On certain classes of mild solutions of the scalar Liénard equation revisited
Main Article Content
Abstract
In this work we revisit the existence, uniqueness and expo- nential decay of some classes of mild solutions which are almost periodic (AP-), asymptotically almost periodic (AAP-) and pseudo almost peri- odic (PAP-) of the scalar Lin ́eard equation by employing the notion of Green function and Massera-type principle. First, by changing variable we convert this equation to a system of first order differential equations. Then, we transform the problem into a framework of an abstract parabolic evolution equation which associates with an evolution family equipped an exponential dichtonomy and the corresponding Green function is exponen- tially almost periodic. After that, we prove a Massera-type principle that the corresponding linear equation has a unique AP-, AAP- and PAP- mild solution if the right hand side and the coefficient functions are AP-, AAP- and PAP- functions, respectively. The well-posedness of semilinear equa- tion is proved by using fixed point arguments and the exponential decay of mild solutions is established by using Gronwall’s inequality. Although our works revisits some previous works on well-posedness of these types of mild solutions for the Lin ́eard equation but provide a difference view by using Green function and go further on the aspects of asymptotic behaviour of solutions and the construction of abstract theory. Our abstract results can be also applied to other parabolic evolution equations.
Article Details
References
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[38] Y. Latushkin, T. Randolph and R. Schnaubelt, Exponential Di- chotomy and Mild Solutions of Nonautonomous Equations in Banach Spaces, Journal of Dynamics and Differential Equations 10, 489–510 (1998)
[39] J. Liu, G. M. N’Guerekata and Nguyen Van Minh, Stability and Periodicity for Evolution Equations, World Scientific, New Jersey-London- Singapore-Beijing-Shanghai-Hong Kong-Taipei-Chennai. (2008)
[40] L. Maniar and R. Schnaubelt, Almost periodicity of inhomoge- neous parabolic evolution equations, in Book: Evolution Equations, Edited By G.R. Goldstein, R. Nagel and S. Romanelli, First Published 2003, eBook Published 2019, Pub. Location Boca Raton, CRC Press. https://doi.org/10.1201/9780429187599
[41] T. Naito and N.V. Minh, Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations, J. Differential Equations 152 (1999), pages 358-376.
[42] W. M. Ruess and V.Q. Phong, Asymptotically almost periodic so- lutions of evolution equations in Banach spaces, J. Differential Equations 122(1995), pages 282-301.
[43] V.Q. Phong, Stability and almost periodicity of trajectories of periodic processes, J. Differential Equations 115 (1995), pages 402-415.
[44] L.Q. Peng and W.T. Wang, Positive almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument, Electron. J. Qual. Theory Differ. Equ., 49 (2010), 1-12.
[45] Zhu Huan-Ran, The Almost Periodic Solution on a Lienard Equation, Chinese Quarterly Journal of Mathematics, 2004, (2): 164 -171 .
[46] P.T. Xuan, N.T. Van and B. Quoc, On Asymptotically Almost Peri- odic Solution of Parabolic Equations on real hyperbolic Manifolds, J. Math. Anal. Appl., Vol. 517, Iss. 1 (2023), pages 1-19.
[47] P.T. Xuan, N.T. Van and T.V. Thuy, On Asymptotically Almost Periodic Solutions of the parabolic-elliptic Keller-Segel system on real hy- perbolic Manifolds, Dynamical System: An international journal (2024), 18 pages. https://doi.org/10.1080/14689367.2024.2360204
[48] C. Xu and M. Liao, Existence and uniqueness of pseudo almost peri- odic solutions for Li ́enard-type system with delays, Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 170, pp. 1-8.
[49] Y. Xu, Positive almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument, Electron. J. Qual. Theory Differ. Equ., 49 (2012), 1-9.
[50] A. Yagi, Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II, Funkcial. Ekvac. 33 (1990), 139-150.
[51] R. Yazgan, On the Pseudo Almost Automorphic Solutions for Li ́enard- Type Systems with Multiple Delays, Journal of Mathematical Extension, Vol. 16, No. 9, (2022), pages 1-13.
[52] R. Yazgan, On the weight pseudo almost periodic solutions for Li ́enard -type systems with variable delays, Mugla Journal of Science and Technol- ogy,(2020) 6. 10.22531/muglajsci.734586.
[53] C.Y. Zhang, Pseudo almost periodic solutions of some differential equa- tions, J. Math. Anal. Appl. 181 (1) (1994) 62–76.
[54] C.Y. Zhang, Pseudo almost periodic solutions of some differential equa- tions II, J. Math. Anal. Appl. 192 (2) (1995) 543–561.
