i 1 α i < 1, and f ∈ C 1 R\{0}, R ∩ C R, R satisfies f s s > 0 for s / 0, and f 0 ∞, where f 0 lim |s| → 0 f s /s. We investigate the global structure of nodal solutions by using the Rabinowitz's global bifurcation theorem.

In this paper, we consider the nonlinear eigenvalue problems u + λh(t)f (u) = 0, 0 < t < 1, u(0) = 0, u(1) = m−2 X i=1 α i u(η i), where m ≥ 3, η i ∈ (0, 1) and α i > 0 for i = 1,. .. , m−2, with P m−2 i=1 α i η i < 1; h ∈ C([0, 1], [0, ∞)) and h(t) ≥ 0 for t ∈ [0, 1] and h(t 0) > 0 for t 0 ∈ [0, 1]; f ∈ C([0, ∞), [0, ∞)) and f (s) > 0 for s > 0, and f 0 = ∞, where f 0 = lim s→0 + f (s)/s. We investigate the global structure of positive solutions by using the nonlinear Krein-Rutman Theorem.

This work presents a new model of the fractional Black-Scholes equation by using the right fractional derivatives to model the terminal value problem. Through nondimensionalization and variable replacements, we convert the terminal value problem into an initial value problem for a fractional convection diffusion equation. Then the problem is solved by using the Fourier-Laplace transform. The fundamental solutions of the derived initial value problem are given and simulated and display a slow anomalous diffusion in the fractional case. KEYWORDS asset pricing models, Black-Scholes equation, fractional derivative, initial value problem, mathematical finance, terminal value problem Math Meth Appl Sci. 2018;41 697-704.wileyonlinelibrary.com/journal/mma

In this article, we study the expansion of the first Melnikov function appearing by perturbing an integrable and reversible system with a homoclinic loop passing through a nilpotent singular point, and obtain formulas for computing the first coefficients of the expansion. Based on these coefficients, we obtain a lower bound for the maximal number of limit cycles near the homoclinic loop. Moreover, as an application of our main results, we consider a type of integrable and reversible polynomial systems, obtaining at least 3, 4, or 5 limit cycles respectively.