On Impulsive Delay Integrodifferential Equations with Integral Impulses
- PDF / 444,912 Bytes
- 22 Pages / 439.37 x 666.142 pts Page_size
- 36 Downloads / 259 Views
On Impulsive Delay Integrodifferential Equations with Integral Impulses Kishor D. Kucche and Pallavi U. Shikhare Abstract. The present research paper is devoted to investigating the existence, uniqueness of mild solutions for impulsive delay integrodifferential equations with integral impulses in Banach spaces. We also investigate the dependence of solutions on initial conditions, parameters and the functions involved in the equations. Our analysis is based on semigroup theory, fixed point technique and an application of Pachpatte’s type integral inequality with integral impulses. An example is provided in support of existence result. Mathematics Subject Classification. 37L05, 47D60, 34A37, 34G20, 35B30, 35A23. Keywords. Impulsive integrodifferential equation, impulsive integral inequality, integral jump condition, fixed point, dependence of solutions.
1. Introduction Numerous real-world physical phenomena cannot be modelled as differential equations with classical initial conditions, specially the issues in the part of biology and medical sciences, for instances, the dynamics of populations subject to sudden changes (e.g. diseases, harvesting and so forth.) and few problems in the field of physics, chemical technology, population dynamics, medicine, mechanics, biotechnology and economics. But, such physical phenomena can be modelled by means of differential equations with impulsive conditions. It very well may be noticed that the impulsive conditions are the blends of classical initial conditions and the short-term perturbations whose span can be insignificant in correlation with the span of the procedure. Therefore, the differential and integrodifferential equations with impulse effect have importance in the modelling many physical phenomena in which sudden changes occurs. The details relating to the theory and applications of impulsive differential equations (IDEs) and its impact in further development can be found in the monograph by Bainov and Simeonov [1], Lakshmikantham et al. [2], and Samoilenko and Perestyuk [3] and in the interesting papers [4–13]. 0123456789().: V,-vol
103
Page 2 of 22
K. D. Kucche and P. U. Shikhare
MJOM
In the beginning, Frigon and O’regan [5], using the fixed point approach determined variety of existence principles for the IDE w (t) = f (t, w(t)) , 0 < t < b, t = tk , − (1.1) w(t+ k ) = Ik w(tk ) , k = 1, 2, . . . , m with the initial condition w(0) = w0 , where Ik maps R into R for k = − 1, 2, . . . , m, w(t+ k ) = lim→0+ w(tk + ) and w(tk ) = lim w(tk + ). In ad→0−
dition, utilizing the idea of upper and lower solutions, authors have derived existence results for the IDE (1.1) with boundary condition w(0) = w(b). First, Wang et al. [4] presented the idea of Ulam’s type stabilities for ordinary IDE and derived these stability results by means of integral inequality of Gronwall type for piecewise continuous functions. Liu [14] have extended the theory to nonlinear impulsive evolution equation ⎧ ⎪ ⎨w (t) = A w(t) + f (t, w(t)) , 0 < t < b, t = tk , (1.2) Δw(tk ) = Ik (w(tk ))
Data Loading...
