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In this paper, a stochastic avian influenza model with two different pathogenic human–avian viruses is studied. The model analyzes the spread of the avian influenza virus from poultry populations to human populations in a random environment. The dynamic behavior of the stochastic avian influenza model is analyzed. Firstly, the existence and uniqueness of a global positive solution are obtained. Secondly, under the condition of high pathogenic virus extinction, the persistence in the mean and extinction of the infected avian population with a low pathogenic virus is analyzed. Thirdly, the sufficient conditions for the existence and uniqueness of the ergodic stationary distribution in the stochastic avian influenza model are derived. We find the threshold of the stochastic model to determine whether the disease spreads when the white noise is small. The analysis results show that random white noise is effective for disease control. Finally, the theoretical results are verified by numerical simulation, and the numerical simulation analysis is carried out for the cases that cannot be theoretically deduced.

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 Ito^ 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.

This paper studies a class of integral boundary value problem of fractional q-difference equations. We first give an explicit expression for the associated Green’s function and obtain an important property of the function. The new property allows us to prove sufficient conditions for the existence of positive solutions based on the associated parameter. The results are derived from the application of a fixed point theorem on order intervals.

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 k1 consecutive external shocks, the size of at least d1 or k2 consecutive external shocks the size of at least d2 occur, where d1k2. 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.

In this paper, the 3IM+1CM theorem with a general difference polynomial Lz,f will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of Lz,f is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of Lz,f is equal to zero is also a good extension for recent results. However, it is new for the case that the sum of the coefficient of Lz,f is not equal to zero. In fact, the main difficulty of proof is also from this case, which causes the traditional theorem invalid. On the other hand, it is more interesting that the nonconstant finite-order meromorphic function f can be exactly expressed for the case f≡−Lz,f. Furthermore, the sharpness of our conditions and the existence of the main result are illustrated by examples. In particular, the main result is also valid for the discrete analytic functions.

Abstract In this paper,we are interested in the existence and uniqueness of positive solutions for integral boundary value problem with fractional q-derivative: Dqαu(t)+f(t,u(t),u(t))+g(t,u(t))=0,0

Abstract This paper is concerned with the two-point boundary value problems of a nonlinear fractional q-difference equation with dependence on the first order q-derivative. We discuss some new properties of the Green function by using q-difference calculus. Furthermore, by means of Schauder’s fixed point theorem and an extension of Krasnoselskii’s fixed point theorem in a cone, the existence of one positive solution and of at least one positive solution for the boundary value problem is established.

We study a two-point Boundary Value Problem depending on two parameters that represents a mathematical model arising from the combustion theory. Applying fixed point theorems for concave operators, we prove uniqueness, existence, upper, and lower bounds of positive solutions. In addition, we give an estimation for the value of λ* such that, for the parameter λ∈[λ*,λ*], there exist exactly three positive solutions. Numerical examples are presented to illustrate various cases. The results complement previous work on this problem.

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.

We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on fixed point index based on parameters that are related to the definitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator.

A class of boundary value problems of Caputo fractional q-difference equation is introduced. Green’s function and its properties for this problem are deduced. By applying these properties and the Leggett-Williams fixed-point theorem, existence criteria of three positive solutions are obtained. At last, some examples are given to illustrate the validity of our main results.

We investigate the traveling wave solutions of a competitive integrodifference system without comparison principle. In the earlier conclusions, a threshold of wave speed is defined while the existence or nonexistence of traveling wave solutions remains open when the wave speed is the threshold. By constructing generalized upper and lower solutions, we confirm the existence of traveling wave solutions when the wave speed is the threshold. Our conclusion completes the known results and shows the different decay behavior of traveling wave solutions compared with the case of large wave speeds.

This paper is concerned with the two-point boundary value problems of nonlinear finite discrete fractional differential equations. On one hand, we discuss some new properties of the Green function. On the other hand, by using the main properties of Green function and the Krasnoselskii fixed point theorem on cones, some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established.

Based on the fixed-point index theory for a Banach space, positive periodic solutions are found for a system of delay difference equations. By using such results, the existence of nontrivial periodic solutions for delay difference equations with positive and negative terms is also considered.

We consider the existence of positive solutions to the nonlinear fractional differential equation boundary value problem D0+αCut+fut,u't=0, t∈0,1, u0=u1=u″0=0, where f:0,+∞×R→0,+∞ is continuous, α∈2,3, and D0+αC is the standard Caputo differentiation. By using fixed point theorems on cone, we give some existence results concerning positive solutions. Here the solutions especially are the interior points of cone.

We investigate the existence of solutions for a Caputo fractional difference equation boundary value problem. We use Schauder fixed point theorem to deduce the existence of solutions. The proofs are based upon the theory of discrete fractional calculus. We also provide some examples to illustrate our main results.

We derive several sufficient conditions for monotonicity of eventually positive solutions on a class of second order perturbed nonlinear difference equation. Furthermore, we obtain a few nonexistence criteria for eventually positive monotone solutions of this equation. Examples are provided to illustrate our main results.

We will consider the existence of multiple positive periodic solutions for a class of abstract difference equations by using the well-known fixed point theorem (due to Krasnoselskii).

By using fixed point theorems in cones, the existence of multiple positive solutions is considered for nonlinear m-point boundary value problem for the following second-order boundary value problem on time scales (ϕ(uΔ))∇+a(t)f(t,u(t))=0, t∈(0,T), ϕ(uΔ(0))=∑i=1m−2aiϕ(uΔ(ξi)), u(T)=∑i=1m−2biu(ξi), where ϕ:R→R is an increasing homeomorphism and homomorphism and ϕ(0)=0. Some new results are obtained for the existence of twin or an arbitrary odd number of positive solutions of the above problem by applying Avery-Henderson and Leggett-Williams fixed point theorems, respectively. In particular, our criteria generalize and improve some known results by Ma and Castaneda (2001). We must point out for readers that there is only the p-Laplacian case for increasing homeomorphism and homomorphism. As an application, one example to demonstrate our results is given.

This paper deals with the existence and uniqueness of periodic solutions for the first-order functional differential equation 𝑦(𝑡)=−𝑎(𝑡)𝑦(𝑡)+𝑓1(𝑡,𝑦(𝑡−𝜏(𝑡)))+𝑓2(𝑡,𝑦(𝑡−𝜏(𝑡))) with periodic coefficients and delays. We choose the mixed monotone operator theory to approach our problem because such methods, besides providing the usual existence results, may also sometimes provide uniqueness as well as additional numerical schemes for the computation of solutions.

This paper deals with the second-order nonlinear systems of difference equations, we obtain the existence theorems of periodic solutions. The theorems are proved by using critical point theory.