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

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

for every $t,s\in C$ with $f(t)\ne f(s)$, where $C$ is a closed subset of $[0,\mathrm{\infty })$, $\alpha ,\sigma \in (0,1)$ are constants, and $\ell :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a function satisfying the condition $\underset{t>0}{inf}\frac{\ell (t)}{t^{\rho }}>0$ 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 $t_{n+1}=f(t_n)$, $n\ge 0$, 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.