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.
- 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/S1048953304405012
- 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.
- 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
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.