Differential and Integral Inequalities: Ordinary by V. Lakshmikantham; S. Leela

By V. Lakshmikantham; S. Leela

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Additional info for Differential and Integral Inequalities: Ordinary Differential Equations v. 1: Theory and Applications: Ordinary Differential Equations

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For collateral reading on differential and integral inequalities, see Szarski [l] and Walter [3]. 0. Introduction T h e most important techniques in the theory of differential equations involve a systematic use of the theory of differential inequalities, or what may be termed as the integration of differential inequalities. We present, in this chapter, a number of varied results depending essentially on this approach. Many of our theorems and their proofs will involve estimates with respect to some convenient norm.

2. 6) t f to. 7) u(t,J = 0 . t, . Proof. 7) is identically zero. Suppose, on the contrary, that there exists a n, t , CT -< t, a, such that ~ ( o> ) 0. < g ( t , r ( t ) ) , If p ( t ) is the minimal solution of t,

I. 10) 48 CHAPTER 2 where A ( t )is a continuous, n x n matrix for t E J and f E C [ J x R", R"]. ,] g(t, u) is monotone nondecreasing in u for each fixed t , and the scalar differential equation u' = -0u -1- kg(t,u ) , has a solution u ( t ) existing for t u(t,) = uo >0 3 to such that lim u ( t ) = 0. 10) defined for t to such that lim x ( t ) = 0. 2, Uniqueness A simple criterion that implies the uniqueness of solutions is the following Perron's condition. 1. Assume that (i) the function g(t, u)is continuous and nonnegative for t,, t to a, 0 u 2b, and, for every t , , to .

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