Boundary Value Problems for Quasilinear Second Order Differential Equations

Abstract This project is concerned with the review of some boundary value problems for nonlinear ordinary differential equations using topological and variational methods. A more classical boundary value problems for ordinary differential equations (like the boundary value problems on a ball, initial value problems, problems on annular domains and positone problems) which represent the main interest of a wide number of researchers in the world is studied.

Contents

Certification ii

Dedication iii

Acknowledgement iv

Abstract viii

1 Introduction 1

2 Literature Review 3

3 Preliminaries 6

3.1 Homeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 Leray-Schauder Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . 14

3.5 Quasilinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5.1 Quasilinear Equation Of Second Order . . . . . . . . . . . . . . . . 16

4 Main Result 18

vi

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2 Tools Of Analysis And Organization . . . . . . . . . . . . . . . . . . . . . 20

4.3 Boundary Value Problems On a Ball . . . . . . . . . . . . . . . . . . . . . 20

4.3.1 Equivalent Integral Equation . . . . . . . . . . . . . . . . . . . . . . 20

4.3.2 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.3 On The Principal Eigenvalues . . . . . . . . . . . . . . . . . . . . . 23

4.3.4 On The Principal Eigenvalue of The p - Laplacian . . . . . . . . . . 28

4.3.5 On The Higher Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 28

4.4 On Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.5 Problems Of Annular Domain . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.5.1 Fixed Point Formulation . . . . . . . . . . . . . . . . . . . . . . . . 32

4.6 Positone Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Reference 39