Existence and Ulam Stability of Solutions for Conformable Impulsive Differential Equations
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Existence and Ulam Stability of Solutions for Conformable Impulsive Differential Equations Wanzheng Qiu1 · JinRong Wang1,2
· Donal O’Regan3
Received: 14 October 2019 / Revised: 19 December 2019 / Accepted: 27 December 2019 © Iranian Mathematical Society 2020
Abstract In this article, we use mathematical induction to derive the representation of the solution of conformable impulsive linear differential equations with constant coefficients. We present the existence of solutions to impulsive nonlinear differential equations with constant coefficients under mild conditions on the nonlinear term. In addition, we consider the concepts of Ulam stability for this type of equation and give Ulam– Hyers and Ulam–Hyers–Rassias stability results. Finally, we give examples to verify our theoretical results. Keywords Conformable impulsive differential equations · Existence of solutions · Ulam–Hyers and Ulam–Hyers–Rassias stability Mathematics Subject Classification 34A12 · 34A37
Communicated by Majid Gazor. This work is partially supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), and Major Research Project of Innovative Group in Guizhou Education Department ([2018]012).
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JinRong Wang [email protected] Wanzheng Qiu [email protected] Donal O’Regan [email protected]
1
Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China
2
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China
3
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
123
Bulletin of the Iranian Mathematical Society
1 Introduction A concept of conformable derivative (CD) was introduced in [1–3] and used to extend Newton mechanics [4], logistic models [5], and cobweb models [6]. The qualitative analysis for linear, semilinear, nonlinear and delay differential equations with the CD was studied in [7–18] and equations of the Caputo derivative were studied in [19–21]. Impulsive differential equations with a CD have not been studied until now. In this paper, we study existence and Ulam stability of solutions for the following impulsive differential equations with a CD: ⎧ a ⎨ Dγ x(t) = λx(t) + f (t, x(t)), λ = 0, t ∈ I := [a, b], b > a ≥ 0, 0 < γ < 1, x(ti ) = bi x(ti ), ti < ti+1 , i ∈ M := {1, 2, . . . , m}, t0 := a, tm+1 = b, bi ∈ R, ⎩ x(a) = xa , (1.1) where Daγ x is called the CD with lower index a of the function x, x(ti ) := x(ti+ ) − x(ti− ), and we set x(ti ) = x(ti− ), f : I × R → R and λ ∈ R\{0}. The paper is organized as follows. In Sect. 2, we recall the basic definitions of the CD and derive the representation of the solution of the impulsive linear problem: ⎧ a ⎨ Dγ x(t) = λx(t) + f (t), λ = 0, t ∈ I , 0 < γ < 1, f ∈ C(I , R), x(ti ) = bi x(ti ), ti < ti+1 , i ∈ M := {1, 2, . . . , m}, t0 := a, tm+1 = b, bi ∈ R, ⎩ x(a) = xa , (1.2) using mathematical induction. In Sect. 3, we give the existence results for (1.1) by
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