- Research Article
- Open Access
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
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:
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 ).
Lemma 1.3 (see ).
Definition 1.4 (see ).
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
In view of (2.5), (2.6) it is obvious that a solution of (2.10) determines a solution of (2.4).
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).
The author was supported by the Council of Czech Government Grants MSM 00216 30503 and MSM 00216 30529.
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