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Some fixed point theorems for multifunctions with applications in game theory

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Cuōng B. C.

Instytut Matematyczny Polskiej Akademii Nauk

Introduction The main result of this paper is concerned with the conditions which guarantee that a multifunction $f: C → 2^X$ defined on an arbitrary subset C of a topological vector space X admits a point x of C such that x∈f(x). First, we give some definitions and propositions which are associated with semicontinuous multifunctions (Part 1). Next, in Part 2, we present a global convergence criterion on variable dimension algorithms for finding an approximate solution of the equation x∈f(x), and then we consider some fixed point theorems for multifunctions defined in finite-dimensional spaces. Part 3 contains fixed point theorems for quasi upper semicontinuous multifunctions defined on arbitrary domains of topological vector spaces which generalize the theorems with boundary conditions. Part 4 is devoted to some fixed point theorems for strongly lower semicontinuous multifunctions and thus here we are first concerned with fixed point theorems under boundary conditions for this class of multi-functions. The last part shows how we can apply the results obtained to existence problem of equilibrium situations in the theory of non-cooperative games.

CONTENTS Introduction..........................................................................................................5 1. Some classes of semicontinuous multifunctions...............................................5 2. A remark on the convergence of variable dimension algorithms.......................9 3. Some fixed point theorems.............................................................................15 4. Fixed point theorems for strongly lower semicontinuous multifunctions..........24 5. Some applications in game theory..................................................................29 References.........................................................................................................34

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1 Cuōng B. C.

1 1985

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Some fixed point theorems for multifunctions with applications in game theory