Abstract
In this work, we consider boundary-value problems of the form
, where the scalar function f(t, x, p, q) may be singular at x = 0. As far as we know, the solvability of the singular boundary-value problems of this form has not been treated yet. Here we try to fill in this gap. Examples illustrating our main result are included.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. P. Agarwal and D. O’Regan, “Boundary value problems with sign changing nonlinearities for second order singular ordinary differential equations,” Appl. Anal., 81, 1329–1346 (2002).
R. P. Agarwal, D. O’Regan, V. Lakshmikantham, and S. Leela, “An upper and lower solution theory for singular Emden-Fowler equations,” Nonlinear Anal.: Real World Appl., 3, 275–291 (2002).
R. P. Agarwal, D. O’Regan, and S. Stanek, “Singular Lidstone boundary value problem with given maximal values for solutions,” Nonlinear Anal., 55, 859–881 (2003).
R. P. Agarwal, D. O’Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht (1998).
L. E. Bobisud and Y. S. Lee, “Existence of monotone or positive solutions of second-order sublinear differential equations,” J. Math. Anal. Appl., 159, 449–468 (1991).
P. M. Fitzpatrick, “Existence results for equations involving noncompact perturbation of Fredholm mappings with applications to differential equations,” J. Math. Anal. Appl., 66, 151–177 (1978).
W. Ge and J. Mawhin, “Positive solutions to boundary value problems for second order ordinary differential equations with singular nonlinearities,” Results Math., 34, 108–119 (1998).
M. K. Grammatikopoulos, P. S. Kelevedjiev, and N. I. Popivanov, “On the solvability of a Neumann boundary value problem,” Nonlinear Anal., To appear.
A. Granas, R. B. Guenther, and J. W. Lee, Nonlinear Boundary Value Problems for Ordinary Differential Equations, Dissnes Math., Warszawa (1985).
Y. Guo, Y. Gao, and G. Zhang, “Existence of positive solutions for singular second order boundary value problems,” Appl. Math. E-Notes, 2, 125–131 (2002).
Q. Huang and Y. Li, “Nagumo theorems of nonlinear singular boundary value problems,” Nonlinear Anal., 29, 1365–1372 (1997).
D. Jiang, P. Y. H. Pang, and R. P. Agarwal, “Nonresonant singular boundary value problems for the one-dimensional p-Laplacian,” Dynam. Systems Appl., 11, 449–457 (2002).
P. Kelevedjiev, “Existence of positive solutions to singular second order boundary value problems,” Nonlinear Anal., 50, 1107–1118 (2002).
P. Kelevedjiev and N. Popivanov, “Existence of solutions of boundary value problems for the equation f(t, x, x′, x″) = 0 with fully nonlinear boundary conditions,” Annuaire Univ. Sofia Fac. Math. Inform., 94, 65–77 (2000).
H. Maagli and S. Masmoudi, “Existence theorems of nonlinear singular boundary value problems,” Nonlinear Anal., 46, 465–473 (2001).
Y. Mao and J. Lee, “Two-point boundary value problems for nonlinear differential equations,” Rocky Mauntain J. Math., 26, 1499–1515 (1996).
S. A. Marano, “On a boundary value problem for the differential equation f(t, x, x′, x″) = 0,” J. Math. Anal. Appl., 182, 309–319 (1994).
S. K. Ntouyas and P. K. Palamides, “The existence of positive solutions of nonlinear singular second-order boundary value problems,” Math. Comput. Modelling, 34, 641–656 (2001).
D. O’Regan, Theory of Singular Boundary Value Problems, World Scientific, Singapore (1994).
P. K. Palamides, “Boundary-value problems for shallow elastic membrane caps,” IMA J. Appl. Math., 67, 281–299 (2002).
I. Rachunková, “Singular Dirichlet second-order BVPs with impulses,” J. Differential Equations, 193, 435–459 (2003).
I. Rachunková and S. Stanek, “Sturm-Liouville and focal higher order BVPs with singularities in phase variables,” Georgian Math. J., 10, 165–191 (2003).
W. V. Petryshyn, “Solvability of various boundary value problems for the equation x″ = f(t, x, x′, x″) − y,” Pacific J. Math., 122, 169–195 (1986).
W. V. Petryshyn and P. M. Fitzpatrick, “Galerkin method in the constructive solvability of nonlinear Hammerstein equations with applications to differential equations,” Trans. Amer. Math. Soc., 238, 321–340 (1978).
W. V. Petryshyn and Z. S. Yu, “Periodic solutions of nonlinear second-order differential equations which are not solvable for the highest-order derivative,” J. Math. Anal. Appl., 89, 462–488 (1982).
W. V. Petryshyn and Z. S. Yu, “Solvability of Neumann BV problems for nonlinear second order ODE’s which need not be solvable for the highest order derivative,” J. Math. Anal. Appl., 91, 244–253 (1983).
A. Tineo, “Existence of solutions for a class of boundary value problems for the equation x″ = F(t, x, x′, x″),” Comment. Math. Univ. Carolin., 29, 285–291 (1988).
Z. Zhang and J. Wang, “On existence and multiplicity of positive solutions to singular multi-point boundary value problems,” J. Math. Anal. Appl., 295, 502–512 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 16, Differential and Functional Differential Equations. Part 2, 2006.
Rights and permissions
About this article
Cite this article
Grammatikopoulos, M.K., Kelevedjiev, P.S. & Popivanov, N.I. On the solvability of a singular boundary-value problem for the equation f(t, x, x′, x″) = 0. J Math Sci 149, 1504–1516 (2008). https://doi.org/10.1007/s10958-008-0079-z
Issue Date:
DOI: https://doi.org/10.1007/s10958-008-0079-z