Admissible Mappings Research Articles

KKM type theorems with applications to generalized vector equilibrium problems in FC-spaces

A new coincidence theorem and some KKM type theorems for better admissible set-valued mappings are established in noncompact FC-space. As applications, some new existence theorems of solutions for several classes of generalized vector equilibrium problems are established in noncompact FC-spaces. The results represented in this paper unify and extend some corresponding known results in the literature.

EIGENVALUES OF COUNTABLY CONDENSING ADMISSIBLE MAPS

Applying a fixed point theorem for compact admissible maps due to Gorniewicz, we prove that under certain conditions each count ably condensing admissible maps in Frechet spaces has a positive eigenvalue. This result has many consequences, including the well-known theorem of Krasnoselskii.

Relative Nielsen theory for noncompact spaces and maps

In this note, we generalize the various existing local and relative Nielsen type numbers to the setting of maps of noncompact ANR-pairs. Then we introduce general classes of admissible maps for which these numbers are well-defined. An application of these relative Nielsen numbers to differential equations is also given.

COINCIDENCE THEOREMS IN PRODUCT G-CONVEX SPACES

By applying the technique of continuous partition of unity, some new coincidence theorems for a better admissible mapping and a family of set-valued mappings defined on the product G-convex spaces are proved. Theorems of this paper improve, unify and generalize many important coincidence theorems and collectively fixed point theorems in recent literature.

Homotopy and normalization properties for admissible maps

New normalization and homotopy properties are presented for a subclass of the Bk-admissible maps of Park. In addition new random fixed point results are given.

A Note on kappa-Diagonal Surface States

We classify all twist-even squeezed states in string field theory which are diagonal in the kappa-basis and simultaneously surface states. For this purpose, we derive a consistency condition for the maps defining kappa-diagonal surface states. It restricts these maps to a two-parameter family of Jacobi sine functions. Not all of them are admissible maps for surface states; standard requirements single out two one-parameter families containing the generalized butterfly states and the wedge states.

Fixed points of condensing multivalued maps in topological vector spaces

With the aid of the simplicial approximation property, we show that every admissible multivalued map from a compact convex subset of a complete metric linear space into itself has a fixed point. From this fact we deduce the fixed point property of a closed convex set with respect to pseudocondensing admissible maps.

Positive eigenvalues of countably contractive maps

With the aid of degree theory for admissible maps due to Väth [1], we prove that a countably contractive self-map defined on a normal cone in a real Banach space has a positive eigenvalue under certain condition. This includes a result of Massabo and Stuart [2] for set contractions.

Fixed point theorems for condensing admissible maps in topological vector spaces

Introducing the notion of c-measure of noncompactness due to S. Hahn, we give a homotopy extension theorem and the fixed point property for pseudo-condensing admissible maps in general topological vector spaces. Moreover, these results are applied to obtain a Leray-Schauder type fixed point theorem for pseudo-condensing admissible maps in topological vector spaces which includes many known results.

Some results on positive eigenvalues of admissible maps

The Birkhoff–Kellogg eigenvalue theorem is extended to a class of compact admissible set-valued maps with not necessarily convex values, where the proof is based on fixed point theorems of Jerofsky and uses elementary methods. Moreover, we prove a result on positive eigenvalues of condensing admissible maps which generalizes known results and has many consequences.

KKM TYPE THEOREMS AND COINCIDENCE THEOREMS ON L-CONVEX SPACES

Some KKM theorems and coincidence theorems involving admissible set-valued mappings and the set-valued mappings with compactly local intersection property are proved in L-convex spaces. As applications, some new fixed point theorems are obtained in L-convex spaces. These theorems improve and generalize many important known results in recent literature.

A generalized Nielsen number and multiplicity results for differential inclusions

The Nielsen number is defined for a rather general class of multivalued maps on compact connected ANRs, including, e.g., admissible maps (in the sense of Górniewicz (1976); compare also Górniewicz (1995)) on tori. Since the Poincaré maps generated by the Marchaud vector fields are of this type (see (Andres, 1997)), we can obtain in such a way multiplicity results for differential inclusions. More precisely, the nontrivial Nielsen number gives a lower estimate of coincidence points (in particular, fixed points) corresponding to the desired solutions.

