Fractional differential equations of Bagley-Torvik and Langevin type

Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for fractional integrals of a certain type are proved. These are used to rigorously prove equivalences between solutions of initial value problems for the fractional derivative equations and solutions of the corresponding integral equations in the space of continuous functions. A novelty is that it is not assumed that the nonlinear term is continuous but that it satisfies the weaker L p-Carathéodory condition. Existence of solutions on an interval [0, T ] in cases where T can be arbitrarily large, so-called global solutions, are proved, obtaining the necessary a priori bounds by using recent fractional Gronwall and fractional Bihari inequalities

Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type

New criteria are established for the existence of multiple positive solutions of a Hammerstein integral equation of the form $$u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))ds \equiv Au(t)$$ where $k$ can have discontinuities in its second variable and $g \in L^{1}$. These criteria are determined by the relationship between the behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the principal (positive) eigenvalue of the linear Hammerstein integral operator $$Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)ds.$$ We obtain new results on the existence of multiple positive solutions of a second order differential equation of the form $$u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1],$$ subject to general separated boundary conditions and also to nonlocal $m$-point boundary conditions. Our results are optimal in some cases. This work contains several new ideas, and gives a {\it unified} approach applicable to many BVPs