Ebook: Existence Theory for Nonlinear Ordinary Differential Equations

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.

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This book presents an up-to-date account of many topics of current interest in the theory of nonlinear ordinary differential equations. Topics include fixed point theory, periodic problems, lower and upper surfaces, positone and semi-positone problems, singular equations, limit circle problems, finite and infinite interval problems, and impulsive differential equations. In addition many important applications are presented to complement the theory.
Audience: This work will be of interest to theoretical and applied mathematicians, and graduate students in the above disciplines.

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This book presents an up-to-date account of many topics of current interest in the theory of nonlinear ordinary differential equations. Topics include fixed point theory, periodic problems, lower and upper surfaces, positone and semi-positone problems, singular equations, limit circle problems, finite and infinite interval problems, and impulsive differential equations. In addition many important applications are presented to complement the theory.
Audience: This work will be of interest to theoretical and applied mathematicians, and graduate students in the above disciplines.
Content:
Front Matter....Pages i-vii
Introduction, notation and preliminaries....Pages 1-3
Fixed point theory....Pages 4-10
Initial value problems....Pages 11-21
First order periodic problems....Pages 22-33
Existence principles for second order boundary value problems....Pages 34-45
Boundary value problems without growth restrictions....Pages 46-57
Positone boundary value problems....Pages 58-66
Semi-positone boundary value problems....Pages 67-81
Differential equations singular in the solution variable....Pages 82-104
Existence principle for singular boundary value problems....Pages 105-115
Nonresonance problems in the limit circle case....Pages 116-132
Resonance problems in the limit circle case....Pages 133-155
Boundary value problems on the half line....Pages 156-163
Existence theory for ordinary differential equations on compact and noncompact intervals....Pages 164-173
Impulsive differential equations....Pages 174-185
Differential equations in abstract spaces....Pages 186-194
Back Matter....Pages 195-200

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This book presents an up-to-date account of many topics of current interest in the theory of nonlinear ordinary differential equations. Topics include fixed point theory, periodic problems, lower and upper surfaces, positone and semi-positone problems, singular equations, limit circle problems, finite and infinite interval problems, and impulsive differential equations. In addition many important applications are presented to complement the theory.
Audience: This work will be of interest to theoretical and applied mathematicians, and graduate students in the above disciplines.
Content:
Front Matter....Pages i-vii
Introduction, notation and preliminaries....Pages 1-3
Fixed point theory....Pages 4-10
Initial value problems....Pages 11-21
First order periodic problems....Pages 22-33
Existence principles for second order boundary value problems....Pages 34-45
Boundary value problems without growth restrictions....Pages 46-57
Positone boundary value problems....Pages 58-66
Semi-positone boundary value problems....Pages 67-81
Differential equations singular in the solution variable....Pages 82-104
Existence principle for singular boundary value problems....Pages 105-115
Nonresonance problems in the limit circle case....Pages 116-132
Resonance problems in the limit circle case....Pages 133-155
Boundary value problems on the half line....Pages 156-163
Existence theory for ordinary differential equations on compact and noncompact intervals....Pages 164-173
Impulsive differential equations....Pages 174-185
Differential equations in abstract spaces....Pages 186-194
Back Matter....Pages 195-200
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