On conjugacy of some systems of functions (Q5943225)

Let \((X,\rho)\) be a complete metric space, \(f_k:[0,1] \to [0,1]\) for \(0\leq k\leq n-1\) continuous strictly increasing mappings, \(f_0(0)=0\), \(f_{n-1}(1)=1\), \(f_{k+1}(0)=f_k(1)\) for \(0\leq k\leq n-2\) and \(F_k:X\times [0,1] \to X\) given functions for \(0\leq k\leq n-1\). The author investigates the bounded and continuous solutions \(\varphi: [0,1]\to X\) of the system of functional equations \[ \varphi\bigl(f_k(t) \bigr)=F_k\bigl( \varphi(t), t\bigr),\;t\in[0,1],\;0\leq k\leq n-1.\tag{1} \] If \(F_k(x,.),\;x\in X\), \(0\leq k\leq n-1\), are bounded on \([0,1]\) and there exists an increasing function \(\alpha: [0,+\infty) \to[0,+\infty)\) such that its sequence of iterates tends pointwise to zero on \([0,+\infty)\) and \[ \rho\bigl(F_k(x,t), F_k(y,t)\bigr) \leq\alpha \bigl(\rho(x,y) \bigr),\;(x,y)\in X\times X,\;t\in [0,1] \] for \(0\leq k\leq n-1\), then for every system \((c_k)_{0\leq k\leq n-1}\) of elements of \(X\) there exists a unique bounded solution \(\varphi:(0,1)\to X\) of system (1) such that \(\varphi (f_k(0))=c_k\), \(1\leq k\leq n-1\). Moreover, if \(F_0(.,0)\) has a unique fixed point \(a\in X\) and \(F_{n-1}(.,1)\) has a unique fixed point \(b\in X\) and \(F_{k+1}(a,0)= F_k(b,0)\) for \(0\leq k\leq n-2\) then there exists a unique bounded continuous solution \(\varphi: [0,1]\to X\) of (1). The author studies the special case \(f_k(x)=n^{-1} (x+k)\), \(0\leq k\leq n-1\), some peculiar properties of curves determined by the continuous solutions of (1), the case when \(X\) is a closed interval in \(\mathbb{R}\), the problem of existence of homeomorphic solutions of (1).