Coincidence degree methods in almost periodic differential equations

Liangping Qi, Rong Yuan



We consider the existence of almost periodic solutions to differential equations by using coincidence degree theory. A new equivalent spectral condition for the compactness of integral operators on almost periodic function spaces is established. It is shown that semigroup conditions are crucial in applications.


Almost periodic solutions; coincidence degree theory; compact integral operators; spectra; semigroups

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A.I. Alonso, R. Obaya and R. Ortega, Differential equations with limit-periodic forcings, Proc. Amer. Math. Soc. 131 (2003), no. 3, 851–857.

J.O. Alzabut, G.T. Stamov and E. Sermutlu, Positive almost periodic solutions for a delay logarithmic population model, Math. Comput. Modelling 53 (2011), no. 1–2, 161–167.

A.S. Besicovitch, Almost Periodic Functions, Dover, New York, 1955.

J. Blot, Almost periodic solutions of forced second order Hamiltonian systems, Ann. Fac. Sci. Toulouse Math. (5) 12 (1991), no. 3, 351–363.

J. Blot, Almost periodically forced pendulum, Funkcial. Ekvac. 36 (1993), no. 2, 235–250.

W. Bogdanowicz, On the existence of almost periodic solutions for systems of ordinary nonlinear differential equations in Banach spaces, Arch. Rational Mech. Anal. 13 (1963), 364–370.

H. Bohr, On almost periodic functions and the theory of groups, Amer. Math. Monthly 56 (1949), 595–609.

W.A. Coppel, Almost periodic properties of ordinary differential equations, Ann. Mat. Pura Appl. (4) 76 (1967), 27–49.

C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, New York, 2009.

C. Corduneanu, A scale of almost periodic functions spaces, Differential Integral Equations 24 (2011), no. 1–2, 1–27.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.

A.M. Fink, Compact families of almost periodic functions and an application of the Schauder fixed-point theorem, SIAM J. Appl. Math. 17 (1969), no. 6, 1258–1262.

A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377, Springer, Berlin, 1974.

A.M. Fink and G. Seifert, Nonresonance conditions for the existence of almost periodic solutions of almost periodic systems, SIAM J. Appl. Math. 21 (1971), no. 2, 362–366.

R.E. Gaines and J.L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics, Vol. 568, Springer, Berlin, 1977.

Y. Hino, T. Naito, N.V. Mihn and J.S. Shin, Almost Periodic Solution of Differential Equations in Banach Spaces, Stability and Control: Theory, Methods and Applications, Vol. 15, Taylor & Francis, London, 2002.

T.W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer, New York, 1980.

B.M. Levitan and V.V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, 1982.

X. Li, Y. Li and C. He, Positive almost periodic solutions for a time-varying fishing model with delay, Int. J. Differential Equations 2011 (2011), 12 pp.

Y. Li and C. Wang, Positive almost periodic solutions for state-dependent delay Lotka–Volterra competition systems, Electron. J. Differential Equations 2012 (2012), no. 91, 10 pp.

Y. Li, L. Yang and W. Wu, Almost periodic solutions for a class of discrete systems with Allee-effect, Appl. Math. 59 (2014), no. 2, 191–203.

Y. Li and Y. Ye, Multiple positive almost periodic solutions to an impulsive nonautonomous Lotka–Volterra predator-prey system with harvesting terms, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), no. 11, 3190–3201.

Y. Li and K. Zhao, Positive almost periodic solutions of non-autonomous delay competitive systems with weak Allee effect, Electron. J. Differential Equations 2009 (2009), no. 100, 11 pp.

J. Liu and C. Zhang, Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations, Adv. Difference Equ. 11 (2013), 21 pp.

J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636.

J. Mawhin, Leray–Schauder degree: a half century of extensions and applications, Topol. Methods Nonlinear Anal. 14 (1999), no. 2, 195–228.

G.M. N’Guerekata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001.

R. Ortega, Degree theory and almost periodic problems, In: Differential Equations, Chaos and Variational Problems, Progr. Nonlinear Differential Equations Appl., Vol. 75, Birkhäuser, Basel, 2008, 345–356.

R. Ortega, The pendulum equation: from periodic to almost periodic forcings, Differential Integral Equations 22 (2009), no. 9–10, 801–814.

W. Rudin, Functional Analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991.

W.M. Schmidt, Diophantine Approximation, Lecture Notes in Mathematics, Vol. 785, Springer, Berlin, 1980.

L. Wang and M. Yu, Favard’s theorem of piecewise continuous almost periodic functions and its application, J. Math. Anal. Appl. 413 (2014), no. 1, 35–46.

L. Wang and H. Zhang, Almost periodic solution of an impulsive multispecies logarithmic population model, Adv. Difference Equ. 96 (2015), 11 pp.

Y. Xie and X. Li, Almost periodic solutions of single population model with hereditary effects, Appl. Math. Comput. 203 (2008), no. 2, 690–697.

R. Yuan, Existence of almost periodic solutions of neutral functional-differential equations via Liapunov–Razumikhin function, Z. Angew. Math. Phys. 49 (1998), no. 1, 113–136.

R. Yuan, On Favard’s theorems, J. Differential Equations 249 (2010), no. 8, 1884–1916.

C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press, Beijing, 2003.

Q. Zhou and J. Shao, A note on Arzela–Ascoli’s lemma in almost periodic problems, Math. Methods Appl. Sci. 40 (2017), no. 1, 274–278.


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