By Stephen R. Bernfeld

ISBN-10: 0120931508

ISBN-13: 9780120931507

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**Extra resources for An Introduction to Nonlinear Boundary Value Problems**

**Example text**

3. I n t h e preceding r e s u l t conditions a r e imposed on f ( t , x , y ) which a r e s u f f i c i e n t t o imply t h a t lower solutions a r e subf'unctions and, therefore, t h a t solutions of BVP's, when they I n t h e next r e s u l t we take t h e uniqueness exist, a r e unique. of BVP's as one of t h e hypotheses. 3. 1) has a s o l u t i o n which extends throughout [a,b]. Suppose further t h a t solutions of boundary value problems X" = f(t,X,X'), x ( t l ) = xl, ~ ( 6 =~x2, ) [tl,t2] C J, when they e x i s t , a r e unique.

Assume t h a t f o r each b > a , f ( t , x , x ' ) s a t i s f i e s Nagumofs condition on [a,b] r e l a t i v e t o the p a i r a,B E C(l)[[a,m),R] with u ( t ) 5 B(t) on [a,-). 7. 1) on [a,m), respectively. 1) xtt = f(t,x,x'), he8 a solution x on E x(a) = c, C(2)[[a,m),R] - such t h a t a ( t ) c x ( t ) < p(t) [a,m). 1, it follows that f o r each n 1 1 there i s a solution xn E C(2)[[a,a+n],R] such t h a t xn(a) = c, xn(a+n) = p ( a + n ) , and a ( t ) 5 x n ( t ) 5 p ( t ) on [ a , a + n l .

On J and l e t c J. Then define >0 Let a,p C(l)[JJR] E be such t h a t with Icy'(t) I, I p l ( t ) I a(t) < p(t) < c on 2 c, f(t,x,c) for XI f(t,X,X') for IX'I f(tJx,-c) for x' 5 CJ 5 -c, and F(t,x,x') = F{ The function F*( t ,B (t),XI) + [x-B(t) ] / 1 + x2 for F*(tJx,x') for a ( t ) < x < B ( t ) , for x < a ( t ) . * (t , a ( t ) , x ' ) + [ x - a ( t ) ] / 1 + x 2 F(t,x,x') J w i l l be called the modification of f ( t J x J x ' ) associated with t h e t r i p l e follows from the d e f i n i t i o n t h a t on J x R x R x >p(t) and t h a t 18 a ( t ) , p(t), c.

### An Introduction to Nonlinear Boundary Value Problems by Stephen R. Bernfeld

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