A Note on Lacunary Sequence Spaces of Fractional Difference Operator of Order$\left(\alpha ,\beta \right)$

Abstract

In the present paper, we defined lacunary sequence spaces of fractional difference operator of order α , β over n -normed spaces via Musielak-Orlicz function M = I k . Our aim in this paper is to study some topological properties and inclusion relation between the spaces I − N α β A , M , u , θ , Δ γ , · , ⋯ , · 0 , I − N α β A , M , u , θ , Δ γ , · , ⋯ , · , and I − N α β A , M , u , θ , Δ γ , · , ⋯ , · ∞ .

Optimum solution of $(k,\gimel )$-Hilfer FDEs by $\mathcal{A}$-condensing operators and the incorporated measure of noncompactness

“…The DEs of the HFDE, ψ-HFDE, and (k, ‫-)ג‬HFDE types, because they provide significant generalizations, are very useful in solving different types of differential equations. For more details of these types of generalizations and results of controllability involving ψ-HFDEs refer to [15][16][17][18][19][20][21][22] and references therein. The work referenced above inspired us to propose a most generalized version of the Hilfer derivative, the so-called (k, ‫-)ג‬Hilfer fractional derivative.…”