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1

Sur un principe géométrique en analyse convexe

100%

Granas A., Lassonde M.

Studia Mathematica

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1991-1992

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tom 101

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nr 1

1-18

EN

In this note we present we present a new elementary approach in the theory of minimax inequalities. The proof of the main result (called the geometric principle) uses only some simple properties of convex functions. The geometric principle (which is equivalent to the well-known lemma of Klee [13]) is shown to have numerous applications in different areas of mathematics.

2

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Nonlinear boundary value problems for ordinary differential equations

88%

Granas A., Guenther R., Lee J.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Comments............................................................................................................................5 CHAPTER I Introduction § 1. Elementary theory of second order differential equations...........................................12 § 2. Topological preliminaries.............................................................................................14 § 3. The maximum principle................................................................................................16 § 4. Existence and a priori bounds-examples.....................................................................19 § 5. Problems with other boundary conditions....................................................................25 CHAPTER II The Bernstein theory of the equation y" = f(t, y, y') § 1. The homogeneous Dirichlet, Neumann, and periodic problems...................................28 § 2. The homogeneous Sturm-Liouville problem................................................................34 § 3. Inhomogeneous boundary conditions..........................................................................35 § 4. Examples and remarks................................................................................................39 § 5. Bernstein-Nagumo growth conditions..........................................................................44 § 6. Nonlinear boundary conditions....................................................................................50 § 7. Uniqueness..................................................................................................................52 CHAPTER III Applications § 1. Steady-state temperature distributions........................................................................56 § 2. The Thomas-Fermi problem........................................................................................59 § 3. Singular boundary value problems..............................................................................62 § 4. Osmotic flow.................................................................................................................64 § 5. Positive solutions to diffusion equations......................................................................70 CHAPTER IV Other second order boundary value problems § 1. Periodic solutions to differential equations of Nirenberg type......................................76 § 2. The Dirichlet problem for y" = f(y') and the Neumann problem for y" = f(t,y,y').............85 § 3. Upper and lower solutions...........................................................................................94 CHAPTER V Even order systems and higher order equations § 1. General existence theorems........................................................................................99 § 2. Second order systems...............................................................................................102 § 3. Third and fourth order problems................................................................................108 § 4. Higher even order equations......................................................................................111 CHAPTER VI Numerical solution of boundary value problems § 1. Newton’s method........................................................................................................113 § 2. The shooting method for the Dirichlet problem..........................................................115 § 3. The shooting method for the Neumann problem........................................................120 § 4. Quasilinearization for boundary value problems........................................................121 References.......................................................................................................................125

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Théorème de minimax sans topologie ni convexité

81%

Granas A., Lee J.-R., Liu F.-C.

Colloquium Mathematicum

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1992

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tom 63

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nr 2

141-144

FR

Dans cete note, nous présentons un théorème de minimax (Théorème A) formulé seulement en langage de la théorie des ensembles. Ce résultat permet de déduire de façon immédiate (en utilisant un lemme de topologie générale) plusieurs théorèmes de minimax bien connus.

4

Le théorème de Lefschetz pour les ANR approximatifs

70%

Gauthier G., Granas A.

Colloquium Mathematicum

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1982

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tom 46

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nr 2

197-202

5

Sur une certaine alternative non-linéaire en analyse convexe

60%

Deguire P., Granas A.

Studia Mathematica

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1986

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tom 83

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nr 2

127-138

Ograniczanie wyników

2 Colloquium Mathematicum

2 Studia Mathematica

5 Granas A.

1 Deguire P.

1 Gauthier G.

1 Guenther R.

1 Lassonde M.

1 Lee J.

1 Lee J.-R.

1 Liu F.-C.

1 1992

1 1991

1 1986

1 1985

1 1982

Sur un principe géométrique en analyse convexe

Nonlinear boundary value problems for ordinary differential equations

Théorème de minimax sans topologie ni convexité

Le théorème de Lefschetz pour les ANR approximatifs

Sur une certaine alternative non-linéaire en analyse convexe