Browsing by Subject "maximum principle"

Now showing 1 - 10 of 10

Access status: Open Access ,
Double phase problems: a survey of some recent results
(Wydawnictwa AGH, 2022) Papageorgiou, Nikolaos Socrates
We review some recent results on double phase problems. We focus on the relevant function space framework, which is provided by the generalized Orlicz spaces. We also describe the basic tools and methods used to deal with double phase problems, given that there is no global regularity theory for these problems.
Access status: Open Access ,
Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems
(Wydawnictwa AGH, 2022) Zeddini, Noureddine; Sari, Rehab Saeed
Let $D$ be a bounded $C^{1,1}$-domain in $\mathbb{R}^d$, $d \geq 2$. The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions $K(D)$ that was defined by N. Zeddini for $d=2$ and by H. Mâagli and M. Zribi for $d \geq 3$ and adapted to study some nonlinear elliptic problems in $D$. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants $\lambda$ and $\mu$ to the following system $\Delta u=\lambda f(x,u,v)$, $\Delta v=\mu g(x,u,v)$ in $D$, $u=\phi_{1}$ and $v=\phi_{2}$ on $\partial D$, where $\phi_{1}$ and $\phi_{2}$ are nontrivial nonnegative continuous functions on $\partial D$. The functions $f$ and $g$ are nonnegative and belong to a class of functions containing in particular all functions of the type $f(x,u,v)=p(x)u^{\alpha}h_{1}(v)$ and $g(x,u,v)=q(x)h_{2}(u)v^{\beta} with $\alpha \geq 1$, $\beta \geq 1$, $h_1$, $h_2$ are continuous on $[0,\infty)$ and $p$, $q$ are nonnegative functions in $K(D)$.
Access status: Restricted ,
Istnienie i jednoznaczność rozwiązania klasycznego układu ewolucyjnych równań różniczkowych
(Data obrony: 2017-07-10) Światłowska, Julita
Wydział Matematyki Stosowanej
Access status: Open Access ,
On a linear-quadratic problem with Caputo derivative
(2016) Idczak, Dariusz; Walczak, Stanisław
In this paper, we study a linear-quadratic optimal control problem with a fractional control system containing a Caputo derivative of unknown function. First, we derive the formulas for the differential and gradient of the cost functional under given constraints. Next, we prove an existence result and derive a maximum principle. Finally, we describe the gradient and projection of the gradient methods for the problem under consideration.
Access status: Open Access ,
On a Robin (p,q)-equation with a logistic reaction
(Wydawnictwa AGH, 2019) Papageorgiou, Nikolaos Socrates; Vetro, Calogero; Vetro, Francesca
We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a $p$-Laplacian and of a $q$-Laplacian ($(p,q)$-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $\lambda \gt 0$ varies. Also, we show that for every admissible parameter $\lambda \gt 0$, the problem admits a smallest positive solution.
Access status: Open Access ,
Optimization of a fractional Mayer problem - existence of solutions, maximum principle, gradient methods
(2014) Idczak, Dariusz; Walczak, Stanisław
In the paper, we study a linear-quadratic optimal control problem of Mayer type given by a fractional control system. First, we prove a theorem on the existence of a solution to such a problem. Next, using the local implicit function theorem, we derive a formula for the gradient of a cost functional under constraints given by a control system and prove a maximum principle in the case of a control constraint set. The formula for the gradient is used to implement the gradient methods for the problem under consideration.
Access status: Open Access ,
Region of existence of multiple solutions for a class of Robin type four-point BVPs
(Wydawnictwa AGH, 2021) Verma, Amit K.; Urus, Nazia; Agarwal, Ravi P.
This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as $\begin{gathered} -u''(x)=\psi(x,u,u'), \quad x\in (0,1),\\ u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta),\end{gathered}$ where $I=[0,1]$, $0\lt\xi\leq\eta\lt 1$ and $\lambda_{1},\lambda_{2} \gt 0$. The nonlinear source term $\psi\in C(I\times\mathbb{R}^2,\mathbb{R})$ is one sided Lipschitz in $u$ with Lipschitz constant $L_1$ and Lipschitz in $u'$ such that $|\psi(x,u,u')-\psi(x,u,v')|\leq L_2(x)|u'-v'|$. We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter $k$ equivalent to $\max_u\frac{\partial \psi}{\partial u}$. We compute the range of $k$ for which iterative sequences are convergent.
Access status: Open Access ,
The maximum principle for viscosity solutions of elliptic differential functional equations
(2013) Karpowicz, Adrian
This paper is devoted to the study of the maximum principle for the elliptic equation with a deviated argument. We will consider viscosity solutions of this equation.
Access status: Restricted ,
Zastosowania zasady maksimum w rozwiązywaniu pewnych problemów matematyki finansowej
(Data obrony: 2011-07-14) Rybka, Anna
Wydział Matematyki Stosowanej
Access status: Restricted ,
Zastosowania zasady maksimum w rozwiązywaniu pewnych problemów matematyki finansowej
(Data obrony: 2011-07-14) Rybka, Anna
Wydział Matematyki Stosowanej