×

Weakly nonlinear hyperbolic differential equation of the second order in Hilbert space. (English) Zbl 07959971

Summary: We consider nonlinear perturbations of the hyperbolic equation in the Hilbert space. Necessary and sufficient conditions for the existence of solutions of boundary-value problem for the corresponding equation and iterative procedures for their finding are obtained in the case when the operator in linear part of the problem hasn’t inverse and can have nonclosed set of values. As an application we consider boundary-value problem for van der Pol equation in a separable Hilbert space.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
35L90 Abstract hyperbolic equations
Full Text: DOI

References:

[1] M. Attari, M. Haeri and M.S. Tavazoei, Analysis of a fractional order Van der Pol-like oscillator via describing function method, Nonlinear Dyn. 61 (2010), 265-274. · Zbl 1204.70018
[2] Z. Bai, W. Li and W. Ge, Existence and multiplicity of solutions for four-point boundaryvalue problems at resonance, Nonlinear Anal. 60 (2005), 1151-1162. · Zbl 1070.34026
[3] Z. Balanov, M. Farzamirad and W. Krawcewicz, Symmetric systems of van der Pol equations, Topol. Methods Nonlinear Anal. 27 (2006), 29-90. · Zbl 1141.34025
[4] P.J. Beek, R.C. Schmidt, A.W. Morris, M.Y. Sim and M.T. Turvey, Linear and nonlinear stiffness and friction in biological rythmic movements, Biol. Cybern. 73 (1995), 499-507. · Zbl 1066.92501
[5] A. Boichuk and O. Pokutnyi, Solutions of the Schrodinger equation in a Hilbert space, Bound. Value Probl. 2014 (2014). · Zbl 1309.34023
[6] A.A. Boichuk and A.A. Pokutnyi, Perturbation theory of operator equations in the Frechet and Hilbert spaces, Ukrainian Math. J. 67 (2016), 1327-1335. · Zbl 1498.47111
[7] A.A. Boichuk and O.O. Pokutnyi, Bifurcation of solutions of the second order boundary-value problems in Hilbert spaces, Miskolc Math. Notes 20 (2019), 139-152. · Zbl 1438.34201
[8] A.A. Boichuk and A.M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, second edition, De Gruyter, Berlin; Boston, 2016. · Zbl 1394.47001
[9] A. Cwiszewski and P. Kokocki, Periodic solutions of nonlinear hyperbolic evolution systems, J. Evol. Equ. 10 (2010), 677-710. · Zbl 1239.34039
[10] Y. Deng and F. Pusateri, On the global behavior of weak null quasilinear wave equations, Comm. Pure Appl. Math. 73 (2020), 1035-1099. · Zbl 1445.35252
[11] I.P. Gavrilyuk, V.L. Makarov and N.V. Mayko, Weighted estimates of the Cayley transform method for abstract differential equations, Comput. Methods Appl. Math. 21 (2021), 53-68. · Zbl 1473.65082
[12] I.P. Gavrilyuk, V.L. Makarov and V.B. Vasylyk, Exponentially Convergent Algorithms for Abstract Differential Equations, Birkhauser, Basel, 2011. · Zbl 1225.47001
[13] Z.M. Ge and S.-Y. Li, Chaos generalized synchronization of new Mathieu-Van der Pol systems with new Duffing-Van der Pol systems as functional system by GYC partial region stability theory, Appl. Math. Model. 35 (2011), 5245-5264. · Zbl 1228.93097
[14] V.I. Gorbachuk and M.L. Gorbachuk, Boundary-Value Problems for Operator Differential Equations, Springer, Dordrecht, 1991. · Zbl 0751.47025
[15] M.C. Gouveia and R. Puystjens, About the group inverse and Moore-Penrose inverse of a product, Linear Algebra Appl. 150 (1991), 361-369. · Zbl 0721.15001
[16] Y.J. Huang and H.K. Liu, A new modification of the variational iteration method for Van der Pol equations, Appl. Math. Model. 37 (2013), 8118-8130. · Zbl 1426.65123
[17] B.Z. Kaplan, I. Gabay, G. Sarafian and D. Sarafian, Biological applications of the “filtered” Van der Pol oscillator, J. Franklin Inst. 345 (2008), 226-232. · Zbl 1167.93352
[18] T. Kawahara, Coupled Van der Pol oscillators — A model of excitatory and inhibitory neural interactions, Biol. Cybern. 39 (1980), 37-43. · Zbl 0448.92009
[19] D.A. Klyushin, S.I. Lyashko, D.A. Nomirovskii, Yu.I. Petunin and V.V. Semenov, Generalized Solutions of Operator Equations and Extreme Elements, Springer, New York, 2012.
[20] S.G. Krein, Linear Equations in Banach Spaces, Birkhauser, Boston, 1982. · Zbl 0535.47008
[21] A. Kumar, M. Muslim and R. Sakthivel, Controllability of the second-order nonlinear differential equations with non-instantaneous impulses, J. Dyn. Control Syst. 24 (2018), 325-342. · Zbl 1391.34100
[22] N. Levinson, A second order differential equation with singular solutions, Ann. Math. 50 (1949), 127-153. · Zbl 0041.42311
[23] J. Limaco, M.R. Nunez-Chavez and D.N. Huaman, Exact controllability for nonlocal and nonlinear hyperbolic PDEs, Nonlinear Anal. 214 (2022), 112569. · Zbl 1476.35135
[24] M. Modanli and A. Akgul, On solutions to the second-order partial differential equations by two accurate methods, Numer. Methods Partial Differential Equations 34 (2018), 1678-1692. · Zbl 1407.65121
[25] S.V. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal. 43 (2005), 2060-2076. · Zbl 1103.65058
[26] C.M.A. Pinto and J.A.T. Machado, Complex order Van der Pol oscillator, Nonlinear Dyn. 65 (2011), 247-254.
[27] G.D. Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press, Cambridge, 2002. · Zbl 1012.35001
[28] R.E. Showalter, Hilbert Space Methods for Partial Differential Equations, TX: Southwest Texas State University, San Marcos, 1994. · Zbl 0991.35001
[29] B. Van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev. Sel. Sci. Pap. 1 (1960), 754-762.
[30] B. Van der Pol and J. Van der Mark, Frequency demultiplication, Nature 120 (1927), 363-364.
[31] B. Van der Pol and J. Van der Mark, The heartbeat considered as a relaxation oscillation, and an electrical model of the heart, Philos. Mag. 6 (1929), 763-775.
[32] C. Zhang, B. Zheng and L. Wang, Multiple Hopf bifurcations of three coupled Van der Pol oscillators with delay, Appl. Math. Comput. 217 (2011), 7155-7166. · Zbl 1223.34102
[33] M. del Pino, P. Drabek, R. Manasevich. The Fredholm Alternative at the first Eigenvalue for the one dimensional \(p\)-Laplacian, J. Differential Equations 151 (1999), 386-419. · Zbl 0931.34065
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.