[2] Amerio,L, Prouse,G Almost Periodic Functions and Functional Equa- tions, Springer, NewYork (1971)
[3] B. Amir and L. Maniar, Composition of pseudo-almost periodic func- tions and Cauchy problems with operator of nondense domain, Ann. Math. Blaise Pascal 6 (1) (1999), pages 1–11.
[4] W. Arendt and Ch.J.K. Batty, Asymptotically Almost Periodic Solu- tions of Inhomogeneous Cauchy Problems on The Half-Line, Bulletin of the London Mathematical Society, Vol. 31, Iss. 3 (1999), pages 291–304
[5] M. Baroun, S. Boulite, T. Diagana and L. Maniar, Almost periodic solutions to some semilinear nonautonomous thermoelastic plate equations, J. Math. Anal. Appl. 349 (1) (2009), pages 74–84.
[6] M. Baroun, S. Boulite, G.M. N’Guerekata ad L. Maniar, Almost automorphy of semilinear parabolic evolution equations, Electron. J. Dif- ferential Equation (60) (2008), pages 1–9.
[7] C.J.K. Batty, W. Hutter and F. R ̈abiger, Almost Periodicity of Mild Solutions of Inhomogeneous Periodic Cauchy Problems, Journal of Differ- ential Equations 156, 309327 (1999)
[8] Besicovitch, AS Almost Periodic Functions Dover, NewYork (1954).
[9] Bochner, S A new approach to almost periodicity ,Proc. Natl. Acad. Sci.
USA 48 2039-2043 (1962).
[10] H. Bohr, Zur Theorie der fastperiodischen Funktionen I, Acta Math. 45, 29-127 (1925)
[11] H. Bohr, Zur Theorie der fastperiodischen Funktionen II, Acta Math. 46, 101-204 (1925)
[12] H. Bohr, Zur Theorie der fastperiodischen Funktionen III, Acta Math. 47, 237-281 (1926)
[13] D.N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York (2009).
[14] T. Caraballo and D. Cheban, Almost periodic and asymptotically almost periodic solutions of Li ́enard equations, Discrete and Continuous Dynamical Systems - B, 2011, 16(3): 703-717.
[15] C. Cuevas and M. Pinto, Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with nondense domain, Nonlinear Anal. 45 (1) (2001), pages 73–83.
[16] P. Cieutat, On the structure of the set of bounded solutions on an almost periodic Li ́enard equation, Nonlinear Analysis, 58 (2004), 885-898
[17] E.H.A. Dads, P. Cieutat and L. Lachimi, Structure of the set of bounded solutions and existence of pseudo almost periodic solutions of a Lienard equation, Differential and Integral Equations, 20 (2007).
[18] P. Cieutat, Almost periodic solutions of forced vectorial Li ́enard equa- tions, Journal of Differential Equations, Vol. 209, Iss. 2 (2005), Pages 302- 328
[19] Ju.L. Daleckii, M.G. Krein, Stability of Solutions of Differential Equa- tions in Banach Spaces, Transl. Amer. Math. Soc., Providence, RI, 1974.
[20] T. Diagana , Almost periodic solutions for some higher-order nonau- tonomous differential equations with operator coefficients, Mathematical and Computer Modelling, 54(11-12), 2672–2685.
[21] T. Diagana, Almost Automorphic Type and Almost Periodic Type Func- tions in Abstract Spaces, Springer International Publishing Switzerland 2013, DOI 10.1007/978-3-319-00849-3.
[22] T. Diagana, E. Hernandez and M. Rabello, Pseudo almost peri- odic solutions to some nonautonomous neutral functional differential equa- tions with unbounded delay, Math. Comput. Model. 45 (9–10) (2007), pages 1241–1252.
[23] T. Diagana, Pseudo almost periodic solutions to some differential equa- tions, Nonlinear Anal. 60 (7) (2005), pages 1277–1286.
[24] T. Diagana and E. Hernandez, Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional- differential equations and applications, J. Math. Anal. Appl. 327 (2) (2007), pages 776–791.
[25] A. M. Fink, Almost-Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Vol. 377, 1974.
[26] A.M.Fink, Convergence and almost periodicity of solutions of forced Lie ́enard equations,SIAM J. Appl. Math. 26(1) (1974) 26–34.
[27] A. Gasull and A. Guillamon, Non-existence, uniqueness of limit cycles and center problems in a system that includes predator-prey systems and generalised Li ́enard equations, Differential Equations Dynam. Systems, 3 (1995), 345-366.
[28] H. Gao and B. Liu, Almost periodic solutions for a class of Li ́enard- type systems with multiple varying time delays, Appl. Math. Model., 34 (2010), pages 72–79.
