Some dynamic integral inequalities with mixed nonlinearities on time scales

Abstract

The objective of this paper is to study some dynamic integral inequalities on time scales, which provide explicit bounds on unknown functions. Our results include many known ones in the literature and can be used as tools in the study of qualitative theory of certain classes of dynamic equations with mixed nonlinearities on time scales. MSC: Primary 26D10; 26D15; secondary 34C11; 34N05

Some New Half-Linear Integral Inequalities on Time Scales and Applications

“…In the present paper, we study some new half-linear integral inequalities on time scales. Our results not only complement the results established in [37] in the sense that the results can be applied in cases when 0 < < < or 0 < < < , but also furnish a handy tool for the study of qualitative properties of solutions of some complicated dynamic equations.…”

“…Because dynamic inequalities play an important role in the study of qualitative properties of solutions of dynamic equations on time scales, many authors have expounded on various classes of dynamic inequalities in recent years; see [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] and the references cited therein. For instance, in 2013, Sun and Hassan [32] investigated the nonlinear integral inequality on time scales…”

Some Nonlinear Delay Volterra–Fredholm Type Dynamic Integral Inequalities on Time Scales

“…Remark 10. In [14], we investigated the integral inequalities with mixed nonlinearities on time scales; however, the delay terms are not considered; furthermore, the inequalities considered in this paper are Volterra-Fredholm type.…”

“…In recent years, many authors have been devoted to studying different kinds of integral inequalities and their applications [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], especially the application of Volterra-Fredholm integrodifferential system [25][26][27][28]. To mention a few, in [8], Gu and Meng considered the nonlinear dynamic integral inequalities on time scales and applied the theoretical results to Volterra-Fredholm integrodifferential system, and Liu [9] investigated the linear delay Volterra-Fredholm type dynamic integral inequalities which generalized the main results of [8].…”

Boundedness of the Solutions to Certain Delay Dynamic Integrodifferential Systems on Time Scales

“…The theory of time scales was established and developed by Hilger [1] and Bohner and Peterson [2,3]. At present, different kinds of integral inequalities and their applications in differential, integral, and integrodifferential equations have become the research focus; see the papers [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. To list a few, Ma and Pečarić [10] established an integral inequality on time scales to study the boundedness of solutions of the delay dynamic differential system.…”