Competition of spatial and temporal instabilities under time delay near the codimension-two Turing-Hopfbifurcations is studied in a reaction-diffusion equation.The time delay changes remarkably the oscillation frequen...Competition of spatial and temporal instabilities under time delay near the codimension-two Turing-Hopfbifurcations is studied in a reaction-diffusion equation.The time delay changes remarkably the oscillation frequency,theintrinsic wave vector,and the intensities of both Turing and Hopf modes.The application of appropriate time delaycan control the competition between the Turing and Hopf modes.Analysis shows that individual or both feedbacks canrealize the control of the transformation between the Turing and Hopf patterns.Two-dimensional numerical simulationsvalidate the analytical results.展开更多
Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
Reaction–diffusion systems are mathematical models which link to several physical phenomena.The most common is the change in space and time of the meditation of one or more materials.Reaction–diffusion modeling is a...Reaction–diffusion systems are mathematical models which link to several physical phenomena.The most common is the change in space and time of the meditation of one or more materials.Reaction–diffusion modeling is a substantial role in the modeling of computer propagation like infectious diseases.We investigated the transmission dynamics of the computer virus in which connected to each other through network globally.The current study devoted to the structure-preserving analysis of the computer propagation model.This manuscript is devoted to finding the numerical investigation of the reaction–diffusion computer virus epidemic model with the help of a reliable technique.The designed technique is finite difference scheme which sustains the important physical behavior of continuous model like the positivity of the dependent variables,the stability of the equilibria.The theoretical analysis of the proposed method like the positivity of the approximation,stability,and consistency is discussed in detail.A numerical example of simulations yields the authentication of the theoretical results of the designed technique.展开更多
This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure: kink and pulse, a mode an...This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure: kink and pulse, a mode analysis method is proposed to approximate the solutions of these localized basic modes and to analyse their stabilities. Using this method, it reconstructs the whole stationary patterns. The cellular mode structures (kink and pulse) in bistable media fundamentally differ from stationary patterns in monostable media showing spatial periodicity induced by a diffusive Taring bifurcation.展开更多
In this paper,we study a semi-linear reaction-diffusion system with a weighted nonlocal source,subject to the null Dirichlet boundary condition.Under certain conditions,we prove that the classical solution exists glob...In this paper,we study a semi-linear reaction-diffusion system with a weighted nonlocal source,subject to the null Dirichlet boundary condition.Under certain conditions,we prove that the classical solution exists globally and blows up in finite time respectively,and then obtain the uniform blow-up rate in the interior.展开更多
The complex Ginzburg-Landau equation (CGLE) has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to t...The complex Ginzburg-Landau equation (CGLE) has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation, and is not valid when a RD system is away from the onset. To test this, we study spirals and anti-spirals in the chlorite-iodide-malonic acid (CIMA) reaction and the corresponding OGLE. Numerical simulations confirm that the OGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.展开更多
Most biochemical processes in cells are usually modeled by reaction-diffusion (RD) equations. In these RD models, the diffusive process is assumed to be Gaussian. However, a growing number of studies have noted that...Most biochemical processes in cells are usually modeled by reaction-diffusion (RD) equations. In these RD models, the diffusive process is assumed to be Gaussian. However, a growing number of studies have noted that intracellular diffusion is anomalous at some or all times, which may result from a crowded environment and chemical kinetics. This work aims to computationally study the effects of chemical reactions on the diffusive dynamics of RD systems by using both stochastic and deterministic algorithms. Numerical method to estimate the mean-square displacement (MSD) from a deterministic algorithm is also investigated. Our computational results show that anomalous diffusion can be solely due to chemical reactions. The chemical reactions alone can cause anomalous sub-diffusion in the RD system at some or all times. The time-dependent anomalous diffusion exponent is found to depend on many parameters, including chemical reaction rates, reaction orders, and chemical concentrations.展开更多
By the degree theory on positive cone together with the technique of a priori estimate, the nontrivial equilibrium solutions of a strong nonlinearity and weak coupling reaction diffusion system and the structure of t...By the degree theory on positive cone together with the technique of a priori estimate, the nontrivial equilibrium solutions of a strong nonlinearity and weak coupling reaction diffusion system and the structure of the equilibrium solutions are discussed.展开更多
