Existence and Stability Results on Hadamard Type Fractional Time-Delay Semilinear Differential Equations

Nazim Mahmudov,Areen Al-Khateeb

doi:10.3390/math8081242

Abstract

A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed. This combines the classic Mittag–Leffler type matrix function with a logarithm and delayed Mittag–Leffler type matrix function. With the help of this introduced delayed perturbation of the Mittag–Leffler type matrix function with a logarithm, we provide an explicit form for solutions to non-homogeneous Hadamard-type fractional time-delay linear differential equations. We also examine the existence, uniqueness, and Ulam–Hyers stability of Hadamard-type fractional time-delay nonlinear equations.

Highlights

PreliminariesLet 0 < a < b < ∞ and C [ a, b] be the Banach space of all continuous functions y : [ a, b] → Rn with the norm kykC := max {ky (t)k : t ∈ [ a, b]}
A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed
The unification of differential equations with delay and differential equations with fractional derivatives is provided by differential equations including both delay and non-integer derivatives, so called time-delay fractional differential equations

Summary

Preliminaries

Let 0 < a < b < ∞ and C [ a, b] be the Banach space of all continuous functions y : [ a, b] → Rn with the norm kykC := max {ky (t)k : t ∈ [ a, b]}. M-L type matrix function with two parameters eα,β (A; t) : R → Rn×n is defined by. We introduce a definition of delayed M-L type matrix function Eh,α,β (ln t) : R+ → Rn×n with logarithm generated by B. Two parameters delayed M-L type matrix function Eh,α,β (ln t) : R+ → Rn×n with logarithm generated by B is defined by. Our definition of the two-parameter delayed M-L type matrix function with logarithm differs substantially from the definition given in [33]. In order to give a definition of delayed perturbation of the M-L type matrix functions with logarithm, we introduce the following matrices Yα,β,k , k = 0, 1, 2,. According to Lemma 3 in the case AB = BA delayed M-L type function Yh,α,β (t, s) has a simple form: Ai i =0. Combining Theorems 1 and 2 together we get the following result

Existence Uniqueness and Stability

Existence Result

Example

Conclusions