×

zbMATH — the first resource for mathematics

Existence result for a singular nonlinear boundary value problem at resonance. (English) Zbl 1134.34013
Summary: We study the existence of solutions of the second-order boundary value problem
\[ \begin{aligned} &u''(t)+ \pi^2u(t)+ a(t)g(u(t))= h(t) \quad \text{a.e. }t\in (0,1),\\ &u'\in AC_{\text{loc}}(0,1),\\ &u(0)=u(1)=0, \end{aligned} \]
where \(g:\mathbb R\to\mathbb R\) is continuous, \(a,h\in\{z\in L_{\text{loc}}^1(0,1)\mid \int_0^1 t|z(t)|\,dt< \infty\}\). The proof of the main result is based upon the Lyapunov-Schmidt procedure and the connectivity properties of the solution set of parametrized families of compact vector fields.

MSC:
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal, R.; O’Regan, D., Singular differential and integral equations with applications, (2003), Kluwer Academic Publishers London
[2] Agarwal, R.; O’Regan, D., Some new results for singular problems with sign changing nonlinearities. fixed point theory with applications in nonlinear analysis, J. comput. appl. math., 113, 1-2, 1-15, (2000) · Zbl 0943.34014
[3] Ahmad, S., Multiple nontrivial solutions of resonant and non-resonant asymptotically linear problems, Proc. amer. math. soc., 96, 3, 405-409, (1986) · Zbl 0634.35029
[4] Ambrosetti, A.; Mancini, G., Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance. the case of the simple eigenvalue, J. differential equations, 28, 2, 220-245, (1978) · Zbl 0393.35032
[5] Asakawa, H., Nonresonant singular two-point boundary value problems, Nonlinear anal., TMA, 44, 6, 791-809, (2001) · Zbl 0992.34011
[6] Bobisud, L.E.; O’Regan, D., Positive solutions for a class of nonlinear singular boundary value problems at resonance, J. math. anal. appl., 184, 2, 263-284, (1994) · Zbl 0805.34019
[7] Costa, D.G.; Gonçalves, J.V.A., Existence and multiplicity results for a class of nonlinear elliptic boundary value problems at resonance, J. math. anal. appl., 84, 2, 328-337, (1981) · Zbl 0479.35037
[8] Gupta, C.P., Solvability of a boundary value problem with the nonlinearity satisfying a sign condition, J. math. anal. appl., 129, 2, 482-492, (1988) · Zbl 0638.34015
[9] Habets, P.; Zanolin, F., Upper and lower solutions for a generalized emden – fowler equation, J. math. anal. appl., 181, 3, 684-700, (1994) · Zbl 0801.34029
[10] Habets, P.; Zanolin, F., Positive solutions for a class of singular boundary value problems, Boll. un. mat. ital., 9, 2, 273-286, (1995) · Zbl 0841.34017
[11] Natanson, I.P., Theory of functions of a real variable, vol. 2, (1955), Ungar New York, Translated Russian into Chinese by Xu Ruiyun · Zbl 0064.29102
[12] Landesman, E.M.; Lazer, A.C., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. math. mech., 19, 609-623, (1969/1970) · Zbl 0193.39203
[13] Iannacci, R.; Nkashama, M.N., Unbounded perturbations of forced second order ordinary differential equations at resonance, J. differential equations, 69, 3, 289-309, (1987) · Zbl 0627.34008
[14] Iannacci, R.; Nkashama, M.N., Nonlinear two-point boundary value problems at resonance without landesman – lazer condition, Proc. amer. math. soc., 106, 4, 943-952, (1989) · Zbl 0684.34025
[15] Ma, R., Multiplicity results for a third order boundary value problem at resonance, Nonlinear anal., 32, 4, 493-499, (1998) · Zbl 0932.34014
[16] Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025
[17] O’Regan, D., Singular Dirichlet boundary value problems. II. resonance case, Czechoslovak math. J., 48, 123, 269-290, (1998) · Zbl 0957.34016
[18] Taliaferro, S.D., A nonlinear singular boundary value problem, Nonlinear anal., 3, 6, 897-904, (1979) · Zbl 0421.34021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.