Continuity, Convexity, Analyticity And Stability Of Solutions For Some Iterative Functional Equations

Iteration is an extensive phenomenon in nature and human life. Iterative equations are those equations which involve iteration as a basic operation. In recent years, iterative equations play an important role in experimental sciences and engineerings and closely link with differential equations, difference equations, integral equations and dynamical systems. Some kinds of iterative functional equations together with some basic results are introduced in Chapter 1.Chapter 2 is devoted to existence of continuous solutions of some iterative equations. Existence, uniqueness and continuous dependence of C0 solutions for an iterative functional equation related to invariant curves of functional differential equations with piecewise constant arguments are given under weaker conditions than that known results of C1 solutions. Symmetry is also considered so that some obtained results are generalized to RN. Moreover, existence, uniqueness and continuous dependence of decreasing solutions and non-monotonic solutions for a linear iterative functional equation are discussed. Some corresponding results are generalized to a quasi-linear iterative equation.In Chapter 3, existence of quasi-convex, quasi-concave, convex and concave solutions of a linear iterative functional equation are studied. Although some results on convex and concave iterative roots are known, there are no results about convexity for more general iterative equations. In this chapter, convexity of both increasing solutions and decreasing solutions is investigated by the divided difference theory and fixed point theory.In Chapter 4, existence of analytic solutions of some iterative equations are studied. As we know, many results on analytic solutions of iterative equations are obtained by constructing a majorant series. For technical reasons, in previous works an indeterminate constant a, as the eigenvalue of the linearization of unknown function at its fixed point, is required to be off the unit circle or lie on the circle with the Diophantine condition. In this chapter, existence of analytic solutions of an iterative differential equation with state-dependent delays is studied by using a similar method. We breakthrough the restriction of the Diophantine condition and study the case that the constant a is a unity root, which offends the Diophantine condition. Moreover, existence of analytic solutions for a linear iterative equation with variable coefficients is given by reducing to an auxiliary equation and using Schauder's fixed point theorem.In Chapter 5, Hyers-Ulam stability of functional equations in single variable is studied. We summarize some known results on Hyers-Ulam stability of functional equations in single variable, simplify conditions in three senses for the generalized gamma functional equation. Furthermore, we discuss Hyers-Ulam stability of a nonlinear iterative equation and prove existence and uniqueness of solutions near its approximate solutions.