# American Institute of Mathematical Sciences

February  2020, 25(2): 617-633. doi: 10.3934/dcdsb.2019257

## 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

Received  February 2019 Revised  May 2019 Published  November 2019

Fund Project: 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

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.

Citation: Rubén Figueroa, Rodrigo López Pouso, Jorge Rodríguez–López. Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 617-633. doi: 10.3934/dcdsb.2019257
##### 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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. Figueroa, R. L. Pouso and J. Rodríguez-López, Degree theory for discontinuous operators, Fixed Point Theory, accepted. Google Scholar [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.  Google Scholar [8] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar [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.  Google Scholar [10] I. Rachůnková, Upper and lower solutions and multiplicity results, J. Math. Anal. Appl., 246 (2000), 446-464.  doi: 10.1006/jmaa.2000.6798.  Google Scholar [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.  Google Scholar [12] I. Rachůnková and M. Tvrdý, Impulsive periodic boundary value problem and topological degree, Functional Differential Equations, Israel Seminar, 9 (2002), 471-498.   Google Scholar [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.   Google Scholar [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.   Google Scholar [15] H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th Ed., Boston, Prentice Hall, 2010. Google Scholar [16] B. Rudolf, An existence and multiplicity result for a periodic boundary value problem, Math. Bohem., 133 (2008), 41-61.   Google Scholar [17] J. R. L. Webb, On degree theory for multivalued mappings and applications, Bolletino Un. Mat. Ital., 9 (1974), 137-158.   Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. Figueroa, R. L. Pouso and J. Rodríguez-López, Degree theory for discontinuous operators, Fixed Point Theory, accepted. Google Scholar [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.  Google Scholar [8] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar [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.  Google Scholar [10] I. Rachůnková, Upper and lower solutions and multiplicity results, J. Math. Anal. Appl., 246 (2000), 446-464.  doi: 10.1006/jmaa.2000.6798.  Google Scholar [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.  Google Scholar [12] I. Rachůnková and M. Tvrdý, Impulsive periodic boundary value problem and topological degree, Functional Differential Equations, Israel Seminar, 9 (2002), 471-498.   Google Scholar [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.   Google Scholar [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.   Google Scholar [15] H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th Ed., Boston, Prentice Hall, 2010. Google Scholar [16] B. Rudolf, An existence and multiplicity result for a periodic boundary value problem, Math. Bohem., 133 (2008), 41-61.   Google Scholar [17] J. R. L. Webb, On degree theory for multivalued mappings and applications, Bolletino Un. Mat. Ital., 9 (1974), 137-158.   Google Scholar [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.  Google Scholar
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