| The dynamical system studies the most basic laws of the objects move-ments. It not only studies the existence and stability of basic movements, suchas the equilibrium state and periodic state, but also emphasizes the topolog-ical structure of trajectory. It pays great attention to the structural stabilityof moving equation and tries to find out that the destruction of structures isthe qualitative changes of moving forms. So, the dynamical system is observ-ing the objects with the views of geometry and topological structure, and it isalso handling the problems with the analytical and algebraic ways. It has bothabundant really backgrounds and deep mathematically foundations.The Iteration which is an important phenomenon in the nature, becomesthe focal points of physics, chemistry, astronomy, ,mechanics and project and soon. Under this circumstance, the dynamical system theory has been developedrapidly. This thesis discusses the existence and continuation of holomorphicsolutions for a functional equation by the means of iteration, and gets somerelative conclusions.The functional equation f(p(z))+f(z) = z,p(z) = zn+c, with backgroundof nonlinear curves of di?erential equations with piecewise constant arguments,and its general form is discussed. In chapter 2, we investigate functional equa-tion f(p(z)) + f(z) = z,p(z) = zn + c in the case when c = 2. In§2.1, weprove the existence of holomorphic solutions in an appropriate region of C by considering the complex dynamics of iteration of p. In§2.2, we discuss theparameter c∈C of p(z) = zn +c which makes the equation have a holomorphicsolution. In§2.3, the continuation of holomorphic solutions is studied by in-vestigating relation between the natural boundary of solution and the Julia setof p. Furthermore, in chapter 3, we make a discussion about the more generalfunctional equation f(p(z)) + f(z) = q(z), where z∈C, p(z) is a polynomialor an entire function and q(z) is any known entire function. |
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