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Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces

Guillaume Vigeral (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider some discrete and continuous dynamics in a Banach space involving a non expansive operator J and a corresponding family of strictly contracting operators Φ (λ, x): = λ J( 1 - λ λ x) for λ ∈ ] 0,1] . Our motivation comes from the study of two-player zero-sum repeated games, where the value of the n-stage game (resp. the value of the λ-discounted game) satisfies the relation vn = Φ( 1 n , v n - 1 ) (resp.  v λ = Φ(λ, v λ )) where J is the Shapley operator of the game. We study the evolution equation u'(t) =...

Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms

Thomas Lorenz (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarly to quasidifferential equations of Panasyuk, the so-called mutational equations of Aubin provide a generalization of ordinary differential equations to locally compact metric spaces. Here we present their extension to a nonempty set with a possibly nonsymmetric distance. In spite of lacking any linear structures, a distribution-like approach leads to so-called right-hand forward solutions. These extensions are mainly motivated by compact subsets of the Euclidean space...

Evolution equations with parameter in the hyperbolic case

Jan Bochenek, Teresa Winiarska (1996)

Annales Polonici Mathematici

The purpose of this paper is to give theorems on continuity and differentiability with respect to (h,t) of the solution of the initial value problem du/dt = A(h,t)u + f(h,t), u(0) = u₀(h) with parameter h ∈ Ω ⊂ ℝ m in the “hyperbolic” case.

Evolution inclusions in non separable Banach spaces

Francesco Saverio De Blasi, Giulio Pianigiani (1999)

Commentationes Mathematicae Universitatis Carolinae

We study a Cauchy problem for non-convex valued evolution inclusions in non separable Banach spaces under Filippov type assumptions. We establish existence and relaxation theorems.

Evolution inclusions of the subdifferential type depending on a parameter

Dimitrios A. Kandilakis, Nikolaos S. Papageorgiou (1992)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study evolution inclusions generated by time dependent convex subdifferentials, with the orientor field F depending on a parameter. Under reasonable hypotheses on the data, we show that the solution set S ( λ ) is both Vietoris and Hausdorff metric continuous in λ ∈ Λ . Using these results, we study the variational stability of a class of nonlinear parabolic optimal control problems.

Evolution Problems and Minimizing Movements

Ugo Gianazza, Massimo Gobbino, Giuseppe Savarè (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We recall the definition of Minimizing Movements, suggested by E. De Giorgi, and we consider some applications to evolution problems. With regards to ordinary differential equations, we prove in particular a generalization of maximal slope curves theory to arbitrary metric spaces. On the other hand we present a unifying framework in which some recent conjectures about partial differential equations can be treated and solved. At the end we consider some open problems.

Exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems

Shao-Yuan Huang, Ping-Han Hsieh (2023)

Czechoslovak Mathematical Journal

We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems - [ φ ( u ' ) ] ' = λ u p 1 - u N in ( - L , L ) , u ( - L ) = u ( L ) = 0 , where p > 1 , N > 0 , λ > 0 is a bifurcation parameter, L > 0 is an evolution parameter, and φ ( u ) is either φ ( u ) = u or φ ( u ) = u / 1 - u 2 . We prove that the corresponding bifurcation curve is ⊂ -shape. Thus, the exact multiplicity of positive solutions can be obtained.

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