Bounded positive solutions of an iterative three-point boundary-value problem with integral boundary condtions

"Positive periodic solutions for a class of first-order iterative differential equations with an application to a hematopoiesis model"

“…[10] recently studied the maximal and minimal nondecreasing bounded solutions for a first order iterative differential equation. The situation is the same with the higher order equations as very few contributions that have been made so far (see [4,5,6,8,13,14]).…”

On the existence and uniqueness of positive periodic solutions of neutral differential equations

“…− H(...), where A(.) = f (t, x(t − τ(t))) − a(t)x(t), and H(...) = H(t, x(t), x [2] (t), ..., x [n] (t)), and they proved two new results in relation to the EUPPSs such that the solution depends upon the functions of the NDE above.…”

“…where x ∈ R, τ i > 0, τ i ∈ R, H(.) = H(t, x(t), x [2] (t), ..., x [n] (t)), x [n] (t) is the iterative term and stands for x composed with itself n times, for example, x [2] (t) = x(x(t)), c i ∈ C([0, T ], (0, 1)), a ∈ C(R, (0, ∞)), and f ∈ C([0, T ] × R 2 , (0, ∞)) and H ∈ C([0, T ] × R n , (0, ∞)) are periodic functions, i.e., c i (t + T ) = c i (t), a(t + T ) = a(t), f i (t + T, x(t), x(t − τ i )) = f i (t, x(t), x(t − τ i )) and H(t + T, x(t), x [2] (t), ..., x [n] (t)) = H(t, x(t), x [2] (t), ..., x [n] (t)). Furthermore, we presume that the functions f i and H(...) satisfy the Lipschitz condition in their respective arguments;…”

Existence, uniqueness, continuous dependence and Ulam stability of mild solutions for an iterative fractional differential equation

“…Iterative differential equations often arise in the modeling of a wide range of natural phenomena such as disease transmission models in epidemiology, two-body problem of classical electrodynamics, population models, physical models, mechanical models and other numerous models. This kind of equations which relates an unknown function, its derivatives and its iterates, is a special type of the so-called differential equations with state-dependent delays, see [5,9,19] and the references therein.…”