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Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities
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 |
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.
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. |
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. |
[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. |
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