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Differential equations in metric spaces

Jacek Tabor (2002)

Mathematica Bohemica

We give a meaning to derivative of a function u ℝ → X , where X is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space 𝒯 x X of x ∈ X . Let u , v [ 0 , 1 ) → X , u ( 0 ) = v ( 0 ) be continuous at zero. Then by the definition u and v are in the same equivalence class if they are tangent at zero, that is if lim h → 0 + d ( u ( h ) , v ( h ) ) h = 0 . By 𝒯 x X we denote...

Exponential expansiveness and complete admissibility for evolution families

Mihail Megan, Bogdan Sasu, Adina Luminiţa Sasu (2004)

Czechoslovak Mathematical Journal

Connections between uniform exponential expansiveness and complete admissibility of the pair ( c 0 ( ℕ , X ) , c 0 ( ℕ , X ) ) are studied. A discrete version for a theorem due to Van Minh, Räbiger and Schnaubelt is presented. Equivalent characterizations of Perron type for uniform exponential expansiveness of evolution families in terms of complete admissibility are given.

Galerkin approximations for nonlinear evolution inclusions

Shouchuan Hu, Nikolaos S. Papageorgiou (1994)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study the convergence properties of the Galerkin approximations to a nonlinear, nonautonomous evolution inclusion and use them to determine the structural properties of the solution set and establish the existence of periodic solutions. An example of a multivalued parabolic p.d.ei̇s also worked out in detail.

Identification problems for degenerate parabolic equations

Fadi Awawdeh, Hamed M. Obiedat (2013)

Applications of Mathematics

This paper deals with multivalued identification problems for parabolic equations. The problem consists of recovering a source term from the knowledge of an additional observation of the solution by exploiting some accessible measurements. Semigroup approach and perturbation theory for linear operators are used to treat the solvability in the strong sense of the problem. As an important application we derive the corresponding existence, uniqueness, and continuous dependence results for different...

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