© Hindawi Publishing Corp. NONLINEAR VARIATIONAL INEQUALITIES ON REFLEXIVE BANACH SPACES AND TOPOLOGICAL VECTOR SPACES

The purpose of this paper is to introduce and study a class of 
nonlinear variational inequalities in reflexive Banach spaces and 
topological vector spaces. Based on the KKM technique, the 
solvability of this kind of nonlinear variational inequalities is 
presented. The obtained results extend, improve, and unify the 
results due to Browder, Carbone, Siddiqi, Ansari, Kazmi, Verma, 
and others.


Introduction.
It is well known that variational inequality theory has significant applications in various fields of mathematics, physics, economics, and engineering science.Siddiqi et al. [6] considered the solvability of a class of nonlinear variational inequality problems in nonempty closed convex subsets and nonempty compact convex subsets of reflexive Banach spaces and locally convex spaces, respectively.Carbone [3] extended the results of Siddiqi et al. [6].Recently, Verma [7] presented the existence and uniqueness of solutions for a class of nonlinear variational inequality problems involving a combination of operators of p-monotone and p-Lipschitz types, which generalizes a result due to Browder [1].On the other hand, Carbone [4] established the existence of solutions of a class of nonlinear variational inequality problems in nonempty convex subsets of topological vector spaces.
Inspired and motivated by research works [1,3,4,6,7], in this paper, we study the solvability of a new class of nonlinear variational inequality problems in nonempty closed convex subsets and nonempty convex subsets of reflexive Banach spaces and topological vector spaces, respectively.The obtained results extend, improve, and unify the corresponding results in [1,3,4,6,7] and others.

Preliminaries.
Throughout this paper, let R=(−∞, +∞) and R + =[0, +∞).Let X, X * be a dual system of topological vector spaces or of Banach spaces, let the bilinear form •, • be continuous, and let K be a nonempty convex set in X.Let A, B, C : K → X * , M : X * × X * × X * → X * , and h : K × K → K be mappings and let f : K → R be a convex functional.Assume that h is linear with respect to the first argument.We consider the nonlinear variational inequality problem: (2.1) ) is equivalent to the following problem: for each given w ∈ X * , find u ∈ K such that Remark 2.1.For a suitable choice of M, A, B, C, h, w, and f , problem (2.1) includes a few kinds of known variational inequalities as special cases (see [1,3,4,6,7] and the references therein).
Recall that a multivalued mapping F : X → 2 X is called the KKM mapping if, for every finite subset i=1 Fu i , where conv(A) denotes the convex hull of A. Lemma 2.2 [5].Let K be a nonempty subset of a Hausdorff topological vector space X and let F : K → 2 X be a KKM mapping.If Fx is closed in X for any x ∈ K and there exists at least a point u ∈ K such that Fu is compact, then x∈K Fx ≠ ∅.
(2.3) Lemma 2.3 [2].Let C be a nonempty convex subset of a topological vector space X.Let A ⊂ C × C and g : C → C such that the following conditions are satisfied: In the rest of this section, let (X, • ) be a Banach space, X * the topological dual space of X, and •, • the dual pair between X and X * .Definition 2.4.A mapping A : K → X * is called h − φ monotone with respect to the first argument of M : X * ×X * ×X * → X * if there exist mappings φ : R + → R + with φ(0) = 0, h : In a similar way, we can define the h − φ monotonicity of the mapping C : K → X * with respect to the third argument of M.

