A note on convex solutions to an equation on open intervals (Q6590677)

Let \(S\) denote a nonempty open interval. For a function \(f:S\to S\) its \(k\)-th iterate is denoted by \(f^{k}\). The author investigates the iterative functional equation \N\[\lambda_1H_1 f(x)+\lambda_2H_2 f^{2}(x) +\dots+\lambda_nH_n f^{n}(x) = F(x), \tag{1}\] \Nwhere \(H_1=\mathrm{id}\) and \(F,H_2,\dots, H_n:S\to S\) are continuous nondecreasing convex functions; \(\lambda_1,\dots ,\lambda_n\) are reals.\N\NTwo theorems are proved. \N\NTheorem 1. Assume that \(\lambda_1<0\) and \(\lambda_j>0\) for \(2\le j\le n\). Let \(g,h:S\to S\) be continuous nondecreasing convex functions such that \(g(x)\le h(x)\), \N\[ \lambda_1H_1 g(x) +\lambda_2H_2 g^{2}(x) +\dots+\lambda_nH_n g^{n}(x) \le F(x), \] and \N\[ \lambda_1H_1 h(x) +\lambda_2H_2 h^{2}(x) +\dots+\lambda_nH_n h^{n}(x) \ge F(x) \] for all \(x\in S\).\NThen Equation (1) has a continuous nondecreasing convex solution \(f:S\to S\) such that \(g(x)\le f(x)\le h(x)\,\, x\in S\).\N\NTheorem 2 contains results of similar nature as Theorem 1 but for higher order convex solutions of Equation (1).\N\NExamples for both theorems show the applicability of the results. The theorems of the author extend and generalize results of \textit{T. Trif} [Aequationes Math. 79, No. 3, 315--325 (2010; Zbl 1223.39014)].