Abstract
In this paper, we consider the existence and uniqueness for the controllability of a dynamical system. Here, measure of non-compactness of set was employed to examine the conditions for darbo’s fixed point theorem which is used to established the existence and uniqueness solution for nonlinear integro-differential equation with implicit derivatives.
Highlights
Let be the class of normalized analytic functions *| | + with ( ) ( )and of the form in the open unit disc ( )and the class of all functions in that are univalent in
The inverse exists but the inverse may not be defined in the entire unit disc
It was established by Koebe in his one quarter theorem, that every univalent function maps the unit disc to a disc of radius
Summary
The class of all functions in that are univalent in. This class of functions is defined on the entire complex plane, one- to-one and onto. The study of the coefficient of bi-univalent functions began with the work of Jahangiri and Hamidi [1], Lewin [2], while the coefficient bounds of bi-univalent functions started with the work of Brannan and Taha [3]. Motivated by the work of Mustafa [11], using the linear operator in (10) we define a new subclass of bi-univalent functions and obtained the initial coefficient estimates for the subclass. For the linear operator of the form (10), we obtain with simple calculations, the inverse function given by: Where We define our subclass of analytic functions and obtain the estimates of the initial coefficients
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