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Hyers-Ulam stability for nonlocal differential equations

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In this paper, we present a result on Hyers-Ulam stability for a class of nonlocal differential equations in Hilbert spaces by using the theory of integral equations with completely positive kernels together with a new Gronwall inequality type. The paper develops some recent results on fractional differential equations to the ones involving general nonlocal derivatives. Instead of Mittag-Leffler functions, we must utilize the characterization of relaxation function. | HNUE JOURNAL OF SCIENCE DOI 10.18173 2354-1059.2020-0041 Natural Science 2020 Volume 65 Issue 10 pp. 3-9 This paper is available online at http stdb.hnue.edu.vn HYERS-ULAM STABILITY FOR NONLOCAL DIFFERENTIAL EQUATIONS Nguyen Van Dac1 and Pham Anh Toan2 1 Faculty of Computer Science and Engineering Thuyloi University 2 Nguyen Thi Minh Khai High School Hanoi Abstract. In this paper we present a result on Hyers-Ulam stability for a class of nonlocal differential equations in Hilbert spaces by using the theory of integral equations with completely positive kernels together with a new Gronwall inequality type. The paper develops some recent results on fractional differential equations to the ones involving general nonlocal derivatives. Instead of Mittag-Leffler functions we must utilize the characterization of relaxation function. Keywords nonlocal differential equation mild solution Hyers-Ulam stability. 1. Introduction Let H be a separable Hilbert space. Consider the following equation k t u t Au t f t u t t J 0 T . 1.1 where the unknown function u takes values in H the kernel k L1loc R A is an inbounded linear operator and f J H H is a Rgiven function. Here the t notation denotes the Laplace convolution i.e. k v t 0 k t s v s ds. In 1 authors introduced a result on the existence regularity and stability for mild solutions to 1.1 where f depends only on u and the initial condition is given by u 0 u0 . 1.2 Our goal in this paper is to consider the Hyers-Ulam stability for 1.1 . The Hyers-Ulam stability for functional equations was founded in 1940 by S.M Ulam in a talk at Wisconsin University see 2 and by D. H Hyers answer a year later for additive functions defined on Banach spaces see 3 . However the first result on the Hyers-Ulam stability of a differential equation was addressed by C.Alsina and R. Ger in 1998 see 4 . In this remarkable work they proved that if a differentiable function Received October 2 2020. Revised October 23 2020. Accepted October 30 2020. .