Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation

We study uniqueness of a solution for an inverse source problem arising in linear time-fractional diffusion equations with time dependent coefficients. New uniqueness results are formulated in Theorem 3.1. We also show optimality of the conditions under which uniqueness holds by explicitly constructing counterexamples, that is by constructing more than one solution in the case when the conditions for uniqueness are violated.     

A COMPETITIVE AND PREY-PREDATOR MODEL WITH STAGE-STRUCTURE

The uniform persistence is proved for a non-autonomous competitive and prey-predator model with ratio-dependent functional response and stage-structure. By constructing a Liapunov functional, we establish the conditions of existence and uniqueness for the positive periodic solution, which is globally asymptotically stable. We get a unique almost periodic solution for an almost periodic system as well under corresponding conditions .by means of the Razumikhin function method.     

Matrix Linearization of Functional-Differential Equations of Point Type and Existence and Uniqueness of Periodic Solutions

We study ω-periodic solutions of a functional-differential equation of point type that is ω-periodic in the independent variable. In terms of the right-hand side of the equation, we state easy-to-verify sufficient conditions for the existence and uniqueness of an ω-periodic solution and describe an iteration process for constructing the solution. In contrast to the previously considered scalar linearization, we use a more complicated matrix linearization, which permits extending the class of equations for which one can establish the existence and uniqueness of an ω-periodic solution.     

Periodic solutions of functional-differential equations of point type

We study periodic solutions of a functional-differential equation of point type. We state conditions for the existence and uniqueness of an ω-periodic solution of the original nonlinear functional-differential equation of point type. An iterative process for constructing such a solution is described, and its convergence rate is estimated.     

Inverse Problem for Quasi-Periodic Differential Pencils with Jump Conditions Inside the Interval

Non-self-adjoint second-order differential pencils on a finite interval with non-separated quasi-periodic boundary conditions and jump conditions are studied. We establish properties of spectral characteristics and investigate the inverse spectral problem of recovering the operator from its spectral data. For this inverse problem we prove the corresponding uniqueness theorem and provide an algorithm for constructing its solution.     

Nonlocal boundary-value problem for linear partial differential equations unsolved with respect to the higher time derivative

We study the well-posedness of the problem with general nonlocal boundary conditions in the time variable and conditions of periodicity in the space coordinates for partial differential equations unsolved with respect to the higher time derivative. We establish the conditions of existence and uniqueness of the solution of the considered problem. In the proof of existence of the solution, we use the method of divided differences. We also prove metric statements on the lower bounds of small denominators appearing in constructing the solution of the problem. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 370–381, March, 2007.     

Global exponential stability and periodicity of reaction–diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions

In this paper, global exponential stability and periodicity of a class of reaction–diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions are studied by constructing suitable Lyapunov functionals and utilizing some inequality techniques. We first prove global exponential convergence to 0 of the difference between any two solutions of the original neural networks, the existence and uniqueness of equilibrium is the direct results of this procedure. This approach is different from the usually used one where the existence, uniqueness of equilibrium and stability are proved in two separate steps. Secondly, we prove periodicity. Sufficient conditions ensuring the existence, uniqueness, and global exponential stability of the equilibrium and periodic solution are given. These conditions are easy to verify and our results play an important role in the design and application of globally exponentially stable neural circuits and periodic oscillatory neural circuits.     

Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems

In this paper we establish the existence and the uniqueness of positive solutions for Dirichlet boundary value problems of nonlinear elliptic equations with singularity. We obtain the existence and the uniqueness by using the mixed monotone method in the cone theory. Moreover, we give an iterative method of constructing the solution. The rate of convergence of the iterative sequence is analyzed.     

ASYMPTOTIC PROPERTY FOR A LOTKA-VOLTERRA COMPETITIVE SYSTEM WITH DELAYS AND DISPERSION

An n-species nonautonomous Lotka-Volterra competition and diffusion model with time delays is investigated. It is shown that the system is uniformly persistent under some appropriate conditions, and by using the skill of constructing an appropriate Lyapunov function, the new sufficient conditions are obtained for the global asymptotic stability and the uniqueness of the positive periodic solution.     

Large solutions for quasilinear elliptic equation with nonlinear gradient term

We study the existence, asymptotic behavior near the boundary and uniqueness of large solutions for a class of quasilinear elliptic equation with a nonlinear gradient term. By constructing the suitable blow-up upper and lower solutions, we obtain the existence and the asymptotic behavior of radial large solutions of the problem in balls and then derive the existence of solutions in a general domain by a comparison argument. By using a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of any nonnegative solution of it near the boundary. The uniqueness is shown by a standard argument.     

