝑦,𝑥≥𝛼𝑦−𝑥,∀𝑦∈𝐾, (12) and then 𝑦(𝑥)≥𝑦(𝑥)𝑇𝑦,𝜂𝑦,𝑥+𝑓−𝑓𝑇𝑥,𝜂𝑦,𝑥+𝛼𝑦−𝑥+𝑓−𝑓≥𝛼𝑦−𝑥,∀𝑦∈𝐾. (13) Therefore, 𝑥∈𝐾 is a solution of problem (10).
Conversely, suppose that 𝑥∈𝐾 is a solution of problem (10) and 𝑦 is any point in 𝐾 with 𝑓(𝑦)<∞. For 𝑡∈(0,1), we let 𝑦_𝑡∶=(1−𝑡)𝑥+𝑡𝑦. From (10), we get that 𝑓(𝑥)<∞. It follows from 𝐾 being convex that 𝑦_𝑡∈𝐾. From (10), we have 𝑇𝑦_𝑡,𝜂𝑦_𝑡,𝑥+𝑓𝑦_𝑡−𝑓(𝑥)≥𝛼𝑦_𝑡𝑡,−𝑥=𝛼𝑦−𝑥 (14) and thus 𝑓𝑦_𝑡−𝑓(𝑥)≥𝛼𝑡𝑦−𝑥−𝑇𝑦_𝑡,𝜂𝑦_𝑡,𝑥. (15) The convexity of 𝑓 implies that 𝑓𝑦_𝑡(𝑥)()(𝑥)𝑓𝑦(𝑥).−𝑓=𝑓1−𝑡𝑥+𝑡𝑦−𝑓≤𝑡−𝑓 (16) From (15) and (16), we get 𝑡𝑓𝑦−𝑓(𝑥)≥𝛼𝑡𝑦−𝑥−𝑇𝑦_𝑡,𝜂𝑦_𝑡,𝑥. (17) By the assumption (b), we have 𝑇𝑦_𝑡,𝜂𝑦_𝑡,𝑥=𝑇𝑦_𝑡()≤(,𝜂1−𝑡𝑥+𝑡𝑦,𝑥1−𝑡)𝑇𝑦_𝑡,𝜂(𝑥,𝑥)+𝑡𝑇𝑦_𝑡,𝜂𝑥,𝑦=𝑡𝑇𝑦_𝑡𝑇().,𝜂𝑦,𝑥=𝑡1−𝑡𝑥+𝑡𝑦,𝜂𝑦,𝑥 (18) It follows from (17) and (18) that 𝑡𝑇()𝑓𝑦(𝑥)𝑡,1−𝑡𝑥+𝑡𝑦,𝜂𝑦,𝑥+𝑡−𝑓≥𝛼𝑦−𝑥 (19) that is 𝑇()𝑦(𝑥)≥𝛼𝑡1−𝑡𝑥+𝑡𝑦,𝜂𝑦,𝑥+𝑓−𝑓𝑦−𝑥𝑡, (20) for all 𝑦∈𝐾. Taking 𝑡→0⁺ in the previous inequality and using 𝜂-hemicontinuity of 𝑇, we get 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥lim_𝑡→0+𝛼𝑡𝑦−𝑥𝑡. (21) From (6), we get lim_𝑡→0⁺(𝛼(𝑡(𝑦−𝑥))/𝑡) is indeterminate form. Using L’ Hôpital’s rule, we obtain that 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥lim_𝑡→0+(𝑑/𝑑𝑡)𝛼𝑡𝑦−𝑥1. (22) By property (7) of weakly relaxed 𝜂-𝛼 monotone of 𝑇, we have 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥0, (23) for all 𝑦∈𝐾 with 𝑓(𝑦)<∞. In case of 𝑓(𝑦)=∞, the relation 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥0 (24) is trivial. Therefore, 𝑥∈𝐾 is also a solution of problem (9).

