Transversality conditions for the existence of solutions of first-order discontinuous functional differential equations
DOI:
https://doi.org/10.12775/TMNA.2021.001Keywords
Discontinuous differential equation, functional differential equation, monotone iterative methodAbstract
We are concerned with the existence of extremal solutions to a large class of first-order functional differential problems under weak regularity assumptions. Our technique involves multivalued analysis and the method of lower and upper solutions in order to obtain a new existence result to a scalar Cauchy problem. As a consequence of this result and a monotone iterative method for discontinuous operators, we derive our main existence result which is illustrated by several examples concerning well-known models: a generalized logistic equation or second-order problems in the presence of dry friction.References
J. Andres and L. Jüttner, Periodic solutions of discontinuous differential systems, Nonlinear Anal. Forum 6 (2001), no. 2, 391–407.
V.I. Bogachev, Measure Theory, Vol. I, Springer, 2007.
D. Bothe, Periodic solutions of non-smooth friction oscillators, Z. Angew. Math. Phys. 50 (1999), no. 5, 779–808.
A. Bressan and W. Shen, On discontinuous differential equations, Differential Inclusions and Optimal Control (J. Andres, L. Górniewicz and P. Nistri, eds.), Julius Schauder Center, Lect. Notes Nonlinear Anal. 2 (1998), 73–87.
A. Cabada, An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl. 2011 (2011), 893753.
A. Cabada, M. Grossinho and F. Minhós, On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions, J. Math. Anal. Appl. 285 (2003), no. 1, 174–190.
J.Á. Cid, On extending existence theory from scalar ordinary differential equations to infinite quasimonotone systems of functional equations, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2661–2670.
J.Á. Cid and R. López Pouso, Ordinary differential equations and systems with timedependent discontinuity sets, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 4, 617–637.
C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, vol. 205, Elsevier, 2006.
R. Figueroa and R. López Pouso, Discontinuous first-order functional boundary value problems, Nonlinear Anal. 69 (2008), 2142–2149.
R. Figueroa, R. López Pouso and J. Rodrı́guez-López, Extremal solutions for secondorder fully discontinuous problems with nonlinear functional boundary conditions, Electron. J. Qual. Theory Differ. Equ. 29 (2018), 1–14.
D. Franco and R. López Pouso, Nonresonance conditions and extremal solutions for first–order impulsive problems under weak assumptions, ANZIAM J. 44 (2003), 393–407.
L. Górniewicz, Topological fixed point theory of multivalued mappings, Springer, 2 Ed. (2006).
S. Heikkilä and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, New York, 1994.
S. Hu, Differential equations with discontinuous right-hand sides, J. Math. Anal. Appl. 154 (1991), 377–390.
R. López Pouso and J. Rodrı́guez-López, Existence and uniqueness of solutions for systems of discontinuous differential equations under localized Bressan–Shen transversality conditions, J. Math. Anal. Appl. 492 (2020), 124425, 16 pp.
R. López Pouso and J. Tomeček, First- and second-order discontinuous functional differential equations with impulses at fixed moments, Nonlinear Anal. 67 (2007), 455–467.
R. Ma, Existence of periodic solutions of a generalized friction oscillator, Nonlinear Anal. 11 (2010), 3316–3322.
R. Precup and J. Rodrı́guez-López, Positive solutions for discontinuous problems with applications to φ-Laplacian equations, J. Fixed Point Theory Appl. 20 (2018), no. 156, 1–17.
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