Conditions For Boundedness Of Solutions Research Articles

Higher Lyapunov Functions Derivatives and Total Poisson Boundedness of Solutions

Using the method of higher Lyapunov functions derivatives we obtain sufficient conditions for total Poisson boundedness of solutions, partial total Poisson boundedness of solutions, and partial total Poisson boundedness of solutions with partially controlled initial conditions.

Poisson Total Boundedness of Solutions of Systems of Differential Equations and Lyapunov Vector Functions

We introduce the notions of Poisson total boundedness of solutions, partial Poisson total boundedness of solutions, and partial Poisson total boundedness of solutions with partly controlled initial conditions. We use the Lyapunov vector function method to obtain sufficient conditions for the Poisson total boundedness of solutions, the partial Poisson total boundedness of solutions, and the partial Poisson total boundedness of solutions with partly controlled initial conditions. As a consequence, we obtain sufficient conditions for the above-mentioned kinds of Poisson total boundedness of solutions based on the Lyapunov function method.

Permanence and periodic solution for a modified Leslie-Gower type predator-prey model with diffusion and non constant coefficients

In this paper we study a predator-prey system, modeling the interaction of two species with diffusion and T-periodic environmental parameters. It is a Leslie-Gower type predator-prey model with Holling-type-II functional response. We establish some sufficient conditions for the ultimate boundedness of solutions and permanence of this system. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Numerical simulations are presented to illustrate the results.

A Condition for Boundedness of Solutions of Bidimensional Switched Affine Systems With Multiple Foci and Centers

AbstractThis paper studies boundedness of solutions of bidimensional switched affine linear systems. Every subsystem of the systems has a single stable/ unstable focus/ center and all the equilibria pairwise differ. By using the multiple polar coordinate systems method, this paper proposes a condition of boundedness of solutions of such switched systems under periodic/ quasi‐periodic switching paths. The condition is also shown a sufficient condition of global asymptotic region stability of such switched systems with respect to a region containing all multiple equilibria. A global asymptotic region‐stabilizing control, a periodic/ quasi‐periodic switching path, and a corresponding algorithm are all designed for such switched control systems. An illustrative example demonstrates the effectiveness and practicality of our new results.

Boundedness of solutions of second order nonlinear dynamic equations

This paper deals with the boundedness of the solutions of the following dynamic equations $$(r(t)x^\Delta (t))^\Delta + a(t)f(x^\sigma (t)) + b(t)g(x^\sigma (t)) = 0$$ and $$(r(t)x^\Delta (t))^\Delta + a(t)x^\sigma (t) + b(t)f(x(t - \tau (t))) = e(t)$$ on a time scale \(\mathbb{T}\). By using the Bellman integral inequality, we establish some sufficient conditions for boundedness of solutions of the above equations. Our results not only unify the boundedness results for differential and difference equations but are also new for the q-difference equations.

Sharp conditions for boundedness in linear discrete Volterra equations

If all solutions of a discrete linear nonconvolution Volterra equation satisfying some positivity conditions are bounded, it is found that the kernel must obey certain restrictions. These are closely related to known sufficient conditions for boundedness of solutions. Applications to systems of Volterra difference equations are provided.

BOUNDEDNESS OF IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH VARIABLE IMPULSIVE PERTURBATIONS

AbstractIn the present paper an initial value problem for impulsive functional differential equations with variable impulsive perturbations is considered. By means of piecewise continuous functions coupled with the Razumikhin technique, sufficient conditions for boundedness of solutions of such equations are found.

A Boundedness Theorem for a Certain Third Order Nonlinear Differential Equation

In this study, we establish conditions for ultimate boundedness of solutions for a certain third order nonlinear differential equation using a complete Yoshizawa function. The result includes and extends the earlier results in the literature.

Global asymptotic behavior in chemostat-type competition models with delay

The asymptotic behavior of solutions of a chemostat-type model in which two species compete for a limiting nutrient supplied at a constant rate is considered. The model incorporates a general nutrient uptake function and two distributed delays. The first delay models the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition and the second indicates that the growth of the species depends on the past concentration of the nutrient. Furthermore, it is assumed that there is interspecific competition between the two species as well as intraspecific competition within each species. Conditions for boundedness of solutions and existence of nonnegative equilibria are given. By constructing appropriate Liapunov-like functionals, some sufficient conditions for global attractivity of the positive equilibrium is obtained. The combined effects of the two different delays are studied. The main results of Freedman and Xu [H.I. Freedman, Y. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. Math. Biol. 31 (1993) 513–527] and Ruan and He [S. Ruan, X.-Z. He, Global stability in chemostat-type competition models with nutrient recycling, SIAM J. Appl. Math. 58 (1) (1998) 170–192] are improved and extended.

Some further remarks on good sets

We show that in n-fold cartesian product, n ≥ 4, a related component need not be a full component. We also prove that when n ≥ 4, uniform boundedness of lengths of geodesics is not a necessary condition for boundedness of solutions of (1) for bounded function f.

