×

Existence theory for functional \(p\)-Laplacian equations with variable exponents. (English) Zbl 1029.34018

The authors consider the solvability of functional \(p\)-Laplacian equations subjected to general boundary conditions, which include as particular cases the Dirichlet and periodic conditions, and also cover a wide class of nonlinear conditions as well a functional ones. The existence of solutions to problems with bounded nonlinear parts is proved. Illustrative examples are presented, too.

MSC:

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

References:

[1] Adje, A., Sur et sous-solutions généralisées et problèmes aux limites du second ordre, Bull. Soc. Math. Bel. Sér. B, 42, 347-368 (1990) · Zbl 0724.34018
[2] Cabada, A.; Habets, P.; Pouso, R. L., Optimal existence conditions for \(φ\)-Laplacian equations with upper and lower solutions in the reversed order, J. Differential Equations, 166, 2, 385-401 (2000) · Zbl 0999.34011
[3] Cabada, A.; Pouso, R. L., Existence result for the problem \((φ(u\)′))′=\(f(t,u,u\)′) with nonlinear boundary conditions, Nonlinear Anal., 35, 221-231 (1999) · Zbl 0920.34029
[4] Cabada, A.; Pouso, R. L., Extremal solutions of strongly nonlinear discontinuous second order equations with nonlinear functional boundary conditions, Nonlinear Anal., 42, 8, 1377-1396 (2000) · Zbl 0964.34016
[5] Cherpion, M.; De Coster, C.; Habets, P., Monotone Iterative methods for boundary value problems, Differential Integral Equations, 12, 3, 309-338 (1999) · Zbl 1015.34009
[6] De Coster, C., Pairs of positive solutions for the one-dimensional \(p\)-Laplacian, Nonlinear Anal., 23, 669-681 (1994) · Zbl 0813.34021
[7] Del Pino, M.; Elgueta, M.; Manásevich, R., A homotopic deformation along \(p\) of a Leray-Schauder degree result and existence for (|\(u\)′|\(^{p−2}u\)′)′+\(f(t,u)=0, u(0)=u(T)=0, p>1\), J. Differential Equations, 80, 1-13 (1989) · Zbl 0708.34019
[8] Garcı́a-Huidobro, M.; Manásevich, R.; Zanolin, F., A Fredholm-like result for strongly nonlinear second order ODE’s, J. Differential Equations, 144, 132-167 (1994) · Zbl 0835.34028
[9] S. Heikkilä, S. Seikkala, Maximum principles and uniqueness results for phi-Laplacian boundary value problems, J. Inequalities Appl. 6 (3) (2000) 339-357.; S. Heikkilä, S. Seikkala, Maximum principles and uniqueness results for phi-Laplacian boundary value problems, J. Inequalities Appl. 6 (3) (2000) 339-357. · Zbl 1002.34008
[10] Lloyd, N. G., Degree Theory (1978), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0367.47001
[11] Manásevich, R.; Mawhin, J., Periodic solutions for nonlinear systems with \(p\)-Laplacian-like operators, J. Differential Equations, 145, 2, 367-393 (1998) · Zbl 0910.34051
[12] Manásevich, R.; Zanolin, F., Time-mappings and multiplicity of solutions for the one-dimensional \(p\)-Laplacian, Nonlinear Anal., 21, 269-291 (1993) · Zbl 0792.34021
[13] McShane, E. J., Integration (1967), Princeton University Press: Princeton University Press Princeton · Zbl 0146.07202
[14] 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
[15] O’Regan, D., Existence theory for \((φ(y\)′))′=\( qf (t,y,y\)′), \(0<t<1\), Comm. Appl. Anal., 1, 33-52 (1997) · Zbl 0887.34019
[16] 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
[17] Wang, J.; Gao, W., Existence of solutions to boundary value problems for a nonlinear second order equation with weak Carathéodory functions, Differential Equations Dynamics Systems, 5, 2, 175-185 (1997) · Zbl 0891.34022
[18] Wang, J.; Gao, W.; Lin, Z., Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem, Tôhoku 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.