Let B0(s, t) be a Brownian pillow with continuous sample paths, and let h, u : [0, 1] 2 → R be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability ψ(u;h) := P{B0(s, t) + h(s, t) ≤ u(s, t), ∀s, t ∈ [0, 1]}. Further we investigate the asymptotic behaviour of ψ(u; γh) with γ tending to ∞, and solve a related minimisation problem.

jurassic deposits are well exposed in the bazehowz area, south west of mashhad city, east alborz, iran. it contains plant macrofossilsbelonging to eighteen species of eleven genera of various orders such as equisetales, filicales, bennettitales, cycadales,corystospermales, caytoniales, ginkgoales and pinales. two biozones were recognized in the type section of bazehowz formation.biozone i is an...

Using the method of upper and lower solutions, we prove that the singular boundary value problem, −u = f(u) u in (0, 1), u(0) = 0 = u(1) , has a positive solution when 0 < α < 1 and f : R → R is an appropriate nonlinearity that is bounded below; in particular, we allow f to satisfy the semipositone condition f(0) < 0. The main difficulty of this approach is obtaining a positive subsolution, whi...

We will always consider a Carathéodory set-valued map F : R×RN RN with nonempty, compact, and convex values. We recall that F is said to be a Carathéodory multifunction if F(·,x) is measurable for each x ∈ RN , and F(t,·) is upper semicontinuous for almost all (a.a.) t ∈ R. For the definitions of these standard notions, see, for example, [30]. Our main result is Theorem 4.2. It states the exist...

We study the existence of solutions anH-system for a revolution surface without boundary for H depending on the radius f . Under suitable conditions we prove that the existence of a solution is equivalent to the solvability of a scalar equation N(a) = L/√2, where N : ⊂ R+ → R is a function depending on H . Moreover, using the method of upper and lower solutions we prove existence results for so...

In this paper we show the existence of solutions to a nonlinear singular second order ordinary differential equation, u(t) = a t u(t) + λf(t, u(t), u(t)), subject to periodic boundary conditions, where a > 0 is a given constant, λ > 0 is a parameter, and the nonlinearity f(t, x, y) satisfies the local Carathéodory conditions on [0, T ] × R × R. Here, we study the case that a well-ordered pair o...

The purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of Sturm-Liouville problem with equation of the form w??=(?2f(x)?R(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in C2(a,b which has a single zero (so called turning point) at point 0x=x and R(x) is a continuously differentiable function. ...

This paper is motivated by the maximization of k-th eigenvalue Laplace operator with Neumann boundary conditions among domains $${{\mathbb {R}}}^N$$ prescribed measure. We relax problem to class (possibly degenerate) densities in $$\mathbb {R}^N$$ mass and prove existence an optimal density. For $$k=1,2$$ , two problems are equivalent maximizers known be one equal balls, respectively. $$k \ge 3...

Let X be a real separable Banach space. The boundary value problem x′ ∈ A(t)x + F (t, x), t ∈ R+, Ux = a, (B) is studied on the infinite interval R+ = [0,∞). Here, the closed and densely defined linear operator A(t) : X ⊃ D(A)→ X, t ∈ R+, generates an evolution operator W (t, s). The function F : R+×X → 2X is measurable in its first variable, upper semicontinuous in its second and has weakly co...