and Applied Analysis 3 In the last section of this paper,we study the system 1.2 . Again, we introduce a notion of solution-tube of 1.2 which generalizes the notion of lower and upper solutions used by Atici et al. 8 . This notion generalizes also condition 1.5 and the notion of solution-tube of systems of second-order differential equations introduced in 10 . In addition, we assume that f sati...

Let σ(n) = ∑ d|nd be the usual sum-of-divisors function. In 1933, Davenport showed that n/σ(n) possesses a continuous distribution function. In other words, the limit D(u) := limx→∞(1/x) ∑ n≤x,n/σ(n)≤u1 exists for all u ∈ [0, 1] and varies continuously with u. We study the behaviour of the sums ∑n≤x,n/σ(n)≤u f (n) for certain complex-valued multiplicative functions f . Our results cover many of...

Let X be a separable Banach space, σ > 0 and Cσ := C([−σ, 0],X) the Banach space of the continuous functions from [−σ, 0] into X, K a locally closed set in X and F : [a, b)×Cσ→ 2 X a closed valued and locally integrable bounded multifunction, withF (., φ) measurable and F (t, .) Lipschitz continuous in theHausdorff–Pompeiumetric. In this paper we establish some sufficient conditions in order th...

We continue from “part I” our address of the following situation. For a Tychonoff space Y, the “second epi-topology” σ is a certain topology on C(Y), which has arisen from the theory of categorical epimorphisms in a category of lattice-ordered groups. The topology σ is always Hausdorff, and σ interacts with the point-wise addition + on C(Y) as: inversion is a homeomorphism and + is separately c...

Abstract: Let X be a Banach space. We define the concept of a bi-parameter semigroup on X and its first and second generators. We also study bi-parameter semigroups on Banach algebras. A relation between uniformly continuous bi-parameter semigroups and σ-derivations is also established. It is proved that if {αt,s}t,s 0 is a uniformly continuous bi-parameter semigroup on a Banach algebra X , who...

Brief Summary of Basic Notions For a b R , ∈ , a b < , let ([ ] ) n C a b R , , be the Banach space of the continuous functions from the interval [ ] a b , into n R equipped with the supremum norm | ⋅ | . In the case 0 a r = − < , 0 b = , we shall denote this space as r C , that is, ([ 0] ) n r C C r R := − , , and put 0 sup ( ) r r σ φ φ σ − ≤ ≤ || || = | | for r C φ∈ . If R σ ∈ , 0 A ≥ and ([...

the aim of this paper is to introduce and study a new concept of strong double δ ( a ) σ -convergence sequences with respect to an orlicz function, and some properties of the resulting sequencespaces were also examined. in addition, we define the δ ( a ) σ -statistical convergence and establish someconnections between the spaces of strong double δ ( a ) σ -convergence sequences and the space of...

This work presents a general method to design continuous-time Σ∆ modulators with sine-shaped feedback DACs. A discrete time to continuous time transformation technique is used to compute the continuous-time loop filter coefficients, taking into account the sinusoidal feedback. Examples of high order lowpass and bandpass continuous-time Σ∆ modulators are given as an illustration for the proposed...

where the horizon T (the upper bound for τ and σ above) may be either finite or infinite (it is assumed that G1(XT ) = G2(XT ) if T is finite and lim inft→∞G2(Xt) ≤ lim supt→∞G1(Xt) if T is infinite). If X is right-continuous, then the Stackelberg equilibrium holds, in the sense that V ∗(x) = V∗(x) for all x with V := V ∗ = V∗ defining a measurable function. If X is right-continuous and left-co...

We show that a dissipative, ergodic measure preserving transformation of a σ-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these. §0 Introduction Let (X,B,m, T ) be an invertible, ergodic measure preserving transformation of a σ-finite measure space, then there are no other σ-finite, m-abso...