Global solutions for leading coefficient problem of a general iterative equation
The paper deals with the iterative equation \[ G(f(x),f^2(x),\dots,f^n(x))=F(x), \leqno(*) \] where \(x \in I:=[a,b]\) and \(f:I \to J\), \(J\) a real interval. The product of \(n\) intervals \(I\) is denoted by \(I^n\) and \(\mathbf a=(a,a,\dots,a)\), \(\mathbf b=(b,b,\dots,b)\), \(\mathbf x=(x_1,x_2,\dots,x_n)\); \(G_i^{\prime}(\mathbf x):=\frac{\partial{G}}{\partial{x_i}}(\mathbf x)\). The following properties on the given function \(G\) are requested. {\parindent=10mm \begin{itemize}\item[(G1)] \(G\) is increasing with respect to each variable \(x_i\), \(i=1,2,\dots,n-1\), and strictly increasing with respect to \(x_n\); \item[(G2)] for every \(\mathbf x, \mathbf y \in I^n\), it is \[ 0\leq G_i^{\prime}(\mathbf x)\leq G_i^{\prime}(\mathbf a) \quad \text{and} \quad |G_i^{\prime}(\mathbf x)-G_i^{\prime}(\mathbf y)|\leq \sum_{j+1}^n N_{ij}| x_j-y_j|, \] where \(N_{ij}\) are nonnegative constants; \item[(G3)] For every \(\mathbf x, \mathbf y \in I^n\), it is \[ 0\leq G_i^{\prime}(\mathbf x)\leq G_i^{\prime}(\mathbf b) \quad \text{and} \quad |G_i^{\prime}(\mathbf x)-G_i^{\prime}(\mathbf y)|\leq \sum_{j+1}^n M_{ij}| x_j-y_j|, \] where \(M_{ij}\) are nonnegative constants. \end{itemize}} Put \[ \mathcal G_1(I):=\{G\mid G:I^n \to \mathbb R \text{ satisfies (G1) and (G2)} \} \] and \[ \mathcal G_2(I):=\{G\mid G:I^n \to \mathbb R \text{ satisfies (G1) and (G3)} \}. \] Similarly, the following properties on \(F\) are needed. Define \[ \mathcal H_+(I,J):=\{f:I \to J\mid f \text{ is a strictly increasing homeomomorphism} \}; \] \[ \mathcal A_G[I]:=\{F \in \mathcal H_+(I,J)\mid F(x)<G(x,x,\dots,x), \quad x \in (a,b) \}; \] \[ \mathcal B_G[I]:=\{F \in \mathcal H_+(I,J)\mid F(x)>G(x,x,\dots,x), \quad x \in (a,b) \}; \] {\parindent=10mm \begin{itemize}\item[(F1)] \(F \in \mathcal A_G[I]\); \item[(F2)] \(F^{\prime}\) is Lipschitz on \(I\) and \(F^{\prime}(x)\geq F^{\prime}(a)>0\) for all \(x \in I\); \item[(F3)] \(F \in \mathcal B_G[I]\); \item[(F4)] \(F^{\prime}\) is Lipschitz on \(I\) and \(F^{\prime}(x)\geq F^{\prime}(b)>0\) for all \(x \in I\). \end{itemize}} Finally, let \[ \mathcal F_1(I,J):=\{F\mid F:I\to J \text{ satisfies (F1) and (F2)} \} \] and \[ \mathcal F_2(I,J):=\{F\mid F:I\to J \text{ satisfies (F3) and (F4)} \}. \] The main result of the paper is the following. Theorem. For a given \(G \in \mathcal G_1(I)\) and \(F \in \mathcal F_1(I,J)\) with the following conditions \[ G(\mathbf a)=F(a), \quad G(\mathbf b)=F(b), \quad \sum_{i=1}^n G_i^{\prime}(\mathbf a)>F^{\prime}(a), \] or for a given \(G \in \mathcal G_2(I)\) and \(F \in \mathcal F_2(I,J)\) with the following conditions \[ G(\mathbf a)=F(a), \quad G(\mathbf b)=F(b), \quad \sum_{i=1}^n G_i^{\prime}(\mathbf b)>F^{\prime}(b), \] equation \((*)\) has infinitely many strictly increasing continuous solutions, which fix the endpoints \(a\) and \(b\).
scientific article; zbMATH DE number 6485806
scientific article