YAN Dong-yan， ZHAO Hou-yu

(School of Mathematics， Chongqing Normal University， Chongqing 401331， China)

Abstract: Schauder’s fixed point theorem and the Banach contraction principle are used to study the polynomial-like iterative functional equation with variable coefficients λ1(x)f(x)+λ2(x)f2(x)+...+λn(x)fn(x)=F(x). We give sufficient conditions for the existence， uniqueness，and stability of the periodic and continuous solutions. Finally， some examples were considered by our results. The results enrich and extend the theory about polynomial-like iterative functional equation.

Keywords: Iterative functional equation; periodic solutions; fixed point theorem

1 Introduction

LetSbe a non-empty set andf:S →Sbe a self-map. For each positive integern，fn， then-fold composition offwith itself， also known as then-th iterate off， is defined recursively byf1=f， andfn=f ?fn-1.

LetSbe a non-empty subset of the real line R andF:S →R be given. The polynomiallike iterative equation

whereλi，i= 1，2，...，nare constants， has been discussed in many papers under various settings. Si [1] obtained results onC2solutions withS=[a，b]， a finite closed interval， andF，aC2self-map on[a，b]，F(xiàn)(a)=a，F(xiàn)(b)=b. His work is based on Zhang’s paper[2]in which the results cover the existence，the uniqueness，and the stability of the differentiable solutions.In [3] invertible solutions are obtained in some local neighbourhoods of the fixed points of the functions. Monotonic solutions and convex solutions are discussed by Nikodem，Xu，and Zhang in [4， 5]. Recently， Ng and Zhao [6] studied the periodic and continuous solutions of Eq.(1.1). For some properties of solutions for polynomial-like iterative functional equations，we refer the interested readers to [7]-[10].

In 2000， Zhang and Baker [11] studied the continuous solutions of

Later， Xu [12] considered the analytic solutions of Eq.(1.2). In this paper， we continue to consider the continuous and periodic solutions of Eq.(1.2). In fact， ifλi(x)，i= 1，2，...，nare constants， then the conclusions in [6] can be derived by our results.

Notations， preliminaries， and the current setting.

Assume thatλi ∈PT(Li，Mi)，|λ1(x)|≥k1＞0，?x ∈[0，T]andF ∈PT(L′，M′)， and we seek solutionsf ∈PT(L，M).

2 Sufficient Conditions for the Existence of Solutions

We shall find sufficient conditions on the constantsL′，M′，LandMunder which the existence of a solution is assured. The Schauder fixed point theorem being a main tool， its statement is included， and will be applied with Ω:=PT(L，M) in the Banach spacePT.

Theorem 2.1 (Schauder([13])) Let Ω be a closed convex and nonempty subset of a Banach space (B，‖·‖). Suppose thatAmaps Ω into Ω and is compact and continuous.Then there existsz ∈Ω withz=Az.

LetA:PT(L，M)→PTbe defined by

The spacePTis closed under composition. The range ofAis clearly contained inPT. Fixed points ofAcorrespond to the solutions of (1.2). Hence we seek conditions under which the assumptions in Schauder’s theorem are met.

Lemma 2.2(see [2]) For anyf，g ∈PT(L，M)，x，y ∈R， the following inequalities hold for every positive integern.

Lemma 2.3 Supposeλi ∈PT(Li，Mi)，|λ1(x)|≥k1＞0，?x ∈[0，T]， then operatorAis continuous and compact onPT(L，M).

Proof For anyf，g ∈PT(L，M)，x ∈R， by (2.3) we get

Thus

This proves thatAis continuous.PT(L，M)is closed，uniformly bounded and equicontinuous on R. By the Arzel`a-Ascoli theorem ([14]， page 28)， applied to a sufficiently large closed interval of R and taking the periodicity of the functions into account， it is compact. Since continuous functions map compact sets to compact sets，Ais a compact map.

Theorem 2.4 Suppose thatF ∈PT(L′，M′)，λi ∈PT(Li，Mi)，i= 1，2，...，nand|λ1(x)|≥k1＞0，?x ∈[0，T]. If the constantsL′，M′，L，Msatisfy the conditions

then (1.2) has a solution inPT(L，M).

Proof By (2.1)， for allx ∈R， we have

Thus the first condition of (2.5) assures

For allx，y ∈R， by (2.1)， (2.2) ， we have

The second condition of (2.5) assures

ThereforeAf ∈PT(L，M). This proves thatAmapsPT(L，M) into itself. All conditions of Schauder’s fixed point theorem are satisfied. Thus there exists anfinPT(L，M) such thatf=Af. This is equivalent to thatfis a solution of (1.2) inPT(L，M).

3 Uniqueness and Stability

In this section， uniqueness and stability of (1.2) will be proved.

Theorem 3.1 (i) Suppose thatAis defined by (2.1) and

ThenAis contractive with contraction constantα ＜1. It has at most one fixed point and(1.2) has at most one solution inPT(L，M).

(ii) Suppose that (2.5) and (3.1) are satisfied. Then (1.2) has a unique solution inPT(L，M).

Proof (i) According to(2.4)，‖Af-Ag‖≤α‖f-g‖and soαis a contraction constant forA. Letf，g ∈PT(L，M) be fixed points ofA. Then‖f-g‖=‖Af-Ag‖ ≤α‖f-g‖.Henceα ＜1 yields‖f-g‖= 0， resulting inf=g. This proves that there is at most one fixed point. (ii) The existence is cared for by Theorem 2.4.

Theorem 3.2 LetPT(L，M) andPT(L′，M′) be fixed and allowF(x) andλi(x) in(1.2) to vary. The unique solution obtained in Theorem 3.1， part (ii)， depends continuously onF(x) andλi(x) (i=1，2，...，n).

Proof Under the assumptions of Theorem 3.1，part(ii)，we consider any two functionsF，GinPT(L′，M′) and a parallel pair of uniquef，ginPT(L，M) satisfying

4 Examples

In this section， some examples are provided to illustrate that the assumptions of Theorem 2.4 and 3.1 do not self-contradict.

Example 4.1 Consider the equation

(3.1) is satisfied， hence by Theorem 3.1， we know the continuous periodic solution is the unique one inP2π(1，1).

Example 4.2 Consider the equation