Strictly Cooperative Systems with a First Integral

Mierczyński, Janusz

doi:10.1137/0518049

Cited by 40 publications

(28 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 1. We could also have proved global asymptotic stability using results of Smillie [26] or even of Mierczynski [22]. But these require verification of stronger monotonicity properties of the flow, which has been avoided here.…”

Section: A Chemical Reaction Networkmentioning

confidence: 99%

Monotone Chemical Reaction Networks

2006

View full text Add to dashboard Cite

We analyze certain chemical reaction networks and show that every solution converges to some steady state. The reaction kinetics are assumed to be monotone but otherwise arbitrary. When diffusion effects are taken into account, the conclusions remain unchanged. The main tools used in our analysis come from the theory of monotone dynamical systems. We review some of the features of this theory and provide a self-contained proof of a particular attractivity result which is used in proving our main result.

show abstract

Section: A Chemical Reaction Networkmentioning

confidence: 99%

Monotone Chemical Reaction Networks

2006

View full text Add to dashboard Cite

show abstract

“…For example, Nakajima [9] proved the asymptotic periodicity of bounded solutions for periodic and cooperative gross-substitute systems with a linear first integral, and this convergence result was further generalized to the almost periodic case by Sell and Nakajima [13]. Mierczynski [8] proved the convergence of bounded solutions for autonomous and strictly cooperative systems with a first integral, and Jiang [6] obtained the same result in the case where these systems are cooperative. In 1993, Tang, Kuang and Smith [17] considered strictly cooperative systems on R n + with a general class of first integrals in R n + , and conjectured that every bounded solution converges to an almost periodic solution.…”

Section: Introductionmentioning

confidence: 80%

Convergence in almost periodic cooperative systems with a first integral

Shen

Zhao

2004

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. This paper is to investigate the asymptotic dynamics in almost periodic cooperative systems with a first integral. By appealing to the theory of skew-product semiflows we establish the asymptotic almost periodicity of bounded solutions to such systems, which extends the existing convergence results for time independent and periodic cooperative systems with a first integral and proves a conjecture of

show abstract

“…Functional conditions are a useful tool in the theory of quasimonotone increasing dynamical systems since in applications they lead to conditions which are often easy to deal with. For a survey on the subject we refer to [3], [5], [7], [9], [10], [11], [12], and the references given there.…”

Section: Remarksmentioning

confidence: 99%

On Quasimonotone Homeomorphisms in Ordered Banach Spaces

Herzog¹

2003

Demonstratio Mathematica

View full text Add to dashboard Cite

Abstract. Let E be a Banach space ordered by a cone K, and let / : E -• E be locally Lipschitz continuous and quasimonotone increasing such that ^(f(y)-f(x))< -L^(y-x) (x < y) for a linear positive functional ^ and L > 0. We prove, under suitable conditions on K, f and "i, that / is a homeomorphism with decreasing and Lipschitz continuous inverse. IntroductionLet (E, || • ||) be a real Banach space, ordered by a cone K. A cone K is a closed convex subset of E with AK C K (A > 0), and Kf)(-K) = {0}. As usual x < y :y -x € K. We will always assume that K is reproducing, that is K -K = E. Then, the set K* = {

0 (x > 0)} is a cone in the space of all continuous linear functionals E*, the dual cone.A functional \I/ 6 K* is called norrning if there are constants 0 < a < ¡3 such thatThe aim of this paper is to prove the following result: THEOREM 1. Let f : E -» E be locally Lipschitz continuous, bounded on bounded subsets of E, and quasimonotone increasing. Let there exist a norrning functional6 K* and L > 0 such that Then f : E -> E is a homeomorphism, and f 1 : E -> E is monotone decreasing and Lipschitz continuous. Moreover each initial value problemis uniquely solvable on [0, oo), and the solution satisfiesfor a constant M > 0. Remarks:1. In particular Theorem 1 applies to linear mappings: Let A : E -• E be linear and continuous, let A* : E* -* E* be its adjoint, and let $ € K* be norming. If . A*\& < -L^/ for some L > 0, then A is an isomorphism. A related result for cones with nonempty interior can be found in [4].2. A finite dimensional version of Theorem 1 is due to the author [6]. In this result it is assumed that K has nonempty interior and that / is merely continuous. In the result above K may have empty interior.3. Functional conditions are a useful tool in the theory of quasimonotone increasing dynamical systems since in applications they lead to conditions which are often easy to deal with. For a survey on the subject we refer to Examples of ordered Banach spaces with reproducing cone and norming functionals are:REMARK: The cone in 4. is reproducing since it contains the reproducing cone Ko = {u : u(l) > > 0}, which is discussed in section 4. The following example shows that, in general, condition (1) in Theorem 1 does not lead to a bijective mapping in case that K is only assumed to be total, that is K -K = E. Consider E = co(N,M) endowed with the maximum norm and ordered by the coneThe cone K is total. To see this, recall that the finite sequences are dense in co(N, R).

show abstract

Strictly Cooperative Systems with a First Integral

Cited by 40 publications

References 10 publications

Monotone Chemical Reaction Networks

Monotone Chemical Reaction Networks

Convergence in almost periodic cooperative systems with a first integral

On Quasimonotone Homeomorphisms in Ordered Banach Spaces

Contact Info

Product

Resources

About