This paper investigates the nonlinear eigenvalue problems (NEPs) for a class of singular Katugampola fractional differential systems with four-point coupled boundary conditions. Firstly, we establish the Green's functions of the aforementioned NEPs and their fundamental analytic properties. Secondly, utilizing some classical fixed-point theorems, some explicit eigenvalue interval-dependent criteria are derived for the existence of at least one positive solution to the addressed NEPs. As applications, some examples are presented to illustrate the feasibility and effectiveness of our main results.
In this paper, we will show the general solution of the Bessel differential equation given by x(2)y"+xy' + (x(2)-v(2))y = 0, where v,x is an element of R and x>0, but only when v= 2m-1/ 2 with m is an element of N. Moreover, contrary to what we found in the literature, our general solution does not depend on a series of functions, our algorithm provides the exact general solution.
We establish the existence of solution for the Navier-Stokes equations with a general damping term, we give examples. We also derive estimates on the asymptotic behavior of the solution, for instance, energy estimate decay in time and extinction in time. We construct a sequence of solutions of auxiliary equations in finite dimension that converges to a genuine solution of the original equations.
