Existence Theory for Nonlinear Ordinary Differential Equations

1. Front Matter

   Pages i-vii

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2. Introduction, notation and preliminaries

   • Donal O’Regan

   Pages 1-3

3. Fixed point theory

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   Pages 4-10

4. Initial value problems

   • Donal O’Regan

   Pages 11-21

5. First order periodic problems

   • Donal O’Regan

   Pages 22-33

6. Existence principles for second order boundary value problems

   • Donal O’Regan

   Pages 34-45

7. Boundary value problems without growth restrictions

   • Donal O’Regan

   Pages 46-57

8. Positone boundary value problems

   • Donal O’Regan

   Pages 58-66

9. Semi-positone boundary value problems

   • Donal O’Regan

   Pages 67-81

10. Differential equations singular in the solution variable

    • Donal O’Regan

    Pages 82-104

11. Existence principle for singular boundary value problems

    • Donal O’Regan

    Pages 105-115

12. Nonresonance problems in the limit circle case

    • Donal O’Regan

    Pages 116-132

13. Resonance problems in the limit circle case

    • Donal O’Regan

    Pages 133-155

14. Boundary value problems on the half line

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    Pages 156-163

15. Existence theory for ordinary differential equations on compact and noncompact intervals

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    Pages 164-173

16. Impulsive differential equations

    • Donal O’Regan

    Pages 174-185

17. Differential equations in abstract spaces

    • Donal O’Regan

    Pages 186-194

18. Back Matter

    Pages 195-200

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We begin our applications of fixed point methods with existence of solutions to certain first order initial initial value problems. This problem is relatively easy to treat, illustrates important methods, and in the end will carry us a good deal further than may first meet the eye. Thus, we seek solutions to Y'. = I(t,y) (1. 1 ) { yeO) = r n where I: I X R n ---+ R and I = [0, b]. We shall seek solutions that are de­ fined either locally or globally on I, according to the assumptions imposed on I. Notice that (1. 1) is a system of first order equations because I takes its values in Rn. In section 3. 2 we will first establish some basic existence theorems which guarantee that a solution to (1. 1) exists for t > 0 and near zero. Familiar examples show that the interval of existence can be arbi­ trarily short, depending on the initial value r and the nonlinear behaviour of I. As a result we will also examine in section 3. 2 the dependence of the interval of existence on I and r. We mention in passing that, in the results which follow, the interval I can be replaced by any bounded interval and the initial value can be specified at any point in I. The reasoning needed to cover this slightly more general situation requires minor modifications on the arguments given here.

• Boundary value problem
• differential equation
• ordinary differential equation
• ordinary differential equations

`I warmly recommend it to everyone interested in ordinary differential equations and its applications.'
Acta Sci. Math., 64 (1998)

• Department of Mathematics, University College Galway, Galway, Ireland

  Donal O’Regan

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