@article{Aydi2024, 
author = {Hassen Aydi and Bessem Samet and Manuel De la Sen},
title = {A fixed point theorem for non-negative functions},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {10},
pages = {29018-29030},
keywords = {fixed points, non-negative functions, convex functions, concave functions, Hermite-Hadamard inequalities},
url = {https://www.sciopen.com/article/10.3934/math.20241408},
doi = {10.3934/math.20241408},
abstract = {In this paper, we are concerned with the study of the existence and uniqueness of fixed points for the class of functions  f:C→C satisfying the inequality   ℓ(αf(t)+(1−α)f(s))≤σℓ(αt+(1−α)s)for every  t,s∈C with  f(t)≠f(s), where  C is a closed subset of  [0,∞),  α,σ∈(0,1) are constants, and  ℓ:[0,∞)→[0,∞) is a function satisfying the condition  inft>0ℓ(t)tρ>0 for some constant  ρ>0. Namely, under a weak continuity condition imposed on  f, we show that  f possesses a unique fixed point, and for every  t0∈C, the Picard sequence defined by  tn+1=f(tn),  n≥0, converges to this fixed point. Next, we study the special cases when  C is a closed interval and  ℓ 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.}
}