Previous |  Up |  Next


Dirichlet problem
We investigate the existence and stability of solutions for higher-order two-point boundary value problems in case the differential operator is not necessarily positive definite, i.e. with superlinear nonlinearities. We write an abstract realization of the Dirichlet problem and provide abstract existence and stability results which are further applied to concrete problems.
[1] A. Boucherif, Nawal Al-Malki: Solvability of a two point boundary value problem. Int. J.  Differ. Equ. Appl. 8 (2003), 129–135. MR 2068777
[2] D. Delbosco: A two point boundary value problem for a second order differential equation with quadratic growth in the derivative. Differ. Integral Equ. 16 (2003), 653–662. MR 1973273 | Zbl 1048.34044
[3] G. Dinca, P. Jeblean: Some existence results for a class of nonlinear equations involving a duality mapping. Nonlinear Anal., Theory Methods Appl. 46 (2001), 347–363. MR 1851857
[4] I. Ekeland, R. Temam: Convex Analysis and Variational Problems. North-Holland, Amsterdam, 1976. MR 0463994
[5] M. Galewski: New variational principle and duality for an abstract semilinear Dirichlet problem. Ann. Pol. Math. 82 (2003), 51–60. MR 2041397
[6] M. Galewski: Stability of solutions for an abstract Dirichlet problem. Ann. Pol. Math. 83 (2004), 273–280. MR 2111714 | Zbl 1097.47053
[7] M. Galewski: The existence of solutions for a semilinear abstract Dirichlet problem. Georgian Math. J. 11 (2004), 243–254. MR 2084987 | Zbl 1083.47048
[8] D. Idczak: Stability in semilinear problems. J.  Differ. Equations 162 (2000), 64–90. MR 1741873 | Zbl 0952.35050
[9] D. Idczak, A. Rogowski: On a generalization of Krasnoselskii’s theorem. J.  Aust. Math. Soc. 72 (2002), 389–394. MR 1902207
[10] T. Kato: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin-Heidelberg-New York, 1980. Zbl 0435.47001
[11] Y. Li: Positive solutions of fourth order periodic boundary value problem. Nonlinear Anal., Theory Methods Appl. 54 (2003), 1069–1078. MR 1993312
[12] Y. Liu, W. Ge: Solvability of a two point boundary value problem at resonance for high-order ordinary differential equations. Math. Sci. Res. J., 7 (2003), 406–429. MR 2020490
[13] Y. Liu, W. Ge: Solvability of a two point boundary value problems for fourth-order nonlinear differential equations at resonance. Z.  Anal. Anwend. 22 (2003), 977–989. MR 2036940
[14] A. Lomtatidze, L. Malaguti: On a two-point boundary value problem for the second order ordinary differential equations with singularities. Nonlinear Anal., Theory Methods Appl. 52 (2003), 1553–1567. MR 1951507
[15] J. Mawhin: Problemes de Dirichlet variationnels non linéaires. Presses Univ. Montréal, Montréal, 1987. (French) MR 0906453 | Zbl 0644.49001
[16] A. Nowakowski, A. Rogowski: Dependence on parameters for the Dirichlet problem with superlinear nonlinearities. Topol. Methods Nonlinear Anal. 16 (2000), 145–130. MR 1805044
[17] A. Nowakowski, A. Rogowski: On the new variational principles and duality for periodic of Lagrange equations with superlinear nonlinearities. J.  Math. Anal. Appl. 264 (2001), 168–181. MR 1868335
[18] D. R.  Smart: Fixed Point Theorems. Cambridge University Press, London-New York, 1974. MR 0467717 | Zbl 0297.47042
[19] S. Walczak: On the continuous dependence on parameters of solutions of the Dirichlet problem. Part I.  Coercive case, Part  II. The case of saddle points. Bull. Cl. Sci., VII.  Sér., Acad. R. Belg. 6 (1995), 247–273. MR 1427337
[20] S. Walczak: Continuous dependence on parameters and boundary data for nonlinear P.D.E. coercive case. Differ. Integral Equ. 11 (1998), 35–46. MR 1607976 | Zbl 1042.35004
Partner of
EuDML logo