Best Proximity Points of Generalized Semicyclic Impulsive Self-Mappings: Applications to Impulsive Differential and Difference Equations

This paper is devoted to the study of convergence properties of distances between points and the existence and uniqueness of best proximity and fixed points of the so-called semicyclic impulsive self-mappings on the union of a number of nonempty subsets in metric spaces. The convergences of distances between consecutive iterated points are studied in metric spaces, while those associated with convergence to best proximity points are set in uniformly convex Banach spaces which are simultaneously complete metric spaces. The concept of semicyclic self-mappings generalizes the well-known one of cyclic ones in the sense that the iterated sequences built through such mappings are allowed to have images located in the same subset as their pre-image. The self-mappings under study might be in the most general case impulsive in the sense that they are composite mappings consisting of two self-mappings, and one of them is eventually discontinuous. Thus, the developed formalism can be applied to the study of stability of a class of impulsive differential equations and that of their discrete counterparts. Some application examples to impulsive differential equations are also given.

The Long Run Demand for Lighting: Elasticities and Rebound Effects in Different Phases of Economic Development

25 p.

Magnetic Cross-Linked Enzyme Aggregates (mCLEAs) of Candida antarctica Lipase: An Efficient and Stable Biocatalyst for Biodiesel Synthesis

Enzyme-catalyzed production of biodiesel is the object of extensive research due to the global shortage of fossil fuels and increased environmental concerns. Herein we report the preparation and main characteristics of a novel biocatalyst consisting of Cross-Linked Enzyme Aggregates (CLEAs) of Candida antarctica lipase B (CALB) which are covalently bound to magnetic nanoparticles, and tackle its use for the synthesis of biodiesel from non-edible vegetable and waste frying oils. For this purpose, insolubilized CALB was covalently cross-linked to magnetic nanoparticles of magnetite which the surface was functionalized with –NH2 groups. The resulting biocatalyst combines the relevant catalytic properties of CLEAs (as great stability and feasibility for their reutilization) and the magnetic character, and thus the final product (mCLEAs) are superparamagnetic particles of a robust catalyst which is more stable than the free enzyme, easily recoverable from the reaction medium and reusable for new catalytic cycles. We have studied the main properties of this biocatalyst and we have assessed its utility to catalyze transesterification reactions to obtain biodiesel from non-edible vegetable oils including unrefined soybean, jatropha and cameline, as well as waste frying oil. Using 1% mCLEAs (w/w of oil) conversions near 80% were routinely obtained at 30°C after 24 h of reaction, this value rising to 92% after 72 h. Moreover, the magnetic biocatalyst can be easily recovered from the reaction mixture and reused for at least ten consecutive cycles of 24 h without apparent loss of activity. The obtained results suggest that mCLEAs prepared from CALB can become a powerful biocatalyst for application at industrial scale with better performance than those currently available.