Infinite Interval Problems for Differential, Difference and Integral Equations, 2001

Infinite interval problems abound in nature and yet until now there has been no book dealing with such problems. The main reason for this seems to be that until the 1970's for the infinite interval problem all the theoretical results available required rather technical hypotheses and were applicable only to narrowly defined classes of problems. Thus scientists mainly offer~d and used special devices to construct the numerical solution assuming tacitly the existence of a solution. In recent years a mixture of classical analysis and modern fixed point theory has been employed to study the existence of solutions to infinite interval problems. This has resulted in widely applicable results. This monograph is a cumulation mainly of the authors' research over a period of more than ten years and offers easily verifiable existence criteria for differential, difference and integral equations over the infinite interval. An important feature of this monograph is that we illustrate almost all the results with examples. The plan of this monograph is as follows. In Chapter 1 we present the existence theory for second order boundary value problems on infinite intervals. We begin with several examples which model real world phenom­ ena. A brief history of the infinite interval problem is also included. We then present general existence results for several different types of boundary value problems. Here we note that for the infinite interval problem only two major approaches are available in the literature.

1 Second Order Boundary Value Problems.- 1.1. Introduction.- 1.2. Some Examples.- 1.3. Preliminary Results.- 1.4. Existence Theory for Problems (1.1.1) – (1.1.3).- 1.5. Existence Theory for Problems of Type (1.1.4).- 1.6. Existence Theory for Problems of Type (1.1.5).- 1.7. Existence Theory for Problems of Type (1.1.6).- 1.8. Existence Theory for Problems of Type (1.1.7).- 1.9. Existence Theory for Problems of Type (1.1.8).- 1.10. Existence Theory for Problems of Type (1.1.9).- 1.11. Existence Theory for Singular Problems of Type (1.1.1) – (1.1.3).- 1.12. Existence Theory for Problems (1.1.10) and (1.1.11).- 1.13. Notes and Remarks.- 1.14. References.- 2 Higher Order Boundary Value Problems.- 2.1. Introduction.- 2.2. Preliminary Results.- 2.3. Existence Theory for Conjugate Type Problems.- 2.4. Existence Theory for Right Focal Type Problems.- 2.5. Notes and Remarks.- 2.6. References.- 3 Continuous Systems.- 3.1. Introduction.- 3.2. Linear Problems (3.1.3), (3.1.2).- 3.3. Nonlinear Problems (3.1.1), (3.1.2).- 3.4. Nonlinear Problems (3.1.4), (3.1.2).- 3.5. Nonlinear Problems (3.1.5), (3.1.6).- 3.6. Notes and Remarks.- 3.7. References.- 4 Integral Equations.- 4.1. Introduction.- 4.2. Existence Theory for (4.1.1) and (4.1.2).- 4.3. Existence Theory for (4.1.3) and (4.1.4).- 4.4. Existence Theory for (4.1.5).- 4.5. Existence Theory and Behaviour of Solutions to (4.1.6).- 4.6. Existence Theory for (4.1.7) and (4.1.8).- 4.7. Existence and Approximation for (4.1.9).- 4.8. Abstract Volterra Equations.- 4.9. Periodic and Almost Periodic Solutions to (4.1.10).- 4.10. Periodic Solutions to (4.1.11).- 4.11. Notes and Remarks.- 4.12. References.- 5 Discrete Systems.- 5.1. Introduction.- 5.2. Linear Problems (5.1.3), (5.1.2).- 5.3. Nonlinear Problems (5.1.1), (5.1.2).- 5.4.Nonlinear Problems (5.1.4), (5.1.2).- 5.5. Second Order Problems (5.1.5), (5.1.7).- 5.6. Summary Discrete Systems (5.1.8).- 5.7. Urysohn Discrete Equations (5.1.9).- 5.8. Notes and Remarks.- 5.9. References.- 6 Equations in Banach Spaces.- 6.1. Introduction.- 6.2. Continuous Equations.- 6.3. Discrete Equations.- 6.4. Continuous and Discrete Equations.- 6.5. Notes and Remarks.- 6.6. References.- 7 Multivalued Equations.- 7.1. Introduction.- 7.2. Existence Theory for (7.1.1).- 7.3. Solution Set of (7.1.2).- 7.4. Existence Theory for (7.1.3).- 7.5. Existence Theory for (7.1.4) and (7.1.5).- 7.6. Notes and Remarks.- 7.7. References.- 8 Equations on Time Scales.- 8.1. Introduction.- 8.2. Existence Theory for (8.1.1).- 8.3. Notes and Remarks.- 8.4. References.