Bol. Soc. Paran. Mat. (3s.) v. 2023 (41) : 1–9.
©SPM –ISSN-2175-1188 on line ISSN-0037-8712 in press
SPM: www.spm.uem.br/bspm doi:10.5269/bspm.62662
On Periodic Solutions of a Recruitment Model with Iterative Terms and a Nonlinear
Harvesting
Lynda Mezghiche and Rabah Khemis
abstract: We consider a first-order delay differential equation involving iterative terms. We prove the
existence of positive periodic and bounded solutions by utilizing the Schauder’s fixed point theorem combined
with the Green’s functions method. Furthermore, by virtue of the Banach contraction principle, the uniqueness
and stability of the solution are also analyzed. Our new results are illustrated with two examples that show
the feasibility of our main findings.
Key Words: Existence, uniqueness, periodic solution, iterative functional differential equation, con-
tinuous dependence.
Contents
1 Introduction 1
2 Relevant preliminaries 2
3 Main results 3
3.1 Existence ............................................. 4
3.2 Uniqueness ............................................ 6
3.3 Stability .............................................. 6
4 Examples 8
5 Conclusion 9
1. Introduction
Our foremost concern in this work is to establish some sufficient criteria that assure the existence,
uniqueness and stability of positive periodic and bounded solutions to the following first-order differential
equation with a time-varying delay and iterative terms:
y
′
(t)+ k (t) y (t)= ay
[2]
(t) − b
y
[2]
(t)
2
− qy
[2]
(t) E (t, y (t) ,y (t − τ (t))) , (1.1)
where y
[2]
(t)= y (y (t)) is the second iterate of y, a,b,q > 0,k ∈ C (R, (0, +∞)) ,τ ∈ C (R, (0, +∞))
are two w-periodic functions and E ∈ C
(
R
3
, (0, +∞)
)
is a w-periodic function with respect to the first
variable and satisfies the following Lipschitz condition:
|E (t, y
1
,y
2
) − E (t, z
1
,z
2
)|≤ ℓ
1
|y
1
− z
1
| + ℓ
2
|y
2
− z
2
| . (1.2)
It is worth noting here that equation (1.1) which involves the second iterate of the state variable can be
seen as a special type of the following delayed differential equation:
y
′
(t)+ k (t) y (t)= ay (t − τ
1
(t, y (t))) − b (y (t − τ
1
(t, y (t))))
2
− qy (t − τ
1
(t, y (t))) E (t, y (t) ,y (t − τ (t))) ,
where τ
1
(t, y (t)) = t − y (t) can denote the gestation period, the life cycle, the time taken from birth
to maturity or between oviposition and eclosion of adults and so on. The dependence on the population
2010 Mathematics Subject Classification: 34K13, 34A12, 47H10.
Submitted February 23, 2022. Published May 22, 2022
1
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