In this paper, the nonlocal nonlinear reaction-diffusion singularly perturbed problems with two parameters are studied. Using a singular perturbation method, the structure of the solutions to the problem is discussed ...In this paper, the nonlocal nonlinear reaction-diffusion singularly perturbed problems with two parameters are studied. Using a singular perturbation method, the structure of the solutions to the problem is discussed in relation to two small parameters. The asymptotic solutions of the problem are given.展开更多
In this paper, positiveness theorems of solutions for several differential inequalities are proved and are used to prove the existence of traveling wave front solutions of reaction-diffusion systems. As an application...In this paper, positiveness theorems of solutions for several differential inequalities are proved and are used to prove the existence of traveling wave front solutions of reaction-diffusion systems. As an application, two examples are given.展开更多
This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem a...This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem and so on, the authors prove the global existence and uniqueness of a. smooth. solution for this IBVP under some appropriate conditions.展开更多
The study of rumor propagation dynamics is of great significance to reduce.false news and ensure the authenticity of news information.In this paper,a SI reaction-diffusion rumor propagation model with nonlinear satura...The study of rumor propagation dynamics is of great significance to reduce.false news and ensure the authenticity of news information.In this paper,a SI reaction-diffusion rumor propagation model with nonlinear saturation incidence is studied.First,through stability analysis,we obtain the conditions for the existence and local stability of the positive equilibrium point.By selecting suitable variable as the control parameter,the critical value of Turing bifurcation and the existence theorem of Turing bifurcation are obtained.Then,using the above theorem and multi-scale standard analysis,the expression of amplitude equation around Turing bifurcation point is obtained.By analyzing the amplitude equation,different types of Turing pattern are divided such as uniform steady-state mode,hexagonal mode,stripe mode and mixed structure mode.Further,in the numerical simulation part,by observing different patterns corresponding to different values of control variable,the correctness of the theory is verified.Finally,the effects of different network structures on patterns are investigated.The results show that there are significant differences in the distribution of users on different network structures.展开更多
In this paper we consider the initial boundary value problem for a class of reaction-diffusion system with time delay in population dynamics. We prove the existence and uniqueness of bounded nonnegative solution for ...In this paper we consider the initial boundary value problem for a class of reaction-diffusion system with time delay in population dynamics. We prove the existence and uniqueness of bounded nonnegative solution for the initial boundary value problem and discuss the asymptotic behaviour of the solution.展开更多
In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion system (u<sub>i</sub><sup>αi</sup>t=u<sub>ixx</sub>...In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion system (u<sub>i</sub><sup>αi</sup>t=u<sub>ixx</sub>-multiply from j=1 to N u<sub>j</sub><sup>mij</sup>, x∈R, t】0,i=1,. . . ,N (Ⅰ) is studied. where 0【a<sub>i</sub>【1. mij≥0 and sum from j=1 to N mij】0, i, j=1, . . . ,N .Necessary and sufficient conditions on existence and large time behaviours of FTWs of (Ⅰ) are obtained by using the matrix theory. Schauder’s fixed point theorem, and upper and lower solutious method.展开更多
In this paper,a reaction-diffusion system is proposed to investigate avian-human influenza.Two free boundaries are introduced to describe the spreading frontiers of the avian influenza.The basic reproduction numbers r...In this paper,a reaction-diffusion system is proposed to investigate avian-human influenza.Two free boundaries are introduced to describe the spreading frontiers of the avian influenza.The basic reproduction numbers rF0(t)and RF0(t)are defined for the bird with the avian influenza and for the human with the mutant avian influenza of the free boundary problem,respectively.Properties of these two time-dependent basic reproduction numbers are obtained.Sufficient conditions both for spreading and for vanishing of the avian influenza are given.It is shown that if rF0(0)<1 and the initial number of the infected birds is small,the avian influenza vanishes in the bird world.Furthermore,if rF0(0)<1 and RF0(0)<1,the avian influenza vanishes in the bird and human worlds.In the case that rF0(0)<1 and RF0(0)>1,spreading of the mutant avian influenza in the human world is possible.It is also shown that if rF0(t0)>1 for any t0>0,the avian influenza spreads in the bird world.展开更多
