Monotone continuous solutions of an equation in linear combination of alternative iterates (Q6633390)

The authors present a discussion about diverse cases of monotonicity of functions. They start discussing monotone Lipschitzian solutions of the functional equation \N\[\N\sum_{i=1}^{n} \lambda_{i} \mathcal{A}_{\sigma (i), \sigma (i) +i-1} f(x)=F(x), \qquad x\in I,\N\]\Nwhere \(F\in C(I, I)\) is given, \(f\in C(I, I)\) is unknown, \(\lambda_{i}\; (1\leq i\leq n)\) are all real numbers, and \(\sigma : \{1, 2, \dots, n\} \rightarrow \mathbb{Z}\) is a given map, which indicates the start of the ``alternative'' iteration. The \((n-k+1)\)-th alternative iteration \(\mathcal{A}_{k,n}: C(I, I) \rightarrow C(I, I)\) is defined as \(\mathcal{A}_{k,n}:f\rightarrow \alpha_{n}\circ f\circ \cdots\circ\alpha_{k+1} \circ f\circ\alpha_{k}\circ f\), \(f\in C(I, I)\), where \(k\), \(n\) are integers such that \(k \leq n\) and \(\mathcal{A} :=\{\alpha_{i}\}_{i\in \mathbb{Z}} \subset C(I, I)\).\N\NThe authors introduce divided differences under alternative iteration to find out the complicated monotonicity of the linear combination of increasing and decreasing functions and discuss their properties under alternative iteration by constructing several lemmas. After this, they use these properties to provide rigorously the proof of the existence, uniqueness and continuous dependence for monotone Lipschitzian solutions of the above equation on a compact interval.\N\NThe authors consider the whole \(\mathbb{R}\) and discuss results, including proofs for bounded monotone Lipschitzian solutions of the above equation with bounded monotone \(F\). Further, they describe results for unbounded monotone Lipschitzian solutions of the above equation with unbounded monotone \(F\).\N\NIn the last section of the paper, the authors discuss the following examples to demonstrate their obtained results:\N\N\begin{itemize}\N\item Consider the alternative iterative equations \N\[\N\frac{1}{16} \mathcal{A}_{\sigma (3), \sigma (3) +2} f(x) + \frac{1}{16} \mathcal{A}_{\sigma (2), \sigma (2) +1} f(x) - f(x) = F_{1}(x), \qquad x\in [0, 1],\N\]\N\[\N-\frac{1}{10}\mathcal{A}_{\sigma (3), \sigma (3) +2} f(x) + \frac{1}{15} \mathcal{A}_{\sigma (2), \sigma (2) +1} f(x) + f(x) = F_{2}(x), \qquad x\in [0, 1],\N\]\Nwhere \(\alpha_{\sigma (1)}(x)=x\) for \(x\in [0, 1]\), \(\alpha_{\sigma (i)+j}(x)=24x/23\) for \(x\in [0, 1/2]\) or \(22x/23 + 1/23\) for \(x \in (1/2, 1]\), \(i=2, 3\) and \(0\leq j<i\), and \(F_{1}(x)=x-\frac{15}{16},\; x\in [0, 1]\),\N\begin{align*}\NF_{2}(x)= \begin{cases} \frac{9}{10}-\frac{11}{12}x & x\in [0, 9/10], \\\N-\frac{1}{12}x+\frac{3}{20} & x\in (9/10, 1]. \end{cases}\N\end{align*}\N\N\item Consider the equation \N\[\Nf(x)+(-1)^{n}\frac{1}{8^{n}}\mathcal{A}_{\sigma (n), \sigma (n) +n-1} f(x)=F(x), \qquad x\in \mathbb{R},\N\]\Nwhere \(\alpha_{\sigma (1)}(x)=x\) for \(x\in \mathbb{R}\) and \(0\leq j <n\), \(\alpha_{\sigma (n)+j}(x)=2x\) for \(x\in \mathbb{R}\) and\N\begin{align*}\NF(x)= \begin{cases} 1 & x \in (-\infty, -1),\\\N-x & x \in [-1, 1],\\\N-1 & x \in (1, +\infty). \end{cases}\N\end{align*}\N\N\item Consider the equation \N\[\N(1+\frac{1}{32^{n}}) \alpha_{\sigma (1)}\circ f(x)+(-1)^{n}\frac{1}{32^{n}}\mathcal{A}_{\sigma (n), \sigma (n) +n-1} f(x)=F(x), \qquad x\in \mathbb{R},\N\]\Nwhere \(\alpha_{\sigma (1)}(x)=x + \sin x/2\) for \(x\in \mathbb{R}\) and \(0\leq j <n\), \(\alpha_{\sigma (n)+j}(x)=x +\cos x\) for \(x\in \mathbb{R}\) and \(F(x)=-x+\sin x\) for \(x\in \mathbb{R}\).\N\N\item Consider the equation \N\[\N\alpha_{\sigma (2)+1}\circ f \circ\alpha_{\sigma (2)}\circ f(x) +\alpha_{\sigma (1)}\circ f(x)=F(x), \qquad x\in [0, 1],\N\]\Nwhere\N\begin{align*}\N\alpha_{\sigma (1)}=\alpha_{\sigma (2)}=\alpha_{\sigma (2)+1}= \begin{cases} \frac{4}{5}x & x \in [0, 5/8),\\\N\frac{1}{2} & x \in [5/8, 7/8],\\\N4x-3 & x \in (7/8, 1], \end{cases}\N\end{align*}\Nand \(F(x)=1-x/2\), \(x\in [0, 1]\).\N\N\item Consider the equation \N\[\N\alpha_{\sigma (2)+1}\circ f \circ\alpha_{\sigma (2)}\circ f(x) +\alpha_{\sigma (1)}\circ f(x)=F(x), \qquad x\in \mathbb{R},\N\]\Nwhere\N\begin{align*}\N\alpha_{\sigma (1)}=\alpha_{\sigma (2)}=\alpha_{\sigma (2)+1}= \begin{cases} x+\pi & x \in (-\infty, -\pi),\\\N0 & x \in [-\pi, 0],\\\Nx & x \in (0, +\infty), \end{cases}\N\end{align*}\Nand \(F(x)=-\pi/2-\arctan x\) for \(x\in \mathbb{R}\).\N\end{itemize}