GENERALIZED g-QUASIVARIATIONAL INEQUALITY

Suppose that X is a nonempty subset of a metric space E and Y is a nonempty subset of a topological vector space F. Let g:X→Y and ψ:X×Y→ℝ be two functions and let S:X→2Y and T:Y→2F∗ be two maps. Then the generalized g-quasivariational inequality problem (GgQVI) is to find a point x¯∈X and a point f∈T(g(x¯)) such that g(x¯)∈S(x¯) and supy∈S(x¯){Re⁡〈f,y−g(x¯)〉


Introduction and preliminaries
The quasivariational inequality has proven to be useful in different areas such as mathematical physics, nonlinear optimization, optimal control theory, and mathematical economics (see Arrow and Debreu [2], Aubin [3], Aubin and Ekeland [6], Mosco [17], and Shafer and Sonnenschein [21]).Many researchers attempted to generalize this inequality by weakening the conditions of existence of a solution.Among these researchers, we can mention Shih and Tan [22], Tian and Zhou [23,24], Zhou and Chen [26], and Nessah and Chu [19].Our work follows this direction of reseach.In this paper, we introduce the generalized g-quasivariational inequality (GgQVI) and provide sufficient conditions for the existence of its solution.
Let E be a metric space and let F be a topological vector space.Let X and Y be nonempty subsets of E and F, respectively, and let 2 X be the family of all nonempty subsets of X.We will denote by F * the continuous dual of F, by Re f , y the real part of pairing between F * and F for f ∈ F * and y ∈ F. Given the functions g : X → Y and ψ : X × Y → R and the maps S : X → 2 Y and T : Y → 2 F * , the generalized g-quasivariational inequality problem (GgQVI) is to find a point x ∈ X, g(x) ∈ S(x), and a point f ∈ T(g(x)) such that sup y∈S(x) {Re f , y − g(x) + ψ(x, y)} = ψ(x,g(x)).
Some particular cases of the (GgQVI) were introduced before: by Chan and Pang [9] in 1982 in the case where E = F = R n , g = id X , and ψ = 0, by Shih and Tan [22] in 1985 in the case where E = F is infinite dimensional, g = id X , ψ = 0, and by Chowdhury and Tarafdar [10] in the case where E = F, g = id X , and ψ = 0.
Gwinner [14], Ansari et al. [1], Ding et al. [12], and Nessah [18] introduced and studied the following nonlinear inequality problem of finding x ∈ X such that where •, • is the pairing between F * and F, in the case where E = F, X = Y , g = id X and C(x) = Y , for all x ∈ X.This problem is equivalent to the problem of solving the GgQVI, where T(y) = 0, for all y ∈ Y and ψ(x, y) = φ(x, y) ≤ Re f , y − g(x) .It is to be noted that in all the previous works, it is assumed that the function φ(x, y) is defined on the cartesian product X × X of the same set X.In contrast, in GgQVI, the function φ(x, y) is defined on the cartesian product of two different sets X × Y .This generalization opens more possibilities for applications of the quasivariational inequalities.One of the potential areas of application of the GgQVI is game theory.Indeed, the existence of some equilibria like the strong Berge equilibrium [16] requires a function φ(x, y) defined on the product of two different sets.
Let us consider the following notations.Let Y be a subset of a topological vector space.Let K be a subset of Y and x ∈ K.
(1) The tangent cone of K in x is defined by (2) The normal cone of K in x is defined by Note that A is the closure of the subset A and ∂A is its boundary.Consider X a nonempty subset of a metrical space E, Y a nonempty subset of a locally convex space F. Let 2 Y be the set of all the parts of Y .
A map A function f : Y → R is said to be upper semicontinuous if for all y 0 ∈ Y , for all λ > f (y 0 ), there is a neighborhood v of y 0 such that for all y ∈ v, λ ≥ f (y); f is said to be continuous if f and − f are upper semicontinuous.We say that f is quasiconcave if for any y 1 , y 2 in Y and for any θ ∈ [0,1], we have min{ f (y 1 ), f (y 2 )} ≤ f (θy 1 + (1 − θ)y 2 ); f is said to be quasiconvex if − f is quasiconcave.