We consider the approximation of systems of reaction-diffusion equations, with the finite element method. The highest derivative in each equation is multiplied by a parameter ε∈ (0, 1], and as ε → 0 the solution ...We consider the approximation of systems of reaction-diffusion equations, with the finite element method. The highest derivative in each equation is multiplied by a parameter ε∈ (0, 1], and as ε → 0 the solution of the system will contain boundary layers. We extend the analysis of the corresponding scalar problem from [Melenk, IMA J. Numer. Anal. 17(1997), pp. 577-601], to construct a finite element scheme which includes elements of size O(εp) near the boundary, where p is the degree of the approximating polynomials. We show that, under the assumption of analytic input data, the method yields exponential rates of convergence, independently of ε, when the error is measured in the energy norm associated with the problem. Numerical computations supporting the theory are also presented, which also show that the method yields robust exponential convergence rates when the error in the maximum norm is used.展开更多
We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge(proportion of both the species)and harvesting of prey species in this contribution...We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge(proportion of both the species)and harvesting of prey species in this contribution.Criteria for asymptotic stability(local and global)and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point.For partial differential equation(PDE),the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated.The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species.A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space,that is,spots,stripe-spot mixtures,labyrinthine,holes,stripe-hole mixtures and stripes replication.Finally,we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.展开更多
This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existen...This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.展开更多
This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on Rn, where n 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar...This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on Rn, where n 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t →∞. The convergence is uniform in Rn. Moreover, the stability of planar waves in reaction-diffusion equations with nonlocal delays is also established by transforming the delayed equations into a non-delayed reaction-diffusion system.展开更多
Boundary control for a class of partial integro-differential systems with space and time dependent coefficients is consid- ered. A control law is derived via the partial differential equation (PDE) backstepping. The...Boundary control for a class of partial integro-differential systems with space and time dependent coefficients is consid- ered. A control law is derived via the partial differential equation (PDE) backstepping. The existence of kernel equations is proved. Exponential stability of the closed-loop system is achieved. Simulation results are presented through figures.展开更多
基金Supported by the Fundamental Research Funds for the Central Universities under Grant No. 09ML56the Foundation for Young Teachers of the North China Electric Power University, China under Grant No. 200611029
文摘Competition of spatial and temporal instabilities under time delay near the codimension-two Turing-Hopfbifurcations is studied in a reaction-diffusion equation.The time delay changes remarkably the oscillation frequency,theintrinsic wave vector,and the intensities of both Turing and Hopf modes.The application of appropriate time delaycan control the competition between the Turing and Hopf modes.Analysis shows that individual or both feedbacks canrealize the control of the transformation between the Turing and Hopf patterns.Two-dimensional numerical simulationsvalidate the analytical results.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
基金The authors declare that they have no funding for the present study。
文摘Reaction–diffusion systems are mathematical models which link to several physical phenomena.The most common is the change in space and time of the meditation of one or more materials.Reaction–diffusion modeling is a substantial role in the modeling of computer propagation like infectious diseases.We investigated the transmission dynamics of the computer virus in which connected to each other through network globally.The current study devoted to the structure-preserving analysis of the computer propagation model.This manuscript is devoted to finding the numerical investigation of the reaction–diffusion computer virus epidemic model with the help of a reliable technique.The designed technique is finite difference scheme which sustains the important physical behavior of continuous model like the positivity of the dependent variables,the stability of the equilibria.The theoretical analysis of the proposed method like the positivity of the approximation,stability,and consistency is discussed in detail.A numerical example of simulations yields the authentication of the theoretical results of the designed technique.
基金Project partially supported by the Outstanding Oversea Scholar Foundation of the Chinese Academy of Sciences (Bairenjihua)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure: kink and pulse, a mode analysis method is proposed to approximate the solutions of these localized basic modes and to analyse their stabilities. Using this method, it reconstructs the whole stationary patterns. The cellular mode structures (kink and pulse) in bistable media fundamentally differ from stationary patterns in monostable media showing spatial periodicity induced by a diffusive Taring bifurcation.
