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Convergence results for nonlinear evolution inclusions

Tiziana Cardinali, Francesca Papalini (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider evolution inclusions of subdifferential type. First, we prove a convergence result and a continuous dependence proposition for abstract Cauchy problem of the form u' ∈ -∂⁻f(u) + G(u), u(0) = x₀, where ∂⁻f is the Fréchet subdifferential of a function f defined on an open subset Ω of a real separable Hilbert space H, taking its values in IR ∪ {+∞}, and G is a multifunction from C([0,T],Ω) into the nonempty subsets of L²([0,T],H). We obtain analogous results for the multivalued...

Derivative of the norm of a linear mapping and its application to differential equations

František Tumajer (1992)

Applications of Mathematics

In this paper the notion of the derivative of the norm of a linear mapping in a normed vector space is introduced. The fundamental properties of the derivative of the norm are established. Using these properties, linear differential equations in a Banach space are studied and lower and upper estimates of the norms of their solutions are derived.

Differential Equations in Abstract Cones

Jankowski, Tadeusz (2000)

Serdica Mathematical Journal

We extend the method of quasilinearization to differential equations in abstract normal cones. Under some assumptions, corresponding monotone iterations converge to the unique solution of our problem and this convergence is superlinear or semi–superlinear

Differential equations in banach space and henstock-kurzweil integrals

Ireneusz Kubiaczyk, Aneta Sikorska (1999)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, using the properties of the Henstock-Kurzweil integral and corresponding theorems, we prove the existence theorem for the equation x' = f(t,x) and inclusion x' ∈ F(t,x) in a Banach space, where f is Henstock-Kurzweil integrable and satisfies some conditions.

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...

Differential inclusions and multivalued integrals

Kinga Cichoń, Mieczysław Cichoń, Bianca Satco (2013)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider the nonlocal (nonstandard) Cauchy problem for differential inclusions in Banach spaces x'(t) ∈ F(t,x(t)), x(0)=g(x), t ∈ [0,T] = I. Investigation over some multivalued integrals allow us to prove the existence of solutions for considered problem. We concentrate on the problems for which the assumptions are expressed in terms of the weak topology in a Banach space. We recall and improve earlier papers of this type. The paper is complemented...

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