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

Abstract

Sufficient conditions for the boundedness and square integrability of solutions and their derivatives of certain fourth order nonlinear differential equation are given by means of the Lyapunov's second method. Our results obtained in this work, generalize existing results on fourth order nonlinear differential equations in the literature. For illustration, an example is also given.

Stability, Boundedness, and Square Integrability of Solutions of Neutral Fourth-Order Differential Equations

“…The investigation of qualitative behaviour of the solution of nonlinear delay differential equation of fourth order has received considerable attention and has been subject of many articles in the literature, for instance, Abou-El-Ela et al [1], Bereketoglu [3], Chin [7], Ezeilo [9]- [12], Kang [18], Omeike [19], Rahmane and Fatmi and Remili [21], Rahmane and Remili [22], Remili and Rahmane [28,29,30], Sadek [31], Sinha [32], Tejumola and Tchegnani [33], Tunç [34], Vlček [35], Wu and Xiong [36]. For nonlinear differential equations of neutral type, there are few results of stability, boundedness, and square integrability of solutions.…”

Stability and boundedness of solutions of nonlinear fourth order differential equations with bounded delay

“…However, [4,5,16] include such a useful content about the qualitative behaviors of differential equations without or with delay. To gain much better perspective on the boundedness and stability, see the papers of Ezeilo [6,7], Hara [8], Harrow [9,10], Tunç [22,23,24,25,26], Remili et al [15,17,18], Wu and Xiong [27] and others and theirs references. As motive from references, we obtain some new consequences on the uniformly asymtotically stability and boundedness of the solutions by means of the Lyapunov's functional approach.…”

Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay

“…In particular, for more results on the stability, boundedness, convergence, etc. of ordinary or functional differential equations of fourth order, see the book of Reissig et al [ 3 ] as a good survey for the works done by 1974 and the papers of Burton [ 4 ], Cartwright [ 5 ], Ezeilo [ 6 – 9 ], Harrow [ 10 , 11 ], Tunç [ 12 – 18 ], Remili et al [ 19 – 23 ], Wu [ 24 ] and others and the references therein. This information indicates the importance of investigating the qualitative properties of solutions of retarded functional differential equations of fourth order.…”