Abstract
In this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations \t\t\ta0(t)Dβ2y(t)+a1(t)Dβy(t)+a2(t)y(t)=b(t),t∈I,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$a_{0}(t)D_{\\beta}^{2}y(t)+a_{1}(t)D_{\\beta}y(t)+a_{2}(t)y(t)=b(t),\\quad t \\in I, $$\\end{document}a_{0}(t)neq0, in a neighborhood of the unique fixed point s_{0} of the strictly increasing continuous function β, defined on an interval Isubseteq{mathbb{R}}. These equations are based on the general quantum difference operator D_{beta}, which is defined by D_{beta}{f(t)}= (f(beta(t))-f(t) )/ (beta(t)-t ), beta(t)neq t. We also construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β-difference equation.
Highlights
1 Introduction Quantum calculus allows us to deal with sets of non-differentiable functions by substituting the classical derivative by a difference operator
Quantum calculus has a lot of applications in different mathematical areas such as the calculus of variations, orthogonal polynomials, basic hyper-geometric functions, economical problems with a dynamic nature, quantum mechanics and the theory of scale relativity; see, e.g., [ – ]
T = s, f (s ), t = s, where f : I → X is a function defined on an interval I ⊆ R, X is a Banach space and β : I → I is a strictly increasing continuous function defined on I, which has only one fixed point s ∈ I and satisfies the inequality: (t – s )(β(t) – t) ≤ for all t ∈ I
Summary
Quantum calculus allows us to deal with sets of non-differentiable functions by substituting the classical derivative by a difference operator. In [ ], the existence and uniqueness of solutions of the β-initial value problem of the first order were established. In Section , we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations in a neighborhood of s. If f : I → X is a continuous function at s , the sequence {f (βk(t))}∞ k= converges uniformly to f (s ) on every compact interval V ⊆ I containing s. Where p : I → C is a continuous function at s and both infinite products are convergent to a non-zero number for every t. The β-exponential functions ep,β (t) and E–p,β (t) are, respectively, the unique solutions of the β-initial value problems: Dβ y(t) = p(t)y(t), y(s ) = , Dβ y(t) = –p(t)y β(t) , y(s ) =. M sup(t,y)∈R f (t, y) < ∞, ρ ∈ ( , )
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