| Citation: |
Mohamed El Khayr Boukraa, Mohamed Saad Bouh Elemine Vall, Ahmed Ahmed. DOUBLE PHASE PROBLEMS IN MUSIELAK SPACES INVOLVING THE $ \varphi $-HILFER FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 2126-2153. doi: 10.11948/20250165
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DOUBLE PHASE PROBLEMS IN MUSIELAK SPACES INVOLVING THE $ \varphi $-HILFER FRACTIONAL DERIVATIVE
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Abstract
In this paper, we introduce a functional framework that extends the $ \varphi $-Hilfer space by incorporating the Laplacian operator within a double-phase structure in the setting of Musielak–Orlicz–Sobolev spaces. We define the $ \varphi $-Riemann–Liouville fractional partial integral and derivative, as well as the Hilfer fractional derivative (HFD), within this generalized context. Furthermore, we construct a Musielak–Orlicz–Sobolev $ \mathbb{H} $ space that integrates both the Laplacian operator and variable exponent Lebesgue spaces, which are central to the analysis of double-phase problems. The paper establishes the fundamental propositions, definitions, and theorems, formulates the necessary hypotheses, and demonstrates the existence of weak solutions for a suitable elliptic problem involving the HFD. To illustrate and support our theoretical results, several examples are also presented.
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References
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Mohamed El Khayr Boukraa, Mohamed Saad Bouh Elemine Vall, Ahmed Ahmed. DOUBLE PHASE PROBLEMS IN MUSIELAK SPACES INVOLVING THE $ \varphi $-HILFER FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 2126-2153. doi: 10.11948/20250165
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