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 a general iterative equation

Most known results on polynomial-like iterative equations are concentrated to increasing solutions. Without the uniformity of orientation and monotonicity, it becomes much more difficult for decreasing cases. In this paper, we prove the existence of decreasing solutions for a general iterative equation, which was proposed as an open problem in [J. Zhang, L. Yang, W. Zhang, Some advances on functional equations, Adv. Math. (China) 24 (1995) 385–405] (or [W. Zhang, J.A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math. 73 (2000) 29–36]).     

Convex solutions to polynomial-like iterative equations on open intervals

The paper deals with the polynomial-like iterative functional equation $\lambda_1 f(x)+\lambda_2 f^2(x)+\cdots+\lambda_n f^n(x)=F(x).$ By using Schauder’s fixed point theorem and a version of the uniform boundedness principle for families of convex (respectively higher order convex) functions as basic tools, the existence of nondecreasing convex (respectively higher order convex) solutions to this equation on open (possibly unbounded) intervals is investigated. The results of the paper complement similar ones established by other authors, concerning the existence of monotonic or convex solutions to the above equation on compact intervals. Some examples illustrating their applicability are provided.     

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.     

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     

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.     

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.     

Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations

We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the long time behavior of the solution and the limit value of its energy for symmetric decreasing initial data in L ² under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results between propagation and extinction for monotone families of initial data in the considered general setting.     

Positive decreasing solutions of quasilinear dynamic equations

We consider a quasilinear dynamic equation reducing to a half-linear equation, an Emden–Fowler equation or a Sturm–Liouville equation under some conditions. Any nontrivial solution of the quasilinear dynamic equation is eventually monotone. In other words, it can be either positive decreasing (negative increasing) or positive increasing (negative decreasing). In particular, we investigate the asymptotic behavior of all positive decreasing solutions which are classified according to certain integral conditions. The approach is based on the Tychonov fixed point theorem.     

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...     

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.     

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.     

Block monotone iterative methods for numerical solutions of nonlinear elliptic equations

Summary. Two block monotone iterative schemes for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear elliptic boundary-value problem, are presented and are shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a computational algorithm for numerical solutions as well as an existence-comparison theorem of the system, including a sufficient condition for the uniqueness of the solution. An advantage of the block iterative schemes is that the Thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one-dimensional problems. The block iterative schemes are compared with the point monotone iterative schemes of Picard, Jacobi and Gauss-Seidel, and various theoretical comparison results among these monotone iterative schemes are given. These comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes. Application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given. Received August 1, 1993 / Revised version received November 7, 1994