We establish some bifurcation results for the boundary value problem −∆u = g(u) + λ|∇u| + μf(x, u) in Ω, u > 0 in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R , λ, μ ≥ 0, 0 < p ≤ 2, f is nondecreasing with respect to the second variable, and g is unbounded around the origin. The asymptotic behaviour of the solution around the bifurcation point is also established, provided g(u) behav...

We investigate the kinetic model of relativistic Vlasov–Maxwell–Chern–Simons system, which originates from gauge theory. This system can be seen as an electromagnetic fields (i.e., Maxwell–Chern–Simons fields) perturbation for classical Vlasov equation. By virtue a nondecreasing function and iteration method, uniqueness existence global solutions 1.5D case are obtained.

In a recent paper new upper and lower limits were given, in the context of the Schrödinger or Klein–Gordon equations, for the number N0 of S-wave bound states possessed by a monotonically nondecreasing central potential vanishing at infinity. In this paper these results are extended to the number N of bound states for the th partial wave, and results are also obtained for potentials that are no...

In this supplement we prove the additional technical lemmas stated in Section 7.1 which are used in the proofs of the main results. Lemma 4 The function H−1 defined in Section 2.1 is concave and nondecreasing. It is strictly increasing for all x where H−1(x) < 12 . Moreover for C ≥ 1 it satisfies H−1(Ct) ≤ C 2 3H−1(t). (60) The function K defined in Section 2.1 is also increasing and satisfies ...

We prove that if P (x) is a polynomial with nonnegative nondecreasing coefficients and n is a positive integer, then P (x + n) is unimodal. Applications and open problems are presented.

Let [a, b] be any bounded closed real interval. The class of all continuous, nondecreasing, associative functions M : [a, b] → [a, b] fulfilling the boundary conditions M(a, a) = a and M(b, b) = b is described.

Let m and r be positive integers. Define f(m, r) to be the least positive integer N such that for every coloring of the integers 1, . . . , N with r colors there exist monochromatic subsets B1 and B2 (not necessarily of the same color), each having m elements, such that (a) max(B1)−min(B1) ≤ max(B2)−min(B2), and (b) max(B1) < min(B2). We improve previous upper bounds to determine that f(m, 4) =...

For positive integers m and r define f (m, r) to be the minimum integer n such that for every coloring of {1,2 . . . . . n} with r colors, there exist two monochromatic subsets B 1, B2~{1, 2 . . . . . n} (but not necessarily of the same color) which satisfy: (i)IBll=lB21=m; (ii) The largest number in B 1 is smaller than the smallest number in B2; (iii) The diameter of the convex hull spanned by...