A Mathematical Journal
Vol. 7, No 3, (15 - 26). December 2005.

Some special classes of neutral
functional differential equations

Constantin Corduneanu
The University of Texas at Arlington
Box 19408, UTA, Arlington TX 76019

concord@uta.edu

ABSTRACT
This paper is dedicated to the investigation of existence, mainly local, of so-

lutions of two classes of neutral functional differential equations. A reduction
method and fixed point methods are emphasized.

RESUMEN

Este art́ıculo está dedicado al estudio de existencia, principalmente local, de
soluciones de dos clases de ecuaciones diferenciales funcionales neutrales. Un
método de reducción y de punto fijo son puestos con algún énfasis.

Key words and phrases: functional equation, neutral equation, existence
of solution

Math. Subj. Class.: 34K40

1 Introduction

In several recent papers of the author [3], [4], [5], as well as in some joint papers
with M. Mahdavi [7], [8], certain types of neutral functional equations (including
functional–differential ones) have been investigated in regard to the existence of so-



16 Constantin Corduneanu
7, 3(2005)

lutions. Such equation, in a rather general form, can be written as

(U x)(t) = (V x)(t), t ∈ [0, T ], (1)

or, in functional differential form,

d

dt
(U x)(t) = (V x)(t), t ∈ [0, T ]. (2)

In (1) and (2), U and V stand for some operators acting on convenient function
spaces, whose elements are defined on [0, T ]. The case [0, T ), T ≤ ∞, has been also
discussed, while the solution has been sought in various function spaces (usually, C,
Lp, 1 ≤ p < ∞).

Let us notice that equation (2), by integrating both sides on [0, t], t ≤ T , takes
the form (1):

(U x)(t) = C +
∫ t

0

(V x)(s)ds, t ∈ [0, T ]. (3)

Moreover, when V is a causal operator, the right hand side of (3) is also causal. The
case of causal operators has been dealt with in the author’s recent book [6].

The basic method of investigating the existence of solutions to equations(1) and
(2) consisted in reducing such equations to the simpler form

x(t) = (W x)(t), t ∈ [0, T ], (4)

and then applying existing results available for (4). In our book [2] we provided exis-
tence results for (4), based on the existing literature, while in [6] we have illustrated
how this method works for neutral equations like (1) or (2).

The aim of this paper is to investigate some classes of neutral equations encoun-
tered in the existing literature, by using mainly the above described method. In other
words, to reduce such equations to the form (4), and then to apply known results.
We are not necessarily intended to reobtain results already known in the literature,
but to see what kind of results one obtains by means of the above described method.

We shall particularly refer to the papers of T.A. Burton [1] and Loris Faina [9], in
which various classes of functional differntial equations, of neutral type, are investi-
gated.

2 Reduction of neutral equations to the form (4)

Let us start with the neutral equation

x′(t) = f (t, x(t), x′(t)), t ∈ [0, T ], (5)

investigated by Loris Faina [9] and other authors. This is the simplest form, in which
x and f take scalar values, or values belonging to IRn.

The initial value condition attached to (5) will be

x(0) = x0 ∈ IRn, (6)



7, 3(2005)
Some special classes of neutral functional differential equations 17

if one deals with the vector case.
Formally, let us denote

x′(t) = y(t), (7)

which implies under rather general assumptions (see below)

x(t) = x0 +
∫ t

0

y(s)ds. (8)

The neutral equation (5) can be now written as

y(t) = f
(

t, x0 +
∫ t

0

y(s)ds, y(t)
)

. (9)

Obviously, the right hand side of (9) engages only the values of y on the interval
0 ≤ s ≤ t (t ≤ T ).

This means that equation (9) is an equation of the form (4), in which the right
hand side is a causal operator (in y). There is, therefore, an equivalence between
the initial value problem (5), (6), and the problem (8), (9). This equivalence will be
further discussed when we precise the underlying function spaces. Since equation (9)
contains only the unknown y(t), we shall be able to investigate it in various function
spaces (continuous or measurable functions), by using known results.

