A fixed point theorem for non-negative functions

In this paper, we are concerned with the study of the existence and uniqueness of fixed points for the class of functions $ f: C\to C $ satisfying the inequality

$ \ell\left(\alpha f(t)+(1-\alpha)f(s)\right)\leq \sigma \ell(\alpha t+(1-\alpha)s) $

for every $ t, s\in C $ with $ f(t)\neq f(s) $, where $ C $ is a closed subset of $ [0, \infty) $, $ \alpha, \sigma\in (0, 1) $ are constants, and $ \ell: [0, \infty)\to [0, \infty) $ is a function satisfying the condition $ \inf_{t > 0} \frac{\ell(t)}{t^\rho} > 0 $ for some constant $ \rho > 0 $. Namely, under a weak continuity condition imposed on $ f $, we show that $ f $ possesses a unique fixed point, and for every $ t_0\in C $, the Picard sequence defined by $ t_{n+1} = f(t_n) $, $ n\geq 0 $, converges to this fixed point. Next, we study the special cases when $ C $ is a closed interval and $ \ell $ is a convex or concave function. Namely, making use of the Hermite-Hadamard inequalities, we obtain several new fixed point theorems. To the best of our knowledge, the considered class of functions was never previously investigated in the literature.