A function f : Y → F * is said to be upper hemicontinuous along line segments in Y if for all y 1 , y 2 ∈ Y , the function z → f (z), y 2 − y 1 is upper semicontinuous on the line segment [y 1 , y 2 ].
We say that the map C : Y → 2 Y is upper hemicontinuous if for any p ∈ Y * , function x → σ(C(x), p) = sup y∈C(x) Re p, y is upper semicontinuous on Y .
We say that the map C : X → 2 E satisfies [4] (1) the tangential condition if where X is assumed to be convex, (2) the dual tangential condition if ∀x ∈ X, ∀p ∈ N X (x), then σ C(x),−p ≥ 0. (1.5) We will use the following results.
Lemma 1.2 [15].Let X be a nonempty convex subset of a vector space and let Y be a nonempty compact convex subset of a Hausdorff topological vector space.Suppose that f is a real-valued function on X × Y such that for each x ∈ X, the map y → f (x, y) is lower semicontinuous and convex on Y and for each fixed y ∈ Y , the map Lemma 1.3 [10].Let E be a topological vector space, let X be a nonempty convex subset of E, let h : X → R be convex, and let T : Lemma 1.4 [8].Let C : E → 2 F be a map, where E and F are metric spaces.If the graph of C is compact, then C is upper semicontinuous.
Lemma 1.5.Let X be a nonempty, compact set in a metric space E, let Y be a nonempty convex, compact set in a Hausdorff locally convex space F, let g be a continuous function from X into Y , and let C be an upper hemicontinuous set-valued function from X into Y , with C(x) nonempty, closed, and convex.Suppose that the following conditions are met. ( Proof.Consider the map Υ defined as follows: (1.7) Let us prove that Υ is upper hemicontinuous.Indeed, let g(x) ∈ g(X) and p ∈ Y * , we have Re p, y = sup Re p, y .(1.8) Let z = y + g(x), then we obtain y = z − g(x) and Re p,z − Re p,g(x) . (1.9) Then Since C is upper hemicontinuous and p, g are continuous functions, then we conclude that F is upper hemicontinuous.Thus, the map Υ is upper hemicontinuous with nonempty, closed, and convex values.Since g is continuous on the compact X, then Weierstrass theorem implies that g(X) is compact.Taking into account condition (2) of Lemma 1.5 and the fact that for g(x) ∈ intg(X), we have is convex in a Hausdorff locally convex space, then all the conditions of the zero-map theorem [7] are verified for Υ.From this theorem, we deduce that there exists x ∈ X such that 0 ∈ Υ(g(x)), that is, g(x) ∈ C(x).

Existence of solution
In the following theorem, we establish a sufficient condition for the existence of a solution of the GgQVI.
Theorem 2.1.Let (1) X be a nonempty compact subset of a metrical space E, (2) Y a nonempty convex and compact subset of a locally convex Hausdorff topological vector space F, (3) g : X → Y a continuous function such that g(X) is a compact and convex subset of Y , (4) S an upper hemicontinuous map from X into 2 Y with nonempty, convex, and closed values such that for any g(x) ∈ ∂g(X), [S(x) − g(x)] ∩ T g(X) (g(x)) = ∅, (5) T : Y → 2 F * an upper hemicontinuous along line segments in X with respect to the weak * -topology on F * such that each T(y) is weak * -compact convex and the function 3) for any g(x)∈∂g(X), for any y ∈ Y , and for any q ∈ F * , there is a w w) and Re q, y ≤ Re q,w , (7) V 0 the set which must be open.
Then, there exists an x ∈ X such that Proof.We divide the proof into three steps.
