A generalization of a boundedness theorem for a certain third-order differential equation

Abstract

1. This paper investigates the boundedness, as t → ∞, of the solutions of the real differential equationwhere α is a constant and φ2, φ3, ψ depend on the arguments shown with φ2, φ′3, ψ continuous.

Analytical and approximate solutions to autonomous, nonlinear, third-order ordinary differential equations

“…Mathematical studies of third-order nonlinear ordinary differential equations include those of Tunç [10] who proved the stability and boundedness of solutions of nonlinear vector differential equations by means of Lyapunov's second method, Ezeilo [11][12][13], Rao [14], Reissig et al [15], Tunç and Ateş [16], etc. On the other hand, mathematical modelling of several physical phenomena sometimes result in third-order nonlinear ordinary differential equations.…”

A Boundedness Theorem for a Certain Third Order Nonlinear Differential Equation

“…I n this paper, conditions obtained using a complete Yoshizawa functions (according to Chukwu [5] ) are exact, see for instance [5] , [10] , and [13] .…”

“…Whenever n and n the hypotheses and conclusion of theorem coincide with those of Chukwu [5] , except hypotheses (i) and (iii) of the theorem which are considerable weaker than those in [5] Remark 2: If n˜ n ™ and n then (1) reduces to the case studied by Ezeilo [10] .…”

“…Since 8 is a positive constant, it follows by (5) that 8 : e G now assume that ^) is so small that [ m M Finally, as in [5] , [10] and [11] , (6) yields f 0 ,h,> N On gathering the estimates (16) and (17) [5] , [10] and [13] the limit f r o corresponding to W [ h clearly exists. An elementary calculation from (2) and (3) will infact yield the following equation…”

“…al., [14] which are background and survey books respectively, and Ademola et. al., [1] ,Afuwape [2] , Andres [3] , Bereketo lu and Györi [4] , Chukwu [5] , Ezeilo [ 6,7,8,9,10,11] , Ezeilo and Tejumola [12] , Hara [14] , Swick [16,17] , Tejumola [18] , and Tunç [19] . These works were done with the aid of complete Yoshizawa and incomplete Lyapunov functions.…”