Extensions of Schauder\u27s and Darbo\u27s Fixed Point Theorems

In this paper, some new extensions of Schauder\u27s and Darbo\u27s fixed point theorems are given. As applications of the main results, the existence of global solutions for first-order nonlinear integro-differential equations of mixed type in a real Banach space is investigated

A new result on impulsive differential equations involving non-absolutely convergent integrals

AbstractIn this paper we obtain, as an application of a Darbo-type theorem, global solutions for differential equations with impulse effects, under the assumption that the function on the right-hand side is integrable in the Henstock sense. We thus generalize several previously given results in literature, for ordinary or impulsive equations

Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces

We prove two existence theorems for the integrodifferential equation of mixed type: x'(t)=f(t,x(t),∫0tk1(t,s)g(s,x(s))ds,∫0ak2(t,s)h(s,x(s))ds), x(0)=x0, where in the first part of this paper f, g, h, x are functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil (HK). In the second part f, g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functions f, g, h, x satisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness