A class of hybrid jump diffusions modulated by a Markov chain is considered in this work. The motivation stems from insurance risk models, and emerging applications in production planning and wireless communications. The models are hybrid in that they involve both continuous dynamics and discrete events. Under suitable conditions, asymptotic expansions of the transition densities for the underlying processes are developed. The formal expansions are validated and the error bounds obtained.

A class of hybrid jump diffusions modulated by a Markov chain is considered in this work. The motivation stems from insurance risk models, and emerging applications in production planning and wireless communications. The models are hybrid in that they involve both continuous dynamics and discrete events. Under suitable conditions, asymptotic expansions of the transition densities for the underlying processes are developed. The formal expansions are validated and the error bounds obtained.

In this paper, we study the distorted Ornstein-Uhlenbeck processes associated with given densities on an abstract Wiener space. We prove that the laws of distorted Ornstein-Uhlenbeck processes converge in total variation norm if the densities converge in Sobolev space $D_2^1$.

In this paper, we study the distorted Ornstein-Uhlenbeck processes associated with given densities on an abstract Wiener space. We prove that the laws of distorted Ornstein-Uhlenbeck processes converge in total variation norm if the densities converge in Sobolev space $D_2^1$.

This article focuses on the study of an age structured SEIRS epidemic model with a vaccination program when the total population size is not kept at constant. We first give the explicit expression of the reproduction number $ \cal{R}(\psi,\widehat{\lambda}) $ in the presence of vaccine ($ \widehat{\lambda} $ is the exponent of growth of total population), and show that the infection-free steady state is linearly stable if ${\cal R}(\psi,\widehat{\lambda})<1$ and unstable if $ {\cal R}(\psi,\widehat{\lambda})>1$, then we apply the theoretical results to vaccination policies to determine the optimal age or ages at which an individual should be vaccinated. It is shown that the optimal strategy can be either one- or two-age strategies.

This article focuses on the study of an age structured SEIRS epidemic model with a vaccination program when the total population size is not kept at constant. We first give the explicit expression of the reproduction number $ \cal{R}(\psi,\widehat{\lambda}) $ in the presence of vaccine ($ \widehat{\lambda} $ is the exponent of growth of total population), and show that the infection-free steady state is linearly stable if ${\cal R}(\psi,\widehat{\lambda})<1$ and unstable if $ {\cal R}(\psi,\widehat{\lambda})>1$, then we apply the theoretical results to vaccination policies to determine the optimal age or ages at which an individual should be vaccinated. It is shown that the optimal strategy can be either one- or two-age strategies.

This paper investigates the existence of solutions of periodic boundary value problems for nonlinear second order functional differential equations. By establishing comparison results, criteria on the existence of maximal and minimal solutions are obtained.

This paper investigates the existence of solutions of periodic boundary value problems for nonlinear second order functional differential equations. By establishing comparison results, criteria on the existence of maximal and minimal solutions are obtained.

In this paper, a logarithmic-exponential penalty function with two parameters for integer programming is discussed. We obtain the exact penalty properties and then establish the asymptotic strong nonlinear duality in the corresponding logarithmic-exponential dual formulation by using the obtained exact penalty properties. The discussion is based on the logarithmic-exponential nonlinear dual formulation proposed in [6].

In this paper, a logarithmic-exponential penalty function with two parameters for integer programming is discussed. We obtain the exact penalty properties and then establish the asymptotic strong nonlinear duality in the corresponding logarithmic-exponential dual formulation by using the obtained exact penalty properties. The discussion is based on the logarithmic-exponential nonlinear dual formulation proposed in [6].

In this paper we deal with the positive steady states of a Competitor-Competitor-Mutualist model with diffusion and homogeneous Dirichlet boundary conditions. We first give the necessary conditions, and then establish the sufficient conditions for the existence of positive steady states.

In this paper we deal with the positive steady states of a Competitor-Competitor-Mutualist model with diffusion and homogeneous Dirichlet boundary conditions. We first give the necessary conditions, and then establish the sufficient conditions for the existence of positive steady states.

We establish the optimal rates of decay estimates of global solutions of some abstract differential equations, which include many partial differential equations. We provide a general treatment so that any future problem will enjoy the decay estimates displayed here as long as the general hypotheses are satisfied. The main hypotheses are the existence of global solutions of the equations and some growth control of the Fourier transform of the solutions. We establish the optimal rates of decay of the solutions for initial data in different spaces. The main ingredients and technical tools are the Fourier splitting method, the iteration skill and the energy estimates.

