Abstract
In this paper, we discuss the existence and uniqueness of solutions for Langevinimpulsive q-difference equations with boundary conditions. Our studyrelies on Banach’s and Schaefer’s fixed point theorems. Illustrativeexamples are also presented.MSC: 26A33, 39A13, 34A37.
Highlights
1 Introduction and preliminaries In recent years, the boundary value problems of fractional order differential equations have emerged as an important area of research, since these problems have applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, aerodynamics, viscoelasticity and damping, electro-dynamics of complex medium, wave propagation, blood flow phenomena, etc. [ – ]
The Langevin equation is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [ ]
Nowadays there is a significant increase of activities in the area of q-calculus due to its applications in various fields such as mathematics, mechanics, and physics
Summary
Introduction and preliminariesIn recent years, the boundary value problems of fractional order differential equations have emerged as an important area of research, since these problems have applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, aerodynamics, viscoelasticity and damping, electro-dynamics of complex medium, wave propagation, blood flow phenomena, etc. [ – ]. We define qk-derivative of a function f : Jk → R at a point t ∈ Jk as follows. Let f : Jk → R be a continuous function, we call the second-order qk-derivative D qk f provided Dqk f is qk-differentiable on Jk with D qk f = Dqk (Dqk f ) : Jk → R.
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