In L. Faina [9] there are more general neutral functional differential equations than
(5). For instance, in (5) one assumes that f is a map from R × C(R) × L1(R) into
IRn, while the initial condition (6) is replaced by x(t) = ϕ(t), t ∈ (−∞, t0], t0 ∈ R
fixed. In other terms, the infinite delay is dealt with. This case can be also covered
by the scheme described above, though the procedure is more intricate.

In T.A. Burton [1], the following neutral functional differential equation is studied

x′(t) = f (t, x(t), x′(t − h(t)) + g(t, x(t), x(t − h(t)),

where 0 ≤ h(t) ≤ h0, h0 > 0 being fixed. To the above equation one attaches the
typical initial condition for delay equations, namely x(t) = ϕ(t), t ∈ [−h0, 0].

We shall rewrite Burton’s equation in the form

x′(t) = f (t, x(t), x′(α(t))) + g(t, x(t), x(α(t))), (10)

where α(t) is such that 0 ≤ α(t) ≤ t on some interval [0, T ]. The right hand side in
(10) can be regarded as a causal operator in x. Hence, (10) is also of the form (4).
Moreover, one can use the initial condition (6) for determining a (unique) solution to
(10), (6).

The literature on neutral functional equations is very rich, and a good amount of
references can be found in our book [6]. In many more cases than those illustrated
above, the method of reduction to equations of the form (1), with causal operator in
the right hand side, can be successfully applied.

In what follows, we shall dwell on the equations (5) and (9), trying to apply the
reduction procedure in these cases, as well as other methods.



18 Constantin Corduneanu
7, 3(2005)

3 The equation (9) in the space C([0, T ], IRn)

We shall consider in this section the equation (9) in the space C([0, T ], IRn). The
assumption to be made are of such a nature that the right hand side of this equation
represent a compact operator on C([0, T1], IR

n), with T1 ≤ T. Obviously, the Arzelà-
Ascoli criterion of compactness in C([0, T ], IRn) will be used.

The compactness of the operator

y(t) −→ f
(

t, x0 +
∫ t

0

y(s)ds, y(t)
)

, (11)

on C([0, T ], IRn) can be achieved under various sets of hypotheses. We shall describe
such a set of hypeotheses, which will imply the existence of a local solution to the
equation (9). Such a solution will generate a continuously differentiable solution to
the problem (5), (6).

Before we proceed with the statement of the hypotheses, it is instructive to look
at a simple example for the equation (5). Namely, we will choose f = 2x(t)x′(t) + 1.
This leads to the integral x(t) = x2(t) + t + c, from which we derive x(t) =
1
2

(
1 +

√
1 − 4c − 4t

)
. Choosing the initial value x0 = 1/2, we get c = 1/4, which

means the solution is x(t) =
1
2

(
1 + 2

√
−t

)
. This shows that we have no solutions of

(5) on any [0, T ], T > 0, for x0 = 1/2.
Therefore, a problem of the form (5), (6) may be deprived of (local) solutions,

even though the right hand side of (5) is quite a usual function.
We shall return now to the general problem (5), (6), and provide some conditions

which assure the existence of local solutions. But before getting into details, we
shall modify somewhat the equation (5), in order to encompass a larger category of
situations. Namely, let us rewrite (9) in the form

y(t) = f
(

t, x0 +
∫ t

0

y(s)ds; y
)

, (9)′

and assume the right hand side in (9)′ is defined on the set [0, T ]×IRn×C([0, T ], IRn).
We shall assume continuity of the map y −→ f, but further hypotheses will be for-
mulated. Compared to (9), the equation (9)′ involves now an operator in the right
hand side, defined on the space C([0, T ], IRn).

The following hypotheses will be made, in view of obtaining the existence of solu-
tions to the equation (9)′:

H1 The map

y −→ f
(

t, x0 +
∫ t

0

y(s)ds; y
)

(12)

is continuous from [0, T ] × IRn × C([0, T ], IRn) into IRn, and causal.