[29] G. M. N’Guerekata, Almost Automorphic and Almost Periodic Func- tions in Abstract Spaces, Kluwer/Plenum Academic Publishers, New York- Boston-Dordrecht- London-Moscow (2001)
[30] J.H. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
[31] N.T. Huy and N.Q. Dang, Existence, uniqueness and conditional sta- bility of periodic solutions to evolution equations, J. Math. Anal. Appl., Vol. 433, Iss. (2016), pages 1190-1203
[32] N.T. Huy, V.T.N. Ha and N.T. Loan, Periodic solutions and their conditional stability for partial neutral functional differential equations, J. Evol. Equ. 19, 1091–1110 (2019).
[33] H. Kozono and M. Nakao, Periodic solution of the Navier–Stokes equa- tions in unbounded domains, Tˆohoku Math. J. 48, 33–50 (1996)
[34] C.E.Langenhop and G.Seifert, Almost periodic solutions of second order non linear differential equations with almost periodic forcing, Proc. Am. Math. Soc. 10(1959) 425–432.
[35] Levitan, BM Almost Periodic Functions Gos.Izdat.Tekhn- Teor.Lit.,Moscow 1953) (in Russian)
[36] A. Li ́enard, E ́tude des oscillations entretenues, Rev. G ́en. l’E ́lectric. XXIII, 901 (1928).
[37] B. Liu and C. Tunc, Pseudo almost periodic solutions for a class of non- linear Duffing system with a deviating argument, J. Appl. Math. Comput., 49 (2015), 233-242.
[38] Y. Latushkin, T. Randolph and R. Schnaubelt, Exponential Di- chotomy and Mild Solutions of Nonautonomous Equations in Banach Spaces, Journal of Dynamics and Differential Equations 10, 489–510 (1998)
[39] J. Liu, G. M. N’Guerekata and Nguyen Van Minh, Stability and Periodicity for Evolution Equations, World Scientific, New Jersey-London- Singapore-Beijing-Shanghai-Hong Kong-Taipei-Chennai. (2008)
[40] L. Maniar and R. Schnaubelt, Almost periodicity of inhomoge- neous parabolic evolution equations, in Book: Evolution Equations, Edited By G.R. Goldstein, R. Nagel and S. Romanelli, First Published 2003, eBook Published 2019, Pub. Location Boca Raton, CRC Press. https://doi.org/10.1201/9780429187599
[41] T. Naito and N.V. Minh, Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations, J. Differential Equations 152 (1999), pages 358-376.
[42] W. M. Ruess and V.Q. Phong, Asymptotically almost periodic so- lutions of evolution equations in Banach spaces, J. Differential Equations 122(1995), pages 282-301.
[43] V.Q. Phong, Stability and almost periodicity of trajectories of periodic processes, J. Differential Equations 115 (1995), pages 402-415.
[44] L.Q. Peng and W.T. Wang, Positive almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument, Electron. J. Qual. Theory Differ. Equ., 49 (2010), 1-12.
[45] Zhu Huan-Ran, The Almost Periodic Solution on a Lienard Equation, Chinese Quarterly Journal of Mathematics, 2004, (2): 164 -171 .
[46] P.T. Xuan, N.T. Van and B. Quoc, On Asymptotically Almost Peri- odic Solution of Parabolic Equations on real hyperbolic Manifolds, J. Math. Anal. Appl., Vol. 517, Iss. 1 (2023), pages 1-19.
[47] P.T. Xuan, N.T. Van and T.V. Thuy, On Asymptotically Almost Periodic Solutions of the parabolic-elliptic Keller-Segel system on real hy- perbolic Manifolds, Dynamical System: An international journal (2024), 18 pages. https://doi.org/10.1080/14689367.2024.2360204
[48] C. Xu and M. Liao, Existence and uniqueness of pseudo almost peri- odic solutions for Li ́enard-type system with delays, Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 170, pp. 1-8.
[49] Y. Xu, Positive almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument, Electron. J. Qual. Theory Differ. Equ., 49 (2012), 1-9.
[50] A. Yagi, Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II, Funkcial. Ekvac. 33 (1990), 139-150.
[51] R. Yazgan, On the Pseudo Almost Automorphic Solutions for Li ́enard- Type Systems with Multiple Delays, Journal of Mathematical Extension, Vol. 16, No. 9, (2022), pages 1-13.
[52] R. Yazgan, On the weight pseudo almost periodic solutions for Li ́enard -type systems with variable delays, Mugla Journal of Science and Technol- ogy,(2020) 6. 10.22531/muglajsci.734586.
[53] C.Y. Zhang, Pseudo almost periodic solutions of some differential equa- tions, J. Math. Anal. Appl. 181 (1) (1994) 62–76.
[54] C.Y. Zhang, Pseudo almost periodic solutions of some differential equa- tions II, J. Math. Anal. Appl. 192 (2) (1995) 543–561.