This paper concerns the solvability of a nonlinear fractional boundary value problem at resonance. By using fixed point theorems we prove that the perturbed problem has a solution, then by some ideas from analysis we show that the original problem is solvable. An example is given to illustrate the obatined results.
Melia azedarach L. (Meliaceae) is a botanical species with focal point of global research for its biological properties. The Melia azedarach tree is distinguished by its rapid growth, its adaptation to different temperate zones, as well as its insecticidal properties. All this made us think of exploiting it in biological control against different stages of mosquitoes. To this end, we aim, through the present work, to evaluate the effectiveness of Melia azedarach extracts against Culex pipiens mosquito. More specifically, our study focuses on determining the chemical composition of Melia almond oil, as well as the larvicidal, ovicidal and repellent activities on Culex pipiens L. mosquito as well as the activities of acetylcholinesterase (AChE) and glutathione-S-transferase (GST). Almond oil was extracted by a Soxhlet and subjected to gas chromatography-mass spectrometry (GC/MS). The yield was found to be 35.17%. The chemical composition revealed the presence of various phytoconstituents. A total of 7 compounds were identified, the main ones being 9,11-Octadecadienoic acid, methyl ester, (E,E)- (79.32%), 9-octadecenoic acid (Z)-, methyl ester (13.24%), hexadecanoic acid and methyl ester (3.69%). The larvicidal bioassays were performed according to the protocol recommended by the World Health Organization with concentrations varying from 20 to 80 mg/L depending on the exposure time (24, 48 and 72 hours). The almond oil exhibited remarkable larvicidal activity against fourth instar larvae and the lethal concentrations were determined (LC25= 23.70 mg/L, LC50=35.49 mg/L, LC90=79.61 mg/L). The results also showed that the oil caused an ovicidal activity with a significant effect on egg hatch. The recorded hatching percentages were respectively 88.79% and 72.40% for the LC25 and LC50, and this compared to the control series. Moreover, this oil exhibited significant repellency against adult mosquitoes. Furthermore, the enzymatic measurements performed on LC50 and LC90 treated larvae revealed a neurotoxic activity and a stimulation of the detoxification system as evidenced, respectively, by an inhibition of AChE and induction in GST activity. Overall, our data proved that Melia azedarach almond oil could be considered as a potent biorational alternative to synthetic insecticides for mosquito control.