By Serban Stratila and Laszlo Zsido
ISBN-10: 0856261092
ISBN-13: 9780856261091
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15. 3) has constant coefficients. Œc; d ; R/. t; s/ 2 Œc; d Œc; d : Proof. t c/Ca; . d c d c à c/Ca  b d a . 5 Lower and Upper Solutions The method of lower and upper solutions is a classical tool in the theory of nonlinear boundary value problems. It allows us to ensure the existence of a solution of the considered problem lying between a pair of ordered functions that satisfy some 50 1 Green’s Functions in the Theory of Ordinary Differential Equations suitable inequalities. In particular, we have information not only about the existence of solutions but also about the location of some of them.
B a/ . Define, m aj C1 aj for every j D 0; : : : ; m 1 and t 2 . Then x W J ! xN 1 ; : : : ; xN m /T W J1 ! a/ N C CN x. m 1/ 0 In ! m 1/ 0 Cn 0 ! and 0 1 h B0C B C hN D B : C : @ :: A 0 Proof. The proof follows as a straightforward change of variables and by taking into t u account that x must be a continuous function at points ai ; i D 1; : : : ; m 1. 24. Notice that the matrix function GN W J1 J1 ! 18) gives the values of the function xN on J1 . Once we have obtained the expression of the vector x, N then for any t 2 J , we know that there is j 2 f0; : : : ; m 1g such that t 2 Œaj ; aj C1 .
C; d ; R/. t; s/ 2 Œc; d Œc; d : Proof. t c/Ca; . d c d c à c/Ca  b d a . 5 Lower and Upper Solutions The method of lower and upper solutions is a classical tool in the theory of nonlinear boundary value problems. It allows us to ensure the existence of a solution of the considered problem lying between a pair of ordered functions that satisfy some 50 1 Green’s Functions in the Theory of Ordinary Differential Equations suitable inequalities. In particular, we have information not only about the existence of solutions but also about the location of some of them.