Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations
© Zdeněk Šmarda. 2010
Received: 30 December 2009
Accepted: 10 March 2010
Published: 16 March 2010
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 . 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:
as if there is valid
as if there is valid
is a constant, as as for each
as where is the general solution of the equation .
In the text we will apply the topological method of Wa ewski 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 ).
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 ).
- (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 ).
Definition 1.4 (see ).
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 ).
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 ).
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
for where is a constant, and depends on .
The set is nonempty, convex and closed.
In view of (2.5), (2.6) it is obvious that a solution of (2.10) determines a solution of (2.4).
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.
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
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).
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.
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.
where is the solution of (2.36).
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 .
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.
has the form and , , , , as .
The author was supported by the Council of Czech Government Grants MSM 00216 30503 and MSM 00216 30529.
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