Multivalued Variational Inequalities with 𝐷_𝐽-Pseudomonotone Mappings in Reflexive Banach Spaces

Abstract

This paper is concerned with the study of a class of variational inequalities with multivalued 𝐷_𝐽-pseudomonotone mappings in reflexive Banach spaces by using the 𝐷_𝐽-antiresolvent technique. An application to the multivalued nonlinear 𝐷_𝐽-complementarity problem is also presented. The results coincide with the corresponding results announced by many others for the gradient state.

1. Introduction and Preliminaries

Variational inequalities give a convenient mathematical framework for discussing a large variety of interesting problems appearing in pure and applied sciences. It is well known that the theory of pseudomonotone mappings plays an important part in the study of the above-mentioned variational inequalities.

In recent years, pseudomonotone theory has become an attractive field for many mathematicians (see [1–8]).

In a very recent paper [9], by using the 𝐷_𝐽-antiresolvent technique (where 𝐽 is the duality mapping) devised by the first author, the author introduced a new concept of monotonicity, which is called the 𝐷_𝐽-pseudomonotone type.

In the present paper, the concept of multivalued 𝐷_𝐽-pseudomonotone mappings in reflexive Banach spaces is used to study a wide class of variational inequalities, called the multivalued 𝐷_𝐽-pseudomonotone variational inequalities.

Moreover, the results obtained in this paper can be applied to the multivalued nonlinear 𝐷_𝐽-complementarity problem. This problem contains, in particular, a mathematical model arising in the study of the postcritical equilibrium state of a thin plate resting, without friction on a flat rigid support (see [10–12]). The results coincide with the corresponding results (see [2, 13–15]) in the case of gradient mappings.

Unless otherwise stated, 𝑉 stands for a real reflexive Banach space with norm ‖⋅‖_𝑉 and 𝑉^⋆ stands for the uniformly convex dual of 𝑉 with the dual norm ‖⋅‖_𝑉^⋆. The duality pairing between 𝑉 and 𝑉^⋆ is denoted by ⟨⋅,⋅⟩. The set of all nonnegative integers is denoted by ℕ. The field of real (resp., positive real) numbers is denoted by ℝ (resp., ℝ⁺). Notation “→” stands for strong convergence and “⇀” for weak convergence.

A mapping 𝐽∶𝑉→𝑉^⋆ is said to be a duality mapping (see, e.g., [16]) with gauge function Φ (i.e., Φ is continuous strictly increasing real-valued function satisfying Φ(0)=0 and lim_𝑡→+∞Φ(𝑡)=+∞) if for every 𝑢∈𝑉,⟨𝐽𝑢,𝑢⟩=‖𝐽𝑢‖_𝑉^⋆‖𝑢‖_𝑉=Φ(‖𝑢‖_𝑉)(‖𝑢‖_𝑉). If 𝑉=𝐻 is a Hilbert space, then 𝐽≡𝐼, the identity mapping.

Assume that 𝑉^⋆ has a weakly sequentially continuous duality mapping 𝐽 (i.e., if {𝑢_𝑛}_𝑛∈ℕ is a sequence in 𝑉 which weakly convergent to a point 𝑢, then the sequence {𝐽(𝑢_𝑛)}_𝑛∈ℕ converges to 𝐽(𝑢) (see, e.g., [17])).

Let 𝑔∶𝑉→ℝ∪{+∞} be a function. The domain of 𝑔 is dom𝑔={𝑢∈𝑉∶𝑔(𝑢)<+∞}. When dom𝑔≠𝜙, 𝑔 is called proper (see, e.g., [18]). The interior of the domain of 𝑔 is denoted by intdom𝑔. The function 𝑔 is said to be Gâteaux differentiable at 𝑢∈intdom𝑔 (see, e.g., [18]), if 𝑔^′𝑢,𝜂=lim_𝑡→0𝑔𝑢+𝑡𝜂−𝑔(𝑢)𝑡 (1) exists for all 𝜂∈𝑉.

Let 𝑔 be proper, convex, lower semicontinuous, and Gâteaux differentiable at 𝑢∈intdom𝑔; then the gradient of 𝑔 at 𝑢 is the function ∇𝑔(𝑢) which is defined by ⟨∇𝑔(𝑢),𝜂⟩=𝑔^′(𝑢,𝜂) for any 𝜂∈𝑉. It is known (see, e.g., [19]) that the conjugate 𝑔^⋆∶𝑉^⋆→ℝ∪{+∞} is also proper, convex, and lower semicontinuous.

The convex function 𝑔 is said to be of Legendre type (see, e.g., [20]) if the following conditions hold: (𝐿₁)intdom(𝑔)≠𝜙, 𝑔 is Gâteaux differentiable on intdom(𝑔) and dom∇𝑔=intdom𝑔; (𝐿₂)intdom(𝑔^⋆)≠𝜙, 𝑔^⋆ is Gâteaux differentiable on intdom(𝑔^⋆) and dom∇𝑔^⋆=intdom(𝑔^⋆).

It is well known (see, e.g., [21]) that if 𝑔 is a proper, convex, lower semicontinuous, and Legendre type, then ∇𝑔^⋆=(∇𝑔)⁻¹ and range ∇𝑔^⋆=dom∇𝑔.

Throughout this paper, the function 𝑔∶𝑉→ℝ⁺∪{+∞} is proper, convex, and lower semicontinuous which is also Legendre on intdom(𝑔).

The Bregman distance (see, e.g., [22]) is the function 𝐷_𝑔∶𝑉×intdom(𝑔)→ℝ∪{+∞} defined by 𝐷_𝑔(𝜈,𝑢)=𝑔(𝜈)−𝑔(𝑢)−∇𝑔(𝑢),𝜈−𝑢, (2) with dom𝐷_𝑔=dom𝑔×intdom𝑔. (3) It should be pointed out that if 𝑉=𝐻 is a Hilbert space and 𝑔=(1/2)‖⋅‖²_𝐻, then ∇𝑔=𝐼 (the identity mapping) and 𝐷_𝐽(𝜈,𝑢)=(1/2)‖𝜈−𝑢‖²_𝐻.

For a multivalued mapping 𝑇∶𝑉→2^𝑉⋆, the associated 𝐷_𝐽-antiresolvent (where 𝐽∶𝑉→𝑉^⋆ is the duality mapping) of 𝐽−𝑇 (see [9]) is the mapping 𝑇^𝐽∶𝑉→2^𝑉, defined by 𝑇^𝐽=∇𝑔^⋆(𝐽−𝑇). (4) Such a mapping is known as (see [23]) a 𝐷_𝑔-antiresolvent mapping of 𝑇 when 𝐽=∇𝑔 (in this case, the mapping 𝑇^𝐽 is denoted by 𝑇^𝑔).

In light of the above-mentioned discussion, we note that if 𝐽−𝑇=∇𝑔, then 𝑇^𝐽 is the identity mapping 𝐼.

Following [9], the mapping 𝐽−𝑇∶𝑉→2^𝑉⋆ is said to be 𝐷_𝐽-pseudomonotone, if for every 𝑢,𝜂∈dom𝐽∩dom𝑇,𝜈∈(𝐽−𝑇)(𝑢),𝜔∈(𝐽−𝑇)(𝜂), and every sequence {𝑢_𝑛}_𝑛∈ℕ⊂dom𝐽∩dom𝑇 and 𝜈_𝑛∈(𝐽−𝑇)(𝑢_𝑛) the conditions ∇𝑔^⋆𝜈_𝑛⇀∇𝑔^⋆(𝜈),limsup_𝑛→∞𝜈_𝑛,∇𝑔^⋆𝜈_𝑛−∇𝑔^⋆(𝜈)≤0 (5) imply that liminf_𝑛→∞𝜈_𝑛,∇𝑔^⋆𝜈_𝑛−∇𝑔^⋆(𝜔)≥𝜔,∇𝑔^⋆(𝜈)−∇𝑔^⋆(𝜔). (6) As remarked in [9], the 𝐷_𝐽-pseudomonotonicity of the mapping 𝐽−𝑇 coincides with the pseudomonotonicity (or the 𝐷_𝑔-pseudomonotonicity in the sense of Bregman distance 𝐷_𝑔) of the mapping ∇𝑔, if 𝐽−𝑇=∇𝑔.

The multivalued variational inequality defined by the 𝐷_𝐽-mapping (or multivalued 𝐷_𝐽-variational inequality) 𝐽−𝑇∶𝑉→2^𝑉⋆ and the set 𝐾⊂𝑉 is to find 𝑢∈𝐾 such that (𝑢)∈()𝜂∶∃𝜈∈𝐽−𝑇)(,𝜔𝐽−𝑇𝜈−𝑓,∇𝑔^⋆(𝜔)−∇𝑔^⋆(𝜈)≥0,∀𝜂∈𝐾, (7) where 𝑓∈2^𝑉⋆.