Coincidences for Admissible and Φ★Maps and Minimax Inequalities

A coincidence theory for Φ★and admissible maps is presented in this paper. One can also deduce from this theory new minimax inequalities of von Neumann–Sion type.

On the multivalued Poincaré operators

By Poincare operators we mean the translation operator along the trajectories of the associated differential system and the first return (or section) map defined on the cross section of the torus by means of the flow generated by the vector field. The translation operator is sometimes also called as Poincare–Andronov or Levinson or, simply, T -operator. In the classical theory (see [K], [W], [Z] and the references therein), both these operators are defined to be single-valued, when assuming, among other things, the uniqueness of the initial value problems. At the absence of uniqueness one usually approximates the right-hand sides of the given systems by the locally lipschitzian ones (implying uniqueness already), and then applies the standard limiting argument. This might be, however, rather complicated and is impossible for the discontinuous right-hand sides. On the other hand, set-valued analysis allows us to handle effectively also with such classically troublesome situations. In particular, the class of admissible maps in the sense of [G] has been shown to be very useful with this respect, because generalized topological invariants like the Brouwer degree, the fixed point

Coincidence Theorems for Admissible Multifunctions on Generalized Convex Spaces

We defined admissible classes of maps which are general enough to include composites of maps appearing in nonlinear analysis or algebraic topology, and generalized convex spaces which are generalizations of many general convexity structures. In this paper we obtain a coincidence theorem for admissible maps defined on generalized convex spaces. Our new result is applied to obtain an abstract variational inequality, a KKM type theorem, and fixed point theorems.

On the Solvability of Systems of Inclusions Involving Noncompact Operators

We consider the solvability of a system \[ \left \{ {\begin {array}{*{20}{c}} {y \in \bar F(x,y),} \\ {x \in \bar G(x,y)} \\ \end {array} } \right .\] of set-valued maps in two different cases. In the first one, the map $(x,y) - \circ \bar F(x,y)$ is supposed to be closed graph with convex values and condensing in the second variable and $(x,y) - \circ \bar G(x,y)$ is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case $\bar F$ is as before with compact, not necessarily convex, values and $\bar G$ is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set $x - \circ S(x)$ of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.

On the solvability of systems of inclusions involving noncompact operators

We consider the solvability of a system \[ { y ∈ F ¯ ( x , y ) , x ∈ G ¯ ( x , y ) \left \{ {\begin {array}{*{20}{c}} {y \in \bar F(x,y),} \\ {x \in \bar G(x,y)} \\ \end {array} } \right . \] of set-valued maps in two different cases. In the first one, the map ( x , y ) − ∘ F ¯ ( x , y ) (x,y) - \circ \bar F(x,y) is supposed to be closed graph with convex values and condensing in the second variable and ( x , y ) − ∘ G ¯ ( x , y ) (x,y) - \circ \bar G(x,y) is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case F ¯ \bar F is as before with compact, not necessarily convex, values and G ¯ \bar G is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set x − ∘ S ( x ) x - \circ S(x) of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.

The shuffle algebra and its derivations

The shuffle algebra on a partially commutative alphabet and its derivations are studied. The derivation of the shuffle algebra is characterized in terms of admissible mappings. The module of all derivations of the shuffle algebra is proved to be isomorphic to the free partially commutative power series Lie algebra with the coefficient ring extended. The orthogonality between the Lie elements and the proper shuffles is established.

On the topological degree for A-compact mappings

In [ 1 ] F. E. Browder and W. V. Petryshyn develop a generalized degree theory for A-proper mappings. This class of mappings is wider than com- pact perturbations of the identity, for which the classical Leray-Schauder degree is defined. Recently Werinski [S] defined a fixed point index for another class of mappings, called Admissible, which includes compact map- pings but does not include the class of A-proper mappings. In this paper we will define a topological degree for a new class of mappings, which we denote A-compact mappings. We will prove that this class contains the classes of A-proper mappings and Admissible mappings. As an application we shall show some classical topological results for not necessarily A-proper mappings.

Maps between p-completed classifying spaces

SynopsisLet G and G' be two connected compact Lie groups with maximal tori T and T'. For a space X, let Xp be the p-completion of X. We will associate to each topological map f:(BG)p→(BG')p an “admissible map” ϕ:π1(T)⊗zZp→π1(T′)⊗zZp. We then show that the study of “admissible maps” in the p-complete case may be reduced to their study in the p-local case.