Set differential equations are usually formulated in terms of the Hukuhara differential. As a consequence, the theory of set differential equations is perceived as an independent subject, in which all results are proved within the framework of the Hukuhara calculus. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of ℝd with their support functions. Using this representation, standard existence and uniqueness theorems for ordinary differential equations can be applied to set differential equations. We provide a geometric interpretation of the main result, and demonstrate that our approach overcomes the heavy restrictions that the use of the Hukuhara differential implies for the nature of a solution.

In this paper we determine theHausdorff measure of noncompactness on the sequence space $n\left( \phi \right)$ of $\text{W}\text{.}\,\text{L}\text{.}\,\text{C}.$ Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach sequence spaces $n\left( \phi \right)$ and $m\left( \phi \right)$. Our aim is to present some existence results for infinite systems of differential equations formulated with the help of measures of noncompactness.

We use an intermediate value theorem for quasi-monotone increasing functions to prove the existence of the smallest and the greatest solution of the Dirichlet problem u″ + f(t, u) = 0, u(0) = α, u(1) = β between lower and upper solutions, where f:[0,1] × E → E is quasi-monotone increasing in its second variable with respect to a regular cone.

We prove that Mazur's functional characterization for one-sided estimates can be restricted to smaller classes of functionals in the case in which the functions under consideration are continuous. We apply this result to stability problems for dynamical systems in $l^\infty$, and in the Banach space of all selfadjoint operators on a Hilbert space.

A construction of differential constraints compatible with the Gibbons-Tsarev equation is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations that are used in searching for Lie operators. We introduce the notion of an invariant solution under an involutive distribution and give sufficient conditions for existence of such solutions.

This paper is devoted to the well-posedness of abstract Cauchy problems for quasi-linear evolution equations. The notion of Hadamard well-posedness is considered, and a new type of stability condition is introduced from the viewpoint of the theory of finite difference approximations. The result obtained here generalizes not only some results on abstract Cauchy problems closely related with the theory of integrated semigroups or regularized semigroups but also the Kato theorem on quasi-linear evolution equations. An application to some quasi-linear partial differential equation of weakly hyperbolic type is also given.

We analyse the limit behaviour of a stochastic structured metapopulation model as the number of its patches goes to infinity. The sequence of probability measures associated with the random process, whose components are the proportions of patches with different number of individuals, is tight. The limit of every convergent subsequence satisfies an infinite system of ordinary differential equations. The existence and the uniqueness of the solution are shown by semigroup methods, so that the whole random process converges weakly to the solution of the system.

In this paper we discuss the asymptotic behaviour, as t → ∞, of the integral solution u(t) of the non-linear evolution equation where {A(t)}t≥0 is a family of m-dissipative operators in a Hilbert space H, and g ∈ Lloc (0, ∞ H).We give some sufficient conditions and some sufficient and necessary conditions to ensure that are weakly convergent.