article · Fractal and Fractional
This research investigates the long-term behaviour of solutions to complex mathematical evolution equations. Specifically, it establishes conditions that guarantee the asymptotic periodicity of bounded mild solutions across two distinct mathematical settings. The first category examines equations involving non-densely defined operators, while the second explores equations governed by densely defined operators with fractional derivatives that generate contraction semigroups. The methodology combines the spectral properties of uniformly bounded continuous functions on the positive real semi-axis with extrapolation theory tailored to non-densely defined operators. A concrete example is included to demonstrate the application of these theoretical results.
Many physical, biological, and engineering systems exhibit recurring or periodic patterns over time. Understanding the mathematical conditions under which evolution equations settle into periodic states allows researchers to better analyse and predict the long-term behaviour of complex processes governed by memory effects and non-standard boundaries.
The abstract does not indicate an application pathway, as it presents purely theoretical mathematical research.
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In the present article, we establish conditions for the asymptotic periodicity of bounded mild solutions in two distinct cases of evolution equations. The first class involves non-densely defined operators, while the second class incorporates densely defined operators with fractional derivatives that generate a semigroup of contractions. Our method integrates the theory of spectral properties of uniformly bounded continuous functions defined on the positive real semi-axis. Additionally, we apply extrapolation theory to evolution equations with non-densely defined operators. To illustrate our main results, we provide a concrete example.
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DOI: 10.3390/fractalfract9020085
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