Springer Online Journal Archives 1860-2000
Abstract We consider systems of differential equations (1)x″=g(t, x) together with boundary conditions of the form (2)x(0)=x 0,x(T)=x 1; (3)x(0)=Qx(T), x′(0)=Qx′(T); and (4)B 1 x(0)−B 2 x′(0)=0=C 1 x(T)+C 2 x(T). Herex=(x 1,...,x n )T andg=g(t, x) are realn-vectors andQ, B i ,C i ,i=1, 2, denoten×n matrices withQ nonsingular andB 2,C 2 positive definite. We examine the existence of solutions of (1) satisfying (3) or (4) and which also stay in a certain regionΩin(t, x) space. Conditions in terms of the Jacobian matrixG (t, x)=g x (t, x) and an auxiliary positive definite symmetric matrixP=P(t) ∈C 2 [0,T] are given which yield the existence of the desired solution of (1), (3) or (1), (4).
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