By Stephen R. Bernfeld

ISBN-10: 0120931508

ISBN-13: 9780120931507

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Example text

Clearly d c > 6; f o r otherwise t h e r e would be a s o l u t i o n - x(t) with [c,d]. x(c) = a ( c ) , x(d) = a(d), an% x ( t ) -> a ( t ) This s o l u t i o n would be d i s t i n c t from x ( t ) 0 on contra- d i e t i n g t h e assumption concerning t h e uniqueness of s o l u t i o n s of BVP's. Now f o r each p o s i t i v e i n t e g e r n, let P(n) t h e proposition t h a t t h e r e e x i s t s an i n t e r v a l [cn,dn] with 0 < dn - c < d - c - (n 1)s and a s o l u t i o n n- - x n on E C(2)[[cn,dn],R] such t h a t d i s t i n c t s o l u t i o n with boundary values X" C [c,d] x (c ) = a ( c n ) , xn(d ) = a ( t ) n n n i s t r u e with [cl,dl] = [ c , d ] ( c , d ).

4 . 2. Consider t h e ' following example on J = [-1,11,' 29 1 , METHODS = e-2(x+1) XI' where n INVOLVING DIFFERENTIAL INEQUALITIES - (x')2nJ x(-1) = 0 = x ( l ) , i s a p o s i t i v e integer. B ( t ) = 0. Choose N1 = - 2 Take and a ( t ) = t 2 - 1 and N2 = 2. 2 a r e s a t i s f i e d . t h a t any solution x ( t ) Ixr(t)(5 2 on J. such t h a t Hence we conclude t 2 -15 x ( t ) _< 0 However, we notice t h a t f o r n satisfies > * J Corollary 1 . 4 . 1 is not applicable. 2 which i s more useful i s t h e following r e s u l t .

Furthermore, there i s an Nn > 0 such t h a t I x t ( t ) l < Nn on [ a , a + n l f o r any solution satisfying a ( t ) 5 x ( t ) 5 p ( t ) on > 1, x,(t) is a solution on [a,a+n]. Thus f o r any fixed n [a,a+nI verifying a ( t ) 5 x,(t) 5 p ( t ) and Ix;(t)l 5 N~ on [ a , a + n ] f o r a l l m > n. Consequently, f o r m i n the sequences {xm(t)], {x;(t)] are both uniformly bounded and equicontinuous on [a, a + n ] Then, employing the standard diagonalization arguments, we obtain a subsequence which converges uniformly on a l l compact subintervals of [ a , ~ ) t o a solution x ( t ) .

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

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