40 research outputs found
Some conjectures in elementary number theory
We announce a number of conjectures associated with and arising from a study
of primes and irrationals in . All are supported by numerical
verification to the extent possible.Comment: Unpublishe
Fractional Sturm-Liouville eigenvalue problems, II
We continue the study of a non self-adjoint fractional three-term
Sturm-Liouville boundary value problem (with a potential term) formed by the
composition of a left Caputo and left-Riemann-Liouville fractional integral
under {\it Dirichlet type} boundary conditions. We study the existence and
asymptotic behavior of the real eigenvalues and show that for certain values of
the fractional differentiation parameter , , there is a
finite set of real eigenvalues and that, for near , there may be
none at all. As we show that their number becomes infinite and
that the problem then approaches a standard Dirichlet Sturm-Liouville problem
with the composition of the operators becoming the operator of second order
differentiation
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented