In this paper sufficient second order optimality conditions for optimal control problems subject to stationary variational inequalities of obstacle type are derived. Since optimality conditions for such problems always involve measures as Lagrange multipliers, which impede the use of efficient Newton type methods, a family of regularized problems is introduced. Second order sufficient optimality conditions are derived for the regularized problems...

We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the $L^p\left(ℝ\right)$-spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable non-negative...

Let X be an arbitrary Abelian group and E a Banach space. We consider the difference-operators ∆n defined by induction:(∆f)(x;y) = f(x+y) - f(x), (∆nf)(x;y1,...,yn) = (∆n-1(∆f)(.;y1)) (x;y2,...,yn)(n = 2,3,4,..., ∆1=∆, x,yi belonging to X, i = 1,2,...,n; f: X --> E).Considering the difference equation (∆nf)(x;y1,y2,...,yn) = d(x;y1,y2,...,yn) with independent variable increments, the most general solution is given explicitly if d: X x Xn --> E is a given bounded function. Also the...

We consider the summation equation, for $t\in {[\mu -2,\mu +b]}_{{\mathbb{N}}_{\mu -2}}$, $$\begin{array}{ccc}\hfill y\left(t\right)={\gamma }_1\left(t\right)H_1\left(\sum _{i=1}^n{a}_{i}y\left({\xi }_i\right)\right)& +{\gamma }_2\left(t\right)H_2\left(\sum _{i=1}^m{b}_{i}y\left({\zeta }_i\right)\right)\hfill & \hfill +\lambda \sum _{s=0}^{b}G(t,s)f(s+\mu -1,y(s+\mu -1))\end{array}$$ in the case where the map $(t,s)\mapsto G(t,s)$ may change sign; here $\mu \in (1,2]$ is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that $G$ is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions $H_1$ and $H_2$. Finally, as an application of the abstract existence result,...

Let Y be a Banach space and let $S\subset L_p$ be a subspace of an $L_p$ space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to $S\left(Y\right)\subset L_p\left(Y\right)$. We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and $e^{-tB}$ is a positive contraction on $L_p$ for any...

A lemma of Gelfand-Hille type is proved. It is used to give an improved version of a result of Kalton on sums of idempotents.

We construct time quasi-periodic solutions and prove almost global existence for the energy supercritical nonlinear Schrödinger equations on the torus in arbitrary dimensions. The main new ingredient is a geometric selection in the Fourier space. This method is applicable to other nonlinear equations.

We show that the Angle Criterion for testing supercyclic vectors depends in an essential way on the geometrical properties of the underlying space. In particular, we exhibit non-supercyclic vectors for the backward shift acting on c₀ that still satisfy such a criterion. Nevertheless, if ℬ is a locally uniformly convex Banach space, the Angle Criterion yields an equivalent condition for a vector to be supercyclic. Furthermore, we prove that local uniform convexity cannot be weakened to strict convexity....

An operator (linear and continuous) in a Fréchet space is hypercyclic if there exists a vector whose orbit under the operator is dense. If the scalar multiples of the elements in the orbit are dense, the operator is supercyclic. We give, for Fréchet space operators, a Supercyclicity Criterion reminiscent of the Hypercyclicity Criterion. We characterize the supercyclic bilateral weighted shifts in terms of their weight sequences. As a consequence, we show that a bilateral weighted shift is supercyclic...

We prove a Supercyclicity Criterion for a continuous linear mapping that is defined on the operator algebra of a separable Banach space ℬ. Our result extends a recent result on hypercyclicity on the operator algebra of a Hilbert space. This kind of result is a powerful tool to analyze the structure of supercyclic vectors of a supercyclic operator that is defined on ℬ. For instance, as a consequence of the main result, we give a very simple proof of the recently established fact that certain supercyclic...