Some new Opial dynamic inequalities with weighted functions on time scales

Abstract

Abstract. In this paper we prove some new dynamic inequalities with two weight functions and some new dynamic inequalities with two unknown functions of Opial type on time scales. The main results will be proved by employing Hölder's inequality, the chain rule and some basic algebraic inequalities.Mathematics subject classification (2010): 26A15, 26D10, 26D15, 39A13, 34A40.

Opial and Lyapunov inequalities on time scales and their applications to dynamic equations

“…(1) If one sets p ¼ g þ b and j ¼ t, where t is decreasing on ½a; c T , then (4.13) improves [17, Theorem 3.1] and [30, Theorem 1]; (2) By setting b ¼ 1 and t ¼ oj p=ðgþ1Þ , where o A Wð½a; c T Þ and j is decreasing on ½a; c T , (4.13) reduces to [2, Theorem 3.2.4] (see also [31]); (3) If p ¼ g þ b, then inequality (4.13) becomes [29,Theorem 3.1] which reduces to [22,Theorem 2.4]. Similar consideration applying to (4.12) yields other results given in [17,22,29,30], and [31].…”

“…R is D-di¤erentiable with f ð0Þ ¼ 0, then when f ðaÞ ¼ 0 or/and f ðbÞ ¼ 0, where p > 1, b > 0, g > 0, C 1 and C 2 are constants. For contributions to inequalities (1.2) and (1.3) we refer the readers to [6,14,15,17,22,24,25,26,29,30,31,35,36]. The best reference here is the book by Agarwal, O'Regan and Saker [2, Chapter 3], where the most popular articles on this subject are collected.…”

Ostrowski type integral inequalities, weighted Ostrowski, and trapezoid type integral inequalities with powers

“…Since the integral and discrete inequalities are important in the analysis of qualitative properties of solutions of differential and difference equations, see [15,17,25], we also believe that the dynamic inequalities with weights on time scales will play the same effective role in the analysis of qualitative properties of dynamic equations with boundary conditions like oscillation, nonoscillation and distribution of zeros of solutions, see [8,9]. For related dynamic inequalities on time scales, we refer the reader to the papers [36][37][38][39][40][41][42][43][44][45] and the books [2,3]. In [11] Bohner and Matthews generalized the Montgomery identity on time scales and proved that…”

Weighted Čebyšev-Ostrqwski type integral inequalities with power means

“…Since the integral and discrete inequalities are important in the analysis of qualitative properties of solutions of differential and difference equations, we also believe that the dynamic Hardy type inequalities with weights on time scales will play the same effective role in the analysis of qualitative properties of dynamic equations with boundary conditions like oscillation, nonoscillation and distribution of zeros of solutions. For related dynamic inequalities on time scales, we refer the reader to the papers [8,9,[24][25][26][27][28][29][30][31][32][33] and the books [2,3].…”