Asymptotics of the Solution to the Cauchy Problem for Linear Parabolic Equations of Second Order with Small Diffusion

This paper is devoted to constructing an asymptotics of the solution to the Cauchy problem for a linear parabolic equation of second order with variable coefficients containing a small parameter at the highest derivative. Sufficient conditions for the existence and uniqueness of the “multiplicative” asymptotic expansion of the global solution of the problem are given.     

Classical solution of the first mixed problem for second-order hyperbolic equation in curvilinear half-strip with variable coefficients

The classical solution of the first mixed problem for a second-order hyperbolic equation with variable coefficients in the case of two independent variables in a curvilinear half-strip is considered. The existence and uniqueness of the classical solution under specific smoothness and matching conditions for given functions are proved. A method is proposed for constructing the solution using the method of sequential approximations for a system of integral equations of the second kind.     

Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms

We give an example of an incompressible diffusion equation whose solution is nonunique. It is shown that this equation has an approximation solution as well as another solution that cannot be obtained by approximation. We give sufficient conditions for the uniqueness of a solution as well as for the uniqueness of an approximation solution.     

Existence and global attractivity of periodic solution for enterprise clusters based on ecology theory with impulse

In this paper, a competition and corporation impulsive model of two enterprises is investigated. Applying the continuation theorem of coincidence degree theory, we derive a set of sufficient conditions that guarantee the existence of at least a positive periodic solution, and by constructing a suitable Lyapunov functional, the uniqueness and global attractivity of the positive periodic solution are included.     

Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks

We formulate discrete-time analogues of integrodifferential equations modelling bidirectional neural networks studied by Gopalsamy and He. The discrete-time analogues are considered to be numerical discretizations of the continuous-time networks and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the equilibria of the continuous-time networks. By constructing a Lyapunov-type sequence, we obtain easily verifiable sufficient conditions under which every solution of the discrete-time analogue converges exponentially to the unique equilibrium. The sufficient conditions are identical to those obtained by Gopalsamy and He for the uniqueness and global asymptotic stability of the equilibrium of the continuous-time network. By constructing discrete-time versions of Halanay-type inequalities, we obtain another set of easily verifiable sufficient conditions for the global exponential stability of the unique equilibrium of the discrete-time analogue. The latter sufficient conditions have not been obtained in the literature of continuous-time bidirectional neural networks. Several computer simulations are provided to illustrate the advantages of our discrete-time analogue in numerically simulating the continuous-time network with distributed delays over finite intervals.     

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

Analysis of age and spatially dependent population model: Application to forest growth

In this paper, we consider an age-structured population model with diffusion. We first establish a comparison principle. We then apply the comparison principle to show the existence and uniqueness of solutions by constructing monotone sequences of weak upper and lower solutions. We also use the comparison principle to study the long-time behavior of the solution and present extinction and boundedness results under certain conditions on the parameters in the model. Additionally, as an example of its application, we apply the model to describe the dynamics of a forest population.     

由主子阵和特殊次序缺损特征对构造Jacobi矩阵

In this paper,an inverse eigenvalue problem of constructing a Jacobian matrix from its prescribed specially ordered defective eigenpairs and a principal subma-trix is considered.The necessary and sufficient conditions for the existence and uniqueness of the solution are derived.Two numerical algorithms and two numer-ical examples are given.     

Global dynamics behaviors for a nonautonomous Lotka–Volterra almost periodic dispersal system with delays

A nonautonomous Lotka–Volterra dispersal system with continuous delays and discrete delays is considered. By using a comparison theorem and delay differential equation basic theory, we obtain sufficient conditions for the permanence of the population in every patch. By constructing a suitable Lyapunov functional, we prove that the system is globally asymptotically stable under some appropriate conditions. Using almost periodic functional hull theory, we get sufficient conditions for the existence, uniqueness and globally asymptotical stability for an almost periodic solution. This implies that the population in every patch exhibits stable almost periodic fluctuation. Furthermore, the results show that the permanence and global stability of system, and the existence and uniqueness of a positive almost periodic solution, depend on the delay; then we call it “profitless”.     

Well-Posedness of the Cauchy Problem for Stochastic Evolution Functional Equations

We prove the existence, uniqueness, and continuous dependence on the initial data of the solutions of the Cauchy problem for stochastic evolution functional equations with random coefficients in Hilbert spaces. We propose a method for constructing an approximating sequence for the solution of the Cauchy problem and obtain an estimate for the rate of convergence to the exact solution.