- Research Article
- Open Access

# Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

- Zdeněk Šmarda
^{1}Email author

**2010**:810453

https://doi.org/10.1155/2010/810453

© Zdeněk Šmarda. 2010

**Received:**30 December 2009**Accepted:**10 March 2010**Published:**16 March 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.

## Keywords

- Differential Equation
- Banach Space
- Singular Point
- Point Theorem
- Fixed Point Theorem

## 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

- (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 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 [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]).

- (1)
- (2)
- (3)

The set of all points of egress (strict egress) is denoted by .

Lemma 1.3 (see [19]).

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.

for where is a constant, and depends on .

Proof.

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.

Example 2.2.

has the form and , , , , as .

for .

## Declarations

### Acknowledgment

The author was supported by the Council of Czech Government Grants MSM 00216 30503 and MSM 00216 30529.

## Authors’ Affiliations

## References

- Agarwal RP, O'Regan D, Zernov OE: A singular initial value problem for some functional differential equations. Journal of Applied Mathematics and Stochastic Analysis 2004, (3):261-270. 10.1155/S1048953304405012Google Scholar
- Čečik VA:
**Investigation of systems of ordinary differential equations with a singularity.***Trudy Moskovskogo Matematičeskogo Obščestva*1959,**8:**155-198.Google Scholar - Diblík I:
**Asymptotic behavior of solutions of a differential equation partially solved with respect to the derivative.***Siberian Mathematical Journal*1982,**23**(5):654-662. 10.1007/BF00971283MATHView ArticleGoogle Scholar - Diblík I:
**On the existence of solutions of a real system of ordinary differential equations entering into a singular point.***Ukrainskii Matematicheskii Zhurnal*1986,**38**(6):701-707.MATHMathSciNetGoogle Scholar - Baštinec J, Diblík J:
**On existence of solutions of a singular Cauchy-Nicoletti problem for a system of integro-differential equations.***Demonstratio Mathematica*1997,**30**(4):747-760.MATHMathSciNetGoogle Scholar - Diblík I:
**On the existence of 0-curves of a singular system of differential equations.***Mathematische Nachrichten*1985,**122:**247-258. 10.1002/mana.19851220124MATHMathSciNetView ArticleGoogle Scholar - Diblík J, Nowak C:
**A nonuniqueness criterion for a singular system of two ordinary differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(4):637-656. 10.1016/j.na.2005.05.042MATHMathSciNetView ArticleGoogle Scholar - Diblík J, Růžičková M:
**Existence of positive solutions of a singular initial problem for a nonlinear system of differential equations.***The Rocky Mountain Journal of Mathematics*2004,**34**(3):923-944. 10.1216/rmjm/1181069835MATHMathSciNetView ArticleGoogle Scholar - Diblík J, Růžičková M:
**Inequalities for solutions of singular initial problems for Carathéodory systems via Ważewski's principle.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(12):4482-4495. 10.1016/j.na.2007.11.006MATHMathSciNetView ArticleGoogle Scholar - Šmarda Z:
**On the uniqueness of solutions of the singular problem for certain class of integro-differential equations.***Demonstratio Mathematica*1992,**25**(4):835-841.MATHMathSciNetGoogle Scholar - Šmarda Z: On a singular initial value problem for a system of integro-differential equations depending on a parameter. Fasciculi Mathematici 1995, (25):123-126.Google Scholar
- Šmarda Z:
**On an initial value problem for singular integro-differential equations.***Demonstratio Mathematica*2002,**35**(4):803-811.MATHMathSciNetGoogle Scholar - Šmarda Z:
**Implicit singular integrodifferential equations of Fredholm type.***Tatra Mountains Mathematical Publications*2007,**38:**255-263.MATHMathSciNetGoogle Scholar - Zernov AE, Kuzina YuV:
**Qualitative investigation of the singular Cauchy problem****.***Ukrainskii Matematichnii Zhurnal*2003,**55**(10):1419-1424.MATHMathSciNetGoogle Scholar - Zernov AE, Kuzina YuV:
**Geometric analysis of a singular Cauchy problem.***Nonlinear Oscillator*2004,**7**(1):67-80.MATHMathSciNetGoogle Scholar - Zernov AE, Chaichuk OR:
**Asymptotic behavior of solutions of a singular Cauchy problem for a functional-differential equation.***Journal of Mathematical Sciences*2009,**160**(1):123-127. 10.1007/s10958-009-9491-2MATHMathSciNetView ArticleGoogle Scholar - Feng M, Zhang X, Li X, Ge W:
**Necessary and sufficient conditions for the existence of positive solution for singular boundary value problems on time scales.***Advances in Difference Equations*2009,**2009:**-14.Google Scholar - Gómez González A, Otero-Espinar V:
**Existence and uniqueness of positive solution for singular BVPs on time scales.***Advances in Difference Equations*2009,**2009:**-12.Google Scholar - Hartman P:
*Ordinary Differential Equations*. John Wiley & Sons, New York, NY, USA; 1964:xiv+612.MATHGoogle Scholar - Srzednicki R:
**Ważewski method and Conley index.**In*Handbook of Differential Equations*. Elsevier/North-Holland, Amsterdam, The Netherlands; 2004:591-684.Google Scholar - Zeidler E:
*Applied Functional Analysis: Applications to Mathematical Physics, Applied Mathematical Sciences*.*Volume 108*. Springer, New York, NY, USA; 1995:xxx+479.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.