The multivalued nonlinear complementarity problem defined by the 𝐷_𝐽-mapping (or multivalued nonlinear 𝐷_𝐽-complementarity problem) 𝐽−𝑇∶𝑉→2^𝑉⋆ and the set 𝐾 is to find 𝑢∈𝐾 such that 𝜈−𝑓,∇𝑔^⋆(𝜈)=0,𝜈−𝑓,∇𝑔^⋆(𝜔)−∇𝑔^⋆(𝜈)≥0,∀𝜂∈𝐾, (8) where 𝑓∈2^𝑉⋆, 𝜈∈(𝐽−𝑇)(𝑢), and 𝜔∈(𝐽−𝑇)(𝜂).

The multivalued 𝐷_𝐽-variational inequality and multivalued nonlinear 𝐷_𝐽-complementarity problem are very general in the sense that they include, as special cases, multivalued variational inequality and multivalued nonlinear complementarity problem.

The following definition and results will be used in the sequel.

Definition 1 (see, e.g., [15, p. 84]). The mapping 𝐴∶𝑉→2^𝑉⋆ is continuous on finite dimensional subspaces if for any finite dimensional subspace 𝑉₀⊂𝑉, the restriction of 𝐴 to 𝑉₀∩dom(𝐴) is weakly continuous.

Corollary 2 (see [24]). Let 𝑗∶𝑉₀⊂𝑉→𝑉 be the injection mapping. Let 𝑗^⋆∶𝑉^⋆→𝑉^⋆₀ be its dual mapping. Then, 𝑗^⋆𝐴𝑗∶dom(𝐴)∩𝑗𝑉₀→2^𝑉⋆0 is continuous.

Corollary 3 (see [25]). Let 𝐾 be a nonempty compact convex set of ℝ^𝑛 and let 𝑆∶𝐾→2^𝐾 be continuous. Then 𝑆 admits a fixed point.

2. Main Results

Theorem 4. Let 𝐾 be a closed convex set in 𝐻 and let 𝑇∶𝐾→2^𝐻 be a multivalued mapping. Then the following are equivalent: (1)∇𝑔^⋆𝜈^′∈𝑃𝑟_𝐾(𝑥)=argmin𝐷_𝐼∇𝑔^⋆𝜔^′=1,𝑥2∇𝑔^⋆(𝜔^′)−𝑥²_𝐻∶∇𝑔^⋆𝜔^′∈𝐾,𝜔^′∈(𝐼−𝑇)𝜂,𝜂∈𝐾, (9) the multivalued projection for 𝐾; (2)∇𝑔^⋆𝜈^′∈𝐾∶∇𝑔^⋆𝜈^′−𝑥,∇𝑔^⋆𝜔^′−∇𝑔^⋆𝜈^′≥0∀∇𝑔^⋆𝜔^′∈𝐾,where𝜔^′∈()𝜂,𝜈𝐼−𝑇_′∈(𝐼−𝑇)𝑦,𝑦,𝜂∈𝐾. (10)

Proof. Assume that (1) holds. Let 𝑥∈2^𝐻 and ∇𝑔^⋆(𝜈^′)∈𝑃𝑟_𝐾(𝑥)⊂𝐾. For every ∇𝑔^⋆(𝜔^′)∈𝐾 and 𝑡∈(0,1], we have 𝐷_𝐼𝑥,∇𝑔^⋆𝜈^′≤𝐷_𝐼𝑥,(1−𝑡)∇𝑔^⋆𝜈^′+𝑡∇𝑔^⋆𝜔^′=𝐷_𝐼𝑥,∇𝑔^⋆𝜈^′−𝑡𝑥−∇𝑔^⋆𝜈^′,∇𝑔^⋆𝜔^′−∇𝑔^⋆𝜈^′+𝑡²𝐷_𝐼∇𝑔^⋆𝜔^′,∇𝑔^⋆𝜈^′. (11) This implies 𝑥−∇𝑔^⋆𝜈^′,∇𝑔^⋆𝜔^′−∇𝑔^⋆𝜈^′≤𝑡𝐷_𝐼∇𝑔^⋆𝜔^′,∇𝑔^⋆𝜈^′. (12) Hence, 𝑡→0⁺ implies (2).
On the other hand, assume that (2) holds. For every ∇𝑔^⋆(𝜔^′)∈𝐾, we have 𝐷_𝐼∇𝑔^⋆𝜔^′,𝑥=𝐷_𝐼∇𝑔^⋆𝜔^′,∇𝑔^⋆𝜈^′+∇𝑔^⋆𝜔^′−𝑥,∇𝑔^⋆𝜔^′−∇𝑔^⋆𝜈^′+𝐷_𝐼∇𝑔^⋆𝜈^′,𝑥≥𝐷_𝐼∇𝑔^⋆𝜈^′,𝑥. (13) This implies (1).