基金Project supported by the Research Program of Natural Science of Universities in Jiangsu Province (Grant No.09KJD110008)the Natural Science Foundation of Nanjing Xiaozhuang University (Grant No.005NXY11)
文摘In this paper,we study a semi-linear reaction-diffusion system with a weighted nonlocal source,subject to the null Dirichlet boundary condition.Under certain conditions,we prove that the classical solution exists globally and blows up in finite time respectively,and then obtain the uniform blow-up rate in the interior.
基金Project supported by the National Natural Science Foundation of China (Grant No 10274003) and the Department of Science and Technology of China.Acknowledgement We thank Cheng X, Wang C and Wang S for helpful discussion.
文摘The complex Ginzburg-Landau equation (CGLE) has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation, and is not valid when a RD system is away from the onset. To test this, we study spirals and anti-spirals in the chlorite-iodide-malonic acid (CIMA) reaction and the corresponding OGLE. Numerical simulations confirm that the OGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.
基金supported by the Thailand Research Fund and Mahidol University(Grant No.TRG5880157),the Thailand Center of Excellence in Physics(ThEP),CHE,Thailand,and the Development Promotion of Science and Technology
文摘Most biochemical processes in cells are usually modeled by reaction-diffusion (RD) equations. In these RD models, the diffusive process is assumed to be Gaussian. However, a growing number of studies have noted that intracellular diffusion is anomalous at some or all times, which may result from a crowded environment and chemical kinetics. This work aims to computationally study the effects of chemical reactions on the diffusive dynamics of RD systems by using both stochastic and deterministic algorithms. Numerical method to estimate the mean-square displacement (MSD) from a deterministic algorithm is also investigated. Our computational results show that anomalous diffusion can be solely due to chemical reactions. The chemical reactions alone can cause anomalous sub-diffusion in the RD system at some or all times. The time-dependent anomalous diffusion exponent is found to depend on many parameters, including chemical reaction rates, reaction orders, and chemical concentrations.
文摘By the degree theory on positive cone together with the technique of a priori estimate, the nontrivial equilibrium solutions of a strong nonlinearity and weak coupling reaction diffusion system and the structure of the equilibrium solutions are discussed.
基金supported by the National Natural Science Foundation of China (Nos. 40676016, 40876010)the Knowledge Innovation Program of Chinese Academy of Sciences (No. KZCX2-YW-Q03-08)+1 种基金the LASG State Key Laboratory Special Fundthe E-Institute of Shanghai Municipal Education Commission (No. E03004)
文摘In this paper, the nonlocal nonlinear reaction-diffusion singularly perturbed problems with two parameters are studied. Using a singular perturbation method, the structure of the solutions to the problem is discussed in relation to two small parameters. The asymptotic solutions of the problem are given.
基金Research supported by the National Natural Science Foundation of China (19971004 19871005).
文摘In this paper, positiveness theorems of solutions for several differential inequalities are proved and are used to prove the existence of traveling wave front solutions of reaction-diffusion systems. As an application, two examples are given.
文摘This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem and so on, the authors prove the global existence and uniqueness of a. smooth. solution for this IBVP under some appropriate conditions.
基金supported by the National Natural Science Foundation of China(Grant No.12002135)Young Science and Technology Talents Lifting Project of Jiangsu Association for Science and Technology.
文摘The study of rumor propagation dynamics is of great significance to reduce.false news and ensure the authenticity of news information.In this paper,a SI reaction-diffusion rumor propagation model with nonlinear saturation incidence is studied.First,through stability analysis,we obtain the conditions for the existence and local stability of the positive equilibrium point.By selecting suitable variable as the control parameter,the critical value of Turing bifurcation and the existence theorem of Turing bifurcation are obtained.Then,using the above theorem and multi-scale standard analysis,the expression of amplitude equation around Turing bifurcation point is obtained.By analyzing the amplitude equation,different types of Turing pattern are divided such as uniform steady-state mode,hexagonal mode,stripe mode and mixed structure mode.Further,in the numerical simulation part,by observing different patterns corresponding to different values of control variable,the correctness of the theory is verified.Finally,the effects of different network structures on patterns are investigated.The results show that there are significant differences in the distribution of users on different network structures.
