Convex solutions of polynomial-like iterative equations

In this paper convex solutions and concave solutions of polynomial-like iterative equations are investigated. A result for non-monotonic solutions is given first and applied then to prove the existence of convex continuous solutions and concave ones. Furthermore, another condition for convex solutions, which is weaker in some aspects, is also given. The uniqueness and stability of those solutions are also discussed.     

Continuously decreasing solutions for polynomial-like iterative equations

Most known results on existence,uniqueness and stability for solutions of the polynomial-like iterative equation Σni=1λifi(x)=F(x) were obtained in the case of λ 1 = 0.In this paper,we construct C 0 decreasing solutions of the iterative equation in the case that λ 1 can vanish to answer the Leading Coefficient Problem.Moreover,we also give the necessary and sufficiently condition for uniqueness of solutions.     

Convex solutions of the polynomial-like iterative equation with variable coefficients

In this paper, by applying the Schauder''s fixed point theorem we prove the existence of increasing and decreasing solutions of the polynomial-like iterative equation with variable coefficients and further completely investigate increasing convex (or concave) solutions and decreasing convex (or concave) solutions of this equation. The uniqueness and continuous dependence of those solutions are also discussed     

Leading coefficient problem for polynomial-like iterative equations

Because of difficulties in applying fixed point theorems, most of known results on the polynomial-like iterative equation were given by assuming λ₁>0 and the existence of solutions under the most natural assumption λn>0 is an interesting problem, called “Leading Coefficient Problem”. For this problem locally expansive invertible C¹ solutions are obtained in the expansive case and the non-hyperbolic case in [W. Zhang, On existence for polynomial-like iterative equations, Results Math. 45 (2004) 185-194] and C⁰ increasing solutions are constructed in [B. Xu, W. Zhang, Construction of continuous solutions and stability for the polynomial-like iterative equation, J. Math. Anal. Appl. 325 (2007) 1160-1170]. In this paper we discuss C¹ solutions for more combinations between expansive mappings and contractive ones and combinations between increasing mappings and decreasing ones.     

Decreasing solutions and convex solutions of the polynomial-like iterative equation

In this paper we prove the existence and uniqueness of decreasing solutions for the polynomial-like iterative equation so as to answer Problem 2 in [J. Zhang, L. Yang, W. Zhang, Some advances on functional equations, Adv. Math. (China) 24 (1995) 385-405] (or Problem 3 in [W. Zhang, J.A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math. 73 (2000) 29-36]). Furthermore, we completely investigate increasing convex (or concave) solutions and decreasing convex (or concave) solutions of this equation so that the results obtained in [W. Zhang, K. Nikodem, B. Xu, Convex solutions of polynomial-like iterative equations, J. Math. Anal. Appl. 315 (2006) 29-40] are improved.     

Periodic and continuous solutions of a polynomial-like iterative equation

Aequationes mathematicae - Schauder’s fixed point theorem and the Banach contraction principle are used to study the polynomial-like iterative functional equation $$\begin{aligned} \lambda...     

Construction of continuous solutions and stability for the polynomial-like iterative equation

Most of known results such as existence, uniqueness and stability for polynomial-like iterative equations were given under the assumption that the coefficient of the first order iteration term does not vanish. The existence with a non-zero leading coefficient was therefore raised as an open problem. It was positively answered for local C¹ solutions later. In this paper this problem is answered further by constructing C⁰ solutions. Moreover, we discuss the stability of those C⁰ solutions, which consequently implies a result of the stability for iterative roots.     

Continuous solutions to two iterative functional equations

Aequationes mathematicae - Based on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions $$varphi $$ of the...     

On the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals

The behaviour of a solution to a Fredholm integral equation of the second kind on a union of open intervals is examined. The kernel of the corresponding integral operator may have diagonal singularities, information about them is given through certain estimates. The weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described.     

Homeomorphisms related to the polynomial-like iterative equation on S^1

In this paper we study all homeomorphisms on the unit circle $\mathbb{S}^1$, whose lifts are $C^0$ solutions of a class of nonhomogeneous polynomial-like iterative equation. By an auxiliary equation, we present all those homeomorphisms and illustrate our results by examples.     

Finite iterative solutions to coupled Sylvester-conjugate matrix equations

This paper is concerned with solutions to the so-called coupled Sylveter-conjugate matrix equations, which include the generalized Sylvester matrix equation and coupled Lyapunov matrix equation as special cases. An iterative algorithm is constructed to solve this kind of matrix equations. By using the proposed algorithm, the existence of a solution to a coupled Sylvester-conjugate matrix equation can be determined automatically. When the considered matrix equation is consistent, it is proven by using a real inner product in complex matrix spaces as a tool that a solution can be obtained within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it can be implemented by using original coefficient matrices, and does not require to transform the coefficient matrices into any canonical forms. The algorithm is also generalized to solve a more general case. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

The polynomial-like iterative equation for PM functions

The polynomial-like iterative equation is an important form of functional equations, in which iterates of the unknown function are linked in a linear combination. Most of known results were given for the given function to be monotone. We discuss this equation for continuous solutions in the case that the given function is a PM(piecewise monotone) function, a special class of non-monotonic functions. Using extension method, we give a general construction of solutions for the polynomial-like iterative equation.     

Direct iterative methods for least-squares solutions to singular operator equations

Least-squares consistency and convergence of iterative schemes are investigated for singular operator equations (1) Tx = f, where T is a bounded linear operator from a Banach space to a Hilbert space. A direct splitting of T into T = M ? N is then used to obtain the iterative formula (2) x^(k+1) = M^?Nx^(k) + M^?f, where M^? is a least-squares generalized inverse of M. Cone monotonicity is used to investigate convergence of (2) to a least-squares solutions to (1), extending results given for the matrix case given by Berman and Plemmons.     

Verification of bifurcation diagrams for polynomial-like equations

The results of our recent paper [P. Korman, Y. Li, T. Ouyang, Computing the location and the direction of bifurcation, Math. Res. Lett. 12 (2005) 933–944] appear to be sufficient to justify computer-generated bifurcation diagram for any autonomous two-point Dirichlet problem. Here we apply our results to polynomial-like nonlinearities.     

Multiple nonnegative solutions of nonlinear integral equations on compact and semi-infinite intervals

Existence results are presented which guarantee the existence of multiple nonnegative continuous solutions of a nonlinear integral equation on both a compact interval and semi-infinite interval