Mild Solutions for Impulsive Integro-Differential Equations Involving Hilfer Fractional Derivative with almost Sectorial Operators

In this manuscript, we establish the mild solutions for Hilfer fractional derivative integro-differential equations involving jump conditions and almost sectorial operator. For this purpose, we identify the suitable definition of a mild solution for this evolution equations and obtain the existence results. In addition, an application is also considered.

Analysis on ψ-Hilfer Fractional Impulsive Differential Equations

In this manuscript, we establish the existence of results of fractional impulsive differential equations involving ψ-Hilfer fractional derivative and almost sectorial operators using Schauder fixed-point theorem. We discuss two cases, if the associated semigroup is compact and noncompact, respectively. We consider here the higher-dimensional system of integral equations. We present herewith new theoretical results, structural investigations, and new models and approaches. Some special cases of the results are discussed as well. Due to the nature of measurement of noncompactness theory, there exists a strong relationship between the sectorial operator and symmetry. When working on either of the concepts, it can be applied to the other one as well. Finally, a case study is presented to demonstrate the major theory.

FUNCTIONAL CALCULI FOR SECTORIAL OPERATORS AND RELATED FUNCTION THEORY

We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

Asymptotically periodic behavior of solutions to fractional non-instantaneous impulsive semilinear differential inclusions with sectorial operators

In this paper, we prove two results concerning the existence of S-asymptotically ω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order $1<\alpha <2$ 1 < α < 2 and generated by sectorial operators. In the first result, we apply a fixed point theorem for contraction multivalued functions. In the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval $J=[0,\infty )$ J = [ 0 , ∞ ) . We adopt the fractional derivative in the sense of the Caputo derivative. We provide three examples illustrating how the results can be applied.