Boundedness and square integrability of solutions of certain third-order differential equations. (English) Zbl 1463.34151

Beldjerd, Djamila; Remili, Moussadek

Math. Bohem. 143, No. 4, 377-389 (2018).

In this paper, the authors prove two theorems on the boundedness and square integrability of solutions of the following scalar non-linear differential equation of third-order: \[ (x^{\prime}+g(x))^{\prime\prime}+a(t)x^{\prime\prime}+b(t)x^{\prime}+c(t)h(x) = 0,\tag{1} \] where \(a(t)\), \(b(t)\), \(c(t)\in C([0,\infty))\) are positive functions. It is also assumed that the functions \(g\) and \(h\) are differentiable for all \(x\). In the first theorem, Theorem 2.1, sufficient conditions are given for the solutions of equation (1) to be bounded. Then, in Theorem 3.1, similar certain sufficient conditions for the square integrability of solutions of equation (1) are established. Here, the method of the proofs is based on the second method of Lyapunov. Finally, the established results are also verified via an example. The results of this paper extend and improve some previous works in the literature.

Reviewer: Cemil Tunç (Van)

MSC:

34C11	Growth and boundedness of solutions to ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems

Keywords:

third-order differential equation; boundedness; square integrability; Lyapunov function

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