On the boundedness of solutions of third order differential equations

Abstract

This research investigates the boundedness, as t->c% of solutions o] the third order order di]/erential equation 2.1(1). Special attention is drawn to a method in current usage which rests on the existence o] appropriate Zoshizawa /unction with which ultimate bound. edness is proved ]or all solutions o] 2.1(1) under quite mild restraints on ], g, h and p.

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] .…”

“…Observe that A ) is a further generalization of the incomplete Lyapunov function A in [16] , while A is the same as a complete signum function in [5] . The required properties of A is contained in the following lemmas.…”

“…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.…”