Abstract
In this article, we are concerned with the existence of mild solutions and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators and nonlocal conditions. The existence results are obtained by first defining Green’s function and approximate controllability by specifying a suitable control function. These results are established with the help of Schauder’s fixed point theorem and theory of almost sectorial operators in a Banach space. An example is also presented for the demonstration of obtained results.
Highlights
In current times, the rising interest of researchers in fractional calculus reflects the popularity of this branch [1,2,3, 5,6,7,8,9,10,11,12,13, 15, 19, 30,31,32]
Under some admissible control inputs, exact controllability steers the system to arbitrary final state, while approximate controllability steers the system to the small neighborhood of arbitrary final state
Jaiswal et al [21] proved the existence of mild solutions of Hilfer Fractional differential equations (FDEs) with almost sectorial operators
Summary
The rising interest of researchers in fractional calculus reflects the popularity of this branch [1,2,3, 5,6,7,8,9,10,11,12,13, 15, 19, 30,31,32]. Jaiswal et al [21] proved the existence of mild solutions of Hilfer FDEs with almost sectorial operators. Definition 2.1 ([20]) Hilfer fractional derivative of a continuously differentiable function f of order 0 < ν < 1 and 0 ≤ μ ≤ 1 is defined as. Lemma 2.1 ([2, 20]) The mild solution for the system of equations Eq (1.1) is defined as follows:. – Ψ s, v(s), q(s, ζ )φ s, ζ , v(ζ ) dζ ≤ 2δ N1 + N2wq∗ , which is finite, by the Lebesgue dominated convergence theorem and the continuity of function Ψ , Pyn(t) – Py(t) −→ 0 as n −→ ∞ This completes the proof of Lemma 3.1.
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