文摘In this paper we consider the initial boundary value problem for a class of reaction-diffusion system with time delay in population dynamics. We prove the existence and uniqueness of bounded nonnegative solution for the initial boundary value problem and discuss the asymptotic behaviour of the solution.
基金Project supported by the Postdoctoral Science Foundation of China the Henan Province Natural Science Foundation of China
文摘In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion system (u<sub>i</sub><sup>αi</sup>t=u<sub>ixx</sub>-multiply from j=1 to N u<sub>j</sub><sup>mij</sup>, x∈R, t】0,i=1,. . . ,N (Ⅰ) is studied. where 0【a<sub>i</sub>【1. mij≥0 and sum from j=1 to N mij】0, i, j=1, . . . ,N .Necessary and sufficient conditions on existence and large time behaviours of FTWs of (Ⅰ) are obtained by using the matrix theory. Schauder’s fixed point theorem, and upper and lower solutious method.
基金supported by National Natural Science Foundation of China(Grant No.11071209)Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education,Science and Technology(Grant No.2010-0025700)Natural Science Foundation of the Higher Education Institutions of Jiangsu Province(Grant No.12KJD110008)
文摘In this paper,a reaction-diffusion system is proposed to investigate avian-human influenza.Two free boundaries are introduced to describe the spreading frontiers of the avian influenza.The basic reproduction numbers rF0(t)and RF0(t)are defined for the bird with the avian influenza and for the human with the mutant avian influenza of the free boundary problem,respectively.Properties of these two time-dependent basic reproduction numbers are obtained.Sufficient conditions both for spreading and for vanishing of the avian influenza are given.It is shown that if rF0(0)<1 and the initial number of the infected birds is small,the avian influenza vanishes in the bird world.Furthermore,if rF0(0)<1 and RF0(0)<1,the avian influenza vanishes in the bird and human worlds.In the case that rF0(0)<1 and RF0(0)>1,spreading of the mutant avian influenza in the human world is possible.It is also shown that if rF0(t0)>1 for any t0>0,the avian influenza spreads in the bird world.
文摘We consider the approximation of systems of reaction-diffusion equations, with the finite element method. The highest derivative in each equation is multiplied by a parameter ε∈ (0, 1], and as ε → 0 the solution of the system will contain boundary layers. We extend the analysis of the corresponding scalar problem from [Melenk, IMA J. Numer. Anal. 17(1997), pp. 577-601], to construct a finite element scheme which includes elements of size O(εp) near the boundary, where p is the degree of the approximating polynomials. We show that, under the assumption of analytic input data, the method yields exponential rates of convergence, independently of ε, when the error is measured in the energy norm associated with the problem. Numerical computations supporting the theory are also presented, which also show that the method yields robust exponential convergence rates when the error in the maximum norm is used.
基金the financial support in part from Special Assistance Programme(SAP-III)sponsored by the University Grants Commission(UGC),New Delhi,India(Grant No.F.510/3/DRS-III/2015(SAP-I)).Dr.S.Djilali is partially supported by the DGRSDT of Algeria.
文摘We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge(proportion of both the species)and harvesting of prey species in this contribution.Criteria for asymptotic stability(local and global)and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point.For partial differential equation(PDE),the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated.The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species.A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space,that is,spots,stripe-spot mixtures,labyrinthine,holes,stripe-hole mixtures and stripes replication.Finally,we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.
基金the National Natural Science Foundation of China(No.10471022,10771032)the Natural Science Foundation of Jiangsu province BK2006088.
文摘This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.
文摘This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on Rn, where n 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t →∞. The convergence is uniform in Rn. Moreover, the stability of planar waves in reaction-diffusion equations with nonlocal delays is also established by transforming the delayed equations into a non-delayed reaction-diffusion system.
文摘Boundary control for a class of partial integro-differential systems with space and time dependent coefficients is consid- ered. A control law is derived via the partial differential equation (PDE) backstepping. The existence of kernel equations is proved. Exponential stability of the closed-loop system is achieved. Simulation results are presented through figures.