Remark 5. In the particular situation when 𝐼−𝑇=∇𝑔 Theorem 4 coincides (in gradient setting) with Theorem 2.3 in [15] and also with Proposition 2.1 (1) and (2) in [2].

Theorem 6. In addition to conditions on 𝑉,𝑉^⋆,𝑔^⋆, and 𝐽, one assumes that 𝑉 is separable, 𝐾⊂𝑉 is nonempty closed and convex, 𝑇∶𝐾→2^𝑉⋆,∇𝑔^⋆ are weakly continuous mappings, and either 𝑇 or ∇𝑔^⋆ is continuous. Moreover, assume that the mapping 𝐽−𝑇∶𝐾→2^𝑉⋆ is a bounded 𝐷_𝐽-pseudomonotone mapping and that, for each 𝜂∈𝐾, there exist 𝜂₀∈𝐾,𝜔₀∈(𝐽−𝑇)(𝜂₀),𝜔∈(𝐽−𝑇)(𝜂), and 𝑟>0 such that 𝜔,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜔₀>𝑓,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜔₀,𝑓∈2^𝑉⋆,∇𝑔^⋆(𝜔)_𝑉≥𝑟. (14) Then there exists a solution to the multivalued 𝐷_𝐽-variational inequality (7).

Proof. Suppose that {𝑥₁,𝑥₂,𝑥₃,…} is an infinite dense set in 𝐾 and 𝑉_𝑚,𝑚∈ℕ, is the linear span of {𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑚}.
Let 𝐾_𝑚=con{𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑚}={∑^𝑚_𝑖=1𝜆_𝑖𝑥_𝑖,𝜆_𝑖≥0,∑^𝑚_𝑖=1𝜆_𝑖=1}. Let 𝑗_𝑚∶𝑉_𝑚→𝑉 be the injection mapping and let 𝑗^⋆_𝑚∶𝑉^⋆→𝑉^⋆_𝑚 be its restriction dual. Observe that ∪𝐾_𝑚 is dense in 𝐾, for 𝑚∈ℕ.
Now, fix an integer 𝑚≥1 and consider the finite dimensional problem.
Find 𝑢_𝑚∈𝐾_𝑚 such that for each 𝜂∈𝐾_𝑚, there exist 𝑠_𝑚∈𝑗^⋆_𝑚(𝐽−𝑇)𝑗_𝑚(𝑢_𝑚),𝜈_𝑚∈(𝐽−𝑇)(𝑢_𝑚),  and  𝜔∈(𝐽−𝑇)(𝜂) such that 𝑠_𝑚−𝑗^⋆_𝑚𝑓,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜈_𝑚≥0. (15) The equivalent form of problem (15) is to find 𝑢_𝑚∈𝐾_𝑚 such that for each 𝜂∈𝐾_𝑚, there exist 𝑠_𝑚∈𝑗^⋆_𝑚(𝐽−𝑇)𝑗_𝑚(𝑢_𝑚),𝜈_𝑚∈(𝐽−𝑇)(𝑢_𝑚),   and  𝜔∈(𝐽−𝑇)(𝜂) such that ∇𝑔^⋆𝜈_𝑚,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜈_𝑚≥∇𝑔^⋆𝜈_𝑚+𝑗^⋆_𝑚𝑓−𝑠_𝑚,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜈_𝑚. (16) Using the identification of 𝑉_𝑚 with ℝ^𝑚 and 𝑉^⋆_𝑚 and Theorem 4 (with ℝ^𝑚=𝐻and𝐽=𝐼), we see that (16) is equivalent to ∇𝑔^⋆(𝜈_𝑚)∈𝑃𝑟_𝐾𝑚(∇𝑔^⋆(𝜈_𝑚)+𝑗^⋆_𝑚𝑓−𝑠_𝑚).
Let 𝐵_𝑟(0) be any closed ball containing 𝐾_𝑚. It is well known (see, e.g., [26, p. 54, 224]) that 𝐵_𝑟(0) is compact and convex in 𝐾_𝑚; thus it is weakly closed.
