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In this work, We investigate the existence and uniqueness of μ-pseudo almost automorphic solutions for a class of abstract two-term fractional differential equations in Banach spaces. Such equations play an important role in modeling complex phenomena arising in physics, engineering, and dynamical systems with memory and hereditary effects.
The main objective is to introduce and study the concept of measure μ-Stepanov pseudo almost automorphic functions. A new composition theorem is established for this class of functions, extending several known results in the literature. This framework is more general than that of weighted pseudo almost automorphic functions and provides greater flexibility in the analysis of fractional evolution equations.
By combining the theory of sectorial operators, regularized strongly continuous operator families, the Banach fixed point theorem, and the Leray-Schauder alternative theorem, we derive sufficient conditions ensuring the existence and uniqueness of μ-pseudo almost automorphic mild solutions for two-term fractional differential equations. The obtained results are established both under Lipschitz conditions on the nonlinear term and in a more general setting where such assumptions are not required.
The proposed theorems generalize and complement several previous contributions concerning almost automorphic and pseudo almost automorphic solutions of fractional evolution equations. Furthermore, an illustrative example involving a fractional partial differential equation is presented to demonstrate the applicability of the theoretical findings.
This study highlights the effectiveness of the measure-theoretic approach in the qualitative analysis of fractional differential equations and opens new perspectives for the investigation of nonautonomous dynamical systems with memory effects.
The obtained results contribute to the development of qualitative methods for fractional dynamical systems and provide a useful framework for future investigations of complex evolution equations in Banach spaces. The results extend existing theories and enhance future applications.
Keywords: measure theory, μ-Stepanov pseudo almost automorphic functions, fractional differential equations, sectorial operators, fixed point theorems.