A generalized Volterra–Fredholm integral inequality and its applications to fractional differential equations

Analysis of fractional boundary value problem with non local flux multi-point conditions on a Caputo fractional differential equation

“…These equations capture non local relations in space and time with memory essentials. Due to extensive applications of FDEs in engineering and science, research in this area has grown significantly all around the world., for instance, see [18], [11], [15] and the references cited therein. Recently, much interest has been created in establishing the existence of solutions for various types of boundary value problem of fractional order with nonlocal multi-point boundary conditions.…”

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

“…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].…”

“…In [20], Xu and Ma considered Volterra-Fredholm type integral inequalities in two independent variables and their applications in partial differential equations. Very recently, in [22], Ding and Ahmad studied Volterra-Fredholm type integral inequalities and their applications to fractional differential equations. As is known to us, few authors pay attention to nonlinear delay Volterra-Fredholm type dynamic integral inequalities on time scales.…”

Reducing model complexity by means of the optimal scaling: Population balance model for latex particles morphology formation

“…, 4, depend only on the time-independent constants in (C.10). Since h(t) and k(t) are continuous, non-negative and non-decreasing functions on [0, T ], a Gronwall's inequality (Theorem A in[23]) leads toM 1 (t) ≤ h(t) e t 0 k(ξ) dξ ≤ (k 1 T + k 2 T 2 ) e k 3 T +k 4 T 2 < ∞, ∀t ∈ [0, T ], T < ∞. (C.12)…”