Existence results for the problem \((\varphi(u'))'= f(t,u,u')\) with nonlinear boundary conditions. (English) Zbl 0920.34029

The authors prove an existence result for problems of the form \[ (\phi(u'))'= f(t,u,u')\text{ a.e. }t\in [a,b],\;g(u(a), u'(a), u'(b))= 0,\;h(u(a))= u(b), \] where \(\phi\) is continuous and increasing from \(\mathbb{R}\) onto \(\mathbb{R}\). The main tools are the method of lower and upper solutions, the Nagumo condition and the Schauder fixed point theorem.


34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Cabada, A.; Pouso, R. L., Existence result for the problem \((φ(u\)′))′=\(f(t,u,u\)′) with periodic and Neumann boundary conditions, Nonlinear Anal. T.M.A., 30, 1733-1742 (1997) · Zbl 0896.34016
[2] DeCoster, C., Pairs of positive solutions for the one-dimensionalp-Laplacian, Nonlinear Anal. T.M.A., 23, 669-681 (1994) · Zbl 0813.34021
[3] Fabry, Ch.; Habets, P., Upper and lower solutions for second-order boundary value problems with nonlinear boundary conditions, Nonlinear Anal. T.M.A., 10, 985-1007 (1986) · Zbl 0612.34015
[4] Lloyd, N. G., Degree Theory (1978), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0367.47001
[5] McShane, E. J., Integration (1967), Princeton University Press: Princeton University Press Princeton · Zbl 0146.07202
[6] O’Regan, D., Some general principles and results for \((φ(y\)′))′=\( qf (t,y,y\)′),\(0<t<1\), SIAM J. Math. Anal., 24, 648-668 (1993) · Zbl 0778.34013
[7] O’Regan, D., Existence theory for \((φ(y′))′=qf(t,y,y′),0<t<1\), Commun. Appl. Anal., 1, 33-52 (1997) · Zbl 0887.34019
[8] Wang, M. X.; Cabada, A.; Nieto, J. J., Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions, Ann. Polon. Math., 58, 221-235 (1993) · Zbl 0789.34027
[9] Wang, J.; Gao, W., Existence of solutions to boundary value problems for a nonlinear second order equation with weak Carathéodory functions, Differential Equations and Dyn. Systems, 5, 2, 175-185 (1997) · Zbl 0891.34022
[10] Wang, J.; Gao, W.; Lin, Z., Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem, T^ohoku Math. J., 47, 327-344 (1995) · Zbl 0845.34038
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.