The existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions in a Banach space is investigated in this manuscript. No assumptions about the compactness of a function or the Lipschitz continuity of a nonlinear function are imposed on the nonlocal item and the nonlinear function, respectively. However, we assumed that the nonlocal item is continuous, the nonlinear term is continuous and satisfies some specified assumptions, and the associated semigroup is compact. Our theorems are proved by means of approximate techniques, semigroup methods, and fixed point theorem. These methods are useful for fixing the noncompactness of operators caused by some specified given assumptions on this paper. The results obtained here improve some known results. Finlay, two examples are presented for illustration of our main results.
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 ν01 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.
In this paper, two significant inequalities for the Hilfer fractional derivative of a function in the space ACγ([0,b],Rn), 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.
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