Sentence examples for boundedness for the solutions from inspiring English sources

In order to study the positivity and boundedness for the solutions of system (1.5), we denote the function on the right hand of system (1.5) as (mathbf{G}= xg_{1},yg_{2})) in which begin{aligned}& g_{1} x,y)=rbiggl 1-frac{x}{K}biggr)-frac{lambda(1-beta)^{n}x^{n-1} y}{1+lambda h(1-beta)^{n}x^{n}}, & g_{2}(x,y)=rbiggl 1-frac{x-beta)^{n}{^{n} }{1+lambda h(1-beta)^{n}x^{n}}-d.

Furthermore, a formal proof for boundedness of the solutions of the dynamic compensator remains as a topic of future work.

In the slow system, except for the positivity and boundedness of the solutions, the existence for disease-free, unique endemic and two endemic equilibria are obtained, and the sufficient conditions on the local stability for disease-free and unique endemic equilibria are established.

The boundedness of the state-trajectory solution at sampling instants guarantees the global boundedness of the solutions for all time.

The aim of this paper is to study the global existence and boundedness of the solutions for the parabolic-parabolic chemotaxis system with linear or nonlinear diffusion and logistic source (1.1).

We firstly investigate the global existence and the uniform boundedness of the solutions for (1.8), then prove the global asymptotic stability of the positive equilibrium (u*, v*) of (1.8) by an important lemma from [21].

Then combining equation (4.28) with Lemma 4.4, we see that there exists a positive constant C 2 such that V k + 1 − V k ≤ − C 2 for all  k ≥ 0, and hence V k ≤ V 0 − k C 2 for all  k ≥ 0. This is a contradiction, since lim k → ∞ ( V 0 − k C 2 ) = − ∞ and V k is lower bounded for all k by using the boundedness of the solutions x k i for 1 ≤ i ≤ n. □.

The boundedness of the solutions is studied in Section 4.

Direct applications of fixed point theory to the study of the stability of dynamic systems including the property of ultimate boundedness for the trajectory solutions having mixed non-expansive and expansive properties through time or being subject to impulsive controls have been given in [21] and [24, 25].

Recently, by virtue of the recession method, Ansari established some necessary and/or sufficient conditions for the nonemptiness and boundedness of the solution set for the SGVEP [5].

In addition, some new sufficient conditions are presented for the nonemptiness and boundedness of the solution set for the GVEP are proposed under the condition that it is strictly feasible in the strong sense.