Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov–Razumikhin Method

Abstract

The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method (LRM). Hence, some results that are available in the literature are improved for the FrRIDEs and obtained under weaker conditions via the advantage of the LRM. In order to illustrate the results, two examples are provided.

On the qualitative analyses solutions of new mathematical models of integro‐differential equations with infinite delay

“…In addition to the above information, numerous different forms of IDEs with and without delay have been taken under consideration in previous studies [28][29][30][31][32][33]. Fundamental properties of solutions of those IDEs have been investigated by the LKM or some the other methods.…”

Chaos coordinated neural key synchronization for enhancing security of IoT

“…The IFSMC regulator created by combining SMC and fuzzy logic-based IFLC was intended for hyperchaotic systems to take use of SMC and fuzzy logic-based IFLC. The goal of the work proposed by Tunç et al [35] is to look at a few of the characteristics of FrRIDEs with Caputo fractional differential equations, which are nonlinear fractions delayed Volterra integro-differential equations. The reported results are also supported by the use of the Lyapunov-Razumikhin technique and the definition of an adequate Lyapunov function (LRM).…”

On two structures of the fractional q‐sequential integro‐differential boundary value problems

“…17,37 It is notable that in view of definition of q-number [𝜎 ] q , an applied property Γ q (𝜎 + 1) = [𝜎] q Γ q (𝜎) is valid. 17 In the Algorithm 0, we provide a pseudo-code based on relations (5) and (6) to compute different values of the Gamma function in the q-setting.…”

“…Additionally, on many fractional abstract modelings, the researchers find various aspects of the structures of solutions for general fractional BVPs. In mentioned context, we can point out some works done before in this regards including previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and etc.…”