7, 3(2005)
Some special classes of neutral functional differential equations 19

H2 For each γ > 0, there exist two functions ω1(r) and ω2(r), continuous on [0, ∞),
ω1(0) = ω2(0) = 0, and positive for r > 0, such that∣∣∣∣f

(
t, x0 +

∫ t
0

y(s)ds; y
)
− f

(
u, x0 +

∫ u
0

y(s)ds; y
)∣∣∣∣ ≤

≤ ω1(|t − u|) + ω2(γ|t − u|),
(13)

for arbitrary t, u ∈ [0, T ], and all y ∈ C, with |y|C ≤ γ.
Let us notice the fact that choosing ω1(r) = αr, ω2(r) = βr, α, β > 0, the
condition (13) becomes a Lipschitz type continuity condition.

H3 The map (12) takes bounded sets in C([0, T ], IR
n), into bounded sets of IRn.

We can now prove the following (local) existence theorem for the equation (9)′.

Theorem 1. Consider the functional equation (9)′ in the space C([0, T ], IRn). As-
sume that the map (12) satisfies the hypotheses H1, H2 and H3. Then equation (9)′

has a local solution (i.e. defined on some interval [0, T1], T1 ≤ T ), provided f (0, x0; y)
is independent of y.

Moreover, the equation
x′(t) = f (t, x(t); x′), (5)′

under initial condition (6), has a local solution, which is continuously differentiable.

Remark 1. The localization is possible due to the causality of the operator (12)
(hypothesis H1).

Remark 2. In case f does not depend on the last argument, the equation (5)′

becomes x′(t) = f (t, x(t)), while hypotheses H2 and H3 are automatically satisfied in
case of continuity. The result of Theorem 1 reduces to the classical Peano’s existence
theorem. The existence of the functions ω1(r) and ω2(r) is a simple consequence of
the uniform continuity of f (t, x) on a set of the form [0, T ]×B, with B compact in IRn.
The imposition of hypothesis H2 is motivated by the fact that the last argument in

f

(
t, x0 +

∫ t
0

y(s)ds; y
)

belongs to an infinite dimensional space, i.e. to C([0, T ], IRn).

Proof of Theorem 1. The equation (9)′ is, according to our hypotheses, a functional
equation with causal operaor of the form (4). The hypotheses H1 and H2 assure the
continuity of the map (12) from C([0, T ], IRn) into itself. Based on hypothesis H3, the
map (12) from C([0, T ], IRn) into C([0, T ], IRn) is also compact. Indeed, according to
the hypothesis H3, the image of the ball |y|C ≤ γ is a bounded set in C([0, T ], IRn).
The inequality (13) in hypothesis H2 tells us that the image of the ball |x|C ≤ γ
consists of equicontinuous functions on [0, T ], with values in IRn. Since γ > 0 is an
arbitrary number, we conclude that the operator (12) is compact (takes bounded sets
into relatively compact sets). Hence Theorem 3.1 in [6] applies directly, keeping also



20 Constantin Corduneanu
7, 3(2005)

in mind that the operator (12) enjoys the property of fixed initial value. Consequently,
(9)′ has a local solution in some space C([0, T1], IR

n), with T1 ≤ T.
This result leads immediately to the existence of a local solution for the problem

(5)′, 6. This ends the proof of Theorem 1.

Remark 3. The case of measurable solutions to the equation (9)′, when the cor-
responding solutions to (5)′ will be absolutely continuous functions, can be treated
in the same manner as in the continuous case. One has to use Theorem 3.3 in [6],
instead of Theorem 3.1. We shall leave to the reader the task of formulating existence
results.

4 Existence of solutions to equation (10)

If we denote again x′(t) = y(t), and take into account the initial condition (6), then
equation (10) becomes

y(t) = f
(

t, x0 +
∫ t

0

y(s)ds, y(α(t))
)

+

+g
(

t, x0 +
∫ t

0

y(s)ds, y(α(t))
)

,

(14)

which is precisely of the form (4), with causal operator in the right hand side. This
is due to the assumption on α(t), namely 0 ≤ α(t) ≤ t, t ∈ [0, T ].

