A new geometric property of Banach space kUKK is given, It is proved that Banach space with this property has weak Banach-saks property, Banach space X is kNUC if and only if it is reflexive and has kUKK property. considering the important role of geometric constants in Banach space geometric properties, The definition of the new constant R₂(X) < k is given by the definition of kUKK and proved that when R₂(X)< k, the Banach space X has a weak fixed point property. Finally, the specific values are calculated in the Cesaro sequence space.

In this paper, we introduce and study the property (h), which extends a-Weyl's theorem. We consider its stability under commuting finite rank and nilpotent perturbations. We prove that property (h) on Banach spaces is related to an important property which has a leading role in local spectral theory:the single-valued extension property. From this result we deduce that property (h) holds for several classes of operators.

In this paper, we study the n-widths of a 2π-periodic convolution function class defined by linear differential operators with real coefficient in Orlicz spaces, and obtain the exact values of n-K width, n-G width, n-L width, n-B width of this function class and its corresponding optimal subspaces.

In this paper, basing on Simon's bounded rationality theory, we first prove and construct an approximation theorem for variational inequalities problems, which provide theoretical support for many relevant different algorithms. Simon's bounded rationality is illustrated and bounded rationality is approaching to full rationality as its ultimate goal. Then, by the methods of set-valued analysis, bounded rationality approximation theory is used for the convergence analysis of solutions of variational inequalities problems. In the sense of Baire category, we obtain the generic convergence of the solutions of monotone variational inequalities problems, in both cases that the function disturbance and the function and constraint set disturbance.

In this paper, we study the boundary layer problem for the incompressible MHD equations in a plane-parallel channel. Using the multiscale analysis and the careful energy method, we prove the convergence of the solution of viscous and diffuse MHD equations to that of the ideal MHD equations as the viscosity and magnetic diffusion coefficient tend to zero.

In this pater, we study the existence and multiplicity of the periodic solutions of a class of Hill's type equations with bounded restoring force. We prove the existence of infinite of subharmonic solutions when the weight is positive. We also consider the existence, multiplicity and dense distribution of symmetric periodic solutions in case of even and periodic weight functions.

This paper mainly study the relations between the solution of the discrete Poisson equation and the solution of the discrete heat equation with exponential nonlinear term by monotone iterative method and comparison principle. When the solutions of the discrete Poisson equation exist, we discuss the asymptotic stability of the solutions to the discrete heat equation with exponential nonlinear term.

In this paper, we study a class of mixed time-varying delayed Lasota-Wazewska model with discontinuous harvesting, which is described by a periodic nonsmooth dynamical system. Base on nonsmooth analysis, Kakutani's fixed point method and the generalized Lyapunov method, easily verifiable delay-independent criteria are established to ensure the existence and exponential stability of positive periodic solutions. Finally, we give an example to further illustrate the effectiveness of our main results.

In this paper, the oscillatory behavior of solutions to a nonlinear second-order neutral differential equation is to study. Using double Riccati transformation and the technique of inequations, some new sufficient conditions are obtained for the solutions of all oscillations and the results generalize, improve and unify the oscillation theorems of half linear functional differential equations, nonlinear equations and generalized Emden-Fowler type equations in the literature recently. At last, some examples are given to illustrate the effectiveness of our results.

In this work, we study the oscillation of the second order Nonlinear Neutral Differential Equations with Distributed Delay

$(r(t)|z'(t)|^{\alpha-1}z'(t))'+\int^{d}_{c}f(t, x(\sigma(t, \xi))){\rm d}\xi=0, $

where $t\geq t_{0}, ~z(t)=x(t)+\int^{b}_{a}p(t, \xi)x(\tau(t, \xi)){\rm d}\xi $. we establish some new oscillation criteria for the above equation. These results extend and improve some known results in the cited literature. Also, our results are illustrated with some examples. It is shown that the theorem has some advantages over the existing literature.

This paper deals with the asymptotic properties of a class of nonlinear and nonautonomous neutral type Hopfield neural networks with time-varying delays. By applying the property of nonnegative matrix and an integral inequality, some sufficient conditions are derived to ensure the existence of the global attracting set and the stability in a Lagrange sense for the considered system. Finally, an example is given to demonstrate the effectiveness of our theoretical result.

We investigate difference Riccati equations with rational coefficients and delay differential equations with constant coefficients. For difference Riccati equations with some relation among coefficients, we prove that every transcendental meromorphic solution is of order no less than one. We also consider the rational solutions for delay differential equations.

