Boundary Value Problems for Quasilinear Second Order Differential Equations

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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



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APA

ONUBEDO, A (2022). Boundary Value Problems for Quasilinear Second Order Differential Equations. Afribary. Retrieved from https://afribary.com/works/boundary-value-problems-for-quasilinear-second-order-differential-equations

MLA 8th

ONUBEDO, AUDU "Boundary Value Problems for Quasilinear Second Order Differential Equations" Afribary. Afribary, 16 Oct. 2022, https://afribary.com/works/boundary-value-problems-for-quasilinear-second-order-differential-equations. Accessed 29 Mar. 2024.

MLA7

ONUBEDO, AUDU . "Boundary Value Problems for Quasilinear Second Order Differential Equations". Afribary, Afribary, 16 Oct. 2022. Web. 29 Mar. 2024. < https://afribary.com/works/boundary-value-problems-for-quasilinear-second-order-differential-equations >.

Chicago

ONUBEDO, AUDU . "Boundary Value Problems for Quasilinear Second Order Differential Equations" Afribary (2022). Accessed March 29, 2024. https://afribary.com/works/boundary-value-problems-for-quasilinear-second-order-differential-equations