Abstract

In this paper, we study some common fixed point results in cone 2-metric spaces equipped with a ternary relation ${T}$. A weaker version of weakly compatible mappings, the notions of $g$-contractions with respect to ${T}$ and $g$-$\varphi$-contractions with respect to ${T}$ are introduced. Some common fixed point results for $g$-contractions and $g$-$\varphi$-contractions with respect to an arbitrary ternary relation ${T}$ and a transitive ternary relation respectively, are proved. To justify the newly introduced notions and results several examples are provided.

Keywords: Fixed point; cone 2-metric; $J$-weakly compatible pair; $\alpha$-$\varphi$-contractive mapping; ternary relation.

MSC: 47H10, 54H25

Abstract

In this paper, we study the existence and multiplicity of positive solutions for a nonlinear fourth-order ODE with multi-point boundary conditions and an integral boundary condition. The main tool is Krasnosel'skii fixed point theorem on cones.

Keywords: Positive solutions; Krasnoselskii's fixed point theorem; fourth-order integral boundary value problems; cone.

MSC: 34B15, 34B18

Abstract

In 1995, Paul Erd\H{o}s and András Gyárfás conjectured that for every graph $X$ of minimum degree at least $3$, there exists a non-negative integer $m$ such that $X$ contains a simple cycle of length $2^m$. In this paper, we prove that the conjecture holds for Cayley graphs of order $2p^2$ and $4p$.

Keywords: Erd\H{o}s-Gyárf\'s conjecture; Cayley graphs; cycles of graphs.

MSC: 05C38, 20B25

Abstract

In this presentation, we give necessary and sufficient conditions (lower and upper bounds) for the existence of linear codes capable of identifying the portion of the codeword which is corrupted by errors named as key errors. An example of such a code is provided. Comparisons among the number of parity check digits of linear codes detecting/locating/correcting key errors are provided. A result on minimum weight of key errors in Hamming sense is also included in the paper.

Keywords: Parity check matrix; syndrome; standard array; hamming weight; error locating codes.

MSC: 94B05, 94B65

Abstract

For any fixed integer $d\geq 2$, the $d$-ary increasing tree is a rooted, ordered, labeled tree where the out-degree is bounded by $d$, and the labels along each path beginning at the root increase. Total path length, or search cost, for a rooted tree is defined as the sum of all root-to-node distances and the Sackin index is defined as the sum of the depths of its leaves. We study these quantities in random $d$-ary increasing trees.

Keywords: $d$-ary increasing tree; total path length; Sackin index; covariance.

MSC: 05C05, 60F05

Abstract

In this paper, by making use of Green's identity for double integrals, we establish some integral inequalities for Schur convex functions defined on domains $D\subset R^{2}$ that are symmetric, convex and have nonempty interiors. Examples for squares and disks are also provided.

Keywords: Schur convex functions; double integral inequalities.

MSC: 26D15