Existence of mild solutions to Hilfer fractional evolution equations in Banach space
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Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00095-5 ORIGINAL PAPER
Existence of mild solutions to Hilfer fractional evolution equations in Banach space J. Vanterler da C. Sousa1 · Fahd Jarad2 · Thabet Abdeljawad3,4 Received: 30 June 2020 / Accepted: 5 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, we investigate the existence of mild solutions to semilinear evolution fractional differential equations with non-instantaneous impulses, using the concepts of equicontinuous (𝛼, 𝛽)-resolvent operator function ℙ𝛼,𝛽 (t) and Kuratowski measure of non-compactness in Banach space 𝛺. Keywords Hilfer fractional evolution equations · Mild solution · Existence · Equicontinuous (𝛼, 𝛽)-resolvent operator · Kuratowski measure of non-compactness Mathematics Subject Classification 26A33 · 34K45 · 47D06
1 Introduction We can start the paper with the following questions: Why study fractional calculus? What are the advantages we gain by investigating and proposing new results in the field of fractional calculus? Are the results presented so far important and relevant to the point of contributing to the scientific community? In a simple and clear answer, Communicated by Feng Dai. * J. Vanterler da C. Sousa [email protected] Fahd Jarad [email protected] Thabet Abdeljawad [email protected] 1
Department of Applied Mathematics, State University of Campinas (UNICAMP)-Imecc, Campinas, SP 13083‑859, Brazil
2
Department of Mathematics, Çankaya University, 06790 Ankara, Turkey
3
Department of Mathematics and Physical Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia
4
Department of Medical Research, China Medical University, 40402 Taichung, Taiwan
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it is enough to notice the exponential scientific growth in the area and the impact that the fractional calculus has contributed to mathematics and other diverse sciences, specially, in a shocking way in the context of mathematical modeling [9, 17, 19, 23, 25–28]. The theory of differential equations with non-instantaneous impulses and impulsive evolution equations in Banach spaces has been investigated by many researchers in the last decades [2, 18, 24]. It is noted that investigating the existence, uniqueness, stability of solutions of differential equations of evolution has been object of study and applicability in the scientific community, since it describes processes that experience a sudden change in their states at certain moments [1, 3, 7, 8, 15]. The applicability of the obtained results related to the differential equations, especially with non-instantaneous impulses, can be found in several areas, such as: physics, engineering, economics, biology, medicine and mathematics itself, among others [2, 18]. Pierri et al. [21] investigated the existence of solutions of a class of abstract semilinear differential equations with non-instantaneous impulses using the semigroup analytic theory. In the same year, Hernandez and O’Regan [16], inv
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