Theorem 12. Let 𝐾 be a nonempty bounded closed convex subset of a real reflexive Banach space 𝐸, and let 𝐸^∗ be the dual space of 𝐸. Suppose that 𝑇∶𝐾→𝐸^∗ is an 𝜂-hemicontinuous and weakly relaxed 𝜂-𝛼 monotone mapping and 𝑓∶𝐾→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold: (a)  𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐾,(b)  for any fixed 𝑦,𝑧∈𝐾, the mapping 𝑥↦⟨𝑇𝑧,𝜂(𝑥,𝑦)⟩ is convex and lower semicontinuous function, (c)  𝛼 is weakly lower semicontinuous; that is, for any net {𝑥_𝛽}, we have that 𝑥_𝛽 converges to 𝑥 in 𝜎(𝐸,𝐸^∗) implies that 𝛼(𝑥)≤liminf𝛼(𝑥_𝛽),
then the problem (9) is solvable.

Proof. Consider the set valued mapping 𝐹∶𝐾→2^𝐸 defined by 𝐹𝑦=𝑥∈𝐾∶𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥0, (25) for all 𝑦∈𝐾.
It is easy to see that 𝑥∈𝐾 solves the problem (9); that is, 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥0,∀𝑦∈𝐾, (26) if and only if 𝑥∈⋂_𝑦∈𝐾𝐹(𝑦). Thus, it is sufficient to prove that ⋂_𝑦∈𝐾𝐹(𝑦)≠∅.
Next, we show that 𝐹 is a KKM mapping. Assume the contrary, then there exists {𝑦₁,𝑦₂,…,𝑦_𝑚}⊂𝐾 such that co{𝑦₁,𝑦₂,…,𝑦_𝑚}⊄⋃^𝑚_𝑖=1𝐹(𝑦_𝑖). This implies that there exists 𝑦₀∈co{𝑦₁,𝑦₂,…,𝑦_𝑚} such that 𝑦₀=∑^𝑚_𝑖=1𝑡_𝑖𝑦_𝑖, where 𝑡_𝑖≥0, 𝑖=1,2,…,𝑚 and ∑^𝑚_𝑖=1𝑡_𝑖=1, but 𝑦₀∉⋃^𝑚_𝑖=1𝐹(𝑦_𝑖).
From (25), we have 𝑇𝑦₀,𝜂𝑦_𝑖,𝑦₀+𝑓𝑦_𝑖−𝑓𝑦₀<0,for𝑖=1,2,…,𝑚. (27) By (b) and (25), we obtain that 0=𝑇𝑦₀,𝜂𝑦₀,𝑦₀=𝑇𝑦₀,𝜂^𝑚_𝑖=1𝑡_𝑖𝑦_𝑖,𝑦₀≤^𝑚_𝑖=1𝑡_𝑖𝑇𝑦₀,𝜂𝑦_𝑖,𝑦₀<^𝑚_𝑖=1𝑡_𝑖𝑓𝑦₀−𝑓𝑦_𝑖=𝑓𝑦₀−^𝑚_𝑖=1𝑡_𝑖𝑓𝑦_𝑖≤𝑓𝑦₀−𝑓𝑦₀=0, (28) which is a contradiction. Therefore, 𝐹 is a KKM mapping.
Now, we define another set-valued mapping 𝐺∶𝐾→2^𝐸 by 𝐺𝑦=𝑥∈𝐾∶𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥𝛼𝑦−𝑥 (29) for all 𝑦∈𝐾.
Next, we will claim that 𝐹(𝑦)⊂𝐺(𝑦) for all 𝑦∈𝐾. For each 𝑦∈𝐾, let 𝑥∈𝐹(𝑦); then, 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥0. (30) From the weakly relaxed 𝜂-𝛼 monotonicity of 𝑇, we get 𝑦(𝑥)+𝑦(𝑥.𝑇𝑦,𝜂𝑦,𝑥+𝑓−𝑓≥𝛼𝑦−𝑥𝑇𝑥,𝜂𝑦,𝑥+𝑓−𝑓)≥𝛼𝑦−𝑥 (31) This implies that 𝑥∈𝐺(𝑦) and hence 𝐹(𝑦)⊂𝐺(𝑦) for all 𝑦∈𝐾. So, 𝐺 is also a KKM mapping.
By assumption, 𝑥↦⟨𝑇𝑦,𝜂(𝑥,𝑦)⟩ and 𝑓 are two convex lower-semicontinuous functions. Then it is easy to see that they are both weakly lower semicontinuous. From the definition of 𝐺 and the weakly lower semicontinuity of 𝛼, we get that 𝐺(𝑦) is weakly closed for all 𝑦∈𝐾. Since 𝐾 is closed bounded and convex, it also is weakly compact, and then 𝐺(𝑦) is weakly compact in 𝐾 for each 𝑦∈𝐾. From Lemma 8 and Theorem 11, we obtain that _𝑦∈𝐾𝐹𝑦=_𝑦∈𝐾𝐺𝑦≠∅. (32) So, there exists 𝑥∈𝐾, such that 𝑇𝑥,𝜂𝑦,𝑥+𝑓𝑦−𝑓(𝑥)≥0,∀𝑦∈𝐾; (33) that is, problem (9) has a solution. This completes the proof.

