On generalized fractional differential equation with Sonine kernel on a function space

Abstract

Sonine kernel is a special type of the general kernels, satisfying some integrability and monotonicity conditions and properties. Sonine kernels generate some convolution polynomials (power functions) used in the construction of the general fractional calculus. The general fractional calculus with Sonine kernel is applied in the setting up and development of operational calculus.Consequently, we research on a kind of a generalized fractional differential equation with a Sonine type kernel: (∗D(κ)Cμ)(t)=ϱϑ(t,μ(t)),t∈(0,T],T<∞,with μ(0)=ω a non-negative and bounded initial data. The function ϑ:(0,T]×C−11(0,T]→R is assumed to be Lipschitz continuous on its second variable, ϱ>0 is a constant, ∗D(κ)C is a generalized Caputo fractional derivative operator and κ is a Sonine kernel belonging to a function space with an integrable singularity at the point of origin. Given some defined properties of the kernel, we establish evidence for the well-posedness of the solution via Banach fixed point theorem and reveal that the solution indicates an exponential growth bound. Moreso, we show that the solution is uniformly continuous in time and has continuous dependence on the initial condition.