• PDF / 178,483 Bytes
• 18 Pages / 612 x 792 pts (letter) Page_size
• 34 Downloads / 211 Views

DOWNLOAD

REPORT

---

DOI: 10.1007/s13226-020-0450-4

SUFFICIENT CONDITIONS FOR EXISTENCE OF INTEGRAL SOLUTION FOR NON-INSTANTANEOUS IMPULSIVE FRACTIONAL EVOLUTION EQUATIONS Jayanta Borah and Swaroop Nandan Bora Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781 039, India e-mails: [email protected]; [email protected] (Received 24 April 2018; accepted 30 May 2019) In this article, we establish sufficient conditions for existence and uniqueness of integral solution for some non-densely defined non-instantaneous impulsive evolution equations on a Banach space involving Caputo fractional derivative. The results are obtained by means of characteristic functions based on probability density. Finally, the main results are illustrated through examples. Key words : Fractional evolution equation; integral solution; non-instantaneous impulse; fixed point theorem. 2010 Mathematics Subject Classification : 26A33, 34A08, 35R12.

1. I NTRODUCTION The study of differential equations with abrupt and instantaneous impulse (processes which at certain instants change their state rapidly) has been a subject of great interest due to its wide applications in physics, economics, population dynamics, control theory etc. There exists extensive literature on the existence and qualitative properties of solutions for differential equations with instantaneous impulses. For developments in the study of mild solutions and its qualitative properties to instantaneous impulsive differential equations, we refer the readers to [3, 6, 21, 26, 30] and the references therein. However, the action of instantaneous impulses does not describe dynamics of some physical processes. For example, the introduction of drugs in the bloodstream and the consequent absorption for the body, which are gradual and continuous processes, is not appropriately explained by instantaneous impulses. In fact this situation should be characterized by a new idea of impulsive action, which starts at a certain point and the action continues for some finite interval. Hern´andez and O’Regan [13]

1066

JAYANTA BORAH AND SWAROOP NANDAN BORA

initiated the study of this new class of abstract semilinear impulsive differential equations with noninstantaneous impulse in a PC-normed space of the following form:

x0 (t) = Ax(t) + f (t, x(t)), t ∈ (si , ti+1 ], i = 0, 1, ..., N,

(1.1)

x(t) = gi (t, x(t)), t ∈ (ti , si ], i = 1, 2, ..., N,

(1.2)

x(0) = x0 ,

(1.3)

where A : D(A) ⊂ X −→ X is a densely defined closed linear operator on a Banach space X and 0 = s0 < t1 ≤ s1 ≤ t2 · · · ≤ tN ≤ sN < tN +1 = T is a partition of the interval J = [0, T ], the functions gi ∈ C((ti , si ] × X, X) for each i = 1, 2, ....N and f : [0, T ] × X → X is a suitable function. Pierri et al. [23] studied the existence and uniqueness of (1.1)-(1.3) in the fractional power space using the theory of analytic semigroups. Yu and Wang [32] investigated the existence of solution with non-instantaneous impulses on Banach spaces by using the theory of semigroup and fixed point methods. Chen et al. [8] obta