Step 1.There exists a point x ∈ X such that g(x) ∈ S(x) and (2.3) Suppose that (2.3) is not true.Then for each x ∈ X, either g(x) / ∈ S(x) or sup y∈S(x) According to separation theorem and considering the fact that S(x) is nonempty, convex, and closed, g(x) / ∈ S(x) implies that for all x ∈ X, there exists q ∈ F * such that where σ(S(x), q) = sup y∈S(x) Re −q, y is the support function of S(x).Let V q = x ∈ X such that Re − q,g(x) > σ S(x),−q .
The equality (2.3) implies that X ⊂ V 0 ∪ q∈F * V q .Since X is compact, it is possible to cover it by a finite number n of its subsets {V 0 ,V q1 ,...,V qn }.Let {h i } i=0,...,n be a continuous partition of unity associated with the subcover {V 0 ,V q1 ,...,V qn }.
Let us introduce the function Φ : (2.6) We now show that there is an x ∈ X such that sup y∈Y Φ(x, y) = Φ x,g(x) . (2.7) Assume that ∀x ∈ X, ∃y ∈ Y such that Φ(x, y) > Φ x,g(x) . (2.8) Consider the following set: Then, for all y ∈ Y , θ y is open and X ⊂ y∈Y θ y .Since X is compact, it can be covered by a finite number r of its subsets {θ y1 ,...,θ yr }.Let {l j } j=1,r be a continuous partition of unity associated with the subcover {θ y1 ,...,θ yr }; that is, we have for all x ∈ X, r j=1 l j (x) = 1 and for all j = 1,r, supp l j ⊂ θ yj .
Consider the map defined by where (2.12) We now show that the map M is upper semicontinuous on X, with nonempty, convex, and closed values in Y and satisfying that for all g(x) ∈ ∂g(X), there exists u ∈ X, there exists α > 0 such that αg(u) + (1 − α)g(x) ∈ M(x).
(1) Let us prove that for all x ∈ X, M(x) = ∅.Consider a point x ∈ X, the function λ → r i=1 λ i Φ(x, y i ) is linear on R r .Therefore, it is continuous over the compact set S and according to the theorem of Weierstrass [5], there exists λ ∈ S such that (2.13) Therefore, y i0 ∈ M(x), which implies that M(x) = ∅.
(2) For all x ∈ X, M(x) is closed in Y .Consider x ∈ X and z ∈ M(x).There is a sequence {z k } k≥1 of elements of M(x) which converges to z.
As a consequence of the fact that for all k ≥ 1, z k ∈ M(x), we get Taking into account condition (6.1) of Theorem 2.1 and the fact that Let x ∈ X and let z, z be two elements of M(x) and θ ∈ [0,1].We now show that θz Since z and z are two elements of M(x), we have max λ∈S r i=1 λ i Φ(x, y i ) ≤ Φ(x,z) and max λ∈S r i=1 λ i Φ(x, y i ) ≤ Φ(x,z).Therefore, max λ∈S r i=1 λ i Φ x, y i ≤ min Φ(x,z), Φ(x,z) . (2.16) Taking into account condition (6.2) of Theorem 2.1, the fact that p i ∈ Y * , i = 1,r, and inequality (2.16), we obtain (4) M is upper semicontinuous.
According to Lemma 1.2, it is sufficient to show that the graph of M is closed in the compact set X × Y .
Then, we have Thus, we conclude that there exists x ∈ X such that sup y∈Y Φ(x, y) = Φ(x,g(x)), that is, for all y ∈ Y , we have Inequality (2.24) implies that q = n i=1 h i (x)q i belongs to the normal cone N g(X) (g(x)).According to Lemma 1.1 and condition (4) of Theorem 2.1, we have σ S(x),−q ≥ Re − q,g(x) . (2.25) The fact that h i (x) > 0, i = 1,...,n, implies that x ∈ supph i ⊂ V qi , that is, which contradicts inequality (2.25).We then conclude that h 0 (x) > 0.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation