Collective fixed points and maximal elements with applications to abstract economies

In this paper, we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is Φ-condensing. As an application of our collective fixed points theorem, we derive the coincidence theorem for two families of multivalued maps defined on product spaces. Then we give some existence results for maximal elements for a family of L_S-majorized multivalued maps whose product is Φ-condensing. We also prove some existence results for maximal elements for a family of multivalued maps which are not L_S-majorized but their product is Φ-condensing. As applications of our results, some existence results for equilibria of abstract economies are also derived. The results of this paper are more general than those given in the literature.     

Lp-continuous extreme selectors of multifunctions with decomposable values: Existence theorems

The existence theorems of L _p -continuous selectors that values are extreme points are proved for a class of multivalued maps. Applications to multivalued maps appearing in multivalued differential equations are presented.Supported in part by RFFI Grant 93-011-264.     

L p-Continuous extreme selectors of multifunctions with decomposable values: Relaxation theorems

Some density theorems of L _p-continuous selectors whose values are extreme points are proved for a class of multivalued maps. applications to the Darboux problem for a differential inclusion are presented.     

Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities

In this paper, we establish some fixed point theorems for a family of multivalued maps under mild conditions. By using our fixed point theorems, we derive some maximal element theorems for a particular family of multivalued maps, namely the Φ-condensing multivalued maps. As applications of our results, we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences. Further applications of our results are also given to minimax inequalities for a family of functions.     

Coincidence and fixed points for hybrid strict contractions

We define the property (E.A) for single-valued and multivalued mappings and introduce the notion of T-weak commutativity for a hybrid pair (f,T) of single-valued and multivalued maps. We obtain some coincidence and fixed point theorems for this class of maps and derive, as application, an approximation theorem.     

Existence of Solutions for Nonlinear Fractional q-Difference Inclusions with Nonlocal Robin (Separated) Conditions

In this paper, we show the existence of solutions for nonlinear fractional q-difference inclusions involving convex as well as non-convex valued maps with nonlocal Robin (separated) conditions. Our results are new in the present configuration and are based on some standard principles for multivalued maps. A special case for q-difference inclusions in the given setting is also discussed. Some interesting observations are presented.     

Multivalued essential maps of approximable and acyclic type

This paper discusses multivalued approximable and acyclic closed maps. We show if F is essential and F ≅ G, then G has a fixed point.     

Lp-Continuous extreme selectors of multifunctions with decomposable values: Relaxation theorems

Some density theorems of L _p-continuous selectors whose values are extreme points are proved for a class of multivalued maps. applications to the Darboux problem for a differential inclusion are presented.Supported in part by RFFI Grant 93-011-264.     

Multivalued contractions

LetE be a Banach space,C a closed convex subset ofE, F a multivalued contraction fromC to itself with closed values. Ifx ₀ is a fixed point and ifF(x ₀) is not a singleton, then there exists a fixed pointx ₁ ofF which is different fromx ₀. We prove also that there is in the Euclidean space ?² a multivalued contraction with compact connected values having a nonconnected set of fixed points.     

On Weyl almost periodic selections of multivalued maps

We prove that Weyl almost periodic multivalued maps R∋t→F(t)∈clU have Weyl almost periodic selections, where clU is the collection of non-empty closed sets of a complete metric space U.     

Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

Let F_n: X₁ → X₂ be a sequence of (multivalued) meromorphic maps between compact Kähler manifolds. We study the asymptotic distribution of preimages of points by F_n and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points. Let (Z_n) be a sequence of holomorphic images of ?^s in a projective manifold. We prove that the currents, defined by integration on Z_n, properly normalized, converge to currents which satisfy some laminarity property. We also show this laminarity property for the Green currents, of suitable bidimensions, associated to a regular polynomial automorphism of ?^k or an automorphism of a projective manifold.     

On coincidence point and fixed point theorems for nonlinear multivalued maps

Several characterizations of MT-functions are first given in this paper. Applying the characterizations of MT-functions, we establish some existence theorems for coincidence point and fixed point in complete metric spaces. From these results, we can obtain new generalizations of Berinde-Berinde?s fixed point theorem and Mizoguchi-Takahashi?s fixed point theorem for nonlinear multivalued contractive maps. Our results generalize and improve some main results in the literature.     

An intersection theorem for multivalued maps

LetE be a Banach space and letV, W ? E be two closed subspaces such thatE=V ⊕ W. LetF, G: E-· E be two multivalued maps. The problem we are concerned with is to give conditions onF andG ensuring the nonemptiness ofF(V) ∩ G(W).     

Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory

This paper introduces a vectorial form of equilibrium version of Ekeland-type variational principle. Some equivalent results to our variational principle are given. As applications, we derive the existence of solutions of a vector equilibrium problem in the setting of complete quasi-metric spaces with a W-distance. Caristi-Kirk fixed point theorem for multivalued maps is also established in a more general setting.     

Set-valued system of fractional differential equations with hysteresis

In this paper, we investigate the global solvability in L₁(0,1) of a set-valued system of nonlinear fractional differential equations with hysteresis. Some existence theorems for both single and multivalued systems are proved.     

Generalized Topological Essentiality and Coincidence Points of Multivalued Maps

A concept of generalized topological essentiality for a large class of multivalued maps in topological vector Klee admissible spaces is presented. Some direct applications to differential equations are discussed. Using the inverse systems approach the coincidence point sets of limit maps are examined. The main motivation as well as main aim of this note is a study of fixed points of multivalued maps in Fréchet spaces. The approach presented in the paper allows to check not only the nonemptiness of the fixed point set but also its topological structure.      

A nonlinear evolution system with two subdifferentials and monotone differential games

It is shown that the first order multivalued equation for V = V(t, x, y, z) involving the sum of two subdifferentials composed with the partials of V (V_t +f(t, x, y, z) · ▽_xV + β(V_y) + γ(V_z) + h(t, x, y, z) ? 0 a.e.) has a Lipschitz solution. This solution is shown to be the value of a differential game in which the players are restricted to choosing monotone nondecreasing functions of time. Accordingly, the multivalued equation is interpreted as the corresponding Hamilton-Jacobi equation of the game.     

Fixed point theorems for some generalized multivalued nonexpansive mappings

In this paper, we introduce a condition on multivalued mappings which is a multivalued version of condition (C_λ) defined by Garcia-Falset et al. (2011) [3]. It is shown here that some of the classical fixed point theorems for multivalued nonexpansive mappings can be extended to mappings satisfying this condition. Our results generalize the results in Lim (1974), Lami Dozo (1973), Kirk and Massa (1990), Garcia-Falset et al. (2011), Dhompongsa et al. (2009) and Abkar and Eslamian (2010) [4], [5], [6], [3], [7] and [8] and many others.     

Continuity, compactness, and degree theory for operators in systems involving p-Laplacians and inclusions

The study of weak solutions for systems of nonlinear partial differential equations of elliptic type with inclusions leads to a multivalued operator of superposition type in Sobolev spaces. We show that, under natural assumptions, this operator has the properties which allow to apply degree theory (fixed point index) for multivalued maps. More precisely, this operator is upper semicontinuous and compact with nonempty convex compact values. For the particular case of systems involving p-Laplacians, we show that there is a homeomorphism transforming the whole system to a situation for which a fixed point index is available.     

Pure strategy Nash equilibrium points and the Lefschetz fixed point theorem

A pure strategy Nash equilibrium point existence theorem is established for a class ofn-person games with possibly nonacyclic (e.g. disconnected) strategy sets. The principal tool used in the proof is a Lefschetz fixed point theorem for multivalued maps, due to Eilenberg and Montgomery, which extends their better known. Eilenberg-Montgomery fixed point theorem (EMT) [Eilenberg/Montgomery, Theorem 1, p. 215] to nonacyclic spaces. Special cases of the existence theorem are also discussed.