In this paper, we will use the Krasnosel’skii fixed point theorem to investigate a discrete fractional boundary value problem of the form −Δνy(t)=λh(t+ν−1)f(y(t+ν−1)), y(ν−2)=Ψ(y), y(ν+b)=Φ(y), where 1<ν⩽2, t∈[0,b]N0, f:[0,∞)→[0,∞) is a continuous function, h:[ν−1,ν+b−1]Nν−1→[0,∞), Ψ,Φ:Rb+3→R are given functionals, where Ψ, Φ are linear functionals, and λ is a positive parameter.

In this paper, we studied the following Caputo fractional difference boundary value problem (FBVP): △ C ν y ( t ) = − f ( t + ν − 1 , y ( t + ν − 1 ) ) , y ( ν − 3 ) = △ y ( b + ν ) = △ 2 y ( ν − 3 ) = 0 , where 2 < ν ⩽ 3 is a real number, △ C ν y ( t ) is the standard Caputo difference. By means of cone theoretic fixed point theorems, some results on the existence of one or more positive solutions for the above Caputo fractional boundary value problems are obtained.

In this paper, we investigate a wear and mixed shock model in which the system can fail due to internal aging or external shocks. The lifetime of the system, due to internal wear, follows continuous phase-type (PH) distributions. The external random shocks arrive at the system according to a PH renewal process. The system will fail when the internal failure occurs or $k_1$ consecutive external shocks, the size of at least $d_1$ or $k_2$ consecutive external shocks the size of at least $d_2$ occur, where $d_1<d_2$, $k_1>k_2$. The failed system can be repaired immediately, and the repair times of the system are governed by continuous PH distributions. The system can be replaced by a new and identical one based on a bivariate replacement policy $(L,N)$. The long-run average profit rate for the system is obtained by employing the closure property of the PH distribution. Finally, a numerical example is also given to determine the optimal replacement policy.

We consider the existence of at least two positive solutions for a system of Caputo fractional difference equations Δ C ν j y j ( t ) = − λ j f j ( y 1 ( t + ν 1 − 1 ) , … , y n ( t + ν n − 1 ) ) , subject to boundary conditions y j ( ν j − 3 ) = Δ y j ( ν j + b ) = Δ 2 y j ( ν j − 3 ) = 0 , where 2 < ν j ⩽ 3 , j = 1 , … , n . We use the Krasnosel’skiĭ fixed point theorem to obtain the sufficient conditions of the existence of two positive solutions for this boundary value problem of Caputo fractional difference equations depending on parameters.

In this paper, we study the dynamic behavior of a stochastic tungiasis model for public health education. First, the existence and uniqueness of global positive solution of stochastic models are proved. Secondly, by constructing Lyapunov function and using It formula, sufficient conditions for disease extinction and persistence in the stochastic model are proved. Thirdly, under the condition of disease persistence, the existence and uniqueness of an ergodic stationary distribution of the model is obtained. Finally, the importance of public health education in preventing the spread of tungiasis is illustrated through the combination of theoretical results and numerical simulation.

We use the Krasnosel’skii fixed point theorem to obtain the sufficient conditions of the existence of two positive solutions for the boundary value problem of fractional difference equations depending on parameters.