For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^2$, let $AC(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\left\{0\right\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the...

We establish new existence results for nontrivial solutions of some integral inclusions of Hammerstein type, that are perturbed with an affine functional. In order to use a theory of fixed point index for multivalued mappings, we work in a cone of continuous functions that are positive on a suitable subinterval of $[0,1]$. We also discuss the optimality of some constants that occur in our theory. We improve, complement and extend previous results in the literature.

Let H be a real Hilbert space and T be a maximal monotone operator on H. A well-known algorithm, developed by R. T. Rockafellar [16], for solving the problem (P) ”To find x ∈ H such that 0 ∈ T x” is the proximal point algorithm. Several generalizations have been considered by several authors: introduction of a perturbation, introduction of a variable metric in the perturbed algorithm, introduction of a pseudo-metric in place of the classical regularization, . . . We summarize some of these extensions...

We study a multilinear fixed-point equation in a closed ball of a Banach space where the application is 1-Lipschitzian: existence, uniqueness, approximations, regularity.

We prove that for each linear contraction T : X → X (∥T∥ ≤ 1), the subspace F = {x ∈ X : Tx = x} of fixed points is 1-complemented, where X is a suitable subspace of L¹(E*) and E* is a separable dual space such that the weak and weak* topologies coincide on the unit sphere. We also prove some related fixed point results.

We show that the Porous Medium Equation and the Fast Diffusion Equation, $\dot{u}-\Delta u^m=f$, with $m\in (0,\infty )$, can be modeled as a gradient system in the Hilbert space $H^{-1}\left(\Omega \right)$, and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets $\Omega \subseteq {ℝ}^n$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.

This paper deals with the existence of positive $\omega$-periodic solutions for the neutral functional differential equation with multiple delays $${(u\left(t\right)-cu(t-\delta ))}^{\text{'}}+a\left(t\right)u\left(t\right)=f(t,u(t-{\tau }_1),\cdots ,u(t-{\tau }_n)).$$ The essential inequality conditions on the existence of positive periodic solutions are obtained. These inequality conditions concern with the relations of $c$ and the coefficient function $a\left(t\right)$, and the nonlinearity $f(t,x_1,\cdots ,x_n)$. Our discussion is based on the perturbation method of positive operator and fixed point index theory in cones.

The author applies a generalized Leggett-Williams fixed point theorem to the study of the nonlinear functional differential equation $x^{\text{'}}\left(t\right)=-a(t,x\left(t\right))x\left(t\right)+f(t,x_t)$. Sufficient conditions are established for the existence of multiple positive periodic solutions.

A translation along trajectories approach together with averaging procedure and topological degree are used to derive effective criteria for existence of periodic solutions for nonautonomous evolution equations with periodic perturbations. It is shown that a topologically nontrivial zero of the averaged right hand side is a source of periodic solutions for the equations with increased frequencies. Our setting involves equations on closed convex cones, therefore it enables us to study positive solutions...