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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Existence of solutions of ordinary differential equations with generalized boundary conditions


Authors: Stephen R. Bernfeld and V. Lakshmikantham
Journal: Trans. Amer. Math. Soc. 182 (1973), 261-274
MSC: Primary 34B15
DOI: https://doi.org/10.1090/S0002-9947-1973-0326043-2
MathSciNet review: 0326043
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Abstract: An investigation of the existence of solutions of the nonlinear boundary value problem $ x' = f(t,x,y),y' = g(t,x,y),AV(a,x(a),y(a)) + BW(a,x(a),y(a)) = {C_1},CV(b,x(b),y(b)) + DW(b,x(b),y(b)) = {C_2}$, is made. Here we assume $ g,f:[a,b] \times {R^p} \times {R^q} \to {R^p}$ are continuous, and $ V,W:[a,b] \times {R^p} \times {R^q} \to R$ are continuous and locally Lipschitz. The main techniques used are the theory of differential inequalities and Lyapunov functions.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0326043-2
Keywords: Ordinary differential equations, boundary value problems, comparison theorem, differential inequalities, Lyapunov functions
Article copyright: © Copyright 1973 American Mathematical Society