Functionals strictly subordinate to series and search for solutions of equations (Q403903)

Let \((X,\rho)\) and \((Y,d)\) be metric spaces. Consider a convergent series \[ \sum^\infty_{j=1} c_j,\quad 0<c_{n+1}< c_n\text{ for all }n\in/bbfN.\tag{1} \] Suppose that \(S\) is the sum of this series. We denote by \(\mathbb{R}_+\) the set of all nonnegative real numbers. A~function \(\lambda: \mathbb{R}_+\to \mathbb{R}_+\) is said to be compatible with the series (1) if \(\lambda(t)\leq {c_{k+1}\over c_k}t\) for \(t\in (c_{k+1}, c_k)\) for \(k\geq 1\) and \(\lambda(t)\leq c\) for \(t> c_1\). Put \(S_k= \sum^\infty_{j=k} c_j\) for each positive integer \(k\). A typical result of this note is as follows. Theorem. Let \(\mu,\nu= \mathbb{R}_+\to \mathbb{R}_+\) be nondecreasing commuting functions, positive on \((0,\infty)\) satisfying \(\mu(0)= \nu(0)= 0\). Suppose that \(\lambda(t)= \mu\nu(t)= \nu\mu(t)\) is compatible with the series (1). Suppose that for some \(z\in Y\), the graph \(G(f)\) is \(\{z\}\)-complete (a notion defined in the paper), with the function \(\varphi(x)= d(f(x),z)\), \(x\in X\) such that given \(x\) in \(X\) there exists \(x'\) in \(X\) satisfying \(\rho(x,x')\leq \mu(\varphi(x))\), \(d(f(x), f(x'))\leq \gamma\varphi(x)\) and \(\varphi(x')\leq \nu(\rho(x, x'))\). Here, \(\gamma> 0\). If \(x_0\in x\), \(R>0\), and \(c_{k_0}\) of the series (1) satisfies \[ \mu(\varphi(x_0))\leq c_{k_0}\,\min\Biggl\{1,{R\over S_{k_0}}\Biggr\}, \] then there exists a solution \(\xi\) of the equation \(f(x)= z\) accessible from \(x_0\) satisfying \(\rho(x_0,\xi)\leq R\). There are three more theorems.