Abstract
In this paper, we present some new Volterra-Fredholm-type discrete fractional sum inequalities. These inequalities can be used as handy and powerful tools in the study of certain fractional sum-difference equations. Some applications are also presented to illustrate the usefulness of our results.
Highlights
1 Introduction It is well known that Gronwall-Bellman-type inequalities and their various generalizations have historically great importance in the qualitative analysis of differential equations, difference equations, and fractional differential equations
There has been an increase in study in the theory of discrete fractional calculus, and many interesting researches have been devoted to many topics of the fractional difference equations
), assume that there exists a function f : N → R+ satisfying exp(
Summary
It is well known that Gronwall-Bellman-type inequalities and their various generalizations have historically great importance in the qualitative analysis of differential equations, difference equations, and fractional differential equations. We employ the Riemann-Liouville definition of the fractional difference initiated by Miller and Ross [ , ], and developed by Atici and Eloe [ – , ] to establish some Volterra-Fredholm-type discrete fractional sum inequalities, which are generalizations of Gronwall-Bellman forms. As an application of the inequalities obtained, the boundedness and uniqueness of the solutions of certain Volterra-Fredholm fractional sum-difference equation are established.
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