Volume 45 , issue 1$-$4 ( 1993 ) | back |

On the solutions of the functional equation $x(t)+A(t)x(f(t))=F(t)$ when the function $F$ satisfies some special conditions | 1$-$5 |

**Abstract**

The results of this paper are concerned with the soluiton $x(t)$ of the functional equation $x(t)+A(t)x(f(t))=F(t)$. Using regular summability methods $T$, we derive some necessary and also some sufficient conditions for the $T$-sum $x(t)$ of the series $\sum_{i=0}^{\bb}(- 1)^iF(f^i(t))$ to be a solution of the above mentioned equation under the specific conditions for $F(t)$.

**Keywords:** Functional eaution

**MSC:** 39B22

On the expansion theorem for a certain boundary value problem for a functional differential equation | 11$-$17 |

**Abstract**

The boundary value problem $$ -y''+q(x)y=\lambda y+ınt_0^{\pi}y\,d\sigma(x),\quad y(0)=y(\pi)=0, $$ is concerned, where $qın C[0,\pi]$ and $\sigma$ is a function of bounded variation. It is proved that the system of eigenfunctions of the given problem is complete and minimal in $L^2(0,\pi)$, and also that functions of a certain class can be expanded into uniformly convergent series with respect to the mentioned system.

**Keywords:** Functional differential equation, boundary value problem, expansion
theorem.

**MSC:** 34K10; 47E05

Divisible linearly ordered topological spaces | 19$-$21 |

**Abstract**

We prove that a ccc linearly ordered topological space is metrizable if and only if it is divisible.

**Keywords:** Linearly ordered topological spaces

**MSC:** 54F05

Automatic modification of smalltalk methods | 23$-$27 |

**Abstract**

This article focuses on the issue of making a kind of preprocessor (called Automatic Modifier of Smalltalk Methods---AMSM) for the programming language Smalltalk. A general algorithm, based on the automatic modification of Smalltalk methods, is proposed. All modifications of related methods are realized in Smalltalk itself. The main parts of method-code and relevant classes are presented. Also, some characteristics (and possible) applications of AMSM are exposed.

**Keywords:** Smalltalk, automatic modification, method, environment

**MSC:** 68N15

An inequality related to the uniform convexity in Banach spaces | 29$-$33 |

**Abstract**

We prove an inequality that implies a 2-concave and $p$-convex Banach lattice is ``more'' uniformly convex than $L^p$.

**Keywords:** Banach space, uniform convexity

**MSC:** 46B20; 46B42

A unified theory of perfect and related functions | 35$-$41 |

**Abstract**

A unified theory has been developed on the basis of the similarity in properties of perfect and allied types of functions. The theory intromites as a starting point a certain subset of $\Cal P(X)$, the power set of a nonvoid set $X$, and an operator $\alpha$ on $\Cal P(X)$; a second operator $\beta$ is also brought into action. This theory of $\beta$-perfect functions includes the theories of perfect, $\theta$-perfect and $\delta$-perfect functions and is seen to generate many new types of functions when different pairs of operators take the roles of the pair $(\alpha,\beta)$.

**Keywords:** $\beta$-perfect function, operator $\alpha$, $\beta$-set,
$\gamma$-set, $\beta$-closed function, $\beta$-convergence and
$\beta$-adherence of filter-bases.

**MSC:** 54C10

Sur le $g$-angle dans un espace normé | 43$-$48 |

**Abstract**

En utilisant la fonctionelle $g$ définie par (1), on définit ainsi dit le $g$-angle dans un espace normé réel. On démontre que cet angle remplit les conditions de Menger (Définition~1) dans un espace strictement convexe. Le $g$-angle et l'angle de Wilson sont égaux seulement dans un espace préhilbertien. L'orthogonalité laquelle est engendrée par le $g$-angle et ainsi disant la $g$-orthogonalité ([3]) sont égaux seulement dans un espace préhilbertien. On donne encore quelques caractérisation nouvelles d'espace préhilbertien.

**Keywords:** $g$-angle

**MSC:** 46B20; 46C50

A note on a support of a linear mapping | 49$-$53 |

**Abstract**

In this note a notion of the support of a linear mapping from $C_b(T)$ into a locally convex space is introduced. Some of the properties are established.

**Keywords:** Locally convex space, support of a linear mapping.

**MSC:** 46A03; 46E10

A counterexample for one variant of McIntosh closed graph theorem | 55$-$56 |

**Keywords:** Locally convex spaces, closed graph theorem

**MSC:** 46A30

A note on generalized recurrent Riemannian manifold | 57$-$60 |

**Abstract**

Generalized recurrent Riemannian manifold is a Riemannian manifold whose curvature tensor satisfies the condition (1). In this paper we prove: If the associated $1$-form satisfies the condition (3), where $\gamma\ne1\ne2$, or, in the case $\gamam\ne\text{const}$, $\gamma_sA^s\ne0$, generalized recurrent Riemannian manifold reduces to a recurrent one.

**Keywords:** Generalized reccurent Riemannian manifold

**MSC:** 53C25

Generalized eigenvector expansion for weakly perturbated discrete operators | 61$-$64 |

**Abstract**

In this paper we consider the expansion theorem in generalized eigenvectors of the operator $A=L+T$, where $L$ is a discrete, positive selfadjoint operator in a separable Hilbert space, and $T$ is a closed operator which is subordinated to $L$ in a certain sense.

**Keywords:** Eigenvector expansion, discrete operator, weak perturbations.

**MSC:** 47A70; 47A55

A note on $\alpha$-equivalent topologies | 65$-$69 |

**Abstract**

This paper responds to the question of when are two topologies $\alpha$-equivalent by using some recently introduced classes of sets as well as the classes of regular open sets, nowhere dense sets and dense sets.

**Keywords:** $\alpha$-equivalen topologies

**MSC:** 54A05

Vojin Dajović, 1914--1993 | 71$-$74 |