论文标题
Caputo分数自我接触差方程的初始和边界价值问题
Initial and Boundary Value Problems for the Caputo Fractional Self-Adjoint Difference Equations
论文作者
论文摘要
在本文中，我们开发了\ [\ nabla [p（t+1）\ nabla_ {a*}^νx（t+1）]+q（t+q（t）]+q（t）x（t）x（t）x（t）= 1 $ n $ 0 <use，我们开发了\ [\ nabla [p（t+1）\ nabla_ {a*} a*}^νx（t+1）]+q（t+q（t）+q（t）x（t）x（t）x（t）$ 0 <se，我们开发了包含caputo分数nabla差异的自动化nabla分数差方程的初始和边界值问题的理论。我们介绍具有Caputo分数差异的Nabla分数演算。我们研究了上面给出的特定自我伴侣Nabla分数方程的特性。在适当条件下，我们证明了初始和边界价值问题的存在和唯一定理。我们介绍了Cauchy函数的定义，该函数使我们能够提供多种常数公式来解决初始值问题。然后，我们表明，这种cauchy功能对于通过Sturm-Liouville型边界条件找到边界价值问题的绿色功能很重要。导出了有关某个绿色功能的几种不平等。这些结果对于使用固定点定理证明存在与我们的线性自相关方程相关的非线性分数方程的解决方案的解决方案很重要。
In this paper we develop the theory of initial and boundary value problems for the self-adjoint nabla fractional difference equation containing a Caputo fractional nabla difference that is given by \[ \nabla[p(t+1)\nabla_{a*}^νx(t+1)] + q(t)x(t) = h(t), \] where $0 < ν\leq 1$. We give an introduction to the nabla fractional calculus with Caputo fractional differences. We investigate properties of the specific self-adjoint nabla fractional difference equation given above. We prove existence and uniqueness theorems for both initial and boundary value problems under appropriate conditions. We introduce the definition of a Cauchy function which allows us to give a variation of constants formula for solving initial value problems. We then show that this Cauchy function is important in finding a Green's function for a boundary value problem with Sturm-Liouville type boundary conditions. Several inequalities concerning a certain Green's function are derived. These results are important in using fixed point theorems for proving the existence of solutions to boundary value problems for nonlinear fractional equations related to our linear self-adjoint equation.
