Existence and approximations of solutions to some fractional order functional integral equations

In this paper we shall study a fractional order functional integral equation. In the first part of the paper, we proved the existence and uniqueness of mile and global solutions in a Banach space. In the second part of the paper, we used the analytic semigroups theory oflinear operators and the fixed point method to establish the existence, uniqueness and convergence of approximate solutions of the given problem in a separable Hilbert space. We also proved the existence and convergence of Faedo-Galerkin approximate solution to the given problem. Finally, we give an example.

A comparative study of the early effects of phenobarbital and 3-methylcholanthrene on the synthesis and transport of ribonucleic acid in rat liver.

At 2-3 h after phenobaribtal administration, the drug has no effect on nucleoplasmic RNA synthesis and decreases nucleolar RNA synthesis. However, at this time there is an increase in the labelling of cytoplasmic poly(A)-containing RNA, even though there is decreased labelling of total polyribosomal RNA. The decrease in labelling of nucleolar and total polyribosomal RNA owing to phenobarbital is a transient phenomenon. Under similar conditions, 3-methylcholanthrene has no effect on nucleolar RNA synthesis, but leads to an increase in synthesis of nucleoplasmic and cytoplasmic poly(A)-containing RNA. Cytosol isolated from phenobarbital-treated, but not from 3-methyl-cholanthrene-treated, animals facilitates an enhanced transport of RNA from nuclei. At the time points investigated, 3-methylcholanthrene or its metabolite shows a 10-15-fold higher concentration in the chromatin than that of phenobarbital or its metabolite. It is suggested that the primary effect of phenobarbital is at the cytoplasmic level, promoting the transport of RNA from the nuclei, which can act as a trigger for enhanced transcription at later periods. 3-Methylcholanthrene or its metabolite directly binds to the chromatin and evokes a selective transcriptional response.