Global Existence and Asymptotic Behavior of Solutions of Nonlinear Differential Equations

Abstract

Abstract. The paper addresses two open problems related to global existence of solutions with a ''linear-like'' behavior at infinity. For a class of second-order nonlinear di¤erential equations, we establish global existence of solutions under milder assumption on the rate of decay of the coe‰cient. Furthermore, as opposed to results reported in the literature, we prove for another class of second-order nonlinear di¤erential equations that the region of the initial data for the solutions with desired asymptotic behavior is unbounded and proper.

On the global existence and boundedness of solutions to a nonlinear integro-differential equations of second order

“…For an excellent survey of almost all results up to 1974 we refer the reader in particular to book by Reissig et al [23]. Further results can be found in the books or the papers by Ahmad and Rama Mohana Rao [2], Fonda and Zanolin [4], Graef ([6], [7]), Huang et al [9], Jin [10], Kato [11], Kiguradze and Chanturia [12], Kitano and Kusano [13], Kroopnick [14], Mirzov [15], Mustafa and Tunc [16], Mustafa and Rogovchenko [17], Nápoles Valdés [18,20], Qian [22], Saker [24], Tiryaki and Ağacık [25], Tunc [26][27][28][29][30][31][32][33], Tunc and Tunc [34], Tunc and Tunc [35], Wong [36], Wong and Burton [38], Wu et al [39], Yoshizawa [40]. http://www.ispacs.com/journals/jiasc/2015/jiasc-00083/…”

A boundary value problem on the half-line for higher-order nonlinear differential equations

“…( (4.16) gives c = 25 4 . Hence, we arrive at the following result: the BVP has at least one solution x such that, for every t 0, 5 8 t 2 x(t) 25 8 t 2 , We notice that (4.20) implies condition (4.19). By the use of (4.20), it is not difficult to verify that the formula (4.21) defines a real number c with c > |ξ|.…”

Global Solutions Approaching Lines at Infinity to Second Order Nonlinear Delay Differential Equations

“…This problem has been the subject of many investigations; we restrict ourselves to mention the recent papers [2], [5], [10][11][12][13][14][15][16][17][18][19][20] and [22][23][24][25][26] as well as the older classical articles [8,9] (for a more extensive bibliography, see [17,18]). It is of special interest to investigate global solutions, i.e.…”

“…solutions on the whole given interval, with prescribed asymptotic behavior. On this problem there is an extensive bibliography (see, for example, [2], [5], [8,9], [11][12][13][14][15][16] and [22][23][24][25][26]; for more references, see [15,17,18]). The present work deals with global solutions that are asymptotic at y to lines for second order delay, and especially ordinary, di¤erential equations.…”