×

Anti-periodic solutions of Abel differential equations with state dependent discontinuities. (English) Zbl 1390.34039

Summary: Given \(T>0\), the Abel-like equation \[ \theta'=f_0+\Sigma_{j\in\mathbb{N}}f_j\theta^j \] is generalized to the case where \(\theta\) and \(\theta'\) are real functions on \([0,T]\) subject to given state dependent discontinuities. Each \(f_j\) is a real function of bounded variation for which \(f_j(0)=(-1)^{j+1}f_j(T)\). Under appropriate conditions, this equation is shown to admit a solution of bounded variation on \([0,T]\) which is \(T\)-anti-periodic in the sense that \(\theta(0)=-\theta(T)\). The contraction principle yields a bound for the rate of uniform convergence to the solution of a sequence of iterates.

MSC:

34A36 Discontinuous ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. R. A FTABIZADEH, S. A IZICOVICI AND N. H. P AVEL, {\it On a class of second-order anti-periodic} {\it boundary value problems}, J. Math. Anal. Appl. 171 (1992), 301-320. · Zbl 0767.34047
[2] B. A HMAD AND J. J. N IETO, {\it Existence and approximation of solutions for a class of nonlinear im-} {\it pulsive functional differential equations with anti-periodic boundary conditions}, Nonlinear Analysis: Theory, Methods & Applications 69 (2008), 3291-3298. · Zbl 1158.34049
[3] C. D. A LIPRANTIS AND O. B URKINSHAW, {\it Principles of real analysis. Second edition}, Academic Press, Inc., Boston, MA, 1990. · Zbl 0706.28001
[4] N. M. H. A LKOUMI AND P. J. T ORRES, {\it Estimates on the number of limit cycles of a generalized Abel} {\it equation}, Discrete contin. Dyn. Syst. 31 (2011), 25-34. · Zbl 1372.34075
[5] N. M. H. A LKOUMI AND P. J. T ORRES, {\it On the number of limit cycles of a generalized Abel equation}, Czechoslovak Math. J. 61(136) (2011), 73-83. · Zbl 1224.34097
[6] A. ´A LVAREZ, J. L. B RAVO,AND M. F ERN ANDEZ´, {\it Limit cycles of Abel equations of the first kind}, J. Math. Anal. Appl. 423 (2015), 734-745. · Zbl 1326.34027
[7] A. ´A LVAREZ, J. L. B RAVO,AND M. F ERN ANDEZ´, {\it Existence of non-trivial limit cycles in Abel equa-} {\it tions with symmetries}, Nonlinear Anal. 84 (2013), 18-28. · Zbl 1312.34079
[8] A. ´A LVAREZ, J. L. B RAVO,AND M. F ERN ANDEZ´, {\it Abel-like differential equations with unique limit} {\it cycle}, Nonlinear Anal. 74 (2011), 3694-3702. · Zbl 1228.34060
[9] A. ´A LVAREZ, J. L. B RAVO,AND M. F ERN ANDEZ´, {\it The number of limit cycles for generalized Abel} {\it equations with periodic coefficients of definite sign}, J. Math. Anal. Appl. 360 (2009), 168-189. · Zbl 1282.34040
[10] A. ´A LVAREZ, J. L. B RAVO,AND M. F ERN ANDEZ´, {\it Existence of non-trivial limit cycles in Abel equa-} {\it tions with symmetries}, Commun. Pure Appl. Anal. 8 (2009), 1493-1501. · Zbl 1273.34045
[11] D. B ATENKOV AND G. B INYAMINI, {\it Uniform upper bounds for the cyclicity of the zero solution of} {\it the Abel differential equation}, J. Differential Equations 259 (2015), 5769-5781. · Zbl 1327.34056
[12] J. L. B RAVO AND M. F ERN ANDEZ´, {\it Stability of singular limit cycles for Abel equations}, Discrete Contin. Dyn. Syst. 35 (2015), 1873-1890. · Zbl 1312.34080
[13] T. C HEN AND W. L IU, {\it Anti-periodic solutions for higher-order Li´enard type differential equation} {\it with p-Laplacian operator}, Bull. Korean Math. Soc. 49 (2012), 455-463. · Zbl 1247.34061
[14] Y. C HEN, J. J. N IETO AND D. O’R EGAN, {\it Anti-periodic solutions for fully nonlinear first order} {\it differential equations}, Mathematics and Computer Modelling 46 (2007), 1183-1190. · Zbl 1142.34313
[15] Y. C HEN, J. J. N IETO AND D. O’R EGAN, {\it Anti-periodic solutions for evolution equations associated} {\it with maximal monotone mappings}, Applied Mathematics Letters 24 (2011), 302-307. · Zbl 1215.34069
[16] H. F. D AVIS, {\it Fourier series and orthogonal functions}, Allyn and Bacon, Inc., Boston, MA, 1963.
