Approximate controllability of impulsive Hilfer fractional differential inclusions

Abstract

In this paper, firstly by utilizing the theory of operators semigroup, probability density functions via impulsive conditions, we establish a new PC 1−ν -mild solution for impulsive Hilfer fractional differential inclusions. Secondly we prove the existence of mild solutions for the impulsive Hilfer fractional differential inclusions by using fractional calculus, multi-valued analysis and the fixed-point technique. Then under some assumptions, the approximate controllability of associated system are formulated and proved. An example is provided to illustrate the application of the obtained theory.

Hilfer-type fractional differential switched inclusions with noninstantaneous impulsive and nonlocal conditions

“…By dividing both sides by n and passing to the limit as n → ∞, we obtain from (9) and (12) that 1 (ηυb 1−γ /Γ(α)) ϕ L p (J,R + ) + h + h/Γ(γ), which contradicts (13). Thus there is a natural number n 0 such that R(B n0 ) ⊆ B n0 .…”

“…The authors in [14] established the existence and uniqueness of global solution in the space of weighted continuous functions for a fractional differential equations involving the Hilfer derivative, the authors in [34] discussed the existence of solutions to nonlocal initial value problems for differential equations with the Hilfer fractional derivative, and the authors in [16] obtained some sufficient conditions to ensure the existence of mild solutions of evolution equation with the Hilfer fractional derivative. In [37], the authors investigated the approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions, the authors in [13] established the approximate controllability of impulsive Hilfer fractional differential inclusions, the author in [24] derived an equivalent definition of the Hilfer derivative, and the authors in [33] studied the controllability of Caputo fractional noninstantaneous impulsive differential inclusions without compactness in reflexive Banach spaces.…”

Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators

“…In 2017, Yang et al [41] discussed the approximate controllabil-ity of Hilfer FDEs with nonlocal conditions in a Banach space with the help of semigroup theory, fixed point techniques, and multivalued analysis. Later on, Debbouche et al [14] and Du et al [17] studied the approximate controllability of Hilfer FDEs and semilinear Hilfer FDEs with impulsive control inclusions in Banach spaces, respectively. In 2018, Lv and Yang [34] investigated the approximate controllability of neutral Hilfer FDEs by applying the techniques of stochastic analysis theory and semigroup operator theory in a Hilbert space.…”

Finite approximate controllability of Hilfer fractional semilinear differential equations

“…Gu and Trujillo [9] obtained some sufficient conditions to ensure the existence of mild solutions of evolution equation with the Hilfer fractional derivative. Yang and Wang [28,29] investigated existence of mild solutions of Hilfer evolution equations and the approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions and Du et al [7] established the approximate controllability of impulsive Hilfer fractional differential inclusions. Kamocki [13] derived an equivalent definition of the Hilfer derivative and showed that such a derivative is very useful in practical applications.…”