A Differential Equation with a State-Dependent Queueing Delay

Abstract

We consider a differential equation with a state-dependent delay motivated by a queueing process. The time delay is determined by an algebraic equation involving the length of the queue for which a discontinuous differential equation holds. The new type of state-dependent delay raises some problems that are studied in this paper. We formulate an appropriate framework to handle the system, and show that the solutions define a Lipschitz continuous semiflow in the phase space. The second main result guarantees the existence of slowly oscillating periodic solutions.

Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

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In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equationẋ(t) = f (t, x(t), x [2] (t), ..., x [n] (t)). As n = 2, this equation can be regarded as a mixed-type functional differential equation with state-dependenceẋ(t) = f (t, x(t), x(T (t, x(t)))) of a special form but, being a nonlinear operator, n-th order iteration makes more difficulties in estimation than usual state-dependence.

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“…Introduction. In this paper we consider a general iterative functional differential equation (abbreviated as IFDE) as followṡ x(t) = f (t, x(t), x [2] (t), ..., x [n] (t)),…”

“…x [n] (t) = x(x [n−1] (t)) for all n ∈ N and x [0] (t) = t. It is an important class of functional differential equations (abbreviated as FDE), and has been widely investigated by scholars. For example, in 1968 Andrzej ( [1]) proved the existence and uniqueness of solutions for (1) when the right hand side of (1) is f (t, x(t), x [2] (t)); in 1984 Eder ( [10]) gave the existence, uniqueness and continuous dependence on initial data of analytic solutions for (1) when the right hand side of (1) is x [2] (t); in 1990 Wang ( [38]) gave the existence, nonexistence and continuation of monotonic C 1 solutions for (1) when the right hand side of (1) is f (x [2] (t)); in 1993 Fečkan ([11]) further gave the existence of C 1 solutions for the above equation;…”

“…which is also an important class of FDE and many dynamic phenomena can be characterized by it, see, e.g., population ecology ( [22]), physiology ( [3]), economics ( [18]), engineering ( [19]), materials science ( [29]), electrodynamics ( [8]), neural networks ( [5]) and control theory ( [20]). In particular, when T (t, x(t)) < t in a vicinity of initial time t 0 , SDM-FDE (2) is the well-known state-dependent delay FDE (see, e.g., [2,7,13,16,17,21,24,28,37]). Currently, there also exist many articles on SDM-FDE (2) which mainly study the existence of solutions.…”

“…In the next year, Dunkel ([9]) used the successive approximation to study the existence of solutions for the one-dimensional (2) associated with initial condition x(t 0 ) = t 0 > 0, when the right hand side of (2) is f (x(T (x(t)))). In 1971, Grimm ([14]) proved the existence of solutions for a generalized one-dimensional (2). Later, Gal ([12]) applied the Contraction Mapping Principle to give the existence theorem of abstract (2) with a densely defined and closed linear operator.…”