Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms

Abstract

This paper deals with the boundedness of solutions to a nonlinear differential equation of fourth order. Using the Cauchy formula for the particular solution of nonhomogeneous differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

Sufficient conditions for the boundedness and square integrability of Solutions of fourth-order differential equations

“…In [22], and [25] Tunç established sufficient conditions for the asymptotic stability of the zero solution and the boundedness of the the following equations…”

The multi-level residue harmonic balance solutions of multi-mode nonlinearly vibrating beams on an elastic foundation

“…There have been many solution methods developed for solving various nonlinear partial differential equations of various physical models and engineering problems (e.g. Lakestani et al., 2006; Dehghan and Shokri, 2007; Alipanah and Dehghan, 2008; Dehghan and Shakeri, 2008; Saadatmandi and Dehghan, 2008; Yousefi and Dehghan, 2010; Andrianov et al., 2013; Han et al., 2013; Pandir et al., 2013; Qiu and Wang, 2013; Tunc and Ates, 2013; Wang and Wen, 2013; Wang et al., 2013). Owing to the presence of nonlinearity in the mathematical models, the closed form solutions could rarely be obtained.…”

Event-based generation of approximate solutions of nonlinear differential equations

“…The sampling instants might be specified beforehand or generated by an event-based algorithm that dynamically selects the sampling points so as to maintain the solution of the approximate system close enough to that of the actual system. The problem of generation of approximate solutions for nonlinear equations and the estimation of the corresponding errors has been largely treated in the literature during the last years [1][2][3][4][5][6], [10][11][12] and is a current field of research. The main tool involved in the analysis is an "ad hoc" use of a known preparatory theorem to the celebrated Bernstein´s theorem, [9], which gives an upper-bound for the maximum norm of the error in-between both the true and the approximate solutions.…”