An abstract averaging method with applications to differential equations

Abstract We present a general formulation of the averaging method in the setting of a semilinear equation L x = e N ( x , e ) , being L a linear Fredholm mapping of index zero. Our general approach provides new results even in the classical periodic framework. Among the applications we obtained there are: a partial answer to an open problem related to the Liebau phenomenon, the multiplicity of periodic solutions for a planar system with delay and the existence of solution for a nonlocal chemical reactor.

[1]  Douglas D. Novaes,et al.  Higher order averaging theory for finding periodic solutions via Brouwer degree , 2014 .

[2]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[3]  Gennaro Infante,et al.  Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory , 2014, Commun. Nonlinear Sci. Numer. Simul..

[4]  M. Zima,et al.  A topological approach to periodic oscillations related to the Liebau phenomenon , 2014, 1408.0130.

[5]  Jaume Llibre,et al.  Averaging methods for finding periodic orbits via Brouwer degree , 2004 .

[6]  Douglas D. Novaes,et al.  Improving the averaging theory for computing periodic solutions of the differential equations , 2015 .

[7]  L. Cesari.  Functional analysis and boundary value problems , 1971 .

[8]  P. Torres.  Mathematical Models with Singularities: A Zoo of Singular Creatures , 2015 .

[9]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[10]  F. Verhulst.  Nonlinear Differential Equations and Dynamical Systems , 1989 .

[11]  M. Zima,et al.  Lyapunov stability for regular equations and applications to the Liebau phenomenon , 2018 .

[12]  G. Propst.  Pumping effects in models of periodically forced flow configurations , 2006 .

[13]  J. Mawhin.  Periodic Solutions in the Golden Sixties: The Birth of a Continuation Theorem , 2005 .

[14]  M. Zima,et al.  New results for the Liebau phenomenon via fixed point index , 2016, 1606.03602.

[15]  J. Cronin.  Fixed points and topological degree in nonlinear analysis , 1995 .

[16]  R. Gaines,et al.  Coincidence Degree and Nonlinear Differential Equations , 1977 .

[17]  J. Mawhin.  Topological degree methods in nonlinear boundary value problems , 1979 .

[18]  P. Torres.  Existence and Stability of Periodic Solutions of a Duffing Equation by Using a New Maximum Principle , 2004 .

[19]  J. Hale.  Oscillations in Nonlinear Systems , 1963 .

[20]  J. '. Cid,et al.  On the pumping effect in a pipe/tank flow configuration with friction , 2014 .

[21]  Gerhart Liebau,et al.  Über ein ventilloses Pumpprinzip , 2004, Naturwissenschaften.

[22]  Fang-Fang Liao.  Periodic solutions of Liebau-type differential equations , 2017, Appl. Math. Lett..

[23]  M. Zima,et al.  A topological degree approach to a nonlocal Neumann problem for a system at resonance , 2019, Journal of Fixed Point Theory and Applications.

[24]  Shengjun Li,et al.  Lyapunov stability of periodic solutions of Brillouin type equations , 2020, Appl. Math. Lett..

[25]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.