Olanzapine (OLZ) is classified as a typical antipsychotic drug utilized for the treatment of schizophrenia. Its oral bioavailability is 60% due to its low solubility and pre-systemic metabolism. Hence, the present work aims to formulate and evaluate OLZ nanoparticles dissolving microneedles (MNs) for transdermal delivery to overcome the problems associated with drug administration orally. OLZ nanoparticles were prepared by the nanoprecipitation method. The optimized OLZ nanoparticle formula was utilized for the fabrication of dissolving MNs by loading OLZ nanodispersion into polydimethylsiloxane (PDMS) micromould cavities, followed by casting the polymeric solution of polyvinylpyrrolidone(PVP-K30) and polyvinyl alcohol (PVA) to form

The necessary optimality conditions with Lagrange multipliers  are studied and derived for a new class that includes the system of Caputo–Katugampola fractional derivatives to the optimal control problems with considering the end time free. The formula for the integral by parts has been proven for the left Caputo–Katugampola fractional derivative that contributes to the finding and deriving the necessary optimality conditions. Also, three special cases are obtained, including the study of the necessary optimality conditions when both the final time  and the final state  are fixed. According to convexity assumptions prove that necessary optimality conditions are sufficient optimality conditions.

Mesoporous silica (MPS) nanoparticle was prepared as carriers for drug delivery systems by sol–gel method from sodium silicate as inexpensive precursor of silica and Cocamidopropyl betaine (CABP) as template. The silica particles were characterized by SEM, TEM, AFM, XRD, and N2adsorption–desorption isotherms. The results show that the MPS particle in the nanorange (40-80 nm ) with average diameter equal to 62.15 nm has  rods particle morphology, specific surface area is 1096.122 m²/g, pore volume 0.900 cm³/g, with average pore diameter 2.902 nm, which can serve as efficient carriers for drugs. The adsorption kinetic of Ciprofloxacin (CIP) drug was studied and the data were analyzed and found to match well with