In this manuscript, we are mainly focusing on the existence of mild solutions for the Hilfer fractional evolution system with infinite delay via measures of noncompactness. Fractional calculus and fixed-point approach are taken into consideration to study the primary outcomes. Finally, we extend the results to the existence of a neutral system with nonlocal conditions. Ultimately, a model is presented for illustration of theory. KEYWORDS Hilfer fractional evolution system, measures of noncompactness, mild solutions, neutral system, nonlocal conditions MSC CLASSIFICATION 34K30; 34K40; 47H08; 47H10 where D v, 0 + denotes the Hilfer fractional derivative of type and order v. Also, 0 ≤ v ≤ 1, 1 2 < < 1, and z(•) take value in Banach space Z with || • ||, and the linear operator A is the infinitesimal generator of analytic semigroup {T(t)} t ≥ 0 on Z. The histories z t ∶ (−∞, 0] → h , z t () = z(t +), ≤ 0 associated with phase space h. ∶ N × h → Z is an appropriate function. Recently, many researchers show great interest in the existence and controllability of Hilfer fractional differential systems with or without delay, the authors can refer the previous articles. 33-44 In a previous work, 40 Harrat et al studied the solvability and optimal controls of an impulsive nonlinear Hilfer fractional delay evolution inclusion in Banach spaces by using fractional calculus, fixed-point technique, semigroup theory, and multivalued analysis. In another work, 45 Vijayakumar and Udhayakumar discussed the approximate controllability for nondensely defined Hilfer fractional differential system with infinite delay by employing fractional calculus and Bohnenblust-Karlin's fixed-point theorem. In a previous study, 46 Kavitha et al proved the controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness by using semigroup theory, fractional calculus, the measure of noncompactness, and the Mönch fixed-point theorem. Nonetheless, upmost definitely, the study of the existence of mild solutions for the Hilfer fractional evolution equations and the Hilfer fractional neutral evolution equations with infinite delay discussed in the article has not been contemplated, and this gives the inspiration for the current manuscript. We now subdivide our article into the following sections: In Section 2, we introduce a few essential facts and definitions associated with our study that are employed, which are utilized throughout the discussion of this article. Section 3 is reserved for discussion about the existence of mild solution for (1.1) and (1.2). In Section 5, we continue our discussion to the systems (1.1) and (1.2) with nonlocal conditions. In Section 6, we present an example of the illustration of the obtained theory.
