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:

(1.1)

where Denote

as if there is valid

as if there is valid

The functions will be assumed to satisfy the following.

  1. (i)

    is a constant, as as for each

  2. (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

(1.2)

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. (1)

    There exist functions and such that

    (1.3)
  2. (2)

    holds for the derivatives of the functions , along trajectories of system (1.2) on the set

    (1.4)
  3. (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

(1.6)

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

(2.1)

for where is a constant, and depends on . 

Proof.

() Denote the Banach space of continuous functions on the interval with the norm

(2.2)

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

(2.3)

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

(2.4)

Set

(2.5)
(2.6)

where is a constant and new functions satisfy the differential equation

(2.7)

From (2.3), it follows that

(2.8)

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

(2.9)

Substituting (2.9) into (2.7) we get

(2.10)

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

(2.11)

where

(2.12)

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

(2.13)

we obtain

(2.14)

Since

(2.15)

then there exists a positive constant such that

(2.16)

Consequently,

(2.17)

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

(2.18)

and the second term

(2.19)

is bounded by terms with exponents which are greater than

(2.20)

From here, we obtain

(2.21)

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

(2.22)

Let

(2.23)

where

(2.24)

Using the same method as above, we have

(2.25)

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

(2.26)

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

(2.27)

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

(2.28)

The solution of the equation

(2.29)

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

(2.30)

where

(2.31)

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

(2.32)

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

(2.33)

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

(2.34)

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

(2.35)

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

(2.36)

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

(2.37)

where is the solution of (2.36).

Denote and substituting it into (2.36), we obtain

(2.38)

Let

(2.39)

where

(2.40)

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

(2.41)

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:

(2.42)

In our case a general solution of the equation

(2.43)

has the form and ,  ,  ,  ,   as .

Further

(2.44)

, as and

(2.45)

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

(2.46)

for .