Caputo–Orlicz framework for functions of bounded Ψ-variation and exact fractal dimensions of their graphs

R.G.P. Sudha and R. Uthayakumar

Chaos, Solitons & Fractals, 2026, vol. 208, issue P1

Abstract: Fractional calculus provides a fundamental analytical framework for modeling nonlocal and memory-dependent phenomena in complex and biological systems, where classical smoothness assumptions often fail. In this paper, we develop a unified Caputo–Orlicz framework for the geometric analysis of continuous functions whose Caputo fractional derivatives possess bounded Ψ-variation, with Ψ a superlinear Young function. Using Orlicz–Hölder inequalities and Luxemburg norm techniques, we prove that the Caputo fractional derivative acts as a bounded linear operator on Orlicz variation spaces and preserves controlled modulus of continuity properties. These analytic results enable us to derive sharp bounds and exact formulas for the Hausdorff and box-counting dimensions of function graphs over both smooth and possibly fractal domains. In particular, we show that superlinear Orlicz growth enforces minimal graph dimensionality, revealing a sharp transition between smooth and fractal geometric regimes. This work establishes a rigorous connection between fractional differentiation, generalized variation theory, and fractal geometry, providing a flexible theoretical foundation for analyzing nonlinear fractional models that arise in contemporary applied mathematics.

Keywords: Caputo fractional derivative; Orlicz variation; BVΨspaces; Hausdorff dimension; Box–counting dimension; Superlinear growth; Fractal graph dimension (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:208:y:2026:i:p1:s0960077926002894

DOI: 10.1016/j.chaos.2026.118148

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