In this paper, we study the minimal wave speed of a competitive system. By constructing upper and lower solutions, we confirm the existence of travelling wave solution at the critical wave speed. This completes earlier results found in the literature. Our conclusion implies that the asymptotic decay behaviour of solutions at the critical wave speed is different from that of solutions at larger wave speeds.

We study a discrete boundary value problem involving the Riemann-Liouville fractional operator. In particular, we introduce a discrete fractional summing boundary value problem that parallels the integral boundary value problem for fractional differential equations. By generalizing a commonly used fractional difference operator, we derive the corresponding Green's function and reformulate the problem as a fixed-point equation. The existence and multiplicity of positive solutions are established. Two illustrative examples are provided to demonstrate the applicability of the theoretical results, and numerical simulations are included for further validation.

In this paper, we introduce the application of three fixed point theorem by discussing the existence of three positive solutions for a class of Caputo fractional difference equation boundary value problem. We establish the condition of the existence of three positive solutions for this problem.