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In the paper we indicate an error made in the proof of the main result of the paper [M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112-119]. Moreover, we provide correct proof of a slightly modified version of the mentioned result. The main tool used in our proof is the technique associated with the Hausdorff measure of noncompactness.

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Melatonin may serve as a potential therapeutic free radical scavenger and broad-spectrum antioxidant. It shows neuroprotective properties against hypoxic-ischemic brain injury in animal models. The authors review the studies focusing on the neuroprotective potential of melatonin and its possibility of treatment after perinatal asphyxia. Melatonin efficacy, low toxicity, and ability to readily cross through the blood-brain barrier make it a promising molecule. A very interesting thing is the difference between the half-life of melatonin in preterm neonates (15 hours) and adults (45-60 minutes). Probably, the use of synergic strategies—hypothermia coupled with melatonin treatment—may be promising in improving antioxidant action. The authors discuss and try to summarize the evidence surrounding the use of melatonin in hypoxic-ischemic events in term and preterm babies.

Applying the technique associated with measures of noncompactness, we prove the existence of nondecreasing solutions of a quadratic integral equation of Urysohn type in the space of real functions defined and continuous on a closed hounded interval. (~)