In this paper we prove existence and uniqueness of classical solutions for the non-autonomous inhomogeneous Cauchy problem d dt u(t) = A(t)u(t) + f(t), 0 ≤ s ≤ t ≤ T, L(t)u(t) = Φ(t)u(t) + g(t), 0 ≤ s ≤ t ≤ T, u(s) = x. The solution to this problem is obtained by a variation of constants formula.

Estimates for the rank of AMNV − UA † MN and more general displacement of A † MN are presented,where AMN is the weighted pseudoinverse of a matrix A.The results are applied to the close-to-Toeplitz,close-to-Vandermonde and close-to-Cauchy matrices.We extend the results due to G. Heinig and F. Hellinger in 1994.

In this paper, we treat some fractional differential equations on the sequence Lebesgue spaces ℓp(N0) with p≥1. The Caputo calculus extends usual derivation. operator, associated to Cauchy problem, is defined by a convolution of compact support and belongs Banach algebra ℓ1(Z). We in detail these sequences. use techniques from algebras Functional Analysis explicity check solution problem.

This paper concerns the fourth order differential equation x′′′′ + ax′′′ + f(x′′) + g(x′) + h(x) = p(t). Using the Cauchy formula for the particular solution of non-homogeneous linear differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

This article concerns the fourth order differential equation x + ax′′′ + bx′′ + g(x′) + h(x) = p(t). Using the Cauchy formula for the particular solution of non-homogeneous linear differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

Motivated by Rosenthal’s famous $$l^1$$ -dichotomy in Banach spaces, Haydon’s theorem, and additionally recent works on tame dynamical systems, we introduce the class of locally convex spaces. This is a natural analogue Rosenthal spaces (for which any bounded sequence contains weak Cauchy subsequence). Our approach based bornology subsets turn closely related to eventual fragmentability. leads,...

We show that in the analytic category, given a Riemannian metric g on a hypersurface M ⊂ Z and a symmetric tensor W on M , the metric g can be locally extended to a Riemannian Einstein metric on Z with second fundamental form W , provided that g and W satisfy the constraints on M imposed by the contracted Codazzi equations. We use this fact to study the Cauchy problem for metrics with parallel ...

Let F (z) = z − H(z) with o(H(z)) ≥ 2 be a formal map from Cn to Cn and G(z) the formal inverse of F (z). In this paper, we give two recurrent formulas for the formal inverse G(z). The first formula not only provides an efficient method for the calculation of G(z), but also reduces the inversion problem to a Cauchy problem of a partial differential equation. The second one is differential free ...

This article aims to give an intuitive introduction to operator semi-groups and their generators from a probabilistic perspective. By ‘intuitive’ it is meant that the article relies mainly on heuristics and analogies to make its points. For example the definition of the generator is motivated from an analogy to the Cauchy functional equation g(t + s) = g(t)g(s) for real-valued functions. No pro...

Let V be a maximal globally hyperbolic flat n+1–dimensional space–time with compact Cauchy surface of hyperbolic type. We prove that V is globally foliated by constant mean curvature hypersurfaces Mτ , with mean curvature τ taking all values in (−∞, 0). For n ≥ 3, define the rescaled volume of Mτ by H = |τ | Vol(M, g), where g is the induced metric. Then H ≥ nVol(M, g0) where g0 is the hyperbol...