Abstract
Motivated by the Hilfer fractional derivative (which interpolates the Riemann-Liouville derivative and the Caputo derivative), we consider a new type of fractional derivative (which interpolates the Hadamard derivative and its Caputo counterpart). We prove the well-posedness for a basic Cauchy type fractional differential equation involving this kind of derivative. This is established in an appropriate underlying space after proving the equivalence of this problem with a certain corresponding Volterra integral equation.
Highlights
In this work, we are concerned with the Hadamard derivative (HDαa+ f) (x) = α) (x d x )∫ dx a f (t) dt, t t a < x < b. (1)
We show that the initial condition in (46) holds
First we prove the existence of a unique solution y in the space C1−γ,log[a, b]
Summary
We consider the following fractional derivative (HDαa+,β f) (x) = (Jβa+(1−α)⋅HDαa++β−αβf) (x). This type of fractional derivative interpolates the Hadamard fractional derivative (β = 0) and the Caputo-Hadamard fractional derivative (β = 1). We study the existence and uniqueness of solutions of a basic fractional differential equation (HDαa+,β y) (x) = f (x, y) , x > a > 0. We find that solutions decay to zero at a logarithmic rate as time goes to infinity To this end, we prove an inequality (which is important by itself).
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