Abstract
In this paper, using the optimal control method, we deal with the following boundary value problemy″ + f(x, y)=0,y(0)=c,y(1)=d, under new nonresonance conditions of the form −A ≤f /′ y (x, y) ≤ β(x) ≤ B, where A > 0. We obtain the existence and uniqueness of solutions of the BVP (1).
Similar content being viewed by others
References
Bernfield, S. R. and Lakshmikantham, V.,An Introduction to Nonlinear Boundary Value Problem, Academic Press, New York 1974.
Birkhoff, G. D. and Rota, G. C.,Ordinary Differential Equations, New York 1969.
Gingold, H.,Uniqueness of solutions of boundary value problem of systems of ordinary differential equations, Pacific J. Math.75, 107–136 (1978).
Gingold, H.,Uniqueness criteria for second order nonlinear boundary value problems, J. Math. Analysis & Applications73, 392–410 (1980).
Gingold, H. and Gustafson, G.,Uniqueness for nth order de la Vallée Poussin boundary value problems, Applic. Analysis20, 201–220 (1985).
Hankerson, D. and Henderson, J.,Optimality for boundary value problems for Lipschitz equations, J. Diff. Eqs.77, 392–404 (1989).
Henderson, J.,Best interval lengths for boundary value problems for third order Lipschitz equations, SIAM J. Math. Analysis18, 293–305 (1987).
Henderson, J.,Boundary value problems for nth order Lipschitz equations, J. Math. Analysis & Applications134, 196–210 (1988).
Henderson, J. and McGwier, R. Jr.,Uniqueness, existence, and optimality for fourth-order Lipschitz equations, J. Diff. Eqs.67, 414–440 (1987).
Jackson, L.,Existence and uniqueness of solutions of boundary value problems for Lipschitz equations, J. Diff. Eqs.32, 76–90 (1979).
Jackson, L.,Boundary value problems for Lipschitz equations, InDifferential Equations, eds. S. Ahmed, M. Keener and A. Lazer, Academic Press, New York 1980, pp. 31–50.
Kiguradze, I. T. and Lomtatidze, A. G.,On certain boundary value problems for second-order linear ordinary differential equations with singularities, J. Math. Analysis & Applications101, 325–347 (1981).
Lazer, A. C. and Leach, D. E.,On a nonlinear two-point boundary value problem, J. Math. Analysis & Applications26, 20–27 (1969).
Lees, M.,Discrete methods for nonlinear two-point boundary value problems, InNumerical Solutions of Partial Differential Equations, ed. J. H. Bramble, Academic Press, New York 1966, pp. 59–72.
Melentsova, Yu.,A best possible estimate of the non-oscillation interval for a linear differential equation with coefficients bounded in Lr, Differential'nye Uravneniya13, 1776–1786 (1987) (Russian); English transl., Differential Equations13, 1236–1244 (1977).
Melentsova, Yu. and Mil'shtein, G.,An optimal estimate of the interval on which a multipoint boundary value problem possesses a solution, Differentsial'nye Uravneniya10, 1630–1641 (1974) (Russian); English transl., Differential Equations10, 1257–1265 (1974).
Melentsova, Yu. and Mil'shtein, G.,Optimal estimation of the nonoscillation interval for linear differential equations with boundary coefficients, Differentsial'nye Uravneniya (Russian); English transl., Differential Equations17, 1368–1379 (1981).
Tippett, J.,An existence-uniqueness theorem for two-point boundary value problems, SIAM J. Math. Analysis5, 153–157 (1974).
Troch, J.,On the interval of disconjugacy of linear autonomous differential equations, SIAM J. Math. Analysis12, 78–89 (1981).
Wang, Huaizhong,Existence and uniqueness of solution for the two-point boundary value problem of second order differential equation y″ +f(x, y)=0, Acta Scientiarum Naturalium Universitatis Jilinensis2, 45–52 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wang, H.Z., Li, Y. Existence and uniqueness of solutions to two point boundary value problems for ordinary differential equations. Z. angew. Math. Phys. 47, 373–384 (1996). https://doi.org/10.1007/BF00916644
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00916644