where T is a time scale unbounded above, the variable delays k, : [t0,∞)T → T are nondecreasing with k(t), (t) < t for all t ∈ [t0,∞)T such that limt→∞ k(t), (t)=∞. The coefficient functions p,q : T→R are right-dense continuous with p bounded and q ≥ 0. To clarify some notation, take −1(t) := sup{s : (s) ≤ t}, −(n+1)(t)= −1( −n(t)) for t ∈ [ (t0),∞)T, and n+1(t) = ( n(t)) for t ∈ [ −3(t0),∞)T. ...

It is known that the solution to a Cauchy problem of linear differential equations: x′(t) = A(t)x(t), with x(t0) = x0, can be presented by the matrix exponential as exp( ∫ t t0 A(s) ds)x0, if the commutativity condition for the coefficient matrix A(t) holds: [ ∫ t t0 A(s) ds, A(t) ] = 0. A natural question is whether this is true without the commutativity condition. To give a definite answer to...

Let L(t) represent the equation x1 + x2 + · · · + xt−1 = xt. For k > 1, 0 6 i 6 k − 1, and ti > 3, the generalized Schur number S(k; t0, t1, . . . , tk−1) is the least positive integer m such that for every k-colouring of {1, 2, . . . ,m}, there exists an i ∈ {0, 1, . . . , k − 1} such that there exists a solution to L(ti) that is monochromatic in colour i. In this paper, we report twenty-six p...

We obtain sufficient conditions for oscillation of all solutions of the neutral impulsive difference equation with continuous variable Δτ(y(t) +P(t)y(t−mτ)) +Q(t)y(t− lτ)= 0, t ≥ t0− τ, t = tk, y(tk + τ)− y(tk)= bk y(tk), k ∈N(1), where Δτ denotes the forward difference operator, that is, Δτz(t)= z(t+ τ)− z(t), P(t)∈ C([t0− τ,∞),R ),Q(t)∈ C([t0− τ,∞),(0,∞)),m, l are positive integers, τ > 0 and...

Proof (of Lemma 2). Assume that S `C t t0. By definition we get that S `C t. If (CANCEL) was used to derive S `C t t0 then t has the form t1 :t2 :t3 and t0 has the form t1 :t3, and furthermore we know that S t1 ! S0 t2 ! S0. It follows immediately that S `C t1:t3, and hence that S `C t0 and finalC(S, t) = finalC(S, t 0), as required. On the other hand, if (SWAP) was used to derive S `C t t0 the...

Let X be a scalar diffusion process with drift coefficient pointing towards the origin, i.e. X is mean-reverting. We denote by X∗ the corresponding running maximum, T0 the first time X hits the level zero. Given an increasing and convex loss function , we consider the following optimal stopping problem: inf0≤θ≤T0 E[ (X ∗ T0 −Xθ)], over all stopping times θ with values in [0, T0]. For the quadra...

This paper intends to investigate the existence of uncountably many bounded positive solutions of a third order nonlinear neutral delay differential equation d dt { r1(t) d dt [ r2(t) d dt ( x(t)− f(t, x(t− σ)) )]} + d dt [ r1(t) d dt g(t, x(p(t))) ] + d dt h(t, x(q(t))) = l(t, x(η(t))), t ≥ t0 in the following bounded closed and convex set Ω(a, b) = { x(t) ∈ C([t0,+∞),R) : a(t) ≤ x(t) ≤ b(t), ...

We study the Hilbert manifold structure on T0(1) — the connected component of the identity of the Hilbert manifold T (1). We characterize points on T0(1) in terms of Bers and pre-Bers embeddings, and prove that the Grunsky operators B1 and B4, associated with the points in T0(1) via conformal welding, are Hilbert-Schmidt. We define a “universal Liouville action” — a real-valued function S1 on T...