Über die Invarianz konvexer Mengen und Differentialungleichungen in einem normierten Raume

Volkmann, Peter

doi:10.1007/bf01629254

Cited by 34 publications

(28 citation statements)

References 7 publications

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“…# v 1 ptq´f pvptqq ≤ w 1 ptq´f pwptqq pt P ra, bqq, vpaq ≤ wpaq ùñ vptq ≤ wptq pt P ra, bqq is valid, see [7,Satz 2]. From (3), we obtain (4) mintx 1 , .…”

Section: Preliminariesmentioning

confidence: 99%

Inequalities and Means from a Cyclic Differential Equation

Herzog¹,

Kunstmann²

2014

Demonstratio Mathematica

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Abstract. We prove that the solution of the cyclic initial value problem u 1 k " 1{2´u k {pu k`1`uk`2 q pk P Z{nZq, up0q " x is convergent to an equilibrium µpxqp1, . . . , 1q, and study the properties of the function x Þ Ñ µpxq and its relation to Shapiro's inequality. IntroductionIn 1954 H. S. Shapiro [6] raised the question of provingfor all n P Nzt1, 2u and all x 1 , . . . , x n ą 0. Meanwhile, after the investigations of numerous authors, (1) is known to be false in general for even n ≥ 14 and all n ≥ 25. Moreover (1) Inspired by Shapiro's inequalities, we investigated the following corresponding initial value problem, which turns out to have some interesting aspects in its asymptotic behaviour. In the sequel, let always n ≥ 3, and let all indexes of vectors in R n be understood modulo n, that is from Z n :" Z{nZ. Let D :" tx " px 1 , . . . , x n q P R n : x k ą 0 pk P Z n qu and let f : D Ñ R n be defined by f pxq " pf 1 pxq, . . . , f n pxqq "ˆ1 2´x k x k`1`xk`2˙k PZn .2010 Mathematics Subject Classification: 26D15, 34C12.

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“…# v 1 ptq´f pvptqq ≤ w 1 ptq´f pwptqq pt P ra, bqq, vpaq ≤ wpaq ùñ vptq ≤ wptq pt P ra, bqq is valid, see [7,Satz 2]. From (3), we obtain (4) mintx 1 , .…”

Section: Preliminariesmentioning

confidence: 99%

Inequalities and Means from a Cyclic Differential Equation

Herzog¹,

Kunstmann²

2014

Demonstratio Mathematica

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“…[1] where the one dimensional case is treated, so one only needs to show that u: [0, 1 ] -► C. This is done (Theorem 2) by using results and techniques formerly used by Redheffer and Walter [4] and in [8], [9], [10] in the study of invariance properties of sets relative to initial value problems for first order equations. A final result (Theorem 3) shows that it suffices to assume / to be defined on [0, 1] x C, provided the continuity of/relative to t is uniform with respect to x EC.…”

Section: Klaus Schmitt and Peter Volkmannmentioning

confidence: 99%

“…(For general closed, convex C(18) has been established in [8] for/defined by f(t, x + %p) = f(t, x) -4L%p. That result, however, is not sufficient for our purposes.)…”

Section: Klaus Schmitt and Peter Volkmannmentioning

confidence: 99%

Boundary Value Problems for Second Order Differential Equations in Convex Subsets of a Banach Space

Schimrigk¹,

Volkmann²

1976

Transactions of the American Mathematical Society

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“…The next result that will be used in the proof of Theorem 1 is a comparison theorem for differential inequalities (see [14]): …”

Section: Preliminariesmentioning

confidence: 99%

On Quasimonotone Homeomorphisms in Ordered Banach Spaces

Herzog¹

2003

Demonstratio Mathematica

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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).

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Über die Invarianz konvexer Mengen und Differentialungleichungen in einem normierten Raume

Cited by 34 publications

References 7 publications

Inequalities and Means from a Cyclic Differential Equation

Inequalities and Means from a Cyclic Differential Equation

Boundary Value Problems for Second Order Differential Equations in Convex Subsets of a Banach Space

On Quasimonotone Homeomorphisms in Ordered Banach Spaces

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