Competition between plasmid-bearing and plasmid-free organisms in a chemostat with nutrient recycling and an inhibitor

The asymptotic behavior of solutions of a model for competition between plasmid-bearing and plasmid-free organisms in the chemostat with two distributed delays and an external inhibitor is considered. The model presents a refinement of the one considered by Lu and Hadeler [Z. Lu, K.P. Hadeler, Model of plasmid-bearing plasmid-free competition in the chemostat with nutrient recycling and an inhibitor, Math. Biosci. 167 (2000) p. 177]. The delays model the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition. Furthermore, it is assumed that there is inter-specific competition between the plasmid-bearing and plasmid-free organisms as well as intra-specific competition within each population. Conditions for boundedness of solutions and existence of non-negative equilibrium are given. Analysis of the extinction of the organisms, including plasmid-bearing and plasmid-free organisms, and the uniform persistence of the system are also carried out. By constructing appropriate Liapunov-like functionals, some sufficient conditions of global attractivity to the extinction equilibria are obtained and the combined effects of the delays and the inhibitor are studied.

Some conditions for boundedness of solutions of difference volterra equations

Some conditions for boundedness of solutions of difference volterra equations

Conditions for the Local Boundedness of Solutions of the Navier–Stokes System in Three Dimensions

We show that a weak solution of the Navier-Stokes system is locally bounded if there is some ε>0so that either or

Solution estimates for abstract nonlinear time-variant differential-delay equations

Semilinear time-variant functional–differential equations in a Banach space are considered. A solution estimate is derived. By that estimate, explicit conditions for boundedness of solutions and stability are established.

On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order

There are given sufficient conditions for the ultimate boundedness of solutions and for the existence of periodic solutions of a certain vector differential equation of third-order.

A Generalization of Volterra Models with Continuous Time Delay in Population Dynamics: Boundedness and Global Asymptotic stability

We introduce a generalization of Volterra models with continuous time delay by adding a nonnegative linear vector function of the species. Sufficient conditions for boundedness of solutions and global asymptotic stability of an equilibrium are provided. Applications from population dynamics are presented. One concerns a predator-prey Volterra model with prey-refuges and continuous time delay, and another deals with a Volterra system with currents of immigration for some species. Furthermore, for this kind of model, by a suitable homotopy function, we provide a sufficient condition for the existence of the globally asymptotically stable equilibrium. Finally an example of a simple model in which two predators are competing for one prey which can take shelter is presented.

Global Asymptotic Stability of Lotka–Volterra Diffusion Models with Continuous Time Delay

We consider the global asymptotic stability of diffusion models with multiple species heterogeneous patches and incorporate the effect of continuous time delays for species in each patch. For two different patches connected by diffusion with multiple species, and with patch dynamics governed by Lotka–Volterra models with continuous time delays (and constant influxes), we give sufficient conditions for boundedness of solutions and the existence of a nonnegative equilibrium point, providing also a candidate for a Lyapunov function. By applying the homotopy function technique for different symbiotic patches with time delays, we provide a sufficient condition for the existence of a positive globally asymptotically stable equilibrium point. Further, we discuss the global asymptotic stability for the prey-predator model, where the prey has its shelter from predation and only the predator can diffuse between patches. We consider also the existence condition for the globally asymptotically stable equilibrium for the model with two identical patches and continuous time delays.

Analysis of three species models of mutualism in predator-prey and competitive systems

Mutualism arises in many qualitatively different ways, but previous 2-species models of mutualism correspond to only a few of these. Following Addicott and Freedman (1983), we present two models of mutualism in which interactions among three species lead to mutualism between two of them. One model involves interactions among a predator, a mutualist-prey, and a mutualist, while the other involves interactions among a competitor, a mutualist-competitor, and a mutualist. Given biologically reasonable constraints upon the functions in the model, we present (1) the conditions for boundedness of solutions, (2) the equilibria and their local stability, and (3) the conditions for the existence of small amplitude periodic solutions, a behavior not predicted by 2-species models of mutualism. The models include costs to the mutualist-prey or mutualist-competitor of associating with the mutualist. Analysis of special cases shows that mutualism can cause the extinction of the predator, or the reversal of competitive outcomes.

Sufficient conditions for boundedness of solutions of linear differential equations

In this paper we obtain sufficient conditions for boundedness of every solution and its derivatives of a linear differential equation. We will also show that our conditions are sharp.

Periodic solutions in n dimensions and Volterra equations

If a nonlinear autonomous n-dimensional system of ordinary differential equations has a bounded solution with a certain uniform stability property, this solution approaches an almost periodic solution with the same stability property. (More precisely, the almost periodic solution is in the set of ω-limit points of the given solution.) If the bounded solution has, in addition to the uniform stability property, an asymptotic stability property, then the solution approaches a periodic solution with the same stability properties. Practical (i.e., computable) sufficient conditions for boundedness of solutions are obtained. The results are applied to generalized Volterra equations.