Existence of the mild solution for some fractional differential equations with nonlocal conditions

We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions: D ^q x(t)=Ax(t)+t ⁿ f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ⁺, x(0)+g(x)=x ₀, where 0<q<1, A is the infinitesimal generator of a C ₀-semigroup of bounded linear operators on a Banach space X.     

Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder's fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka-Volterra systems.     

Existence results for a class of semilinear nonlocal evolution equations

In this paper, we study a class of semilinear evolution equations with nonlocal initial conditions and give some new results on the existence of mild solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Persistence of traveling wave solutions in a bio-reactor model with nonlocal delays

By considering the random migration of individuals and the period of consuming the captured nutrient, we first introduce the nonlocal delays into a bio-reactor model. The persistence of nontrivial traveling wave solutions then is proved by combining the geometric singular perturbation theory with the center manifold theorem. From the viewpoint of biology, our results indicate that the nonlocality induced by small average delays is harmless to the growth of the species.     

Existence theorems for a class of nonlocal differential equations

A class of nonlocal second-order ordinary differential equations of the form

y^″(x)=f(x,y(x),(y○λ)(x),y^′(x))     

Traveling waves of a diffusive predator-prey model with nonlocal delay and stage structure

In this paper, a diffusive predator-prey model with nonlocal delay and stage structure is investigated. By using the cross iteration method and Schauder's fixed point theorem, we reduce the existence of traveling wave solutions to the existence of a pair of upper-lower solutions. Numerical simulations are carried out to illustrate the theoretical results.     

Existence results and the moment estimate for nonlocal stochastic differential equations with time-varying delay

This paper considers a class of nonlocal stochastic differential equations with time-varying delay whose coefficients are dependent on the pth moment. By applying the fixed point theorem, the existence and uniqueness of the solution of nonlocal stochastic differential delay equations is studied. Also, a class of moment estimates of solutions is considered. The results are a generalization and continuation of the recent results on this issue. An example is provided to illustrate the effectiveness of our results.     

Periodic solutions in an array of coupled FitzHugh–Nagumo cells

We analyse the dynamics of an array of N²$N^2$ identical cells coupled in the shape of a torus. Each cell is a 2-dimensional ordinary differential equation of FitzHugh–Nagumo type and the total system is _ZN×_ZN${\mathbb{Z}}_N\times {\mathbb{Z}}_N$-symmetric. The possible patterns of oscillation, compatible with the symmetry, are described. The types of patterns that effectively arise through Hopf bifurcation are shown to depend on the signs of the coupling constants, under conditions ensuring that the equations have only one equilibrium state.     

Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal

This paper is concerned with a nonlocal evolution equation which is used to model the spatial dispersal of organisms. We study the existence, uniqueness and stability of the positive steady solution for this nonlocal evolution equation under general conditions. The global dynamics are also investigated and a trichotomy of the global asymptotics is established.     

Asymptotic stability of traveling wave fronts in nonlocal reaction–diffusion equations with delay

This paper is concerned with the asymptotic stability of traveling wave fronts of a class of nonlocal reaction–diffusion equations with delay. Under monostable assumption, we prove that the traveling wave front is exponentially stable by means of the (technical) weighted energy method, when the initial perturbation around the wave is suitable small in a weighted norm. The exponential convergent rate is also obtained. Finally, we apply our results to some population models and obtain some new results, which recover, complement and/or improve a number of existing ones.     

Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition

In this paper we investigates the blow-up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition, we obtain the blow-up condition and its blow-up rate estimate.     

On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In this paper, the problem of solving the one-dimensional wave equation subject to given initial and non-local boundary conditions is considered. These non-local conditions arise mainly when the data on the boundary cannot be measured directly. Several finite difference methods with low order have been proposed in other papers for the numerical solution of this one dimensional non-classic boundary value problem. Here, we derive a new family of efficient three-level algorithms with higher order to solve the wave equation and also use a Simpson formula with higher order to approximate the integral conditions. Additionally, the fourth-order formula is also adapted to nonlinear equations, in particular to the well-known nonlinear Klein–Gordon equations which many physical problems can be modeled with. Numerical results are presented and are compared with some existing methods showing the efficiency of the new algorithms.     

Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation

We consider the damped semilinear viscoelastic wave equation

Existence and global stability of traveling curved fronts in the Allen-Cahn equations

This paper is concerned with existence and stability of traveling curved fronts for the Allen-Cahn equation in the two-dimensional space. By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front.     

A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications

Considerable work has gone into studying the properties of nonlocal diffusion equations. The existence of a principal eigenvalue has been a significant portion of this work. While there are good results for the existence of a principal eigenvalue equations on a bounded domain, few results exist for unbounded domains. On bounded domains, the Krein–Rutman theorem on Banach spaces is a common tool for showing existence. This article shows that generalized Krein–Rutman can be used on unbounded domains and that the theory of positive operators can serve as a powerful tool in the analysis of nonlocal diffusion equations. In particular, a useful sufficient condition for the existence of a principal eigenvalue is given.     

Existence and uniqueness of entire solutions for a reaction-diffusion equation

We consider entire solutions of u_t=u_xx-f(u), i.e. solutions that exist for all (x,t)∈R², where f(0)=f(1)=0<f^′(0). In particular, we are interested in the entire solutions which behave as two opposite wave fronts of positive speed(s) approaching each other from both sides of the x-axis and then annihilating in a finite time. In the case f^′(1)>0, we show that such entire solution exists and is unique up to space-time translations. In the case f^′(1)<0, we derive two families of such entire solutions. In the first family, one cannot be any space-time translation of the other. Yet all entire solutions in the second family only differ by a space-time translation.