We know that the relaxed 𝜂-𝛼 monotonicity implies the weakly relaxed 𝜂-𝛼 monotonicity. Therefore, Theorem 12 can be deduced to the following corollary.

Corollary 13. Let 𝐾 be a nonempty bounded closed convex subset of a real reflexive Banach space 𝐸, and let 𝐸^∗ be the dual space of 𝐸. Suppose that 𝑇∶𝐾→𝐸^∗ is an 𝜂-hemicontinuous and relaxed 𝜂-𝛼 monotone mapping and 𝑓∶𝐾→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold: (a)  𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐾, (b)  for any fixed 𝑦,𝑧∈𝐾, the mapping 𝑥↦⟨𝑇𝑧,𝜂(𝑥,𝑦)⟩ is convex and lower semicontinuous function, (c)  𝛼 is weakly lower semicontinuous,
then the problem (9) is solvable.

Remark 14. Since the monotonicity, 𝑝-monotonicity, and relaxed 𝛼-monotonicity imply relaxed 𝜂-𝛼 monotonicity, we can be applying Corollary 13 to the other problems for the mapping satisfies these property.

Next, we study and prove that result for the case of 𝐾 is unbounded set.

Theorem 15. Let 𝐾 be a nonempty unbounded closed convex subset of a real reflexive Banach space 𝐸, and let 𝐸^∗ be the dual space of 𝐸. Suppose that 𝑇∶𝐾→𝐸^∗ is an 𝜂-hemicontinuous and weakly relaxed 𝜂-𝛼 monotone mapping and 𝑓∶𝐾→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold: (a)  𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐾, (b)  for any fixed 𝑦,𝑧∈𝐾, the mapping 𝑥↦⟨𝑇𝑧,𝜂(𝑥,𝑦)⟩ is convex and lower semicontinuous function, (c)  𝛼 is weakly lower semicontinuous, (d)  𝑇 is weakly 𝜂-coercive with respect to 𝑓; that is, there exists 𝑥₀∈𝐾 such that 𝑇𝑥,𝜂𝑥₀,𝑥+𝑓𝑥₀−𝑓(𝑥)<0, (34) whenever 𝑥∈𝐾 and ‖𝑥‖ large enough,
then the problem (9) is solvable.

