Fractional integro-differentiation by Chen-Hadamard

Mahmadiyor Yahshiboev

Department of Mathematics, Samarkand State University, University Boulevard 15, 703004,Samarkand, Uzbekistan

The fractional differentiation by Chen-Hadamard

D_{c}^{α}f = \frac{1}{μ(α, l)} ∫_{0}^{∞} \frac{(Δ̃_{t}^{l}f_{c+})(x) + (Δ̃_{t^{−1}}^{l}f_{c−})(x)}{|ln t|^{1+α}} \frac{dt}{t},

is discussed, where Δ̃_{t}^{l} = ∑_{k=1}^{l}(−1)^{k}\binom{k}{l}f(x · t^{k}), l > α > 0, c > 0, f ∈ L_{p} (R_{+}^{1}, \frac{dx}{x})

( or f ∈ L_{p}^{loc} (R_{+}^{1}, \frac{dx}{x})),

f_{c+}(x) = \{ \begin{matrix}f(x), & x > c \\ 0, & x < c, \end{matrix} f_{c−}(x) = \{ \begin{matrix}0, & x > c \\ f(x), & x < c \end{matrix} .

Since the Chen construction can be applied to functions with an arbitrary growth when x → ∞ or x → 0, this construction is more convenient when applied to such functions than the integro-differentiation by Hadamard [1] itself. As usual, the fractional derivative is to be treated as a certain limit. To this end, several types of different ”truncation” of the Chen-Hadamard fractional derivative are introduced, denoted by

D_{c,ρ}^{α}f,  D^{∗}_{c,ρ}^{α} f,  D_{c,ρ̃}^{α}f,

where \frac{1}{\sqrt[ℓ]{e}} < ρ < 1, ρ̃ = |ln \frac{x}{c} | ln \frac{1}{ρ}, ℓ > α > 0. The following inversion theorem is valid, where J_{c}^{α}φ stands for the corresponding Chen-Hadamard fractional integral.

Theorem 1. Let f = J_{c}^{α}φ, φ ∈ L_{p} (R_{+}^{1}, \frac{dx}{x}) (or φ ∈ L_{p}^{loc} (R_{+}^{1}, \frac{dx}{x})), 1 ≤ p < ∞, α > 0, c > 0. Then

lim_{ρ→1} D_{c,ρ}^{α}f = φ,  (1)
lim_{ρ→1} D^{∗}_{c,ρ}^{α} f = lim_{ρ→1} D_{c,ρ̃}^{α}f = φ,  (2)

The limits in (1)-(2) can be understood both in L_{p} (R_{+}^{1}, \frac{dx}{x}) or L_{p}^{loc} (R_{+}^{1}, \frac{dx}{x}), correspondingly, 1 ≤ p < ∞, except for the case p = 1 in (2), or almost everywhere.

Theorem 2 For the function f(x) to be presented as f(x) = (J_{c}^{α}φ)(x), where φ ∈ L_{p}^{loc}(R_{+}^{1}, \frac{dx}{x}); α > 0, c > 0, 1 < p < ∞, it is necessary and sufficient that | ln \frac{x}{c}|^{−α}f(x) ∈ L_{p}^{loc}(R_{+}^{1}, \frac{dx}{x}), and

lim_{ρ→1} D_{c,ρ̃}^{α}f

exists in L_{p}^{loc}(R_{+}^{1}, \frac{dx}{x}).

References

[1] S.G. Samko, A.A. Kilbas, O.I. Marichev. Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach. Sci. Publ., London-N.York (1993) (Russian Ed.:Fractional Integrals and Derivatives and Some of Their Applications. Nauka i Tekhnika, Minsk (1987).