Positivity 11 (2007), 523–535
2007 Birkh¨auser Verlag Basel/Switzerland
Positive Solutions of Equations with Nonlinear
Abstract. We consider nonlinear scalar equations with causal mappings. These
equations include diﬀerential, diﬀerential-delay, integral, integro-diﬀerential,
diﬀerence and other traditional equations. Conditions that provide the exis-
tence of positive solutions are established. The suggested approach enables us
to consider various classes of equations from the uniﬁed point of view.
Mathematics Subject Classiﬁcation (2000). 34 K 20; 34 K 99; 93 D 05; 93D25.
Keywords. causal mappings, positive solutions, diﬀerential equations, func-
tional diﬀerential equations, diﬀerence equations with continuous time.
1. Introduction and main deﬁnitions
The present paper is devoted to positive solutions of scalar equations with non-
linear causal mappings (operators). These equations include diﬀerential, diﬀeren-
tial-delay, integro-diﬀerential, diﬀerence equations with continuous time and other
traditional equations. For the details see the excellent book .
Many books and papers are devoted to positive solutions of various con-
crete classes of equations, such as functional diﬀerential equations (see for instance
[1, 2, 3, 8, 14, 16]) and diﬀerence equations, cf. [1, 2, 12] and references given
therein. But to the best of our knowledge the positivity conditions for equations
with causal mappings were not explored in the available literature.
We establish positivity conditions for solutions of a class of equations with
causal mappings. The suggested approach enables us to consider various classes of
equations from the uniﬁed point of view.
The paper is organized as follows. It consists of 7 sections. In this section
we deﬁne the causal mappings. The main result of the paper-Theorem 2.1 on the
existence of positive solutions is stated in Section 2. The proof of this theorem
is divided into a series of lemmas which are presented in Section 3. Sections 4
This research was supported by the Kamea Fund of the Israel.