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Singular Cauchy Initial Value Problem for Certain Classes of IntegroDifferential 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 firstorder differential and integrodifferential 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 righthand 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 Waewski'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 Waewski'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 IntegroDifferential 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