We shall consider now a particular case of equation (14), as far as the function
g is concerned. Instead of the term f , we shall consider another operator-like term.
More precisely, we shall deal with the functional equation

y(t) + g(y(α(t))) = C +
∫ t

0

(W y)(s)ds, (15)

under the following assumptions:

1) g : C([0, T ], IRn) −→ C([0, T ], IRn) is a contraction map on this space:

|g(x) − g(y)|C ≤ λ|x − y|C , λ ∈ [0, 1);

2) W : C([0, T ], IRn) −→ C([0, T ], IRn) is a continuous causal operator, taking
bounded sets of C([0, T ], IRn) into bounded sets;

3) α : [0, T ] −→ [0, ∞) is continuous, and such that α(0) = 0 and 0 ≤ α(t) ≤ t for
t ∈ [0, T ].

Remark 4. The vector C ∈ IRn is arbitrary, but it can be chosen in such a way to
satisfy some kind of initial condition. For instance, if we assign to y the initial value
y0, and assume (without loss of generality) that g(θ) = θ ∈ IRn, then one obtains
C = y0.



7, 3(2005)
Some special classes of neutral functional differential equations 21

In regard to the equation (15), the following existence result can be stated:

Theorem 2. Consider the functional equation (15), under conditions 1), 2), 3) stated
above. Then, there exists a solution y = y(t), defined on some interval
[0, T1] ⊂ [0, T ], for each C ∈ IRn. This solution is such that y(t) + g(y(α(t))) is
continuously differentiable.

Proof. The hypotheses accepted are of such a nature that allow the application of
Theorem 6.1 in [6], which yields the existence result.

The idea of proof is based on the fact that the functional equation y(t)+g(y(α(t))) =
f (t) is uniquely solvable in C([0, T ], IRn). Moreover, y(t) depends continuously of f (t),
which allows to deal with (15) by contraction mapping principle, or by another fixed
point methods.

Details of this approach can be found in our paper [5], where further existence
results are obtained.

5 Further considerations on equation (10)

The idea of proof mentioned above can be adapted to other functional equations. For
an illustration we will consider the equation (10), as well as the auxiliary equation

x′(t) = g(t, x(t), x(α(t))) + f (t), (16)

with f ∈ C([0, T ], IRn). The attached initial condition is (6). The functional integral
equation equivalent to (16), (6) is

x(t) = x0 +
∫ t

0

f (s)ds +
∫ t

0

g(s, x(s), x(α(s)))ds. (17)

Let us assume the following conditions on the data in equation (17):

1) g : [0, T ] × IRn × IRn −→ IRn is continuous, and satisfies the Lipschitz condition

|g(t, x, y) − g(t, x̄, ȳ)| ≤ L(|x − x̄| + |y − ȳ|),

with L > 0;

2) f ∈ C([0, T ], IRn);

3) α(t) is continuous on [0, T ], and α(0) = 0, 0 ≤ α(t) ≤ t.

It is easy to see that the usual process of iteration leads to the following relation-
ship:

x(k+1)(t) − x(k)(t) =

=
∫ t

0

[
g

(
s, x(k)(s), x(k)(α(s))

)
− g

(
s, x(k−1)(s), x(k−1)(α(s))

)]
ds, k ≥ 1,



22 Constantin Corduneanu
7, 3(2005)

with

x(0)(t) = x0 +
∫ t

0

f (s)ds.

We further derive on behalf of condition 1)

|x(k−1)(t) − x(k)(t)| ≤

≤ L
∫ t

0

[
|x(k)(s) − x(k−1)(s)| + |x(k)(α(s)) − x(k−1)(α(s))|

]
ds,

which leads to

sup
0≤s≤t

|x(k+1)(s) − x(k)(s)| ≤ 2L
∫ t

0

sup
0≤u≤s

|x(k)(u) − x(k−1)(u)|ds, (18)

if we keep in mind that 0 ≤ α(t) ≤ t.
The inequality (18) can be processed in the usual manner, and one finds that

lim x(k)(t) = x(t) as k −→ ∞, uniformly on [0, T ], with x(t) satisfying (16). The
uniqueness can be also proven by the standard method, as well as the continuous
dependence of the solution with respect to f ∈ C([0, T ], IRn).

The auxiliary result established above enables us to make some progress in regard
to the equation (10). We rewrite it for the reader’s convenience,

x′(t) = g(t, x(t), x(α(t))) + f (t, x(t), x′(α(t))),

and regard it as a (nonlinear) perturbed equation associated to (16). Of course, we
preserve the initial condition (6).

