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In this paper, the maximal‐weighted Hölder regularity for the fractional abstract Cauchy problem is presented. First, two classes of novel interpolation spaces constructed by resolvent families are portrayed. Afterward, by applying the resolvent method and the properties of interpolation spaces, the maximal‐weighted Hölder regularity of the mild solution is established in weighted Hölder space C0β,γ(X)$C_0^{\beta ,\gamma }(X)$ (0≤γ≤β$0\le \gamma \le \beta$, 0<β<1$0<\beta <1$). Eventually, an instance for the fractional differential equation of 2m‐order multidimensional parabolic type is provided to verify the reasonability of main conclusions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of fractional differential equations with Riemann-Liouville fractional derivatives and infinite delay in Banach spaces. The weighted delay is developed to deal with the case of non-zero initial value, which leads to the unboundedness of the solutions. Existence and uniqueness results are obtained based on the theory of measure of non-compactness, Schaude’s and Banach’s fixed point theorems. As auxiliary results, a fractional Gronwall type inequality is proved, and the comparison property of fractional integral is discussed.

In this paper, two significant inequalities for the Hilfer fractional derivative of a function in the space A C γ ([ 0 , b ] , R n) , 0 ≤ γ ≤ 1 are obtained. We first verified the extremum principle for the Hilfer fractional derivative. In addition, we estimated the Hilfer derivative of a function at its extreme points. Furthermore, we derived and proved a maximum principle for a nonlinear Hilfer fractional differential equation. Finally, we analyzed the solutions of a nonlinear Hilfer fractional differential equation. Our results generalize and extend some obtained theorems on this topic. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the concept of regional enlarged observability (ReEnOb) for fractional differential equations (FDEs) with the Hilfer derivative. To proceed this, we develop an approach based on the Hilbert uniqueness method (HUM). We mainly reconstruct the initial state ν 0 1 on an internal subregion ω from the whole domain Ω with knowledge of the initial information of the system and some given measurements. This approach shows that it is possible to obtain the desired state between two profiles in some selective internal subregions. Our findings develop and generalize some known results. Finally, we give two examples to support our theoretical results. [ABSTRACT FROM AUTHOR]