Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear differential equations

On the Asymptotic Behaviour of the Solutions to a Class of Second Order Nonlinear Differential Equations

“…where a ∈ ‫.ޒ‬ Theorem A generalizes results obtained in [1], [2], [5], [6], [13], [14], and, though it does not ensure that all solutions are asymptotic to lines (see Example 3), it represents a starting point for our approach. = ∞.…”

“…where f ∈ C([1, ∞) × ‫,ޒ‬ ‫)ޒ‬ have been widely investigated by Cohen [1], Constantin [2], Dannan [5], Fan Wei Meng [6], S. Rogovchenko and Y. Rogovchenko [12], Tong [13] and Trench [14]. Using integral inequalities, these papers were concerned with sufficient conditions which ensure that all solutions of (1) will approach those of u = 0.…”

Global Existence and Asymptotic Behavior of Solutions of Nonlinear Differential Equations

“…(1) that possess property (L) only for a part of solutions with initial data satisfying an additional condition. As opposed to the papers by Dannan [3], S. Rogovchenko and Yu. Rogovchenko [12], and Waltman [18], where the regions of linear-like behavior have di¤erent nature but are always bounded, the novelty of Theorem 6 also lies in the fact that for the considered class of equations the region of linear-like behavior is unbounded and proper (that is, it is neither void nor R 2 ).…”

Nonlinear Gronwall–Bellman Type Inequalities and Their Applications

“…These inequalities play an important role in many fields. They are applied to the investigation of the stability, boundedness, global existence, uniqueness, and continuous dependence on the initial or boundary value and parameters of solutions to differential equations, integral equations, as well as difference equations [1][2][3][4][5][6][7][8][9][10][11]. They are also used to study the regularized family of models for homogeneous incompressible two-phase flows [12], the state of the high nonlinear circuit [13], the Cousin problems and the emergence of the Sheaf Concept [14].…”