We investigate the fractional differential equation u″ + A c D α u = f(t, u, c D μ u, u′) subject to the boundary conditions u′(0) = 0, u(T)+au′(T) = 0. Here α ∈ (1, 2), µ ∈ (0, 1), f is a Carathéodory function and c D is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives....

Using a suitable version of Mawhin’s continuation principle, we obtain an existence result for the Floquet boundary value problem for second order Carathéodory differential equations by means of strictly localized $C^2$ bounding functions.

In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator $T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)≔v\left(x\right){ʃ}_0^{\infty }u\left(t\right)f\left(t\right)dt$ when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

We consider the Volterra integral operator $T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)=v\left(x\right){ʃ}_0^{x}u\left(t\right)f\left(t\right)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_n\left(T\right)$ of T are established when 1 < p < ∞. When p = 2 these yield $li{m}_{n\to \infty }n{a}_n\left(T\right)={\pi }^{-1}{ʃ}_0^{\infty }\left|u\left(t\right)v\left(t\right)\right|dt$. We also provide upper and lower estimates for the ${\ell }^{\alpha }$ and weak ${\ell }^{\alpha }$ norms of (an(T)) when 1 < α < ∞.

In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong $A_{\mathrm{\infty }}$ weight with respect to the metric associated with the operator. Roughly speaking, the strong $A_{\mathrm{\infty }}$ condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations....

Necessary and sufficient conditions governing two-weight $L^p$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.

We consider two operator space versions of type and cotype, namely $S_p$-type, $S_q$-cotype and type (p,H), cotype (q,H) for a homogeneous Hilbertian operator space H and 1 ≤ p ≤ 2 ≤ q ≤ ∞, generalizing “OH-cotype 2” of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and $L_p$ spaces, and we investigate the relationship between a homogeneous Hilbertian space H and operator spaces with cotype (2,H). As applications we consider operator space versions of generalized little Grothendieck’s...