Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2016
Volume
Vol. 36
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 36 (2016)
Projects
Pages
Articles
Vertex-weighted Wiener polynomials of subdivision-related graphs
(2016) Azari, Mahdieh; Iranmanesh, Ali; Došlić, Tomislav
Singly and doubly vertex-weighted Wiener polynomials are generalizations of both vertex-weighted Wiener numbers and the ordinary Wiener polynomial. In this paper, we show how the vertex-weighted Wiener polynomials of a graph change with subdivision operators, and apply our results to obtain vertex-weighted Wiener numbers.
Kernel conditional quantile estimator under left truncation for functional regressors
(2016) Helal, Nacéra; Ould Saïd, Elias
Let $Y$ be a random real response which is subject to left-truncation by another random variable $T$. In this paper, we study the kernel conditional quantile estimation when the covariable $X$ takes values in an infinite-dimensional space. A kernel conditional quantile estimator is given under some regularity conditions, among which in the small-ball probability, its strong uniform almost sure convergence rate is established. Some special cases have been studied to show how our work extends some results given in the literature. Simulations are drawn to lend further support to our theoretical results and assess the behavior of the estimator for finite samples with different rates of truncation and sizes.
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.
Positive solutions of boundary value problems with nonlinear nonlocal boundary conditions
(2016) Padhi, Seshadev; Pati, Smita; Hota, D. K.
We consider the existence of at least three positive solutions of a nonlinear first order problem with a nonlinear nonlocal boundary condition given by $\begin{aligned} x^{\prime}(t)& = r(t)x(t) + \sum_{i=1}^{m} f_i(t,x(t)), \quad t \in [0,1],\\ \lambda x(0)& = x(1) + \sum_{j=1}^{n} \Lambda_j(\tau_j, x(\tau_j)),\quad \tau_j \in [0,1],\end{aligned}$ where $r:[0,1] \rightarrow [0,\infty)$ is continuous; the nonlocal points satisfy $0 \leq \tau_1 \lt \tau_2 \lt \ldots \lt \tau_n \leq 1$ the nonlinear function $f_i$ and $\tau_j$ are continuous mappings from $[0,1] \times [0,\infty) \rightarrow [0,\infty)$ for $i = 1,2,\ldots ,m$ and $j = 1,2,\ldots ,n$ respectively, and $\lambda \gt 0$ is a positive parameter.
Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces
(2016) Stăncuţ, Ionela-Loredana; Stîrcu, Iulia Dorotheea
In this paper we consider an eigenvalue problem that involves a nonhomogeneous elliptic operator, variable growth conditions and a potential $V$ on a bounded domain in $\mathbb{R}^N$ ($N\geq 3$) with a smooth boundary. We establish three main results with various assumptions. The first one asserts that any $\lambda\gt 0$ is an eigenvalue of our problem. The second theorem states the existence of a constant $\lambda_{*}\gt 0$ such that any $\lambda\in(0,\lambda_{*}]$ is an eigenvalue, while the third theorem claims the existence of a constant $\lambda^{*}\gt 0$ such that every $\lambda\in[\lambda^{*}, \infty)$ is an eigenvalue of the problem.