• PDF / 495,958 Bytes
• 14 Pages / 439.37 x 666.142 pts Page_size
• 102 Downloads / 265 Views

DOWNLOAD

REPORT

---

JAMC

O R I G I NA L R E S E A R C H

Existence of solutions for set differential equations involving causal operator with memory in Banach space Jingfei Jiang · C.F. Li · H.T. Chen

Received: 11 July 2012 / Published online: 9 October 2012 © Korean Society for Computational and Applied Mathematics 2012

Abstract This paper concerns with the IVP for set differential equations involving causal operators with memory. By using the techniques of the measure of noncompactness, the existence of solutions has been established. The maximal and minimal solutions are obtained by means of Arzela-Ascoli Theorem. Keywords Set differential equation · Causal operator · Measure of non-compactness · Existence of solutions · Maximal and minimal solution Mathematics Subject Classification (2010) 34A12 · 34G20 · 34G25 · 34L30

1 Introduction In this paper, we consider the following IVP for set functional differential equation involving causal operator with memory  DH U (t) = Q(U, Φ0 )(t), t ∈ (t0 , T ], (1) Ut 0 = Φ 0 ∈ C 1 , where Q(U, Φ0 )(t) : E0 → E is a causal or a Volterra Operator with memory, E0 , E are two given Banach spaces. J. Jiang () · C.F. Li School of Mathematics and Computing science, Xiangtan University, Xiangtan, 411105, P.R. China e-mail: [email protected] C.F. Li e-mail: [email protected] H.T. Chen School of Astronautics, Harbin Institute of Technology, P O Box 137, Harbin, 150001, P.R. China

184

J. Jiang et al.

Now the IVP (1) is equivalent to the set Hukuhara integral equation:  t Q(U, Φ0 )(s)ds U (t) = Φ0 (t0 ) +

(2)

t0

and Ut0 = Φ 0

on [t0 − h, t0 ].

In recent years, the study of set differential equations was initiated in a metric space and some results such as existence and uniqueness of the solutions were obtained, see [2, 11, 13–17, 19, 20, 22–25]. Base on these results, the study of set differential equations involving causal operators with memory has been developed; see [4, 5, 7, 8, 12, 21, 26]. The term causal operators are adopted from the engineering literatures and differential equations with finite or infinite delay, some results have been introduced in [1, 18] and [6, 9, 10]. In this paper, we study the IVP for set differential equations involving causal operators with memory. By using the techniques of the measure of non-compactness [3], we obtain the existence of solutions for this problem. The rest of this paper is organized as follows. In Sect. 2, some concepts and preparation results are given. Section 3 is devoted to obtain existence of the solution by the techniques of the measure of non-compactness, then we present the maximal and minimal solutions for set differential equation with causal operator.

2 Preliminaries In this section, some preliminaries which will be used in this paper are given. Let Kc (Rn ) be the collection of all non-empty compact and convex subsets of Rn . Let   D[A, B] = max sup d(x, A), sup d(y, B) , (2.1) x∈B

y∈A

where d(x, A) = infx∈B [d(x, y) : y ∈ A], A, B are arbitrary two bounded sets in Kc (Rn ) and D[·,·] satisfies the following properties: D[A + C