<p>In the present paper, using Schauder’s fixed point theorem and the Banach contraction principle, we discuss the polynomial-like iterative equation <Equation ID="Equ11"> <EquationSource Format="TEX">$\begin{aligned} \lambda _1f(x)+\lambda _2f^2(x)+\cdots +\lambda _nf^n(x)=F(x),\,\,x\ge 0. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <msup> <mi>f</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <msup> <mi>f</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi>x</mi> <mo>≥</mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>The sufficient conditions for the existence, uniqueness, and stability of the asymptotically periodic and continuous solutions are presented. An example of the existence, uniqueness, and stability of asymptotically periodic and continuous solutions of an iterative equation is examined.</p>

Asymptotically periodic and continuous solutions of a polynomial-like iterative equation

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发表日期：

2026年4月11日

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Chao Xia,
Rumeng Huo,
Xi Wang,
Zhinan Xia

摘要

In the present paper, using Schauder’s fixed point theorem and the Banach contraction principle, we discuss the polynomial-like iterative equation $\begin{aligned} \lambda _1f(x)+\lambda _2f^2(x)+\cdots +\lambda _nf^n(x)=F(x),\,\,x\ge 0. \end{aligned}$ $\begin{matrix} λ_{1} f (x) + λ_{2} f^{2} (x) + \dots + λ_{n} f^{n} (x) = F (x), x \geq 0 . \end{matrix}$ The sufficient conditions for the existence, uniqueness, and stability of the asymptotically periodic and continuous solutions are presented. An example of the existence, uniqueness, and stability of asymptotically periodic and continuous solutions of an iterative equation is examined.