Proof. For 𝜖>0, define 𝐾_𝜖∶={𝑦∈𝐾∶‖𝑦‖≤𝜖}. Consider the following problem:find𝑥∈𝐾_𝜖suchthat𝑇𝑥_𝜖,𝜂𝑦,𝑥_𝜖+𝑓𝑦−𝑓𝑥_𝜖≥0,∀𝑦∈𝐾_𝜖. (35)
Since 𝐾_𝜖 is bounded, by Theorem 12, we get that the problem (35) has at least one solution 𝑥_𝜖∈𝐾_𝜖.
For 𝑥₀ in the weakly 𝜂-coercivity condition (d), we fixed 𝜖>‖𝑥₀‖. From (35), we can find that 𝑥_𝜖∈𝐾_𝜖 such that 𝑇𝑥_𝜖,𝜂𝑥₀,𝑥_𝜖+𝑓𝑥₀−𝑓𝑥_𝜖≥0. (36) Since 𝑥_𝜖∈𝐾_𝜖, we have ‖𝑥_𝜖‖≤𝜖. If ‖𝑥_𝜖‖=𝜖 for all 𝜖, we may choose 𝜖 large enough so that the weakly 𝜂-coercivity condition (d) implies that 𝑇𝑥_𝜖,𝜂𝑥₀,𝑥_𝜖+𝑓𝑥₀−𝑓𝑥_𝜖<0, (37) which contradicts (36). Therefore, there exists 𝜖 such that ‖𝑥_𝜖‖<𝜖.
For each 𝑦∈𝐾, we can choose 0<𝑡<1 such that 𝑡𝑦+(1−𝑡)𝑥_𝜖∈𝐾_𝜖. From (35) and the fact that 𝑡𝑦+(1−𝑡)𝑥_𝜖∈𝐾_𝜖, we have 𝑇𝑥_𝜖,𝜂𝑡𝑦+(1−𝑡)𝑥_𝜖,𝑥_𝜖+𝑓𝑡𝑦+(1−𝑡)𝑥_𝜖−𝑓𝑥_𝜖≥0. (38) By the above inequality and the convexity of 𝑓 and mapping in (b), we get 𝑡𝑇𝑥_𝜖,𝜂𝑦,𝑥_𝜖+𝑡𝑓𝑦−𝑡𝑓𝑥_𝜖=𝑡𝑇𝑥_𝜖,𝜂𝑦,𝑥_𝜖+(1−𝑡)𝑇𝑥_𝜖,𝜂𝑥_𝜖,𝑥_𝜖+𝑡𝑓𝑦+(1−𝑡)𝑓𝑥_𝜖−𝑓𝑥_𝜖≥𝑇𝑥_𝜖,𝜂𝑡𝑦+(1−𝑡)𝑥_𝜖,𝑥_𝜖+𝑓𝑡𝑦+(1−𝑡)𝑥_𝜖−𝑓𝑥_𝜖≥0, (39) for all 𝑦∈𝐾. This implies that 𝑇𝑥_𝜖,𝜂𝑦,𝑥_𝜖+𝑓𝑦−𝑓𝑥_𝜖≥0, (40) for all 𝑦∈𝐾. Therefore, 𝑥_𝜖∈𝐾 is a solution of the problem (9). This completes the proof.

It is easy to see that the relaxed 𝜂-𝛼 monotonicity implies the weakly relaxed 𝜂-𝛼 monotonicity. So, Theorem 15 can be deduced to the following corollary.

Corollary 16. Let 𝐾 be a nonempty unbounded closed convex subset of a real reflexive Banach space 𝐸, and let 𝐸^∗ be the dual space of 𝐸. Suppose that 𝑇∶𝐾→𝐸^∗ is an 𝜂-hemicontinuous and relaxed 𝜂-𝛼 monotone mapping and 𝑓∶𝐾→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold: (a)  𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐾, (b)  for any fixed 𝑦,𝑧∈𝐾, the mapping 𝑥↦⟨𝑇𝑧,𝜂(𝑥,𝑦)⟩ is convex and lower semicontinuous function, (c)  𝛼 is weakly lower semicontinuous, (d)  𝑇 is weakly 𝜂-coercive with respect to 𝑓,
then the problem (9) is solvable.

Since the 𝜂-coercivity with respect to 𝑓 implies that the weakly 𝜂-coercivity with respect to 𝑓, we can utilize Corollary 16 to the result of Fang and Huang [16].

Corollary 17 (Theorem 2.3 [16]). Let 𝐾 be a nonempty unbounded closed convex subset of a real reflexive Banach space 𝐸, and let 𝐸^∗ be the dual space of 𝐸. Suppose that 𝑇∶𝐾→𝐸^∗ is an 𝜂-hemicontinuous and relaxed 𝜂-𝛼 monotone mapping and 𝑓∶𝐾→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold: (a)  𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐾, (b)  for any fixed 𝑦,𝑧∈𝐾, the mapping 𝑥↦⟨𝑇𝑧,𝜂(𝑥,𝑦)⟩ is convex and lower semicontinuous function, (c)  𝛼 is weakly lower semicontinuous, (d)  𝑇 is 𝜂-coercive with respect to 𝑓,
then the problem (9) is solvable.

