Abstract
In this paper we initiate the study of quantum calculus on finite intervals. We define the -derivative and -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 -difference equations. MSC:26A33, 39A13, 34A37.
Highlights
Quantum calculus is the modern name for the investigation of calculus without limits
It arose interest due to high demand of mathematics that models quantum computing. q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hyper-geometric functions and other sciences quantum theory, mechanics and the theory of relativity
It has been shown that quantum calculus is a subfield of the more general mathematical field of time scales calculus
Summary
Quantum calculus is the modern name for the investigation of calculus without limits. We define the qk-integral and prove its basic properties. We prove existence and uniqueness results for initial value problems for first- and second-order impulsive q-difference equations. For t ≥ , we set Jt = {tqn : n ∈ N ∪ { }} ∪ { } and define the definite q-integral of a function f : Jt → R by t
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