By Stephen R. Bernfeld
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Extra info for An Introduction to Nonlinear Boundary Value Problems
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 . Proof: Let f (t>XJX') such t h a t o(t) _< x ( t ) 5 B(t), t to E (a,bl be any solution o f x E C(2)[J,R1 such t h a t cp(a,x(a)) E J. 5 x *( a ) 5 q(a,x(a)) x" = and Suppose t h a t t h e r e e x i s t s a x'(to) > q ( t o , x ( t o ) ) . Define 30 15. EXISTENCE IN THE LARGE Then, i n some i n t e r v a l t o t h e l e f t of to.
1) has a solution x E C(*)[J,R] with a ( t ) c x ( t ) C p ( t ) and Ix'(t)l S N on J . 1 it i s possible t o obtain the existence of solutions on i n f i n i t e intervals. 1. 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).
2. Let f E C [ J x R x R , R ] , f(t,x,x') be nondecreasing in x for each (t,x') and satisfy If(t,x,yl)f(t,x,y2)l5~ Iy1-y21 for (t,x> E J X R and yljy2 E R. 4) has a solution. If f is strictly increasing in x, then show that the solution is unique. 7) which may be computed explicitly. Then, using monotony of f, show a,@ are lower, upper solutions. 4. 3. Verify that p(t) = t, a(t) = t 1 - 3t +2 are upper, lower solutions for the problem x'' = - 1x1' + t, x(1) = 0, x'(2) = 1, on J = [1,2]. 2 to this problem.