An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators
KeywordsCyclic feedback systems, coincidence degree, periodic solutions, continuation theorems, $\phi$-Laplacian operators
AbstractUsing Mawhin's coincidence degree theory, we obtain some new continuation theorems which are designed to have as a natural application the study of the periodic problem for cyclic feedback type systems. We also discuss some examples of vector ordinary differential equations with a $\phi$-Laplacian operator where our results can be applied. Our main contribution in this direction is to obtain a continuation theorem for the periodic problem associated with $(\phi(u'))' + \lambda k(t,u,u') = 0$, under the only assumption that $\phi$ is a homeomorphism.
T. Bartsch and J. Mawhin, The Leray–Schauder degree of S 1 -equivariant operators associated to autonomous neutral equations in spaces of periodic functions, J. Differential Equations 92 (1991), 90–99.
P. Benevieri, J.M. do Ó and E.S. de Medeiros, Periodic solutions for nonlinear systems with mean curvature-like operators, Nonlinear Anal. 65 (2006), 1462–1475.
P. Benevieri, J.M. do Ó and E.S. de Medeiros, Periodic solutions for nonlinear equations with mean curvature-like operators, Appl. Math. Lett. 20 (2007), 484–492.
C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded φ-Laplacians, J. Dynam. Differential Equations 22 (2010), 463–471.
A. Boscaggin, G. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree, Trans. Amer. Math. Soc. (to appear).
R.F. Brown, A Topological Introduction to Nonlinear Analysis, 3rd edition, Springer, Cham, 2014.
A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992), 41–72.
A. Capietto, D. Qian and F. Zanolin, Periodic solutions for differential systems of cyclic feedback type, Differential Equations Dynam. Systems 7 (1999), 99–120.
C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper solutions, Math. Sci. Eng., vol. 205, Elsevier, Amsterdam, 2006.
G. Feltrin and F. Zanolin, Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree, J. Differential Equations 262 (2017), 4255–4291.
S. Fučı́k, J. Nečas, J. Souček and V. Souček, Spectral Analysis of Nonlinear Operators, Lecture Notes in Mathematics, vol. 346, Springer, Berlin, 1973.
R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 568, Springer, Berlin, 1977.
P. Hartman, On boundary value problems for systems of ordinary, nonlinear, second order differential equations, Trans. Amer. Math. Soc. 96 (1960), 493–509.
H.-W. Knobloch, On the existence of periodic solutions for second order vector differential equations, J. Differential Equations 9 (1971), 67–85.
Q. Liu, D. Qian and B. Chu, Nonlinear systems with singular vector φ-Laplacian under the Hartman-type condition, Nonlinear Anal. 74 (2011), 2880–2886.
S. Lu and M. Lu, Periodic solutions for a prescribed mean curvature equation with multiple delays, J. Appl. Math. (2014), Art. ID 909252.
J. Mallet-Paret and H.L. Smith, The Poincaré–Bendixson theorem for monotone cyclic feedback systems, J. Dynam. Differential Equations 2 (1990), 367–421.
R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with pLaplacian-like operators, J. Differential Equations 145 (1998), 367–393.
R. Manásevich and J. Mawhin, Periodic solutions for some nonlinear systems with pLaplacian like operators, Proceedings of the IV Catalan Days of Applied Mathematics (Tarragona, 1998), Univ. Rovira Virgili, Tarragona, 1998, pp. 103–122.
R. Manásevich and J. Mawhin, Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators, J. Korean Math. Soc. 37 (2000), 665–685.
J. Mawhin, Équations intégrales et solutions périodiques des systèmes différentiels non linéaires, Acad. Roy. Belg. Bull. Cl. Sci. 55 (1969), 934–947.
J. Mawhin, The solvability of some operator equations with a quasibounded nonlinearity in normed spaces, J. Math. Anal. Appl. 45 (1974), 455–467.
J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 40, American Mathematical Society, Providence, 1979.
J. Mawhin, The Bernstein–Nagumo problem and two-point boundary value problems for ordinary differential equations, Qualitative Theory of Differential Equations, Vol. I, II (Szeged, 1979), Colloq. Math. Soc. János Bolyai, vol. 30, North-Holland, Amsterdam, 1981, pp. 709–740.
J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Mathematics, vol. 1537, Springer, Berlin, 1993, pp. 74–142.
J. Mawhin, Some boundary value problems for Hartman-type perturbations of the ordinary vector p-Laplacian, Nonlinear Anal. 40 (2000), 497–503.
J. Mawhin, Periodic solutions of systems with p-Laplacian-like operators, Nonlinear Analysis and its Applications to Differential Equations (Lisbon, 1998), Progr. Nonlinear Differential Equations Appl., vol. 43, Birkhäuser, Boston, 2001, pp. 37–63.
J. Mawhin, Reduction and continuation theorems for Brouwer degree and applications to nonlinear difference equations, Opuscula Math. 28 (2008), 541–560.
J. Mawhin, Resonance problems for some non-autonomous ordinary differential equations, Stability and Bifurcation Theory for Non-Autonomous Differential Equations, Lecture Notes in Mathematics, vol. 2065, Springer, Heidelberg, 2013, pp. 103–184.
J. Mawhin, Variations on some finite-dimensional fixed-point theorems, Ukrainian Math. J. 65 (2013), 294–301.
J. Mawhin and A. J. Ureña, A Hartman–Nagumo inequality for the vector ordinary p-Laplacian and applications to nonlinear boundary value problems, J. Inequal. Appl. 7 (2002), 701–725.
R.D. Nussbaum, The fixed point index and some applications, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 94, Presses de l’Université de Montréal, Montréal, 1985.
R.D. Nussbaum, The fixed point index and fixed point theorems, Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Mathematics, vol. 1537, Springer, Berlin, 1993, pp. 143–205.
F. Obersnel and P. Omari, The periodic problem for curvature-like equations with asymmetric perturbations, J. Differential Equations 251 (2011), 1923–1971.
N. Rouche and J. Mawhin, Équations différentielles ordinaires. Tome II: Stabilité et solutions périodiques, Masson et Cie, Éditeurs, Paris, 1973.
How to Cite
Number of views and downloads: 0
Number of citations: 0