From Corollary 2, 𝑗^⋆(𝐽−𝑇)𝑗∶𝐵_𝑟(0)→2^𝐵𝑟(0) is continuous; hence the function ∇𝑔^⋆(𝜈)↦𝑃𝑟_𝐵𝑟(0)(∇𝑔^⋆(𝜈)+𝑗^⋆_𝑚𝑓−𝑠_𝑚) is continuous from 𝐵_𝑟(0) into 2^𝐵𝑟(0).
Hence, by Corollary 3, this equation admits a solution. If the closed convex set 𝐾 is assumed to be bounded, then by the reflexivity of 𝑉 it is weakly compact (by employing the Banach -Alaoglu theorem (see, e.g., [16, p. 3]).
Then we have a subsequence denoted by {𝑢_𝑚}_𝑚∈ℕ such that 𝑢_𝑚⇀𝑢∈𝐾. Since 𝐽−𝑇 is bounded, we have ‖𝜈_𝑚‖_𝑉^⋆≤𝑀for all 𝑚∈ℕ. Since 𝐽−𝑇 is weakly continuous and since either 𝐽−𝑇 or ∇𝑔^⋆ is continuous by hypothesis, it follows that 𝑇^𝐽 is weakly continuous by [27, Lemma 1]. So, we have ∇𝑔^⋆(𝜈_𝑚)⇀∇𝑔^⋆(𝜈).
Now, we prove that limsup_𝑚→∞𝜈_𝑚,∇𝑔^⋆𝜈_𝑚−∇𝑔^⋆(𝜈)≤0. (17) For any 𝜖>0, choose 𝑁 so large and 𝑢^∼∈𝐾_𝑁 such that ∇𝑔^⋆(𝜈)−∇𝑔𝜈^⋆∼_𝑉<𝜖for𝜈^∼∈(𝐽−𝑇)𝑢^∼. (18) Therefore, we have 𝜈_𝑚−𝑓,∇𝑔^⋆𝜈_𝑚−∇𝑔𝜈^⋆∼≤0for𝑚≥𝑁. (19) Since 𝐾_𝑁⊂𝐾_𝑚, we have limsup_𝑚→∞𝜈_𝑚,∇𝑔^⋆𝜈_𝑚−∇𝑔^⋆(𝜈)=limsup_𝑚→∞𝜈_𝑚,∇𝑔^⋆𝜈_𝑚−∇𝑔𝜈^⋆∼+𝜈_𝑚,∇𝑔𝜈^⋆∼−∇𝑔^⋆(𝜈≤𝑓_𝑉⋆)+𝑀𝜖. (20) Since 𝜖 is arbitrary, this shows the desired inequality.
By the 𝐷_𝐽-pseudomonotonicity of 𝐽−𝑇, it follows that liminf_𝑚→∞𝜈_𝑚,∇𝑔^⋆𝜈_𝑚−∇𝑔^⋆(𝜔)≥𝜈,∇𝑔^⋆(𝜈)−𝑔^⋆(𝜔) (21) for all 𝜂∈dom𝐽∩dom𝑇 and 𝜔∈(𝐽−𝑇)(𝜂).
If 𝜂∈𝐾_𝑛,𝑚≥𝑛, we have 𝜈_𝑚,∇𝑔^⋆𝜈_𝑚−∇𝑔^⋆(𝜔)≤𝑓,∇𝑔^⋆𝜈_𝑚−∇𝑔^⋆(𝜔). (22) Hence 𝜈,∇𝑔^⋆(𝜈)−∇𝑔^⋆(𝜔)≤𝑓,∇𝑔^⋆(𝜈)−∇𝑔^⋆(𝜔) (23) for every 𝜂 in 𝐾_𝑛,𝑛∈ℕ,𝜔∈(𝐽−𝑇)(𝜂).
Since ∪_𝑛𝐾_𝑛 is dense in 𝐾_𝑛, so we have that 𝑢 is a solution to (7).
Now, to complete the proof, we consider the case when 𝐾 is unbounded.
In this case we consider the set 𝐾_𝜌={𝜂∈𝐾∶‖∇𝑔^⋆(𝜔)‖_𝑉≤𝜌}, where 𝜌=max{‖∇𝑔^⋆(𝜔₀)‖_𝑉,𝑟}.
Since 𝐾_𝜌 is bounded, there exists at least one 𝑢_𝜌∈𝐾_𝜌: 𝜈_𝜌−𝑓,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜈_𝜌≥0 (24) for 𝜈_𝜌∈(𝐽−𝑇)(𝑢_𝜌) and 𝜂∈𝐾_𝜌.
Since 𝜂₀∈𝐾_𝜌, we have 𝜈_𝜌−𝑓,∇𝑔^⋆𝜈_𝜌−∇𝑔^⋆𝜔₀≤0for𝜔₀∈(𝐽−𝑇)𝜂₀. (25) This, together with (14), implies that ‖∇𝑔^⋆(𝜈_𝜌)‖_𝑉<𝜌.
To clarify that 𝑢_𝜌 is also a solution to original problem on 𝐾, for any 𝜂∈𝐾, set ∇𝑔^⋆(𝜔_𝑡) = (1−𝑡)∇𝑔^⋆(𝜈_𝜌)+𝑡∇𝑔^⋆(𝜔) for 𝑡>0 is sufficiently small, where 𝜔_𝑡∈(𝐽−𝑇)(𝜂_𝑡) and 𝜂_𝑡∈𝐾_𝜌. Consequently 𝑢_𝜌∈𝐾_𝜌⊂𝐾∶0≤𝜈_𝜌−𝑓,∇𝑔^⋆𝜔_𝑡−∇𝑔^⋆𝜈_𝜌=𝑡𝜈_𝜌−𝑓,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜈_𝜌for𝜂∈𝐾. (26) This completes the proof.