Main results.
Our main results are as follows.
Theorem 3.1.Let X be a reflexive Banach space, X * its dual space, and K a nonempty convex closed subset of X. Suppose that M : with respect to the first argument and satisfies h(x, x) = 0 for all x ∈ K, and f : K → R is convex functional.Assume that A, B, C : K → X * are continuous from line segments in K to the weak topology of X * such that A is h − φ monotone with respect to the first argument of M, B is h − ψ relaxed monotone with respect to the second argument of M, C is h − ω monotone with respect to the third argument of M, and Then, for each given w ∈ X * , u ∈ K is a solution of problem (2.1) if and only if u ∈ K is a solution of the following problem: find u ∈ K such that Proof.Suppose that u ∈ K is a solution of problem (2.1).Since A is h − φ monotone with respect to the first argument of M, B is h−ψ relaxed monotone with respect to the second argument of M, and C is h − ω monotone with respect to the third argument of M, it follows that Conversely, suppose that (3.2) holds.Let v be an arbitrary point in K.
Note that h is linear with respect to the first argument and h(x, x) = 0 for any x ∈ K. Thus, Since f is convex, by (3.2) and (3.4) we conclude that for all t ∈ (0, 1] and v ∈ K.In view of (3.5), we get This completes the proof.
From Theorem 3.1, we immediately obtain the following result.
Theorem 3.3.Let X, X * , K, h, and f be as in Theorem 3.1.Assume that
Proof.Let the multivalued mappings F,G : K → 2 K be defined as for all v ∈ K, respectively.Now, we claim that F is a KKM mapping on K.If not, then there exist {u i : which is a contradiction.Hence, F is a KKM mapping.
Next, we claim that Fv ⊂ Gv for all v ∈ K. Indeed, let u ∈ Fv.Then, we infer that That is, problem (2.1) has a solution u ∈ K.
Finally, we show the uniqueness of solution.Suppose that v ∈ K is another solution of problem (2.1) with v ≠ u.Then, (3.17)By (3.9) and (3.17), we infer that which is a contradiction.This completes the proof.
As a consequence of Theorem 3.5, we have the following theorem.
Remark 3.7.Theorem 3 of Siddiqi et al. [6] and Theorem 2.2 of Verma [7] are special cases of Theorem 3.6.Theorem 3.8.Let X be a topological vector space, X * the topological dual space of X, K a nonempty convex subset of X, D a nonempty compact convex set in K, and the bilinear form linear with respect to the first argument and continuous with respect to the second argument.If, for each given w ∈ X * , the following condition: is satisfied and the set is compact, then problem (2.1) has a solution in K.
where w is a given point in K. Condition (3.19) ensures that (u, u) ∈ E. For each u ∈ K, the set is closed as can be easily seen from the continuity of M, A, B, C, •, • , and h relative to the second argument and lower semicontinuity of f .Now, we assert that, for each v ∈ K, the set That is, u ∈ K is a solution of problem (2.1).This completes the proof.
From Theorem 3.8, we have the following theorem.Theorem 3.10.Let X, X * , •, • , A, B, C, M, f , and h be as in Theorem 3.8, where K is a nonempty compact convex set in X.If (3.19) holds, then problem (2.1) has a solution in K. Remark 3.11.If X, X * is a dual system of locally convex spaces, Theorem 3.10 reduces to a result which extends [3, Theorem 1] and [6,Theorem 4].

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:
Thus, G is a KKM mapping.It follows from Theorem 3.1 that v∈K Fv = v∈K Gv. (3.13)Since f is lower semicontinuous and h, ϕ, ψ, and ω are continuous, therefore Gv is closed for any v ∈ K.Note that K is bounded closed convex.It is clear that K is weakly compact set in X.Hence, Gv is weakly compact set in K since Gv ⊂ K for each v ∈ K. Lemma 2.2 and (3.13) ensure that 23) is convex.Let x, y ∈ E v and t, s > 0 with t + s = 1.Then, we know that Since K is convex, by (3.25) we obtain that tx + sy ∈ E v .It follows from Lemma 2.3 that there exists u ∈ K such that K ×{u} ⊂ E. This means that Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; edleonel@rc.unesp.brAlexander Loskutov, Physics Faculty, Moscow State University, Vorob'evy Gory, Moscow 119992, Russia; loskutov@chaos.phys.msu.ru