Applications of quantum calculus on finite intervals to impulsive difference inclusions

Sotiris K Ntouyas,Jessada Tariboon

doi:10.1186/1687-1847-2014-262

Abstract

Recently Tariboon and Ntouyas (Adv. Differ. Equ. 2013:282, 2013) introduced the notions of -derivative and -integral of a function on finite intervals. As applications existence and uniqueness results for initial value problems for first- and second-order impulsive -difference equations was proved. In this paper, continuing the study of Tariboon and Ntouyas (Adv. Differ. Equ. 2013:282, 2013), we apply the quantum calculus to initial value problems for impulsive first- and second-order -difference inclusions. We establish new existence results, when the right hand side is convex valued, by using the nonlinear alternative of Leray-Schauder type. Some illustrative examples are also presented. MSC:34A60, 26A33, 39A13, 34A37.

Highlights

Introduction and preliminariesIn [ ] the notions of qk-derivative and qk-integral of a function f : Jk := [tk, tk+ ] → R, have been introduced and their basic properties was proved
We recall the notions of qk-derivative and qk-integral on finite intervals
We define qkderivative of a function f : Jk → R at a point t ∈ Jk as follows

Summary

Introduction

Introduction and preliminariesIn [ ] the notions of qk-derivative and qk-integral of a function f : Jk := [tk, tk+ ] → R, have been introduced and their basic properties was proved. Existence and uniqueness results for initial value problems for first- and second-order impulsive qkdifference equations was proved. ), Dqk f = Dqf , where Dq is the well-known q-derivative of the function f (t) defined by f (t) – f (qt)

Results

Conclusion