On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

On chagnes of signs in infinite series

Доказана теорема. Пус ть {а_n} — последовате льность положительн ых чисел, монотонно стре мящаяся к нулю, причем Σa _n =∞. Тогда найд ется такая последовательность знаковε(п)=±1 (п=1,2, ...), что для каждого натураль ного т и любого целого b с условием 0≦b≦m?1 выполнено равен ство $$\mathop \sum \limits_{n \equiv b(mod m)} \varepsilon (n)a_n = 0$$ . Приведены также неко торые родственные ре зультаты и указаны нерешенные в опросы.     

On absolute summability factors of infinite series

In this paper using δ-quasi-monotone sequences a theorem on summability factors of infinite series, which generalizes a theorem of Bor [4] on summability factors of infinite series, is proved. Also, in the special case this theorem includes a result of Mazhar [8] on |C, 1|_k summability factors.     

On the existence of optimal controls for nonlinear infinite dimensional systems

We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.     

On the existence of universal series by trigonometric system

In this paper we prove the following: let ω(t) be a continuous function, increasing in [0,∞) and ω(+0)=0. Then there exists a series of the form

     

On general summability factor theorem of infinite series

A lower triangular matrix with nonzero principal diagonal entries is called a triangle. In this paper we obtain the sufficient conditions for ∑a_nλ_n to be summable ∣A∣_k whenever ∑a_n is summable ∣T∣_k for a triangle T.     

On the absolute Cesàro summability factors of infinite series

The author has established that if {λ _n | is a convex sequence such that the series\(\sum {\frac{{\lambda _n }}{n}} \) is convergent and if Σa _n is bounded [R, logn, 1] with indexk, then\(\sum {a_n \lambda _n } \) is summable |C, 1|k fork>1. The casek=1 of the theorem is due to Pati [3].