We are concerned with first order set-valued problems with very general boundary value conditions $$\left\{\begin{array}{c}{u}_g^{\text{'}}\left(t\right)\in F(t,u\left(t\right)),{\mu }_g\text{-a.e.}\in [0,T],\hfill \\ L\left(u\right(0),T))=0\hfill \end{array}\right.$$ involving the Stieltjes derivative with respect to a left-continuous nondecreasing function $g:[0,T]\to ℝ$, a Carathéodory multifunction $F:[0,T]\times ℝ\to 𝒫\left(ℝ\right)$ and a continuous $L:{ℝ}^2\to ℝ$. Using appropriate notions of lower and upper solutions, we prove the existence of solutions via a fixed point result for condensing mappings. In the periodic single-valued case, combining an existence theory for the linear case with a recent result involving...

We study a stochastic fractional partial differential equations of order $\alpha \>1$ driven by a compensated Poisson measure. We prove existence and uniqueness of the solution and we study the regularity of its trajectories.

Mathematics Subject Classification: 26A33, 76M35, 82B31A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which are spectral integrals of Levy processes.

MSC 2010: 26A33, 34A08, 34K37

Let (FP) abbreviate the statement that ${\int }_0^1\left({\int }_0^{1}fdy\right)dx={\int }_0^1\left({\int }_0^{1}fdx\right)dy$ holds for every bounded function f: [0,1]² → ℝ whenever each of the integrals involved exists. We shall denote by (SFP) the statement that the equality above holds for every bounded function f: [0,1]² → ℝ having measurable vertical and horizontal sections. It follows from well-known results that both of (FP) and (SFP) are independent of the axioms of ZFC. We investigate the logical connections of these statements with several other strong Fubini type properties...

In dimension one it is proved that the solution to a total variation-regularized least-squares problem is always a function which is "constant almost everywhere" , provided that the data are in a certain sense outside the range of the operator to be inverted. A similar, but weaker result is derived in dimension two.

In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory....

Let ϕ be an arbitrary bijection of ${ℝ}_{+}$. We prove that if the two-place function ${\varphi }^{-1}[\varphi \left(s\right)+\varphi \left(t\right)]$ is subadditive in ${ℝ}_{+}^2$ then $\varphi$ must be a convex homeomorphism of ${ℝ}_{+}$. This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of ${ℝ}_{+}$ are also given. We apply the above results to obtain several converses of Minkowski’s inequality.

For any subanalytic $C^k$-Whitney field (k finite), we construct its subanalytic $C^k$-extension to ${ℝ}^n$. Our method also applies to other o-minimal structures; e.g., to semialgebraic Whitney fields.

We obtain (weighted) Poincaré type inequalities for vector fields satisfying the Hörmander condition for p < 1 under some assumptions on the subelliptic gradient of the function. Such inequalities hold on Boman domains associated with the underlying Carnot- Carathéodory metric. In particular, they remain true for solutions to certain classes of subelliptic equations. Our results complement the earlier results in these directions for p ≥ 1.

Results on integration by parts and integration by substitution for the variational integral of Henstock are well-known. When real-valued functions are considered, such results also hold for the Generalized Riemann Integral defined by Kurzweil since, in this case, the integrals of Kurzweil and Henstock coincide. However, in a Banach-space valued context, the Kurzweil integral properly contains that of Henstock. In the present paper, we consider abstract vector integrals of Kurzweil and prove Substitution...