Generating Flows on Metric Spaces

Abstract

ŽThe notion of an arc field on a locally complete but not necessarily locally . compact metric space is introduced as a generalization of a vector field on a manifold. Generalizing the Cauchy᎐Lipschitz Theorem, sufficient conditions on arc fields are given under which the existence and uniqueness of solution curves and flows are proven. A continuous analog of an iterated function system is given as an example. ᮊ

Differential equations in metric spaces with applications

“…The present construction is similar to that in [4,7,21,23,24]. On one hand, here we need the function ω to estimate the speed of convergence to 0 in (2.7), while in [23, (3.17)] or, equivalently, [22,Condition 4.…”

Infinitesimal calculus in metric spaces

“…a class of continuous maps h : X × [0, 1] → X such that h(x, 0) = x for each x ∈ X. Such an approach is adopted in [5], [6], [19] and in the works using mutational spaces. Let us note however that in such contributions the arcs t → h(x, t) are not supposed to be cadences (nor even rhythmed); on the other hand, strong uniform estimates are required.…”

“…Among the fields which required such generalizations are the following topics: differential equations ( [31]), duality ([50], [54]...), evolution of domains ( [5], [6], [57], [58]), geometry ([34]- [36]), image reconstruction ( [29], [47], [49]), mechanics ([42], [51]), morphogenesis ( [5]), nonlinear analysis and optimization ( [2], [18], [25], [22]), shape optimization ( [1], [5], [11], [13], [24], [37], [27], [28], [62]), stochastic problems ( [45], [56]), viability and invariance ( [26], [59]). Several models exist: Cartesian squares, metric measure spaces ( [2], [21], [38], [39], [40]...), mutational spaces ( [5], [6], [26]- [28]...) and their variants ( [19]...) with various purposes.…”

$\R \times BL^* $ Valued Consumer Resource Model

“…However, R + × M + is a complete metric space and not a Banach Space. With slight modifications of the definitions one could use the techniques of either mutational analysis [15,16,17] or differential equations in metric spaces [18] or arcflows of arcfields [19,20] to generate a semiflow that satisfies the equivalent of the initial value problem in semiflow theory language.…”