Semilinear Differential-Functional Systems in Two Independent Variables

✍ Scribed by Z. Kamont

• Publisher: John Wiley and Sons
• Year: 1986
• Tongue: English
• Weight: 684 KB
• Volume: 127
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19861270117

✦ Synopsis

I. Introduction

Let OcR? be a compact set and AicC(O, R), i =1, ..., n. (We denote by C ( X , Y ) the set of all continuous functions defined in X taking values in Y ; X, Y being arbitrary metric spaces.) Assume that I,= ( ( t , x) : t =0, xE[bo, b J } can. For ( t , x) and t 2 0 we define &(t, x) = max Ai(t, x) and A*(t, x) = min Ai(t, x).

Let w* be the maximum solution of the problem

and o* be the minimum solution of We assume that o* and w* exist on [0, a] and E c D where E = { ( t , x) : tE [ 0 , a ] , w * ( t ) sx ~w * ( t ) } , Let E o = [ --to, O]x[bo, b , ] , t o g O and @=(@pi, ..., On)€C(Eo, Rn).

functional systems of the form The present paper deals with the CAUCHY problem for semilinear differentialu(t, x) =@(t, x) for (t, x ) c E , , where u(t, x) =colon (ut(t, x), ..., un(t, x)), A(t, x) is a diagonal matrix, A