Remark 7. In the particular situation when 𝐽−𝑇=∇𝑔, Theorem 6 coincides with the Brezis Theorem (see, e.g., [13, 14]) for the case of gradient mapping.
We are now in a position to state and prove the following theorem.

Theorem 8. Let all assumptions of Theorem 6 hold, except for condition (14) let it be replaced by the 𝐷_𝐽-coercive condition: for 𝜔∈(𝐽−𝑇)(𝜂), lim_{‖∇𝑔⋆(𝜔)‖𝑉→∞}𝜔,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜔₀∇𝑔^⋆(𝜔)_𝑉𝜔=+∞,₀∈(𝐽−𝑇)𝜂₀,𝜂,𝜂₀∈𝐾. (27) Suppose further that 𝐾 has the following property (W): 𝛼∇𝑔^⋆(𝜔)∈𝐾 for all ∇𝑔^⋆(𝜔)∈𝐾 and 𝛼≥0.
Then for every 𝑓∈2^𝑉⋆ there exist 𝑢∈𝐾, 𝜈∈(𝐽−𝑇)(𝑢) such that 𝜈−𝑓,∇𝑔^⋆(𝜈)=0,𝜈−𝑓,∇𝑔^⋆(𝜔)−∇𝑔^⋆(𝜈)≥0 (28) for all 𝜂∈𝐾,𝜔∈(𝐽−𝑇)(𝜂).

Proof. Let 𝑓∈2^𝑉⋆ satisfy ‖𝑓‖_𝑉^⋆<𝑀 and ∇𝑔^⋆𝜔_0𝑉<2𝑀−𝑓_𝑉⋆𝑓_𝑉⋆𝜌. (29) The 𝐷_𝐽-coercivity of 𝐽−𝑇 implies that there exists 𝜌>0 such that 𝜔,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜔₀≥2𝑀∇𝑔^⋆(𝜔)_𝑉 (30) for 𝜔∈(𝐽−𝑇)(𝜂),𝜂∈𝐾 with ‖∇𝑔^⋆(𝜔)‖_𝑉≥𝜌.
So we conclude 𝜔−𝑓,∇𝑔^⋆(𝜔)−∇𝑔^⋆𝜔₀≥2𝑀∇𝑔^⋆(𝜔)_𝑉−𝑓_𝑉⋆∇𝑔^⋆(𝜔)_𝑉−𝑓_𝑉⋆∇𝑔^⋆𝜔_0𝑉≥2𝑀−𝑓_𝑉⋆𝜌−𝑓_𝑉⋆∇𝑔^⋆𝜔_0𝑉>0for𝜂∈𝐾,𝜔∈(𝐽−𝑇)𝜂𝑤𝑖𝑡ℎ∇𝑔^⋆(𝜔)_𝑉≥𝜌. (31) The second part of (28) thus follows from Theorem 6.
To prove the first part of (28), observe that we can choose a point 𝜂 in 𝐾 and 𝜔∈(𝐽−𝑇)(𝜂) and assume that ∇𝑔^⋆(𝜔)=0.
Therefore, from Theorem 6, we have 𝜈−𝑓,∇𝑔^⋆(𝜈)≤0 (32) for all 𝜈∈(𝐽−𝑇)(𝑢),𝑢∈𝐾.
On the other hand, setting ∇𝑔^⋆(𝜔)=𝛼∇𝑔^⋆(𝜈), where 𝛼>1, we get 0≤𝜈−𝑓,∇𝑔^⋆(𝜔)−∇𝑔^⋆(𝜈)=(𝛼−1)𝜈−𝑓,∇𝑔^⋆(𝜈). (33) This implies 𝜈−𝑓,∇𝑔^⋆(𝜈)≥0. (34) So, 𝜈−𝑓,∇𝑔^⋆(𝜈)=0. (35) This completes the proof of (28).

