Henry–Gronwall type q‐fractional integral inequalities

Abstract

It has been shown that these types of equations have numerous applications in diverse fields and thus have evolved into multidisciplinary subjects. In this paper, the q-fractional integral inequalities of Henry-Gronwall type are presented. By using inequality technique related to the quantum calculus, such integral inequalities are suggested. Furthermore, an applications to q-fractional integral equations is also given.

Estimation of divergences on time scales via the Green function and Fink’s identity

“…Example 1 Choose T = R in Theorem 4, to get the same result as one can obtain from [15, (2.1)] by utilizing (1) and (7).…”

“…The concepts of quantum calculus on finite intervals were given by Tariboon and Ntouyas [37,38], and they obtained certain q-analogues of classical mathematical objects, which motivated numerous researchers to explore the subject in detail. Subsequently, several new results related to quantum counterpart of classical mathematical results have been established, see [7,29,34].…”

The existence of nonnegative solutions for a nonlinear fractional q-differential problem via a different numerical approach

“…In [ 30 ] the investigation is centered around the quantum estimates by utilizing the quantum Hahn integral operator via the quantum shift operator. In [ 20 ] the q -fractional integral inequalities of Henry–Gronwall type are presented.…”

“…Soon afterward, it is further promoted by Al-Salam and Agarwal [9,10], where many outstanding theoretical results are given. Its emergence and development extended the application of interdisciplinary to be further and aroused widespread attention of the scholars; see [11][12][13][14][15][16][17][18][19][20][21][22][23] Then Liang and Zhang [24] studied the existence and uniqueness of positive solutions by properties of the Green function, the lower and upper solution method, and the fixed point theorem for the fractional equation D σ q [k](s) + w(s, k(s)) = 0 for 0 < s < 1 under the boundary conditions k(0) = k (0) = 0 and k (1) = m-2 i=1 i k (ς i ), where 2 < σ ≤ 3, and c D σ q is the Riemann-Liouville fractional derivative. In 2015, Zhang et al [25] through the spectral analysis and fixed point index theorem obtained the existence of positive solutions of the singular nonlinear fractional differential equation…”

A generalized fractional (q, h)–Gronwall inequality and its applications to nonlinear fractional delay (q, h)–difference systems

“…In 2019, Chen et al 10 obtained a nabla discrete Gronwall inequality on , based on this inequality, Chen et al 11 gave some Ulam–Hyers stability results of discrete fractional Caputo equations. In 2020, Makhlouf et al 12 developed some Henry–Gronwall type q –fractional sum inequalities. However, for a nonnegative function x ( t ), the monotonicity of fractional sum function of x ( t ) is not specified in previous works, 6,8,13 which results in the incomplete proofs.…”