[17] W. D ING, Y. X ING AND M. H AN, {\it Anti-periodic boundary value problems for first order impulsive} {\it functional differential equations}, Applied mathematics and Computation 186 (2007), 45-53. · Zbl 1124.34039
[18] D. F RANCO, J. J. N IETO AND D. O’R EGAN, {\it Anti-periodic boundary value problem for nonlinear} {\it first order ordinary differential equations}, Mathematical Inequalities & Applications 6 (2003), 477- 485. · Zbl 1097.34015
[19] A. G ASULL AND Y. Z HAO, {\it On a family of polynomial differential equations having at most three} {\it limit cycles}, Huston J. Math. 39 (2013), 191-203. · Zbl 1283.34030
[20] T. J ANKOWSKI, {\it Ordinary differential equations with nonlinear boundary conditions of antiperiodic} {\it type}, Computers & Mathematics with Applications 6 (2004), 1419-1428. · Zbl 1105.34007
[21] G. J AUME, G. M AITE AND L. J AUME, {\it Universal centres and composition conditions}, Proc. Lond. Math. Soc. 106 (2013), 481-507. · Zbl 1277.34034
[22] Y. K ATZNELSON, {\it An introduction to harmonic analysis}, John Wiley & Sons, Inc., New York, 1968. · Zbl 0169.17902
[23] Y. L I AND L. H UANG, {\it Anti-periodic solutions for a class of Li´enard-type systems with continuously} {\it distributed delays}, Nonlinear Analysis: Real World Applications 10 (2009), 2127-2132. · Zbl 1163.45305
[24] L. L IU AND Y. L I, {\it Existence and uniqueness of anti-periodic solutions for a class of nonlinear n-th} {\it order functional differential equations}, Opuscula Mathematica 31 (2011), 61-74. · Zbl 1241.34078
[25] M. N AKAO AND H. O KOCHI, {\it Anti-periodic solution for u}{\it xx}{\it − }(σ({\it u}{\it x})){\it x}{\it − u}{\it xxt}= {\it f }({\it x,t}), J. Math. Anal. Appl. 197 (1996), 796-809.
[26] H. O KOCHI, {\it On the existence of anti-periodic solutions to a nonlinear evolution equation associated} {\it with odd subdifferential operators}, J. Funct. Anal. 91 (1990), 246-258. · Zbl 0735.35071
[27] H. O KOCHI, {\it On the existence of anti-periodic solutions to nonlinear parabolic equations in non cylin-} {\it drical domains}, Nonlinear Anal. 14 (1990), 771-783. · Zbl 0715.35091
[28] H. O KOCHI, {\it On the existence of periodic solutions to nonlinear abstract parabolic equations}, J. Math. Soc. Japan 40 (1988), 541-553. · Zbl 0679.35046
[29] C. O U, {\it Antiperiodic solutions for high-order Hopfield neural networks}, Computers & Mathematics with Applications 56 (2008), 1838-1844. · Zbl 1152.34378
[30] F. P AHOVICH, {\it Weak and strong composition conditions for the Abel differential equation}, Bull. Sci. math. 138 (2014), 993-998. · Zbl 1315.34040
[31] H. L. R OYDEN, {\it Real Analysis. Third edition}, Macmillan Publishing Company, New York, 1988. · Zbl 0704.26006
[32] W. R UDIN, {\it Real and Complex Analysis. Third edition}, McGraw-Hill Book Company, New York, 1987.
[33] K. W ANG, {\it A new existence result for nonlinear first-order anti-periodic boundary value problems}, Applied Mathematics Letters 21 (2008), 1159-1154. · Zbl 1168.34315
[34] R. W U, {\it An anti-periodic LaSalle oscillatory theorem}, Applied Mathematics Letters 21 (2008), 928- 933. · Zbl 1152.34314
[35] Y. Y IN, {\it Remarks on first order differential equations with anti-periodic boundary conditions}, Nonlin ear Times and Digest 2 (1995), 83-94. · Zbl 0832.34018
[36] Y. Y IN, {\it Monotone iterative technique and quasilinearization for some anti-periodic problems}, Non linear World 3 (1996), 253-266. · Zbl 1013.34015
[37] A. H. Z EMANIAN, {\it Distribution Theory and Transform Analysis}, Dover Publications, Inc., Minneola, N.Y., 1987. (Received September 7, 2016){\it J.-M. Belley} {\it Facult´e des sciences} {\it Universit´e de Sherbrooke} {\it Sherbrooke, QC, J1K 2R1, Canada} {\it e-mail:}jean-marc.belley@usherbrooke.ca {\it Aziz Gueye} {\it Facult´e des sciences} {\it Universit´e de Sherbrooke} {\it Sherbrooke, QC, J1K 2R1, Canada} {\it e-mail:}aziz.gueye@usherbrooke.ca Differential Equations & Applications www.ele-math.com
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.