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Some applications of the topological degree theory to multi-valued boundary value problems

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Pruszko T.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction......................................................................................................5 I. Preliminaries 1. Strong convergence and weak convergence in Banach spaces..................7 2. Compact and weakly compact sets in Banach spaces.................................7 3. Weakly compact sets in the space of integrable functions...........................8 4. Compact sets in the space of continuous functions.....................................9 5. Basic integral and differential inequalities..................................................10 II. Multi-valued mappings 1. Upper semi-continuous, compact and weakly compact mappings..............12 2. L-compact mappings..................................................................................14 3. Caratheodory conditions for convex-valued mappings...............................17 4. Convex-valued, weakly compact selectors.................................................19 5. Compact convex-valued vector fields.........................................................20 III. Multi-valued boundary value problems 1. The degree of the boundary value problem...............................................21 2. Existence theorems....................................................................................23 IV. Boundary value problems for ordinary differential equations 1. Admissible boundary value problems associated with problem (IV.1).........27 2. Existence theorems....................................................................................29 3. First order problems...................................................................................31 4. Second order problems..............................................................................34 V. Boundary value problems for some hyperbolic partial differential equations 1. Multi-valued Darboux problem....................................................................36 2. A multi-valued problem with nonlinear boundary conditions.......................38 VI. Boundary value problems for elliptic partial differential equations 1. Basic function spaces................................................................................40 2. The general boundary value problem........................................................41 References ...................................................................................................44

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Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations

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Gęba L., Pruszko T.

Annales Polonici Mathematici

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1991-1992

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tom 56

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nr 1

49-61

EN

This paper treats nonlinear elliptic boundary value problems of the form (1) L[u] = p(x,u) in $Ω ⊂ ℝ^n$, $u = Du = ... = D^{m-1}u$ on ∂Ω in the Sobolev space $W_0^{m,2}(Ω)$, where L is any selfadjoint strongly elliptic linear differential operator of order 2m. Using both topological degree arguments and minimax methods we obtain existence and multiplicity results for the above problem.

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On the structure of the set of solutions of a functional equation with applications to boundary value problems

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Bielawski D., Pruszko T.

Annales Polonici Mathematici

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1991

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tom 53

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nr 3

201-209

Ograniczanie wyników

2 Annales Polonici Mathematici

3 Pruszko T.

1 Bielawski D.

1 Gęba L.

2 1991

1 1984

Some applications of the topological degree theory to multi-valued boundary value problems

Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations

On the structure of the set of solutions of a functional equation with applications to boundary value problems