Remark 18. Theorems 11, 12, and 15 are improvement of the results of Fang and Huang [16] from the corresponding results of variational-like inequality problems for relaxed 𝜂-𝛼 monotone mapping to weakly relaxed 𝜂-𝛼 monotone mapping. Also, these results are extension of the known results of Hartman and Stampacchia [22] and corresponding results in [14, 19, 23].

4. Variational-Like Inequalities Problems with Weakly Relaxed 𝜂-𝛼 Semimonotone Mapping

Through this section, let 𝐸 be an arbitrary Banach space with its dual space 𝐸^∗, let 𝐸^∗∗ denote the dual space of 𝐸^∗, and let 𝐾 be a nonempty closed convex subset of 𝐸^∗∗.

Definition 19. A mapping 𝐴∶𝐾×𝐾→𝐸^∗∗ is said to be weakly relaxed 𝜂-𝛼 semimonotone if the following conditions hold: (a) for each fixed 𝑢∈𝐾, 𝐴(𝑢,⋅) is weakly relaxed 𝜂-𝛼 monotone; that is, there exist mappings 𝜂∶𝐾×𝐾→𝐸^∗ and 𝛼∶𝐸^∗∗→ℝ such thatlim_𝑡→0+𝛼(𝑡𝑥)=0,lim_𝑡→0+𝑑𝛼()𝑑𝑡𝑡𝑥=0, (41) for all 𝑡>0 and 𝑥∈𝐸^∗∗ and 𝐴(𝑢,𝑣)−𝐴(𝑢,𝑤),𝜂(𝑣,𝑤)≥𝛼(𝑣−𝑤)∀𝑣,𝑤∈𝐾; (42) (b) for each fixed 𝑣∈𝐾, 𝐴(⋅,𝑣) is completely continuous; that is, for any net {𝑢_𝛽} with 𝑢_𝛽 converges to 𝑢₀ in 𝜎(𝐸^∗∗,𝐸^∗), we have that {𝐴(𝑢_𝛽,𝑣)} converges to 𝐴(𝑢₀,𝑣) in the norm topology of 𝐸^∗.

Remark 20. We obtain that relaxed 𝜂-𝛼 semimonotonicity due to Fang and Huang [16] implies weakly relaxed 𝜂-𝛼 semimonotonicity. Therefore, the class of relaxed 𝜂-𝛼 semimonotone mappings is subclass of the class of weakly relaxed 𝜂-𝛼 semimonotone mappings.

Let 𝐴∶𝐾×𝐾→𝐸^∗∗ and 𝜂∶𝐾×𝐾→𝐸^∗ be two mappings and 𝑓∶𝐾→ℝ∪{∞} is a proper convex lower-semicontinuous function. We consider the following problem: find𝑢∈𝐾suchthat𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝑓(𝑣)−𝑓(𝑢)≥0,∀𝑣∈𝐾. (43)

Theorem 21. Let 𝐸 be a real Banach space and let 𝐾⊂𝐸^∗∗ be a nonempty bounded closed convex set. Suppose that 𝐴∶𝐾×𝐾→𝐸^∗∗ is a weakly relaxed 𝜂-𝛼 semimonotone mapping and let 𝑓∶𝐾→ℝ∪{∞} be a proper convex lower semicontinuous function. If the following conditions hold: (a)  𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐾, (b)  for any fixed 𝑦,𝑧,𝑤∈𝐾, the mapping 𝑥↦⟨𝐴(𝑧,𝑤),𝜂(𝑥,𝑦)⟩ is convex and lower semicontinuous function, (c)  for each 𝑥∈𝐾, 𝐴(𝑥,⋅)∶𝐾→𝐸^∗∗ is finite-dimensional continuous; that is, for any finite-dimensional subspace 𝐹⊂𝐸^∗∗, 𝐴(𝑥,⋅)∶𝐾∩𝐹→𝐸^∗∗ is continuous, (d)  𝛼 is convex lower semicontinuous,
then the problem (43) is solvable.

