The Existence of Mild Solutions for Hilfer Fractional Differential Equations with Infinite Delay in Orlicz Space

The Hilfer fractional derivative effectively captures non-locality, historical dependence, and memory effects, making it valuable for modeling real-world systems, and exponential growth can describe explosive growth phenomena in real-world problems. This paper focuses on the existence of mild solutions for infinite-delay differential equations involving Hilfer fractional derivatives, fractional Laplacian operator (−Δ)δ, and exponentially growing functions in Orlicz spaces. First, by utilizing standard Lp-Lq estimates for strongly continuous semigroups generated by fractional Laplacian operator, the existence of global solutions in the Orlicz space expLp(Rd) and the time-weighted Lz(Rd) space is established. Then, by leveraging Hölder’s interpolation inequality, the existence of local solutions in L1(Rd)∩L∞(Rd) is established.