The following fixed point scheme can be attached to the equation (19): for each
continuously differntiable u(t) on [0, T ], with values inIRn, we shall attach the unique
solution x(t) of the equation like (16)

x′(t) = g(t, x(t), x(α(t))) + f (t, u(t), u′(α(t))) (19)

with the initial condition (6). The existence and uniqueness of x(t), under the above
scheme, is guaranteed under the conditions 1), 2) and 3) specified above. Conse-
quently, in the space C([0, T ], IRn), or rather in the space C(1)([0, T ], IRn), we have
defined an opeator u −→ x, where u and x are related by the equation (19), with x
satisfying also (6).

Let us denote by V the operator defined above, i.e.

x(t) = (V u)(t), t ∈ [0, T ], (20)

with u and x as described above. The operator V appears as a compound operator:
first, u −→ f (t, u(t), u′(α(t))), and second f −→ x, with x the solution of (19), (6).

As noticed earlier in this section, the second operator is continuous on C([0, T ], IRn).
The first operator involved, u −→ f (t, u(t), u′(α(t))) can be made continuous, under
adequate hypotheses on the function f . It is obviously continuous from C(1)([0, T ], IRn)
into C([0, T ], IRn) when f (t, u, v) is continuous on [0, T ] × IRn × IRn.



7, 3(2005)
Some special classes of neutral functional differential equations 23

Instead of pursuing the above scheme, which can certainly lead to results of ex-
istence for (10), we shall attempt to apply the contraction mapping principle to the
equation (10), but modifying somewhat the scheme presented above. Namely, we will
consider the scheme described by the following equation, attached to (10):

x′(t) = g(t, x(t), x(α(t))) + f (t, x(t), u′(α(t))). (21)

By means of (21) and (6), we shall define the operator on C(1)([0, T ], IRn), say x(t) =
(W u)(t), in the following manner. Given u ∈ C(1)([0, T ], IRn), the equation (21)
can be solved in x under rather mild assumptions (as seen above, under Lipschitz
condition). The unique solution of (21), (6) will be denoted by x(t) = (W u)(t).
From (2) we derive the following relationship between x = W u and y = W v, where
u, v ∈ C(1)([0, T ], IRn):

x′(t) − y′(t) = g(t, x(t), x(α(t))) − g(t, y(t), y(α(t)))+
+f (t, x(t), u′(α(t))) − f (t, y(t), v′(α(t))).

(22)

Assuming also a Lipschitz condition on f (t, x, y), as we did already on g(t, x, y), we
obtain

|x′(t) = y′(t)| ≤ L(|x(t) − y(t)| + |x(α(t)) − y(α(t))|)+
+M|x(t) − y(t)| + m|u′(α(t)) − v′(α(t))|,

for any t ∈ [0, T ], where L, M and m are positive numbers. The above inequality
yields

sup |x′(t) − y′(t(| ≤ (2L + M ) sup |x(t) − y(t)|+
+m sup |u′(α(t)) − v′(α(t))|,

(23)

with sup taken on [0, T ], or on any [0, T1], T1 ≤ T.
But

x(t) − y(t) =
∫ t

0

[x′(s) − y′(s)]ds, (24)

because x(0) = y(0) = x0, according to (6). From (23) we derive

sup |x(t) − y(t)| ≤ T sup |x′(s) − y′(s)|, (25)

with sup taken on [0, T ]. Taking into account (23), (24) and (25) we obtain

sup |x′(t) − y′(t)| ≤ (2L + M )T sup |x′(t) − y′(t)|+
+m sup |u′(t) − v′(t)|.

