American Institute of Mathematical Sciences

Advanced
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

Rubén Figueroa , Rodrigo López Pouso and  Jorge Rodríguez–López ,

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  February 2020 Early access  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.

Keywords: Discontinuous differential equations, upper and lower solutions, degree theory.
Mathematics Subject Classification: Primary: 34A36, 34B15; Secondary: 47H11.
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

show all references

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

[1]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180
[2]

Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315
[3]

Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89
[4]

Chunyan Ji, Yang Xue, Yong Li. Periodic solutions for SDEs through upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4737-4754. doi: 10.3934/dcdsb.2020122
[5]

João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217
[6]

Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575
[7]

Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014
[8]

Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3567-3582. doi: 10.3934/dcds.2013.33.3567
[9]

Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103
[10]

Józef Banaś, Monika Krajewska. On solutions of semilinear upper diagonal infinite systems of differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 189-202. doi: 10.3934/dcdss.2019013
[11]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012
[12]

Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102
[13]

Cristopher Hermosilla. Stratified discontinuous differential equations and sufficient conditions for robustness. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4415-4437. doi: 10.3934/dcds.2015.35.4415
[14]

Christoph Kawan. Upper and lower estimates for invariance entropy. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 169-186. doi: 10.3934/dcds.2011.30.169
[15]

Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861
[16]

Alberto Cabada, João Fialho, Feliz Minhós. Non ordered lower and upper solutions to fourth order problems with functional boundary conditions. Conference Publications, 2011, 2011 (Special) : 209-218. doi: 10.3934/proc.2011.2011.209
[17]

Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784
[18]

Wenying Feng, Guang Zhang, Yikang Chai. Existence of positive solutions for second order differential equations arising from chemical reactor theory. Conference Publications, 2007, 2007 (Special) : 373-381. doi: 10.3934/proc.2007.2007.373
[19]

B. Coll, A. Gasull, R. Prohens. Center-focus and isochronous center problems for discontinuous differential equations. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 609-624. doi: 10.3934/dcds.2000.6.609
[20]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (193)
  • HTML views (142)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences