Related papers: Problems with mixed boundary conditions in Banach …

Let $1< \alpha <2$ and $A$ be the generator of an $\alpha$-times resolvent family $\{S_\alpha(t)\}_{t \ge 0}$ on a Banach space $X$. It is shown that the fractional Cauchy problem ${\bf D}_t^\alpha u(t) = Au(t)+f(t)$, $t \in [0,r]$; $u(0),…

We give an upper bound for the difference of two solutions of Euler schemes approximating the Cauchy problem \[\begin{cases} \dot{u}(t) + Au(t) \ni f(t) \quad (t \in [0, T]), \\ u(0) = u^0, \end{cases}\] where $A \subseteq X \times X$ is a…

For a control Cauchy problem $$\dot x= {f}(t,x,u,v) +\sum_{\alpha=1}^m g_\alpha(x) \dot u_\alpha,\quad x(a)=\bar x, $$ on an interval $[a,b]$, we propose a notion of limit solution $x,$ verifying the following properties: i) $x$ is defined…

We consider an initial boundary value problem in a bounded domain $\Omega$ over a time interval $(0, T)$ for a time-fractional wave equation where the order of the fractional time derivative is between $1$ and $2$ and the spatial elliptic…

In this paper, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Solutions to these equations are called monopoles. Under some simple topological assumptions, we show that the solution space of all monopoles is a…

In a rectangular domain, a boundary-value problem is considered for a mixed-type equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. Using the method…

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval \begin{equation} u_t+u_x+u_{xxx}+uu_x=0,\qquad u(x,0)=\phi(x), \qquad 0<x<L, \ t>0 \qquad (1) \end{equation} subject…

In this paper we study the nonlinear Neumann boundary value problem of the following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=\lambda f(x,u) in a…

We study the existence of solutions to abstract equations of the form $0 = Au + F(u)$, $u\in K\subset E$, where A is an abstract differential operator acting in a Banach space $E$, $K$ is a closed convex set of constraints being invariant…

This paper deals with the two-dimensional incompressible, laminar, steady-state boundary layer equations.First, we determine a family of velocity distributions outside the boundary layer such that these problems may have similarity…

Two main results are presented: 1) a new class of applied problems that lead to equations with $(p,q)$-Laplace is presented; 2) a method for solving nonlinear boundary value problems involving $(p,q)$-Laplace with measurable unbounded…

We prove an a-priori bound for the Lipschitz constant of a smooth one-phase free boundary graph F(u) in two dimensions. The function u satisfies a fully nonlinear elliptic equation in its positive side, and the gradient of u is equal to 1…

We consider periodic homogenization of boundary value problems for quasilinear second-order ODE systems in divergence form of the type $a(x,x/\varepsilon,u(x),u'(x))'= f(x,x/\varepsilon,u(x),u'(x))$ for $x \in [0,1]$. For small…

The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space $X$, we study the Cluster Value Problem for the ball algebra $A_u(B_X)$, the Banach algebra of all uniformly continuous…

Starting with approximate solutions of the equation $-\Delta u=wu^3$ on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and…

We study boundary value problems with measure data in smooth bounded domains $\Omega$, for semilinear equations involving Hardy type potentials. Specifically we consider problems of the form $-L_V u + f(u) = \tau$ in $\Omega$ and…

Consider a smooth vector field $f\colon \mathbb{R}^n\to\mathbb{R}^n$ and a maximal solution $\gamma\colon \,]a,b[\,\to \mathbb{R}^n$ to the ordinary differential equation $x'=f(x)$. It is a well-known fact that, if $\gamma$ is bounded, then…

This work is devoted to Lipschitz conditions on bounded harmonic functions on the upper half-space in $\mathbb {R}^n$. Among other results we prove the following one. Let $U(x',x_n)$ be a real-valued bounded harmonic function on the upper…

Given a map $f \colon E \longrightarrow F$ between Banach spaces (or Banach lattices), a set $A$ of $E$-valued bounded sequences, ${\bf x} \in A$ and a vector topology $\tau$ on $F$, we investigate the existence of an infinite dimensional…

In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{…