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Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations
Advances in Difference Equations volume 2010, Article number: 810453 (2010)
Abstract
The existence and uniqueness of solutions and asymptotic estimate of solution formulas are studied for the following initial value problem: ,
,
, where
is a constant and
. An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used.
1. Introduction and Preliminaries
The singular Cauchy problem for first-order differential and integro-differential equations resolved (or unresolved) with respect to the derivatives of unknowns is fairly well studied (see, e.g., [1–16]), but the asymptotic properties of the solutions of such equations are only partially understood. Although the singular Cauchy problems were widely considered by using various methods (see, e.g., [1–13, 16–18]), the method used here is based on a different approach. In particular, we use a combination of the topological method of T. Ważewski (see, e.g., [19, 20]) and Schauder's fixed point theorem [21]. Our technique leads to the existence and uniqueness of solutions with asymptotic estimates in the right neighbourhood of a singular point.
Consider the following problem:

where Denote
as
if there is valid
as
if there is valid
The functions will be assumed to satisfy the following.
-
(i)
is a constant,
as
as
for each
-
(ii)
as
where
is the general solution of the equation
.
In the text we will apply the topological method of Waewski and Schauder's theorem. Therefore, we give a short summary of them.
Let be a continuous function defined on an open
-set
,
an open set of
the boundary of
with respect to
and
the closure of
with respect to
. Consider the system of ordinary differential equations

Definition 1.1 (see [19]).
The point is called an egress (or an ingress point) of
with respect to system (1.2) if for every fixed solution of system (1.2),
, there exists an
such that
for
. An egress point (ingress point)
of
is called a strict egress point (strict ingress point) of
if
on interval
for an
.
Definition 1.2 (see [19]).
An open subset of the set
is called a
-subset of
with respect to system (1.2) if the following conditions are satisfied.
-
(1)
There exist functions
and
such that
(1.3) -
(2)
holds for the derivatives of the functions
,
along trajectories of system (1.2) on the set
(1.4) -
(3)
holds for the derivatives of the functions
,
along trajectories of system (1.2) on the set
(1.5)
The set of all points of egress (strict egress) is denoted by .
Lemma 1.3 (see [19]).
Let the set be a
-subset of the set
with respect to system (1.2). Then

Definition 1.4 (see [19]).
Let be a topological space and
Let . A function
such that
for all
is a retraction from
to
in
.
The set is a retract of
in
if there exists a retraction from
to
in
.
Theorem 1.5 (Ważewski's theorem [19]).
Let be some
-subset of
with respect to system (1.2). Let
be a nonempty compact subset of
such that the set
is not a retract of
but is a retract
. Then there is at least one point
such that the graph of a solution
of the Cauchy problem
for (1.2) lies in
on its right-hand maximal interval of existence.
Theorem 1.6 (Schauder's theorem [21]).
Let E be a Banach space and S its nonempty convex and closed subset. If P is a continuous mapping of S into itself and PS is relatively compact then the mapping P has at least one fixed point.
2. Main Results
Theorem 2.1.
Let assumptions (i) and (ii) hold, then for each there exists one solution
of initial problem (1.1) such that

for where
is a constant, and
depends on
.
Proof.
() Denote
the Banach space of continuous functions
on the interval
with the norm

The subset of Banach space
will be the set of all functions
from
satisfying the inequality

The set is nonempty, convex and closed.
() Now we will construct the mapping
. Let
be an arbitrary function. Substituting
instead of
into (1.1), we obtain the differential equation

