Abstract

In this paper, we consider the nonlinear fifth order differential equation $$x^{(v)}+ax^{(iv)}+b\dddot x+f(\ddot{x})+g(\dot{x})+h(x)=p(t; x, \dot{x},\ddot{x},\dddot x,x^{(iv)})$$ and we used the Lyapunov's second method to give sufficient criteria for the zero solution to be globally asymptotically stable as well as the uniform boundedness of all solutions with their derivatives.

Keywords: Boundedness; Lyapunov function; nonlinear fifth order differential equations; stability.

MSC: 34A34, 34D20, 34D23, 34D99

Abstract

Let $A(p)$ be the class of functions $f(z)=z^{p}+\sum_{k=p+1}^{\infty}a_{k}z^{k}$ $(p\in N=\{1,2,\dots\})$ which are analytic in the unit disc $U=\{z:|z|<1\}$. The object of the present paper is to give some properties of Noor integral operator $I_{n+p-1}f(z)$ of $(n+p-1)$-th order, where $I_{n+p-1}f(z)=\left[ \frac{z^{p}}{(1-z)^{n+p}}\right]^{(-1)}*f(z)$ $(n>-p$, $f(z)\in A(p))$ and $*$ denotes convolution (Hadamard product).

Keywords: Analytic function; Noor integral operator; convolution.

MSC: 30C45

Abstract

Let $K$ be either a CW or a metric simplicial complex. We find sufficient conditions for the subspace inequality $$A\subset X, \quad K\in \text{\rm AE}(X)\Rightarrow K\in \text{\rm AE}(A).$$ For the Lebesgue dimension ($K=S^n$) our result is a slight generalization of Engelking's theorem for a strongly hereditarily normal space $X$. As a corollary we get the inequality $$A\subset X\Rightarrow\dim_GA\leq\dim_GB.$$ for a certain class of paracompact spaces $X$ and an arbitrary abelian group $G$. As for the addition theorems $$\gather K\in \text{\rm AE}(A), \;\; L\in\text{\rm AE}(B)\Rightarrow K\ast L\in\text{\rm AE}(A\cup B),\\ \dim_G(A\cup B)\leq\dim_GA+\dim_GB+1, \endgather$$ we extend Dydak's theorems for metrizable spaces ($G$ is a ring with unity) to some classes of paracompact spaces.

Keywords: Dimension; cohomological dimension; absolute extensor; CW-complex; metric simplicial complex; subspace theorem; addition theorem.

MSC: 55M11, 54F45

Abstract

Article dedicated to the eightieth anniversary of professor Dušan Adnadjević, former Editor-in-chief of Matematički Vesnik.