We find conditions ensuring the existence of the one-sided Minkowski content for d-dimensional closed sets in R, in connection with regularity properties of their boundaries. Moreover, we provide a class of sets stable under finite unions for which the one-sided Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positi...

We propose a new concept of generalized di erentiation of setvalued maps that captures rst order information. This concept encompasses the standard notions of Fréchet di erentiability, strict di erentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between th...

In this paper we study the class of differential variational inequality(DVI) in a finite-dimension Euclidean space <n, which is the following form ẋ(t) = f(t, x(t)) + B(t, x(t))u(t) , x(0) = x0 ∈ <n 0 ≤ (ũ− u(t))T [G(t, x(t)) + F (u(t))] for almost all ũ ∈ K u(t) ∈ K We study stability and perturbation of the DVI under the OSL condition. Besides, we establish a Prior Bound Theorem, which is a u...

Monotone iterative technique is employed for studying the existence of solutions to the second-order nonlinear singular boundary value problem − ` p(x)y′(x) ́′ + p(x)f ` x, y(x), p(x)y′(x) ́ = 0 for 0 < x < 1 and y′(0) = y′(1) = 0. Here p(0) = 0 and xp′(x)/p(x) is analytic at x = 0. The source function f(x, y, py′) is Lipschitz in py′ and one sided Lipschitz in y. The initial approximations are...

We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general of the form of a standard Lipschitz “variation”. Part of this investigation, but of interest in its own right, is an example of a nowhere locally Lipschitz mi...

In the paper some known and new extensions of the famous theorem of Filippov (1967) and a theorem of Pliś (1965) for differential inclusions are presented. We replace the Lipschitz condition on the set-valued map in the right-hand side by a weaker onesided Lipschitz (OSL), one-sided Kamke (OSK) or a continuity-like condition (CLC). We prove new Filippov-type theorems for singularly perturbed an...

This paper is devoted to give a simplified proof of the trace theorem for functions of bounded deformation defined on bounded Lipschitz domains of Rn. As a consequence, the existence of one-sided Lebesgue limits on countably H-rectifiable sets is also established.