We establish some results that concern the Cauchy-Peano problem in Banach spaces. We first prove that a Banach space contains a nontrivial separable quotient iff its dual admits a weak*-transfinite Schauder frame. We then use this to recover some previous results on quotient spaces. In particular, by applying a recent result of Hájek-Johanis, we find a new perspective for proving the failure of the weak form of Peano's theorem in general Banach spaces. Next, we study a kind of algebraic genericity...

In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDS epidemic model with an imperfect HIV vaccine, varying total population size and distributed time delay to become infectious due to intracellular delay between initial infection of a cell by HIV and the release of new virions. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique....

A basic mathematical model of the immune response when cancer cells are recognized is proposed. The model consists of six ordinary differential equations. It is extended by taking into account two types of immunotherapy: active immunotherapy and adoptive immunotherapy. An analysis of the corresponding models is made to answer the question which of the presented methods of immunotherapy is better. The analysis is completed by numerical simulations which show that the method of adoptive immunotherapy...

The objective of this work is the application of Krasnosel'skii's fixed point technique to prove the existence of periodic solutions of a system of coupled nonlinear integro-differential equations with variable delays. An example is given to illustrate this work.

This paper is concerned with a second-order functional differential equation of the form $x^{\text{'}\text{'}}\left(z\right)=x(az+b{x}^{\text{'}}\left(z\right))$ with the distinctive feature that the argument of the unknown function depends on the state derivative. An existence theorem is established for analytic solutions and systematic methods for deriving explicit solutions are also given.

We study existence of analytic solutions of a second-order iterative functional differential equation $x^{\text{'}\text{'}}\left(z\right)={\sum }_{j=0}^k{\sum }_{t=1}^{\infty }{C}_{t,j}\left(z\right){\left(x^{\left[j\right]}\left(z\right)\right)}^t+G\left(z\right)$ in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form $\alpha ²y^{\text{'}\text{'}}\left(\alpha z\right)y^{\text{'}}\left(z\right)=\alpha y^{\text{'}}\left(\alpha z\right)y^{\text{'}\text{'}}\left(z\right)+\left[y^{\text{'}}\left(z\right)\right]³[{\sum }_{j=0}^k{\sum }_{t=1}^{\infty }{C}_{t,j}\left(y\left(z\right)\right){\left(y\left({\alpha }^{j}z\right)\right)}^t+G\left(y\left(z\right)\right)]$ invertible analytic solutions of the form $y\left(\alpha y^{-1}\left(z\right)\right)$ for the original equation are obtained. Besides the hyperbolic case 0 < |α| < 1, we focus on α on the unit circle S¹, i.e., |α|=1. We discuss not only those α at resonance, i.e. at a root of unity, but also near resonance under the...