On the fixed points of correspondences in modular spaces (Q555229)

Let \(\rho\) be a modular on a real linear space \(X\), and let \(f\) be a correspondence from \(X\) to \(Y\). \(f: C\to C\) is called a \(k\)-contraction \((0\leq k< 1)\) if for all \(x,y\in C\) and \(p\in f(x)\), \(q\in f(y)\), there holds \(\rho(p-q)\leq k\rho(x-y)\). The growth function, defined for \(t\in [0,\infty)\), is defined as \[ w_\rho(t)= \text{inf}\{w: \rho(tx)\leq w\rho(x),\, x\in X,\,0<\rho(x)\}, \] and is applied to obtain the following fixed point theorems in modular spaces \((X,\rho)\). 1. Every \(k\)-contraction \(f: B\to B\) with \(w_\rho(2)^2 k<{1\over 2}\) has a fixed point. 2. Let \(X_\rho\) be a complete modular space, \(B\) a compact and starshaped subset of \(X_\rho\). If \(w_\rho(\beta) w^2_\rho(2)<{1\over 2}\) for every \(\beta\in (0,1)\) and \(f\) is nonexpansive, then \(f\) has a fixed point.