Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions

Rubén Figueroa; Rodrigo López Pouso; Jorge Rodríguez–López

doi:10.3934/dcdsb.2019257

Article Contents

2020, Volume 25, Issue 2: 617-633. Doi: 10.3934/dcdsb.2019257

This issue Previous Article Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities Next Article Stability for one-dimensional discrete dynamical systems revisited

Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions

Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782, Facultade de Matemáticas, Campus Vida, Santiago, Spain

^* Corresponding author
^* Corresponding author

Received: January 31, 2019

Revised: April 30, 2019

Early access: November 2019

Published: February 2020

Rodrigo López Pouso was partially supported by Ministerio de Economía y Competitividad, Spain, and FEDER, Project MTM2016-75140-P, and Xunta de Galicia ED341D R2016/022 and GRC2015/004. Jorge Rodríguez-López was financially supported by Xunta de Galicia Scholarship ED481A-2017/178

Abstract Full Text(HTML) Related Papers Cited by

Abstract

We present existence and multiplicity principles for second–order discontinuous problems with nonlinear functional conditions. They are based on the method of lower and upper solutions and a recent extension of the Leray–Schauder topological degree to a class of discontinuous operators.

Keywords:

Mathematics Subject Classification: Primary: 34A36, 34B15; Secondary: 47H11.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal., 11 (1972), 346-384. doi: 10.1016/0022-1236(72)90074-2.
[2]	A. Cabada and R. L. Pouso, Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions, Nonlinear Analysis, 42 (2000), 1377-1396. doi: 10.1016/S0362-546X(99)00158-3.
[3]	A. Cellina and A. Lasota, A new approach to the definition of topological degree for multivalued mappings, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur., 47 (1969), 434-440.
[4]	C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering, 205. Elsevier B. V., Amsterdam, 2006.
[5]	C. De Coster and S. Nicaise, Lower and upper solutions for elliptic problems in nonsmooth domains, J. Differential Equations, 244 (2008), 599-629. doi: 10.1016/j.jde.2007.08.008.
[6]	R. Figueroa, R. L. Pouso and J. Rodríguez-López, Degree theory for discontinuous operators, Fixed Point Theory, accepted.
[7]	R. Figueroa, R. L. Pouso and J. Rodríguez-López, Extremal solutions for second-order fully discontinuous problems with nonlinear functional boundary conditions, Electron. J. Qual. Theory Differ. Equ., (2018), 14 pp.
[8]	A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.
[9]	R. López Pouso, Schauder's fixed-point theorem: New applications and a new version for discontinuous operators, Bound. Value Probl., (2012), Art. ID 2012: 92, 14 pp. doi: 10.1186/1687-2770-2012-92.
[10]	I. Rachůnková, Upper and lower solutions and multiplicity results, J. Math. Anal. Appl., 246 (2000), 446-464. doi: 10.1006/jmaa.2000.6798.
[11]	I. Rachůnková and M. Tvrdý, Existence results for impulsive second order periodic problems, Nonlinear Anal., 59 (2004), 133-146. doi: 10.1016/j.na.2004.07.006.
[12]	I. Rachůnková and M. Tvrdý, Impulsive periodic boundary value problem and topological degree, Functional Differential Equations, Israel Seminar, 9 (2002), 471-498.
[13]	I. Rachůnková and M. Tvrdý, Non-ordered lower and upper functions in second order impulsive periodic problems, Dyn. Contin. Discrete Impuls. Syst., 12 (2005), 397-415.
[14]	I. Rachůnková and M. Tvrdý, Periodic problems with $\phi$-Laplacian involving non-ordered lower and upper functions, Fixed Point Theory, 6 (2005), 99-112.
[15]	H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th Ed., Boston, Prentice Hall, 2010.
[16]	B. Rudolf, An existence and multiplicity result for a periodic boundary value problem, Math. Bohem., 133 (2008), 41-61.
[17]	J. R. L. Webb, On degree theory for multivalued mappings and applications, Bolletino Un. Mat. Ital., 9 (1974), 137-158.
[18]	X. Xian, D. O'Regan and R. P. Agarwal, Multiplicity results via topological degree for impulsive boundary value problems under non-well-ordered upper and lower solution conditions, Bound. Value Probl., (2008), Art. ID 197205, 21 pp. doi: 10.1155/2008/197205.