Set


where is a constant and new functions
satisfy the differential equation

From (2.3), it follows that

Substituting (2.5), (2.6) and (2.8) into (2.4) we get

Substituting (2.9) into (2.7) we get

In view of (2.5), (2.6) it is obvious that a solution of (2.10) determines a solution of (2.4).
Now we will use Waewski's topological method. Consider an open set
. Investigate the behaviour of integral curves of (2.10) with respect to the boundary of the set

where

Calculating the derivative along the trajectories of (2.10) on the set

we obtain

Since

then there exists a positive constant such that

Consequently,

From here and by L'Hospital's rule
for
is an arbitrary real number. These both identities imply that the powers of
affect the convergence to zero of the terms in (2.14), in decisive way.
Using the assumptions of Theorem 2.1 and the definition of we get that the first term
in (2.14) has the form

and the second term

is bounded by terms with exponents which are greater than

From here, we obtain

for sufficiently small depending on
.
The relation (2.21) implies that each point of the set is a strict ingress point with respect to (2.10). Change the orientation of the axis
into opposite. Now each point of the set
is a strict egress point with respect to the new system of coordinates. By Wa
ewski's topological method, we state that there exists at least one integral curve of (2.10) lying in
for
. It is obvious that this assertion remains true for an arbitrary function
Now we will prove the uniqueness of a solution of (2.10). Let be also the solution of (2.10). Putting
and substituting into (2.10), we obtain

Let

where

Using the same method as above, we have

for . It is obvious that
for
Let
be any nonzero solution of (2.14) such that
for
Let
be such a constant that
If the curve
lays in
for
, then
would have to be a strict egress point of
with respect to the original system of coordinates. This contradicts the relation (2.25). Therefore, there exists only the trivial solution
of (2.22), so
is the unique solution of (2.10).
From (2.5), we obtain

where is the solution of (2.4) for
Similarly, from (2.6), (2.9) we have

It is obvious (after a continuous extension of for
that
maps
into itself and
.
() We will prove that
is relatively compact and
is a continuous mapping.
It is easy to see, by (2.26) and (2.27), that is the set of uniformly bounded and equicontinuous functions for
By Ascoli's theorem,
is relatively compact.
Let be an arbitrary sequence functions in
such that

The solution of the equation

corresponds to the function and
for
Similarly, the solution
of (2.10) corresponds to the function
. We will show that
uniformly on
, where
,
is a sufficiently small constant which will be specified later. Consider the region

where

There exists sufficiently small constant such that
for any
,
. Investigate the behaviour of integral curves of (2.29) with respect to the boundary
Using the same method as above, we obtain for trajectory derivatives

for and any
. By Wa
ewski's topological method, there exists at least one solution
lying in
. Hence, it follows that

and is a constant depending on
. From (2.5), we obtain

where is a constant depending on
This estimate implies that
is continuous.
We have thus proved that the mapping satisfies the assumptions of Schauder's fixed point theorem and hence there exists a function
with
The proof of existence of a solution of (1.1) is complete.
Now we will prove the uniqueness of a solution of (1.1). Substituting (2.5), (2.6) into (1.1), we get

Equation (2.7) may be written in the following form:

Now we know that there exists the solution of (1.1) satisfying (2.1) such that

where is the solution of (2.36).
Denote and substituting it into (2.36), we obtain

Let

where

If (2.38) had only the trivial solution lying in then
would be the only solution of (2.38) and from here, by (2.36),
would be the only solution of (1.1) satisfying (2.1) for
.
We will suppose that there exists a nontrivial solution of (2.38) lying in
. Substitute
instead of
into (2.38), we obtain the differential equation

Calculating the derivative along the trajectories of (2.41) on the set
, we get
for
.
By the same method as in the case of the existence of a solution of (1.1), we obtain that in there is only the trivial solution of (2.41). The proof is complete.
Example 2.2.
Consider the following initial value problem:

In our case a general solution of the equation

has the form and
,
,
,
,
as
.
Further

,
as
and

According to Theorem 2.1, there exists for every constant the unique solution
of (2.42) such that

for .
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Acknowledgment
The author was supported by the Council of Czech Government Grants MSM 00216 30503 and MSM 00216 30529.
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Šmarda, Z. Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations. Adv Differ Equ 2010, 810453 (2010). https://doi.org/10.1155/2010/810453
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DOI: https://doi.org/10.1155/2010/810453