Quantum calculus on finite intervals and applications to impulsive difference equations

Abstract

In this paper we initiate the study of quantum calculus on finite intervals. We define the q k -derivative and q k -integral of a function and prove their basic properties. As an application, we prove existence and uniqueness results for initial value problems for first-and second-order impulsive q k -difference equations.

Some Integral Inequalities Using Quantum Calculus Approach

“…These results will be helpful in the development of our main results. These results are mainly due to Tariboon et al [30,31].…”

Some new Hermite-Hadamard type inequalities for h-convex functions via quantum integral on finite intervals

“…Let f : I → R be a continuous function. Then we have 17]). Let f, g : I → R be two continuous functions and α ∈ R. Then, for x ∈ I, we have…”

“…Recently, Tariboon and Ntouyas introduced the quantum calculus on finite intervals in the paper [17]. In [13], Noor et al applied quantum analogue of classical integral identity to establish some quantum estimates for Hermite-Hadamard inequalities for q-differentiable convex functions and q-differentiable quasi convex functions.…”

Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

“…The quantum operators are widely used in mathematic fields such as hypergeometric series, complex analysis, orthogonal polynomials, combinatorics, hypergeometric functions, and the calculus of variations. The quantum calculus is also found in many applications, such as quantum mechanics and particle physics [1][2][3][4][5][6][7][8][9].…”