Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2012
Volume
Vol. 32
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 32 (2012)
Projects
Pages
Articles
Boundary value problems for n-th order differential inclusions with four-point integral boundary conditions
(2012) Bashir, Ahmad Khuda Bakhsh; Ntouyas, Sotiris K.
In this paper, we discuss the existence of solutions for a four-point integral boundary value problem of $n$-th order differential inclusions involving convex and non-convex multivalued maps. The existence results are obtained by applying the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.
On the extended and Allan spectra and topological radii
(2012) Arizmendi, Hugo; Carrillo-Hoyo, Angel Manuel; Roa-Fajardo, Jairo
In this paper we prove that the extended spectrum $\Sigma(x)$, defined by W. Żelazko, of an element $x$ of a pseudo-complete locally convex unital complex algebra $A$ is a subset of the spectrum $\sigma_A(x)$, defined by G.R. Allan. Furthermore, we prove that they coincide when $\Sigma(x)$ is closed. We also establish some order relations between several topological radii of $x$, among which are the topological spectral radius $R_t(x)$ and the topological radius of boundedness $\beta_t(x)$.
A note on a relation between the weak and strong domination numbers of a graph
(2012) Boutrig, Razika; Chellali, Mustapha
In a graph $G=(V,E)$ a vertex is said to dominate itself and all its neighbors. A set $D \subset V$ is a weak (strong, respectively) dominating set of $G$ if every vertex $v \in V-S$ is adjacent to a vertex $u \in D$ such that $d_G(v) \geq d_G(u)$ $d_G(v) \leq d_G(u)$, respectively). The weak (strong, respectively) domination number of $G$, denoted by $\gamma_w(G)$ $(\gamma_s(G)$, respectively), is the minimum cardinality of a weak (strong, respectively) dominating set of $G$. In this note we show that if $G$ is a connected graph of order $n \geq 3$, then $\gamma_w(G) + t\gamma_s(G) \leq n$, where $t=3/(\Delta+1)$ if $G$ is an arbitrary graph, $t=3/5$ if $G$ is a block graph, and $t=2/3$ if $G$ is a claw free graph.
Uniformly continuous composition operators in the space of bounded φ-variation functions in the Schramm sense
(2012) Ereú, Tomás; Merentes Díaz, Nelson José; Sánchez, José Luis; Wróbel, Małgorzata
We prove that any uniformly continuous Nemytskii composition operator in the space of functions of bounded generalized $\Phi$-variation in the Schramm sense is affine. A composition operator is locally defined. We show that every locally defined operator mapping the space of continuous functions of bounded (in the sense of Jordan) variation into the space of continous monotonic functions is constant.
Existence and solution sets of impulsive functional differential inclusions with multiple delay
(2012) Helal, Mohmed; Ouahab, Abdelghani
In this paper, we present some existence results of solutions and study the topological structure of solution sets for the following first-order impulsive neutral functional differential inclusions with initial condition: $\begin{cases}\frac{d}{dt}[y(t)-g(t,y_t)] \in F(t,y_t) + \sum_{i=1}^{n_*} y(t-Ti), & a.e.\, t \in J\setminus\{t_1,...,t_m\} \\ y(t_k^+)-y(t_k^-)=I_k(y(t_k^-)), & k=1,...,m, \\ y(t)=\phi(t), & t \in [-r,0],\end{cases}$ where $J:=[0,b]$ and $0=t_0\lt t_1 \lt ...\lt t_m\lt t_{m+1}=b$ ($m \in \mathbb{N}^*$), $F$ is a set-valued map and g is single map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,...,m$). Our existence result relies on a nonlinear alternative for compact u.s.c. maps. Then, we present some existence results and investigate the compactness of solution sets, some regularity of operator solutions and absolute retract (in short AR). The continuous dependence of solutions on parameters in the convex case is also examined. Applications to a problem from control theory are provided.