Theory Of Differential Inequalities Research Articles

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.

Singularly perturbed solution for nonlinear equations of fourth order with two parameters

A class of singularly perturbed boundary value problem for nonlinear equation of fourth order with two parameters is considered. Under suitable conditions, the existence and asymptotic behavior of solution for boundary value problem are studied using theory of differential inequalities.

The solvability for a class of singularly perturbed quasi-linear differential system

The solvability for a class of singularly perturbed Robin problem of quasi-linear differential system is considered. Using the boundary layer corrective method the formal asymptotic solution is constructed. And using the theory of differential inequality the uniform validity of the asymptotic expansions for solution is proved.

Asymptotic Solution of Activator Inhibitor Systems for Nonlinear Reaction Diffusion Equations

A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.

Singularly perturbed solution for third order nonlinear equations with two parameters

A class of singularly perturbed boundary value problems for nonlinear equation of the third order with two parameters is considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of the solution for boundary value problem are studied.

Singularly perturbed solution for weakly nonlinear equations with two parameters

A class of singularly perturbed boundary value problems of weakly nonlinear equation for fourth order on the interval [a, b] with two parameters is considered. Under suitable conditions, firstly, the reduced solution and formal outer solution are constructed using the expansion method of power series. Secondly, using the transformation of stretched variable, the first boundary layer corrective term near x = a is constructed which possesses exponential attenuation behavior. Then, using the stronger transformation of stretched variable, the second boundary layer corrective term near x = a is constructed, which also possesses exponential attenuation behavior. The thickness of second boundary layer is smaller than the first one and forms a cover layer near x = a. Finally, using the theory of differential inequalities, the existence, uniform validity in the whole interval [a, b] and asymptotic behavior of solution for the original boundary value problem are proved. Satisfying results are obtained.

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation

A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.

Existence of solutions of boundary value problem for 3rd order nonlinear system

By appropriate bounding function pair and modified functions, using the theory of differential inequality and the theory of topological degree, this paper presents the existence criterions of solutions for the system of 3rd order differential equations under nonlinear boundary conditions, and extends many known results.

The nonlinear boundary value problem for a class of integro-differential system

In this paper, using the theory of differential inequalities, we study the nonlinear boundary value problem for a class of integro-differential system. Under appropriate assumptions, the existence of solution is proved and the uniformly valid asymptotic expansions for arbitrary n-th order approximation and the estimation of remainder term are obtained simply and conveniently.

The dynamic behavior of N-species cooperation system with continuous time delays and feedback controls

In this paper, we consider a N-species cooperation system with continuous time delays and feedback controls. For general nonautonomous case, by using differential inequality theory and constructing a suitable Lyapunov functional, we obtain sufficient conditions which guarantee the permanence and the global attractivity of the system. For the periodic case, we obtain sufficient conditions which guarantee the existence, uniqueness and stability of a positive periodic solution. For the almost periodic case, by using the Razumikhin type theorem, we obtain the sufficient conditions which guarantee the existence of a positive almost periodic solution of the system.

Global attractivity in an almost periodic multi-species nonlinear ecological model

A nonlinear almost periodic predator–prey model with n-preys and m-predators is studied in this paper, which can be seen as the modification of the traditional multi-species Lotka–Volterra predator–prey model. For general nonautonomous case, by using the differential inequality theory, we obtain the sufficient conditions which guarantee the uniform persistence and nonpersistence of the system; After that, by constructing a suitable Lyapunov function, some sufficient conditions are obtained which ensure the global attractivity of the system. For almost periodic case, by constructing a suitable Lyapunov function, sufficient conditions which guarantee the existence of an unique globally attractive positive almost periodic solution of the system are obtained. Examples together with their numeric simulations show the feasibility of our main results.

Differential inequalities method to nth-order boundary value problems

By the theory of differential inequality, bounding function method, and the theory of topological degree, this paper presents the existence criterions of solutions for the general th-order differential equations under nonlinear boundary conditions, and extends many existing results.

A Class of Nonlinear Singularly Perturbed Problems for Reaction Diffusion Equations with Boundary Perturbation

A class of nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation are considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems is studied using the theory of differential inequalities.

A class of nonlinear nonlocal singularly perturbed problems for reaction diffusion equations with boundary perturbation

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, first, the outer solution of the original problem was obtained. Secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer was constructed. Finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems was studied, and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation were discussed.

Dynamics of fluids flows under Markovian structural perturbations

In this paper, we investigate dynamics of plane flows and three-dimensional flows of an incompressible viscous Newtonian fluids under Markovian structural perturbations. We employ the theory of random partial differential inequalities together with the concept of vector Lyapunov-like functionals to investigate different kinds of stability and convergence properties of the solution process of the model and characterize the effects of the random structural perturbations on convergence and stability.

The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations with a boundary perturbation

The nonlinear nonlocal singularly perturbed initial boundary value problems for reaction diffusion equations with a boundary perturbation is considered. Under suitable conditions, the outer solution of the original problem is obtained. Using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. And then using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied. Finally the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.

A class of singularly perturbed nonlinear boundary value problem

The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered. Under suitable conditions, the asymptotic behavior of solution for the original problems is studied by using theory of differential inequalities.

Asymptotic behavior of solution for a class of reaction diffusion equations

A class of initial boundary value problems for the reaction diffusion equations are considered. The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.

Nonlinear singularly perturbed predator-prey reaction diffusion systems

A class of nonlinear predator-prey reaction diffusion systems for singularly perturbed problems are considered. Under suitable conditions, by using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems are studied.

Asymptotic estimation for the system of electro-rheological fluids

By the movement model of ball particles in electrorheological(ER) fluids, the asymptotic behavior of the movement of the particles in ER fluids is considered when the mass of the particles is small. Using the theory of differential inequalities, the expression of asymptotic estimation is given and proved.