Gronwall–Bellman type nonlinear delay integral inequalities on time scales

Abstract

In this paper, some Gronwall-Bellman type nonlinear delay integral inequalities on time scales are established, which provide a handy tool in deriving boundedness of solutions of certain delay dynamic equations on time scales. Our results generalize some of the main results in Lipovan (2006) [1], Pachpatte (2000) [2], Ferreira and Torres (2009) [3], Zhang and Meng (2008) [4], Cheung and Ren (2006) [5], Kim (2009) [6], and some of our results unify continuous and discrete analysis in the literature.

Finite‐time stability of a class of oscillating systems with two delays

“…Liang et al extend the study of Lazarević et al and Luo et al to study a second order–delayed system based on the notation of delayed matrix cosine/sine of a polynomial of degree. For more recent contributions, we refer the reader to several previous studies . However, to our best knowledge, finite‐time stability of oscillating systems with 2 delays have not been studied extensively.…”

“…For more recent contributions, we refer the reader to several previous studies. [25][26][27][28][29][30][31][32][33][34][35] However, to our best knowledge, finite-time stability of oscillating systems with 2 delays have not been studied extensively.…”

Oscillation results for a fractional order dynamic equation on time scales with conformable fractional derivative

“…In [13], Hilger initiated the theory of time scale trying to treat continuous and discrete analysis in a consistent way. Since then, the theory of time scale has received a lot of attention in recent years, and various investigations have been done by many authors [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Among these investigations, some authors have taken research in oscillation of dynamic equations on time scales (see [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] and the references therein).…”

Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics

“…The travelling waves solution, peaked solitary wave solutions for equation (1) with g(u) = au m + ku and some α, m, γ were showed in [28,27] and references therein. More related work in this fields can be found in [23,21,19,22,29,30,31,32,25].…”