(26)

Since we want (26) to be a relation showing the fact that the operator W is a con-
traction on C(1)([0, T ], IRn), we see from (26) that a first condition to be imposed
is

(2L + M )T < 1. (27)

If we admit (27), then (26) allows us to write

sup |x′(t) − y′(t)| ≤ m[1 − (2L + M )T ]−1 sup |u′(t) − v′(t)|,



24 Constantin Corduneanu
7, 3(2005)

which really represents a contraction condition for W , as soon as

λ = m[1 − (2L + M )T ]−1 < 1. (28)

It is appropriate to notice the fact that the norm in C(1)([0, T ], IRn) is (by our choice)

|x0| + sup |x′(t)|. (29)

Accordingly, the norm for u − v should be |u0 − v0| + sup |u′(t) − v′(t)|, which is in
advantage of the contraction inequality

|W u − W v|C(1) ≤ λ|u − v|C(1) , (30)

as derived from above.
Therefore, we can now state the following (global) existence result for the problem

(10), (6):

Theorem 3. Consider the problem (10), (6), and assume the following conditions
are verified by the functions f and g:

1) f, g : [0, T ] × IRn × IRn −→ IRn are continuous maps;

2) f and g satisfy the Lipschitz type conditions

|f (t, x, y) − f (t, x̄, ȳ)| ≤ L(|x − x̄| + |y − ȳ|),
|g(t, x, y) − g(t, x̄, ȳ)| ≤ M|x − x̄| + m|y − ȳ|,

with positive constants L, M and m;

3) the inequalities (27) and (28) are satisfied.

Then, there exists a unique solution x(t) ∈ C(1)([0, T ], IRn), which can be approxi-
mated by the scheme described by the equation (21).

The proof of Theorem 3 has been carried out above, before its statement.

Remark 5. It is obvious from the inequalities (27) and (28) that severe restrictions
must be imposed to the constants L, M, m and T .

First, if L, M are fixed, it is obvious that the inequality (27) can be satisfied
provided we choose T small enough: T < (2L + M )−1. This restriction suggests that
we need to confine our investigation, possibly, to a smaller interval than the original
interval [0, T ]. But this kind of restriction is in accordance with the fact we are looking
for local solutions to our problem.

Second, once we choose T such that (27) takes place, there remains the inequality
(28) to be satisfied. If the constants L, M and T are fixed, then the only way to
satisfy (28) is to choose m small enough.

In conclusion, local existence for the problem (10), (6) is always assured by choos-
ing the constant m sufficiently small.



7, 3(2005)
Some special classes of neutral functional differential equations 25

Other neutral equations can be investigated in regard to the existence of their
solutions, using approaches described above. We suggest to the reader to try such
procedures on equations of the form

x′(t) = f (t, x(t), x(t − h)) + g(t, x(t), x′(t − h)),

under an initial condition of the form x(t) = ϕ(t), t ∈ [−h, 0]. Also, similar to the
equation (15), is

d

dt
[x(t) + g(x(t − h))] = (W x)(t),

with initial datum x(t) = ϕ(t), t ∈ [−h, 0].
See our paper [4] for details.

Received: July 2003. Revised: December 2003.

References

[1] T.A. Burton, An existence theorem for neutral equations. Nonlinear Studies
5 (1998), 1-6.

[2] C. Corduneanu, Integral Equations and Applications. Cambridge Univ.
Press, 1991.

[3] C. Corduneanu, Neutral functional equations with abstract Volterra opera-
tors. In ”Advances in Nonlinear Dynamics”, Gordon & Breach, (1997), 229-
235.

[4] C. Corduneanu, Neutral functional equations of Volterra type. Functional
Differential Equations (Israel), (1997), 265-270.

[5] C. Corduneanu, Existence of solutions to neutral functional differential equa-
tions. J. Diff. Equations, 168 (2000), 93-101.

[6] C. Corduneanu, Functional Equations with Causal Operators, Taylor & Fran-
cis, 2002.

[7] C. Corduneanu, M. Mahdavi, On neutral functional differential equations
with causal operators. Proc. Third Workshop of the Int. Inst. General System
Science, Tianjin (China), 1998, 43-48.

[8] C. Corduneanu, M. Mahdavi, On neutral functional differential equations
with causal operators, II. In ”Integral Methods in Science and Engineering”,
Chapman & Hall CRC Press, Research Notes #418 (2000), 102-106.



26 Constantin Corduneanu
7, 3(2005)

[9] Loris Faina, Existence and continuous dependence for a class of neutral func-
tional differential equations. Annales Polonici Mathematici, LXIV.3 (1996),
215-226.