The following proposition gives a characterization of the sum of two 𝐷_𝐽-Pseudomonotone mappings.

Proposition 9. Let 𝑉,𝑉^⋆, and 𝐽 be as above and let 𝑇_𝑖∶𝑉→2^𝑉⋆,𝑖=1,2,and∇𝑔^⋆ be weakly continuous mappings. If 𝐽−𝑇_𝑖∶𝑉→2^𝑉⋆,𝑖=1,2, are 𝐷_𝐽-pseudomonotone mappings such that dom𝐽∩dom𝑇_𝑖≠𝜙,𝑖=1,2, then ∑²_𝑖=1(𝐽−𝑇_𝑖) is 𝐷_𝐽-pseudomonotone.

Proof. Let 𝑦^∼_𝑛∈∑²_𝑖=1(𝐽−𝑇_𝑖)(𝜂_𝑛),𝑦^∼∈∑²_𝑖=1(𝐽−𝑇_𝑖)(𝜂),𝜂_𝑛,𝜂∈dom𝐽∩dom𝑇_𝑖,𝑖=1,2, with ∇𝑔^⋆(𝑦^∼_𝑛)⇀∇𝑔^⋆(𝑦^∼) and limsup𝑦_𝑛→∞^∼_𝑛,∇𝑔𝑦^⋆∼_𝑛−∇𝑔𝑦^⋆∼≤0. (36) Now, we prove for 𝑦_𝑛^(𝑖)∈(𝐽−𝑇_𝑖)(𝜂_𝑛),𝑦^(𝑖)∈(𝐽−𝑇_𝑖)(𝜂),𝑖=1,2, that limsup_𝑛→∞𝑦_𝑛⁽¹⁾,∇𝑔^⋆𝑦_𝑛⁽¹⁾−∇𝑔^⋆𝑦⁽¹⁾≤0,limsup_𝑛→∞𝑦_𝑛⁽²⁾,∇𝑔^⋆𝑦_𝑛⁽²⁾−∇𝑔^⋆𝑦⁽²⁾≤0. (37) If limsup_𝑛→∞𝑦_𝑛⁽²⁾,∇𝑔^⋆𝑦_𝑛⁽²⁾−∇𝑔^⋆𝑦⁽²⁾=𝜖>0, (38) (note that otherwise, by symmetry), then there exists a subsequence {𝑦_𝑛⁽²⁾_𝑘}_𝑘∈ℕ⊂{𝑦_𝑛⁽²⁾}_𝑛∈ℕ such that limsup_𝑘→∞𝑦_𝑛⁽²⁾_𝑘,∇𝑔^⋆𝑦_𝑛⁽²⁾_𝑘−∇𝑔^⋆𝑦⁽²⁾=𝜖. (39) This implies that limsup_𝑘→∞𝑦_𝑛⁽¹⁾_𝑘,∇𝑔^⋆𝑦_𝑛⁽¹⁾_𝑘−∇𝑔^⋆𝑦⁽¹⁾=limsup𝑦_𝑘→∞^∼_𝑛𝑘,∇𝑔𝑦^⋆∼_𝑛𝑘−∇𝑔𝑦^⋆∼−𝑦_𝑛⁽²⁾_𝑘,∇𝑔^⋆𝑦_𝑛⁽²⁾_𝑘−∇𝑔^⋆𝑦⁽²⁾≤0−𝜖. (40) From the 𝐷_𝐽-pseudomonotonicity of 𝐽−𝑇₁, we get for all 𝑦^′∈(𝐽−𝑇₁)(𝜂^′),𝜂^′∈dom𝐽∩dom𝑇₁𝑦⁽¹⁾,∇𝑔^⋆𝑦⁽¹⁾−∇𝑔^⋆𝑦^′≤liminf_𝑘→∞𝑦_𝑛⁽¹⁾_𝑘,∇𝑔^⋆𝑦_𝑛⁽¹⁾_𝑘−∇𝑔^⋆𝑦^′. (41) Letting 𝑦^′=𝑦⁽¹⁾, we obtain 0≤liminf_𝑘→∞𝑦_𝑛⁽¹⁾_𝑘,∇𝑔^⋆𝑦_𝑛⁽¹⁾_𝑘−∇𝑔^⋆𝑦⁽¹⁾≤0−𝜖, (42) a contradiction.
Hence, limsup_𝑘→∞𝑦_𝑛⁽¹⁾_𝑘,∇𝑔^⋆𝑦_𝑛⁽¹⁾_𝑘−∇𝑔^⋆𝑦⁽¹⁾≤0,limsup_𝑘→∞𝑦_𝑛⁽²⁾_𝑘,∇𝑔^⋆𝑦_𝑛⁽²⁾_𝑘−∇𝑔^⋆𝑦⁽²⁾≤0. (43) This holds for any subsequence, so (37) holds and the proof follows immediately by the superadditivity of the liminf.