We establish the optimal rates of decay estimates of global solutions of some abstract differential equations, which include many partial differential equations. We provide a general treatment so that any future problem will enjoy the decay estimates displayed here as long as the general hypotheses are satisfied. The main hypotheses are the existence of global solutions of the equations and some growth control of the Fourier transform of the solutions. We establish the optimal rates of decay of the solutions for initial data in different spaces. The main ingredients and technical tools are the Fourier splitting method, the iteration skill and the energy estimates.

Under a relatively realistic model, this paper theoretically analyzes the road traffic status inside an urban working field, including its radial roads and circular ones. Concretely, the radial and the circular average traveling distances of a car commuter in a tiny ring with wide ${d}x$ are first derived, and then the necessary road area, road area rate distributions, the proportion between the radial and the circular roads to be needed are also calculated. The results presented here and the properties shown through the numerical analysis are considered to be significant at the very beginning stage of designing an urban city.

Under a relatively realistic model, this paper theoretically analyzes the road traffic status inside an urban working field, including its radial roads and circular ones. Concretely, the radial and the circular average traveling distances of a car commuter in a tiny ring with wide ${d}x$ are first derived, and then the necessary road area, road area rate distributions, the proportion between the radial and the circular roads to be needed are also calculated. The results presented here and the properties shown through the numerical analysis are considered to be significant at the very beginning stage of designing an urban city.

Using reflecting function of Mironenko we construct some differential systems which are equivalent to the given differential system. This gives us an opportunity to find out the monodromic matrix of these periodic systems which are not integrable in finite terms.

Using reflecting function of Mironenko we construct some differential systems which are equivalent to the given differential system. This gives us an opportunity to find out the monodromic matrix of these periodic systems which are not integrable in finite terms.

In this paper we show the decay of solutions to the initial-boundary value problem for some nonlinear hyperbolic equation with a nonlinear dissipative term, by using a difference inequality.

In this paper we show the decay of solutions to the initial-boundary value problem for some nonlinear hyperbolic equation with a nonlinear dissipative term, by using a difference inequality.

A group of necessary and sufficient conditions for the nonoscillation of a second order linear delay equation with impulses $$ \left(r(t)u^{\prime}\right)^{\prime}=-p(t)u(t-\tau) $$ are obtained in this paper, where $p(t)=\sum\limits_{n=1}^{\infty}a_n\delta(t-t_{n}),$ $\delta(t) $ is a Dirac $\delta-$function, and for each $n \in \bf N$, $a_n>0,\hskip 0.1cm t_{n}\rightarrow{\infty} $ as $n\rightarrow{\infty}$. Furthermore, the boundedness of the solutions is also investigated if the equation is nonoscillatory. An example is given to illustrate the use of the main theorems.

A group of necessary and sufficient conditions for the nonoscillation of a second order linear delay equation with impulses $$ \left(r(t)u^{\prime}\right)^{\prime}=-p(t)u(t-\tau) $$ are obtained in this paper, where $p(t)=\sum\limits_{n=1}^{\infty}a_n\delta(t-t_{n}),$ $\delta(t) $ is a Dirac $\delta-$function, and for each $n \in \bf N$, $a_n>0,\hskip 0.1cm t_{n}\rightarrow{\infty} $ as $n\rightarrow{\infty}$. Furthermore, the boundedness of the solutions is also investigated if the equation is nonoscillatory. An example is given to illustrate the use of the main theorems.

A Caccioppoli type estimate is established for a class of second order PDEs of divergence type, and its removable singularities of Hausdorff dimension greater than zero is obtained.

For a Carnot group G, we establish the relationship between extended mean values and r-convex functions which is introduced in this paper, which is a class of inequalities of Hadamard type for r-convex function on G.

For a Carnot group G, we establish the relationship between extended mean values and r-convex functions which is introduced in this paper, which is a class of inequalities of Hadamard type for r-convex function on G.

A mathematical model of a predator-prey model with Ivlev's functional response concerning integrated pest management (IPM) is proposed and analyzed. We show that there exists a stable pest-eradication periodic solution when the impulsive period is less than some critical values. Further more, the conditions for the permanence of the system are given. By using bifurcation theory, we show the existence and stability of a positive periodic solution. These results are quite different from those of the corresponding system without impulses. Numerical simulation shows that the system we consider has more complex dynamical behaviors. Finally, it is proved that IPM stragey is more effective than the classical one.

A mathematical model of a predator-prey model with Ivlev's functional response concerning integrated pest management (IPM) is proposed and analyzed. We show that there exists a stable pest-eradication periodic solution when the impulsive period is less than some critical values. Further more, the conditions for the permanence of the system are given. By using bifurcation theory, we show the existence and stability of a positive periodic solution. These results are quite different from those of the corresponding system without impulses. Numerical simulation shows that the system we consider has more complex dynamical behaviors. Finally, it is proved that IPM stragey is more effective than the classical one.

The model of nonlinear differential systems with impulsive effect on random moments is brought forward in this paper. Then, sufficient conditions for (uniform, uniform and ultimate, and uniform and uniformly ultimate) $p-$moment boundedness of the systems are presented. Finally, an example is discussed to show applications of two obtained results.

The model of nonlinear differential systems with impulsive effect on random moments is brought forward in this paper. Then, sufficient conditions for (uniform, uniform and ultimate, and uniform and uniformly ultimate) $p-$moment boundedness of the systems are presented. Finally, an example is discussed to show applications of two obtained results.

In this paper, we investigate the number, location and stability of limit cycles in a class of perturbed polynomial systems with $(2n+1)$ or $(2n+2)$-degree by constructing detection function and using qualitative analysis. We show that there are at most $n$ limit cycles in the perturbed polynomial system, which is similar to the result of Perko in [8] by using Melnikov method. For $n=2$, we establish the general conditions depending on polynomial's coefficients for the bifurcation, location and stability of limit cycles. The bifurcation parameter value of limit cycles in [5] is also improved by us. When $n=3$ the sufficient and necessary conditions for the appearance of 3 limit cycles are given. Two numerical examples for the location and stability of limit cycles are used to demonstrate our theoretical results.

In this paper, we investigate the number, location and stability of limit cycles in a class of perturbed polynomial systems with $(2n+1)$ or $(2n+2)$-degree by constructing detection function and using qualitative analysis. We show that there are at most $n$ limit cycles in the perturbed polynomial system, which is similar to the result of Perko in [8] by using Melnikov method. For $n=2$, we establish the general conditions depending on polynomial's coefficients for the bifurcation, location and stability of limit cycles. The bifurcation parameter value of limit cycles in [5] is also improved by us. When $n=3$ the sufficient and necessary conditions for the appearance of 3 limit cycles are given. Two numerical examples for the location and stability of limit cycles are used to demonstrate our theoretical results.

In this paper the dynamical behaviors of a predator-prey system with Holling Type-IV functional response are investigated in detail by using the analyses of qualitative method, bifurcation theory, and numerical simulation. The qualitative analyses and numerical simulation for the model indicate that it has a unique stable limit cycle. The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddle-node bifurcation, Hopf bifurcation, homoclinic bifurcation and bifurcation of cusp-type with codimension two (ie, the Bogdanov-Takens bifurcation), and we show the existence of codimension three degenerated equilibrium and the existence of homoclinic orbit by using numerical simulation.

In this paper the dynamical behaviors of a predator-prey system with Holling Type-IV functional response are investigated in detail by using the analyses of qualitative method, bifurcation theory, and numerical simulation. The qualitative analyses and numerical simulation for the model indicate that it has a unique stable limit cycle. The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddle-node bifurcation, Hopf bifurcation, homoclinic bifurcation and bifurcation of cusp-type with codimension two (ie, the Bogdanov-Takens bifurcation), and we show the existence of codimension three degenerated equilibrium and the existence of homoclinic orbit by using numerical simulation.

We study the periodic problem for differential inclusions in $\ssize \Bbb{R}^N$. First we look for extremal periodic solutions. Using techniques from multivalued analysis and a fixed point argument we establish an existence theorem under some general hypotheses. We also consider the ``nonconvex periodic problem'' under lower semicontinuity hypotheses, and the ``convex periodic problem'' under general upper semicontinuity hypotheses on the multivalued vector field. For both problems, we prove existence theorems under very general hypotheses. Our approach extends existing results in the literature and appear to be the most general results on the nonconvex periodic problem.

We study the periodic problem for differential inclusions in $\ssize \Bbb{R}^N$. First we look for extremal periodic solutions. Using techniques from multivalued analysis and a fixed point argument we establish an existence theorem under some general hypotheses. We also consider the ``nonconvex periodic problem'' under lower semicontinuity hypotheses, and the ``convex periodic problem'' under general upper semicontinuity hypotheses on the multivalued vector field. For both problems, we prove existence theorems under very general hypotheses. Our approach extends existing results in the literature and appear to be the most general results on the nonconvex periodic problem.

We will study the following problem. Let $ X_t, \ t\in [0,T]$, be an ${\pmb R}^d$--valued process defined on a time interval $t\in [0,T]$. Let $Y$ be a random value depending on the trajectory of $X$. Assume that, at each fixed time $t\leq T$, the information available to an agent (an individual, a firm, or even a market) is the trajectory of $X$ before $t$. Thus at time $T$ , the random value of $Y(\omega )$ will become known to this agent. The question is: how will this agent evaluate $Y$ at the time $t$?\\ We will introduce an evaluation operator ${\mathcal{E}}_t[Y]$ to define the value of $Y$ given by this agent at time $t$. This operator ${\mathcal{E}}% _t[\cdot ]$ assigns an $(X_s)_{0\leq s\leq T}$--dependent random variable $Y$ to an $(X_s)_{0\leq s\leq t}$--dependent random variable ${\mathcal{E}}_t[Y]$. We will mainly treat the situation in which the process $X$ is a solution of a SDE (see equation (3.1)) with the drift coefficient $b$ and diffusion coefficient $\sigma $ containing an unknown parameter $\theta =\theta _t $. We then consider the so called super evaluation when the agent is a seller of the asset $Y$. We will prove that such super evaluation is a filtration consistent nonlinear expectation. In some typical situations, we will prove that a filtration consistent nonlinear evaluation dominated by this super evaluation is a $g$--evaluation. We also consider the corresponding nonlinear Markovian situation.

We will study the following problem. Let $ X_t, \ t\in [0,T]$, be an ${\pmb R}^d$--valued process defined on a time interval $t\in [0,T]$. Let $Y$ be a random value depending on the trajectory of $X$. Assume that, at each fixed time $t\leq T$, the information available to an agent (an individual, a firm, or even a market) is the trajectory of $X$ before $t$. Thus at time $T$ , the random value of $Y(\omega )$ will become known to this agent. The question is: how will this agent evaluate $Y$ at the time $t$?\\ We will introduce an evaluation operator ${\mathcal{E}}_t[Y]$ to define the value of $Y$ given by this agent at time $t$. This operator ${\mathcal{E}}% _t[\cdot ]$ assigns an $(X_s)_{0\leq s\leq T}$--dependent random variable $Y$ to an $(X_s)_{0\leq s\leq t}$--dependent random variable ${\mathcal{E}}_t[Y]$. We will mainly treat the situation in which the process $X$ is a solution of a SDE (see equation (3.1)) with the drift coefficient $b$ and diffusion coefficient $\sigma $ containing an unknown parameter $\theta =\theta _t $. We then consider the so called super evaluation when the agent is a seller of the asset $Y$. We will prove that such super evaluation is a filtration consistent nonlinear expectation. In some typical situations, we will prove that a filtration consistent nonlinear evaluation dominated by this super evaluation is a $g$--evaluation. We also consider the corresponding nonlinear Markovian situation.