Proof. Let 𝐹⊂𝐸^∗∗ be a finite-dimensional subspace with 𝐾_𝐹∶=𝐹∩𝐾≠∅. For each 𝑤∈𝐾, consider the following problem:find𝑢₀∈𝐾_𝐹suchthat𝐴𝑤,𝑢₀,𝜂𝑣,𝑢₀+𝑓(𝑣)−𝑓𝑢₀≥0,∀𝑣∈𝐾_𝐹. (44) It follows from 𝐾_𝐹 being bounded closed and convex and 𝐴(𝑤,⋅) being continuous on 𝐾_𝐹 and weakly relaxed 𝜂-𝛼 monotone for each fixed 𝑤∈𝐾 that from Theorem 12, we obtain that problem (44) has a solution 𝑢₀∈𝐾_𝐹.
Next, define a set-valued mapping 𝑇∶𝐾_𝐹→2^𝐾𝐹 as follows: 𝑇𝑤=𝑢∈𝐾_𝐹∶𝐴(𝑤,𝑢),𝜂(𝑣,𝑢)+𝑓(𝑣)−𝑓(𝑢)≥0,∀𝑣∈𝐾_𝐹,∀𝑤∈𝐾_𝐹. (45) By Theorem 11, we get that, for each 𝑤∈𝐾_𝐹, 𝑇𝑤=𝑢∈𝐾_𝐹∶𝐴()((𝑣)(𝑢)()𝑤,𝑢,𝜂𝑣,𝑢)+𝑓−𝑓≥𝛼𝑣−𝑢,∀𝑣∈𝐾_𝐹. (46)
It is known that every convex lower-semicontinuous function in Banach spaces is weakly lower semicontinuous. Therefore, condition (b) and the proper convex lower semicontinuity of 𝑓 and 𝛼 implies that 𝑇 has nonempty bounded closed and convex values. By the complete continuity of 𝐴(⋅,𝑢), we have that 𝑇 is upper semicontinuous. Using the Kakutani-Fan-Glicksberg fixed-point theorem, we obtain that 𝑇 has a fixed-point 𝑤₀∈𝐾_𝐹 and thus 𝐴𝑤₀,𝑤₀,𝜂𝑣,𝑤₀+𝑓(𝑣)−𝑓𝑤₀≥0,∀𝑣∈𝐾_𝐹. (47)
Now, define ℱ∶=𝐹⊂𝐸^∗∗∶𝐹isfinitedimensionalwith,𝑊𝐹∩𝐾≠∅_𝐹𝐴()((𝑣)(𝑢)()∶=𝑢∈𝐾∶𝑢,𝑣,𝜂𝑣,𝑢)+𝑓−𝑓≥𝛼𝑣−𝑢,∀𝑣∈𝐾_𝐹,∀𝐹∈ℱ. (48) From (47), using Theorem 11, we obtain that 𝑊_𝐹 is nonempty and bounded. Here, we denote 𝑊_𝐹 by the 𝜎(𝐸^∗∗,𝐸^∗)-closure of 𝑊_𝐹 in 𝐸^∗∗ and thus 𝑊_𝐹 is 𝜎(𝐸^∗∗,𝐸^∗)-compact in 𝐸^∗∗.
It is known that, for any 𝐹_𝑖∈ℱ, 𝑖∈ℕ, we have 𝑊_{∩𝑖𝐹𝑖}⊂∩𝑊_𝐹𝑖. Therefore, {𝑊_𝐹∶𝐹∈ℱ} has the finite intersection property; that is, _𝐹∈ℱ𝑊_𝐹≠∅. (49)
Next, we show that, for 𝑢∈⋂_𝐹∈ℱ𝑊_𝐹, 𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝑓(𝑣)−𝑓(𝑢)≥0,∀𝑣∈𝐾. (50) Indeed, for 𝑣∈𝐾, let 𝐹∈ℱ be such that 𝑣∈𝐾_𝐹 and 𝑢∈𝐾_𝐹. Then, there exists a net {𝑢_𝛽} in 𝑊_𝐹 such that 𝑢_𝛽 converges to 𝑢 in 𝜎(𝐸^∗∗,𝐸^∗). From the definition of 𝑊_𝐹, we have 𝐴𝑢_𝛽,𝑣,𝜂𝑣,𝑢_𝛽+𝑓(𝑣)−𝑓𝑢_𝛽≥𝛼𝑣−𝑢_𝛽. (51) By the complete continuity of 𝐴(⋅,𝑣) and the proper convex lower semicontinuity of 𝑓 and 𝛼, we get 𝐴(𝑢,𝑣),𝜂(𝑣,𝑢)+𝑓(𝑣)−𝑓(𝑢)≥𝛼(𝑣−𝑢),∀𝑣∈𝐾. (52) Again, using Theorem 11, we conclude that 𝐴(𝑢,𝑣),𝜂(𝑣,𝑢)+𝑓(𝑣)−𝑓(𝑢)≥0,∀𝑣∈𝐾. (53) This implies that 𝑢 is a solution of the problem (43). This completes the proof.