3. Application to Multivalued Nonlinear 𝐷_𝐽-Complementarity Problem

As applications of Theorem 8 we consider the multivalued nonlinear 𝐷_𝐽-complementarity problem (8) with 𝑇=𝜆𝑇₁+𝑇₂−𝜆𝐽, where 𝑇_𝑖,𝑖=1,2 are two nonlinear multivalued mappings from 𝐾 to 2^𝑉⋆, and 𝜆∈(0,∞).

Theorem 10. Let 𝑉,𝑉^⋆,𝑔^⋆,𝐽, and 𝐾 be the same as in Theorem 6, and suppose that 𝐾 has the property (W). Let the mappings ∇𝑔^⋆ and 𝑇_𝑖,𝑖=1,2 be weakly continuous and let either ∇𝑔^⋆ or both 𝑇_𝑖,𝑖=1,2 be continuous. Let 𝐽−𝑇_𝑖∶𝐾→2^𝑉⋆,𝑖=1,2 be two bounded 𝐷_𝐽-pseudomonotone mappings.

Let 1𝜌₁=inf_{𝑛∈𝐾‖∇𝑔⋆𝜆𝑤1+𝑤2‖𝑉≠0}𝑤₁,∇𝑔^⋆𝜆𝑤₁+𝑤₂−∇𝑔^⋆𝜆𝑤₀⁽¹⁾+𝑤₀⁽²⁾×∇𝑔^⋆𝜆𝑤₁+𝑤₂²_𝑉⁻¹,𝑎=liminf_{𝑛∈𝐾‖∇𝑔⋆𝜆𝑤1+𝑤2‖𝑉→∞}𝑤₂,∇𝑔^⋆𝜆𝑤₁+𝑤₂−∇𝑔^⋆𝜆𝑤₀⁽¹⁾+𝑤₀⁽²⁾×∇𝑔^⋆𝜆𝑤₁+𝑤₂²_𝑉⁻¹,𝑏=liminf_𝑡→∞Φ(𝑡)𝑡 (44) Be such that 𝑎<𝑏, where 𝑤_𝑖∈(𝐽−𝑇_𝑖)(𝜂),𝑤₀^(𝑖)∈(𝐽−𝑇_𝑖)(𝜂₀),𝑖=1,2,𝜂,𝜂₀∈𝐾,𝜌₁>0 and Φ is the gauge function. Then for every 𝜆>𝜌₁(𝑏−𝑎) problem (8) with 𝑇=𝜆𝑇₁+𝑇₂−𝜆𝐽 has a solution in 𝐾.

Proof. By Proposition 9, 𝐽−𝑇=𝜆(𝐽−𝑇₁)+(𝐽−𝑇₂) is 𝐷_𝐽-pseudomonotone for every 𝜆≥0. Set 𝜆>𝜌(𝑏−𝑎). Then liminf_{𝜂∈𝐾‖∇𝑔⋆(𝜆𝑤1+𝑤2)‖𝑉→∞}𝜆𝑤₁+𝑤₂,∇𝑔^⋆𝜆𝑤₁+𝑤₂−∇𝑔^⋆𝜆𝑤₀⁽¹⁾+𝑤₀⁽²⁾×∇𝑔^⋆𝜆𝑤₁+𝑤₂²_𝑉⁻¹≥𝜆𝜌₁+𝑎>𝑏>0. (45) This implies that the mapping 𝐽−𝑇=𝜆(𝐽−𝑇₁)+(𝐽−𝑇₂) is 𝐷_𝐽-coercive.

The conclusion follows from Theorem 8.