Volume 69 , issue 4 ( 2017 ) | back |

Remarks on partial $b$-metric spaces and fixed point theorems | 231$-$240 |

**Abstract**

In this paper, we prove some properties of a partial b-metric space in the sense of Shukla. As applications, we show that fixed point theorems on partial $b$-metric spaces can be implied from certain fixed point theorems on $b$-metric spaces. We also give examples to illustrate the results.

**Keywords:** Fixed point; $b$-metric; partial metric; partial $b$-metric.

**MSC:** 47H10, 54H25, 54D99, 54E99

Quasi-regularity of harmonic maps based on Blaschke products | 241$-$247 |

**Abstract**

The purpose of this paper is to find conditions which guarantee quasiregularity of a harmonic map of the unit disk $D $ of the form $f(z)={Re}B_1(z)+i{Im}B_2(z) $, where $B_1, B_2$ are automorphisms of $D$.

**Keywords:** Quasiconformal maps; harmonic maps; quasiregular maps.

**MSC:** 30C99

The metric dimension of comb product graph | 248$-$258 |

**Abstract**

A set of vertices $W$ \textit{resolves} a graph $G$ if every vertex is uniquely determined by its coordinate of distance to the vertices in $W$. The minimum cardinality of a resolving set of $G$ is called the \textit{metric dimension} of $G$. In this paper, we consider a graph which is obtained by the comb product between two connected graphs. Let $o$ be a vertex of $H$. The \textit{comb product} between $G$ and $H$, denoted by $G\rhd_o H$, is a graph obtained by taking one copy of $G$ and $|V(G)|$ copies of $H$ and identifying the $i$-th copy of $H$ at the vertex $o$ to the $i$-th vertex of $G$. We give an exact value of the metric dimension of $G\rhd_o H$ where $H$ is not a path or $H$ is a path where the vertex $o$ is not a leaf. We also give the sharp general bounds of $\beta(G\rhd_o P_n)$ for $n\geq 2$ where the vertex $o$ is a leaf of $P_n$.

**Keywords:** Basis; comb product; metric dimension; resolving set.

**MSC:** 05C12, 05C76

Linear Hamiltonian system in scale of Hilbert spaces | 259$-$270 |

**Abstract**

We consider an initial value problem for linear Hamiltonian system in the scale of Hilbert spaces and prove an existence and uniqueness theorem. We also prove a version of the Poincaré Recurrence Theorem.

**Keywords:** Linear PDE; infinite dimensional Hamiltonian system.

**MSC:** 35A01, 37K05, 35A02

Existence of three weak solutions for a quasilinear Dirichlet problem | 271$-$280 |

**Abstract**

The aim of this note is to establish the existence of three solutions for a two-point boundary value problem. The approach is based on variational methods. Some particular cases and two concrete examples are then presented.

**Keywords:** Dirichlet boundary condition; variational methods; critical points.

**MSC:** 34B15, 34B18, 58E05

Symmetric tilings in the square lattice | 281$-$295 |

**Abstract**

We apply the method of Gröbner bases to polyomino tilings, following and developing the ideas of Bodini and Nouvel. The emphasis is, in the spirit of the paper M. Muzika Dizdarević, R.~T. Živaljević, \emph{Symmetric polyomino tilings, tribones, ideals and Gröbner bases}, Publ. Inst. Math. \textbf{98} (112) (2015), 1--23., on tiling problems with added symmetry conditions. The main problem studied in the paper covers case of tiling by three-in-line polyominoes, centrally symmetric with respect to the origin.

**Keywords:** Symmetric Z-tiling; ring of invariants; Gröbner bases.

**MSC:** 52C20, 13P10

Existence of one weak solution for $p(x)$-biharmonic equations involving a concave-convex nonlinearity | 296$-$307 |

**Abstract**

In the present paper, using variational approach and the theory of the variable exponent Lebesgue spaces, the existence of nontrivial weak solutions to a fourth order elliptic equation involving a $p(x)$-biharmonic operator and a concave-convex nonlinearity the Navier boundary conditions is obtained.

**Keywords:** Critical points; $p(x)$-biharmonic operator; Navier boundary conditions; concave-convex nonlinearities; Mountain Pass Theorem;
Ekeland's variational principle.

**MSC:** 35J60, 35J48