Theorem 22. Let 𝐸 be a real Banach space and let 𝐾⊂𝐸^∗∗ be a nonempty unbounded closed convex set. Suppose that 𝐴∶𝐾×𝐾→𝐸^∗∗ is a weakly relaxed 𝜂-𝛼 semimonotone mapping and let 𝑓∶𝐾→ℝ∪{∞} be a proper convex lower semicontinuous function. If the following conditions hold: (a)  𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐾, (b)  for any fixed 𝑦,𝑧,𝑤∈𝐾, the mapping 𝑥↦⟨𝐴(𝑧,𝑤),𝜂(𝑥,𝑦)⟩ is convex and lower semicontinuous function, (c)  for each 𝑥∈𝐾, 𝐴(𝑥,⋅)∶𝐾→𝐸^∗ is finite-dimensional continuous, (d)  𝛼 is convex lower semicontinuous, (e)  there exists a point 𝑢₀∈𝐾 such that liminf_{‖𝑢‖→∞}[⟨𝐴(𝑢,𝑢),𝜂(𝑢,𝑢₀)⟩+𝑓(𝑢)−𝑓(𝑢₀)]>0, then the problem (43) is solvable.

Proof. For 𝜖>0, we denote by 𝐵_𝜖 the closed ball with radius 𝜖 and center at 0 in 𝐸^∗∗. By Theorem 21, the problem 𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝑓(𝑣)−𝑓(𝑢)≥0,∀𝑣∈𝐵_𝜖 (54) has a solution 𝑢_𝜖∈𝐵_𝜖.
Let 𝜖 be large enough such that 𝑢₀∈𝐵_𝜖. Therefore, 𝐴𝑢_𝜖,𝑢_𝜖,𝜂𝑢₀,𝑢_𝜖+𝑓𝑢₀−𝑓𝑢_𝜖≥0. (55) By condition (e), we get that {𝑢_𝜖} is bounded. So, we may suppose that 𝑢_𝜖 converges to 𝑢 in 𝜎(𝐸^∗∗,𝐸^∗) as 𝜖→∞. It follows from Theorem 11 that 𝐴𝑢_𝜖,𝑣,𝜂𝑣,𝑢_𝜖+𝑓(𝑣)−𝑓𝑢_𝜖≥𝛼𝑣−𝑢_𝜖,∀𝑣∈𝐾. (56) Letting 𝜖→∞, we have 𝐴(𝑢,𝑣),𝜂(𝑣,𝑢)+𝑓(𝑣)−𝑓(𝑢)≥𝛼(𝑣−𝑢),∀𝑣∈𝐾. (57) Again from Theorem 11, we get 𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝑓(𝑣)−𝑓(𝑢)≥0,∀𝑣∈𝐾. (58) This show that 𝑢∈𝐾 is a solution of the problem (43). This completes the proof.

Remark 23. Theorems 21 and 22 extend and improve Theorems 3.1 and 3.2 of Fang and Huang [16] and Theorems 2.1 to 2.6 of Chen [7].

Acknowledgments

Marwan Amin Kutbi gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. Wutiphol Sintunavarat gratefully acknowledges Thammasat University for the support of this research.