In this paper, based on linear triangular element and improved $L1$ approximation, a fully-discrete scheme is proposed for time fractional diffusion equations with $\alpha$ order Caputo fractional derivative. Firstly, the unconditional stability is proved. Secondly, by employing the properties of the element and Ritz projection operator, superclose analysis for the projection operator is deduced with order $O(h^2+\tau^{2-\alpha})$. Further more, combining with relationship between the interpolation operator and Ritz projection, superclose analysis for the interpolation operator is also investigated with order $O(h^2+\tau^{2-\alpha})$. And then, the superconvergence result is obtained through the interpolated postprocessing technique. Finally, numerical results are provided to show the validity of our theoretical analysis.

Rogers-Ramanujan identities are among the most famous q-series in partition theory and combinatorics, they have been proved and generalized widely. The purpose of this paper is to establish a class of new multisum Rogers-Ramanujan identities by applying the bilateral Bailey lemma and iterating technique.

This paper discusses the contrast structure solution for the singularly perturbed problem with slow-fast layers and discontinuous righthand side. By applying the boundary function method, the asymptotic solution of this problem is constructed. Then using the sewing connection method, the existence of the solution is shown and the asymptotic solution is proved to be uniformly valid. Finally, an example is given to illustrate the main results.

By building a second order adjoint equation, the Rodrigues type representation for the second kind solution of a second order difference equation of hypergeometric type on nonuniform lattices is given. The general solution of the equation in the form of a combination of a standard Rodrigues formula and a "generalized" Rodrigues formula is also established.

Nonlinear parabolic equation is studied by H¹-Galerkin mixed finite element method. The bilinear element and the zero-order Raviart-Thomas elements are utilized to discuss superclose properties of the original variable u in H¹(Ω) and the flux p=▽u in H(div; Ω) under the semi-discrete scheme and Euler fully-discrete scheme. During the process, the splitting technique is used and the regularity of u and p are not improved. The numerical example confirm the theory.

In this paper, a measles epidemic model with partial immunity and environmental transmission is considered, and the basic reproduction number R₀ is obtained. By constructing Lyapunov functions, we prove the global asymptotic stability of the infection-free equilibrium and the endemic equilibrium. When R₀ < 1, the infection-free equilibrium is globally asymptotically stable, which implies that measles dies out eventually; when R₀ > 1, the model has a unique endemic equilibrium, which is globally asymptotically stable, that is the transmission of measles keeps a steady state. Finally, the simulations are carried to verify the rationality of the results. This work has practical significance for guiding us to prevent and control the measles spread.

A martingale measure is constructed by using a mean correcting transform in a general semimartingale market model. It is shown that this measure is the mean correcting martingale measure if the semimartingale exists a continuous local martingale part. Although this measure cannot be equivalent to the physical probability for a pure jump semimartingale process, we show that option price of a European option with a convex payoff function under this measure is still arbitrage free if the arbitrage-free interval can reach universal bounds.

In this paper, we study a class of second-order hyperbolic equations with small parameters. An adaptive moving mesh method for solving the equation with finite differencing scheme is proposed, and the moving mesh algorithm is given. The superiority of the method is verified by numerical experiments, and the result on uniform mesh is improved.

The law of population quantity change is one of the key problems in Animal Ecology and Resource Management. By studying the change of population quantity, we can effectively grasp the population dynamics and living habits, which is of great significance for the rational utilization of resources and the protection of ecology. In this paper, we discuss the three-species predator-prey model from the point of fractal theory. We construct the Julia set of 3D predatorprey model. By studying the property of Julia sets, we discuss the conditions for the model to be stable, and take feedback control terms to realize the transformation from instability to stability. In addition, the effects of single population changes on the other two populations and ecosystems were analyzed. Finally, the nonlinear coupling terms with different parameters are constructed, the response system is transformed into the target system, and the synchronization between different systems is realized. Simulation results show the effectiveness of the method.

The complex dynamics of an intraguild predation (IGP) model is investigated in this paper, and the model incorporates the Holling-Ⅱ functional response functions. The sufficient conditions are obtained for the existence and local stability of boundary equilibria. Then, the numerical simulations are applied to the model under the given values of parameters. The numerical results show that the system may have an attracting invariant torus but no positive equilibrium. Furthermore, the Poincaré map and Fourier transform spectrum analysis are performed to study the complex dynamics of the system on the invariant torus. The results suggest that the dynamics on the invariant torus is almost periodic.

In scientific research, often using the observed data of some objective object for complex systems, estimates that this is actually one of the most basic problems with the scientific, in statistical science is one of the most basic problem, known as the point estimate, of point estimate optimal benign numerous research articles. But because cognitive world philosophy east and west is different, so the history of east and west point estimate calculation and reasoning methods have quite big difference. In this paper, through the comparison of the calculation method of east and west point estimation, illustrate the east as the calculation method of the point estimation in image mathematics has reproducibility, and the west as the calculation method of the point estimation of multivariate statistical analysis has not reproducibility. Reproducibility from the point of view, the oriental image mathematical point estimation method is more scientific.