一类<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo></math></inline-formula>⁃比例Hilfer分数阶演化方程温和解的存在性

一类 $(ρ, k, φ)$ ⁃比例Hilfer分数阶演化方程温和解的存在性

摘要: 在最新引入的 $(ρ, k, φ)$ -比例分数阶算子基础上，研究了一类 $(ρ, k, φ)$ -比例Hilfer分数阶Cauchy问题温和解的存在唯一性和解的连续依赖性。利用概率密度函数、 $(ρ, k, φ)$ -比例Hilfer分数阶导数的性质和半群理论，得到了温和解的定义。通过构造合适的加权空间，根据Banach压缩映射原理，研究了解的存在唯一性。构造广义Gronwall不等式，得到了解的连续依赖性。

Abstract: The existence， uniqueness and continuous dependence of solutions for a class of $(ρ, k, φ)$ ⁃proportional Hilfer fractional Cauchy problems were investigated. The probability density function， properties of the $(ρ, k, φ)$ ⁃proportional Hilfer fractional derivative and semigroup theory were utilized to define mild solutions. A proper weighted space was introduced， and within this space， the Banach contraction principle was applied to discuss the uniqueness of the solutions. The continuous dependence of the data on the Cauchy problem was proven by constructing a generalized Gronwall inequality.