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

By Stephen R. Bernfeld

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Extra resources for An Introduction to Nonlinear Boundary Value Problems

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Let hypotheses ( i ) and ( i i ) of Lemma 1 . 3 . 1 Then there exists a solution x hold. 1. Since f* is bounded, F is also bounded. 4). 21 We now show that 1. 5). Assume t h a t x(a) > @(a) i n which case zero. Then there exists a [a,6). 8). 6) implies, because of the assumption x ( a ) _> p(a), t h a t x ' ( a ) 2 @ ' ( a ) . We claim that there i s a to E [a,6) such that x'(t,) on a 5 t 5 - @ ' ( t o )> 0 . 6. 9), x"(t) - D-B' x'(t) - p'(t)IO we would obtain ( t ) _> L[X' ( t ) - B' (t)1 which, by the theory of d i f f e r e n t i a l inequalities, yields x'(t) - B'(t) 2 This, together with on a

Hence y(a) < @(a). < @(t> on JI in nonempty. Moreover, x(t) E n(d) implies that h(x(b), x'(b)) > 0 . However, inf[x(b) = d: x(t) E D] = do, which leads to a contradiction by a convergence argument and the fact that h(x,y) is nondecreasing in y. Hence the proof is complete. 1 which is the content of the next result. 5. 3 hold. 1), respectively, on J, such that a(&) = @(a) and cp(t,y(t)) 5 ~'(t)I +(t,r(t>), for y(t) 37 = a(t),B(t), t E J. 1. 1) has a solution x E C(2)[J,R] (t,x(t),x'(t>) where E n, t = c = B(a), the such t h a t E J, il i s t h e s e t given by R = [(t,x,x'): Proof: (t,x> E (0 and cp(t,x) Define a modified function 5 x' 5 Jr(t,x)l.

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. Discuss the merits. 1). 4) and yet the assumptions on f are just 33 1. 1). We s h a l l f i r s t prove t h e following. 2. 2) a(b) J _< d g(x,y) i s nondecreasing i n and g(a(a),a'(a)) 2 g(B(a),B'(a)) 0, It i s enough t o show t h a t given Proof: with a,B; C[[a(a),B(a)] xR,R], x, J _< f3(b), there i s a solution x(t,E) 5 0.