Trichotomy and bounded solutions of nonlinear differential equations

summary:The existence of bounded solutions for equations $x'=A(t)x+f(t,x)$ in Banach spaces is proved. We assume that the linear part is trichotomic and the perturbation $f$ satisfies some conditions expressed in terms of measures of noncompactness

On quadratic integral equations in Orlicz spaces

AbstractIn this paper we study the quadratic integral equation of the formx(t)=g(t)+λx(t)∫abK(t,s)f(s,x(s))ds. Several existence theorems for a.e. monotonic solutions in Orlicz spaces are proved for strongly nonlinear functions f. The presented method of the proof can be easily extended to different classes of solutions

Non-compact perturbations of mm-accretive operators in general Banach spaces

summary:In this paper we deal with the Cauchy problem for differential inclusions governed by $m$-accretive operators in general Banach spaces. We are interested in finding the sufficient conditions for the existence of integral solutions of the problem $x'(t)\in -A x(t)+f(t,x(t))$, $x(0)=x_0$, where $A$ is an $m$-accretive operator, and $f$ is a continuous, but non-compact perturbation, satisfying some additional conditions

A note on Peano’s Theorem on time scales

AbstractIn this paper we investigate the dynamic Cauchy problem in Banach spaces. We check how dense a time scale must be in such a way that Peano’s Theorem holds and we present a counterexample to Peano’s Theorem on a time scale with only one right dense point

Differential inclusions and multivalued integrals

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 by a short survey about multivalued integration including Pettis and Henstock-Kurzweil-Pettis multivalued integrals

Monotonic solutions for quadratic integral equations

Using the Darbo fixed point theorem associated with the measure of noncompactness, we establish the existence of monotonic integrable solution on a half-line ℝ₊ for a nonlinear quadratic functional integral equation

Kneser-type theorem for the Darboux problem in Banach spaces

summary:In this paper we study the Darboux problem in some class of Banach spaces. The right-hand side of this problem is a Pettis-integrable function satisfying some conditions expressed in terms of measures of weak noncompactness. We prove that the set of all local pseudo-solutions of our problem is nonempty, compact and connected in the space of continuous functions equipped with the weak topology

Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces

We investigate the structure of the set of solutions of the Cauchy problem x' = f(t,x), x(0) = x₀ in Banach spaces. If f satisfies a compactness condition expressed in terms of measures of weak noncompactness, and f is Pettis-integrable, then the set of pseudo-solutions of this problem is a continuum in $C_{w}(I,E)$, the space of all continuous functions from I to E endowed with the weak topology. Under some additional assumptions these solutions are, in fact, weak solutions or strong Carathéodory solutions, so we also obtain Kneser-type theorems for these classes of solutions