Abstract method of upper and lower solutions and application to singular boundary value problems


The method of upper and lower solutions is presented for the xed point problem associated to operators which are compositions of a linear operator and a nonlinear mapping. Spectral properties of the linear part together with growth and monotonicity properties of the nonlinear part are involved. An application to singular boundary value  problems is included.


Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania


Upper and lower solution; monotone iterative principle; fixed point; cone; positive solution; singular boundary value problem

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R. Precup, Abstract method of upper and lower solutions and application to singular boundary value problems, Studia Univ. Babes-Bolyai Math. 61 (2016), 443-451.


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Studia Universitatis Babes-Bolyai Mathematica

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Babeș-Bolyai University

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[1] Agarwal, R.P., O’Regan, D., Precup, R., Construction of upper and lower solutions with applications to singular boundary value problems, J. Comput. Anal. Appl., 7(2005), 205-221.
[2] Amman, H., Fixed point equations and nonlinear eigenvalues problems in ordered Banach spaces, SIAM Review, 18(1976), 620-709.
[3] Anderson, N., Arthurs, A.M., Analytic bounding functions for diffusion problems with Michaelis-Menten kinetics, Bull. Math. Biol., 47(1985), 145-153.
[4] Brezis, H., Analyse fonctionnelle, Masson, Paris, 1983.
[5] Cabada, A., An overview of the lower and upper solutions method with nonlinear boundary value conditions, Boundary Value Problems, 2011(2011), Article ID 893753, 18 pages.
[6] Carl, S., Heikkila, S., Nonlinear Differential Equations in Ordered Spaces, Chapman & Hall/CRC, Boca Raton, 2000.
[7] Chan, C.Y., Hon, Y.C., Computational methods for generalized Emden Fowler models of neutral atoms, Quart. Appl. Math., 46(1988), 711-726.
[8] De Coster, C., Habets, P., Two-Point Boundary Value Problems: Lower and Upper Solutions, Elsevier, Amsterdam, 2006.
[9] Deimling, K., Nonlinear Functional Analysis, Springer, Berlin, 1985.
[10] Guo, D., Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988.
[11] O’Regan, D., Theory of Singular Boundary Value Problems, World Scientific, Singapore, 1994.
[12] Precup, R., Monotone technique to the initial values problem for a delay integral equation from biomathematics, Studia Univ. Babe¸s-Bolyai Math., 40(1995), no. 2, 63-73.
[13] Precup, R., Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, 2002.
[14] Rus, I.A., Fixed points, upper and lower fixed points: abstract Gronwall lemmas, Carpathian J. Math., 20(2004), 125-134.

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