The asymptotic expansion of the heat kernel $\Theta (t)=\sum\limits_{j=1}^\infty \exp (-t\lambda \Sb \\ j \endSb )$ where $\{\lambda \Sb \\ j \endSb \}\Sb \\ j=1 \endSb ^\infty $ are the eigenvalues of the negative Laplacian $% -\Delta \Sb \\ n \endSb =-\sum\limits_{k=1}^n(\frac \partial {\partial x^k})^2$ in $R^n(n=2$ or $3)$ is studied for short-time $t$ for a general bounded domain $\Omega $ with a smooth boundary $\partial \Omega .$ In this paper, we consider the case of a finite number of the Dirichlet conditions $\phi =0$ on $\Gamma \Sb \\ i \endSb \,\,(i=1,...,J)$ and the Neumann conditions $% \frac{\partial \phi }{\partial \upsilon \Sb \\ i \endSb }=0$ on $\Gamma \Sb \\ i \endSb \,\,(i=J+1,\cdots ,k)$ and the Robin conditions $(\frac \partial {\partial \upsilon \Sb \\ i \endSb }+\gamma \Sb \\ i \endSb )\phi =0$ on $\Gamma \Sb \\ i \endSb \,\,(i=k+1,\cdots ,m)$ where $\gamma \Sb \\ i \endSb $ are piecewise smooth positive impedance functions, such that $\partial \Omega $ consists of a finite number of piecewise smooth components $\Gamma \Sb \\ i \endSb \,\,(i=1,\cdots ,m)$ where $\partial \Omega =\bigcup\limits_{i=1}^m\Gamma \Sb \\ i \endSb .$ We construct the required asymptotics in the form of a power series over $t.$ The senior coefficients in this series are specified as functionals of the geometric shape of the domain $\Omega .$ This result is applied to calculate the one-particle partition function of a ``special ideal gas'', i.e., the set of non-interacting particles set up in a box with Dirichlet, Neumann and Robin boundary conditions for the appropriate wave function. Calculation of the thermodynamic quantities for the ideal gas such as the internal energy, pressure and specific heat reveals that these quantities alone are incapable of distinguishing between two different shapes of the domain. This conclusion seems to be intuitively clear because it is based on a limited information given by a one-particle partition function; nevertheless, its formal theoretical motivation is of some interest.

The asymptotic expansion of the heat kernel $\Theta (t)=\sum\limits_{j=1}^\infty \exp (-t\lambda \Sb \\ j \endSb )$ where $\{\lambda \Sb \\ j \endSb \}\Sb \\ j=1 \endSb ^\infty $ are the eigenvalues of the negative Laplacian $% -\Delta \Sb \\ n \endSb =-\sum\limits_{k=1}^n(\frac \partial {\partial x^k})^2$ in $R^n(n=2$ or $3)$ is studied for short-time $t$ for a general bounded domain $\Omega $ with a smooth boundary $\partial \Omega .$ In this paper, we consider the case of a finite number of the Dirichlet conditions $\phi =0$ on $\Gamma \Sb \\ i \endSb \,\,(i=1,...,J)$ and the Neumann conditions $% \frac{\partial \phi }{\partial \upsilon \Sb \\ i \endSb }=0$ on $\Gamma \Sb \\ i \endSb \,\,(i=J+1,\cdots ,k)$ and the Robin conditions $(\frac \partial {\partial \upsilon \Sb \\ i \endSb }+\gamma \Sb \\ i \endSb )\phi =0$ on $\Gamma \Sb \\ i \endSb \,\,(i=k+1,\cdots ,m)$ where $\gamma \Sb \\ i \endSb $ are piecewise smooth positive impedance functions, such that $\partial \Omega $ consists of a finite number of piecewise smooth components $\Gamma \Sb \\ i \endSb \,\,(i=1,\cdots ,m)$ where $\partial \Omega =\bigcup\limits_{i=1}^m\Gamma \Sb \\ i \endSb .$ We construct the required asymptotics in the form of a power series over $t.$ The senior coefficients in this series are specified as functionals of the geometric shape of the domain $\Omega .$ This result is applied to calculate the one-particle partition function of a ``special ideal gas'', i.e., the set of non-interacting particles set up in a box with Dirichlet, Neumann and Robin boundary conditions for the appropriate wave function. Calculation of the thermodynamic quantities for the ideal gas such as the internal energy, pressure and specific heat reveals that these quantities alone are incapable of distinguishing between two different shapes of the domain. This conclusion seems to be intuitively clear because it is based on a limited information given by a one-particle partition function; nevertheless, its formal theoretical motivation is of some interest.

In this article we introduce the generalized lacunary difference sequence spaces $[N_\theta, M, \Delta^m]_0$, $ [N_\theta, M, \Delta^m]_1$ and $[N_\theta, M, \Delta^m]_\infty$ using $m^{th}-$ difference. We study their properties like completeness, solidness, symmetricity. Also we obtain some inclusion relations involving the spaces $[N_\theta, M, \Delta^m]_0$,$ [N_\theta, M, \Delta^m]_1$ and $[N_\theta, M, \Delta^m]_\infty$ and the Ces\`aro summable and strongly Ces\`aro summable sequences.