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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 918281, 18 pages
Singular Initial Value Problem for a System of Integro-Differential Equations
1Department of Mathematics, Brno University of Technology, 61600 Brno, Czech Republic
2Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Received 29 October 2012; Accepted 15 November 2012
Academic Editor: Juntao Sun
Copyright © 2012 Zdeněk Šmarda and Yasir Khan. 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.
Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem: , , , where , , are constants and . An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used. Particular attention is paid to construction of asymptotic expansions of solutions for certain classes of systems of integrodifferential equations in a right-hand neighbourhood of a singular point.
1. Introduction and Preliminaries
Singular initial value problem for ordinary differential and integro-differential equations 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 initial value problems were widely considered using various methods (see, e.g., [1–13, 16]), our approach to this problem is essentially different from others known in the literature. In particular, we use a combination of the topological method of T. Ważewski  and Schauder’s fixed point theorem . Our technique leads to the existence and uniqueness of solutions with asymptotic estimates in the right-hand neighbourhood of a singular point. Asymptotic expansions of solutions are constructed for certain classes of systems of integrodifferential equations as well.
Consider the following problem: where , are constants, , , , , .
Denote(i) as if there is a right-hand neighbourhood and a constant such that for .(ii) if there is valid .(iii)if there is valid.
Definition 1.1. The sequence of functions is called an asymptotic sequence as if for all .
Definition 1.2. The series , , is called an asymptotic expansion of the function up to th term as if(a) is an asymptotic sequence, (b)
The functions , and will be assumed to satisfy the following:(i), , , as , , as for each , ,(ii), , , as where is the general solution of the equation .
In the text, we will apply topological method of Ważewski 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 , and the closure of . Consider the following system of ordinary differential equations:
Definition 1.3 (see ). The point is called an egress (or an ingress point) of with respect to system (1.5) if for every fixed solution of the problem , 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 1.4 (see ). An open subset of the set is called an subset of with respect to system (1.5) if the following conditions are satisfied. (1)There exist functions , and such that (2) holds for the derivatives of the functions , along trajectories of system (1.5) on the set (3) holds for the derivatives of the functions , along trajectories of system (1.5) on the set The set of all points of egress (strict egress) is denoted by .
Definition 1.6 (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 1.7 (Ważewski’s theorem ). Let be some subset of with respect to system (1.5). 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 (1.5) lies on its right-hand maximal interval of existence.
Theorem 1.8 (Schauder’s theorem ). Let be a Banach space and its nonempty convex and closed subset. If is a continuous mapping of into itself and is relatively compact then the mapping has at least one fixed point.
2. Main Results
Proof. (1) Denote the Banach space of vector-valued continuous functions on the interval with the norm
The subset of Banach space will be the set of all functions from satisfying the inequality
The set is nonempty, convex, and closed.
(2) Now we will construct the mapping . Let be an arbitrary function. Substituting , instead of , into (1.1), we obtain the following differential equation: Put where is a constant and new functions , satisfy the differential equations as From (2.3), it follows Substituting (2.5), (2.6), and (2.8) into (2.4), we get Substituting (2.9) into (2.7), we get In view of (2.5) and (2.6), it is obvious that a solution of (2.10) determines a solution of (2.4).
Now we use Ważewski’s topological method. Consider an open set . Denote . Define an open subset as follows: where Calculating the derivatives , along the trajectories of (2.10) on the set , , we obtain Since then there exists a positive constant such that Consequently, 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 (2.13), in a decisive way.
Using the assumptions of Theorem 2.1 and the definition of , , , we get that the first term in (2.13) has the following form: and the second term is bounded by terms with exponents which are greater than , . From here, we obtain for sufficiently small , depending on , , .
It is obvious that .
Change the orientation of the axis t into opposite. Then, with respect to the new system of coordinates, the set is the subset with respect to system (2.10). By Ważewski’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 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 Define where Using the same method as above, we have for sufficiently small , . It is obvious that for . Let be any nonzero solution of (2.10) such that for . Let be such a constant that . If the curve lay in for , then would have to be a strict egress point of with respect to the original system of coordinates. This contradicts the relation (2.24). Therefore there exists only the trivial solution of (2.21), so is the unique solution of (2.10).
From (2.5) we obtain where is the solution of (2.4) for . Similarly, from (2.6) and (2.9), we have It is obvious (after a continuous extension of for , ) that maps into itself and .
(3) We will prove that is relatively compact and is a continuous mapping.
It is easy to see, by (2.25) and (2.26), that is the set of uniformly bounded and equicontinuous functions for . By Ascoli’s theorem, is relatively compact.
Let be an arbitrary sequence vector-valued functions in such that The solution of the following equation: 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 following region: where There exists sufficiently small constant such that for any . Investigate the behaviour of integral curves of (2.28) with respect to the boundary , . Using the same method as above, we obtain the following trajectory derivatives: for and any . By Ważewski’s topological method, there exists at least one solution lying in , . Hence, it follows that , are constants depending on , . From (2.5), we obtain where , are constants 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) and (2.6) into (1.1), we get Equation (2.7) may be written in the following form:
Now we know that there exists the solution of (1.1) satisfying (1.2) such that where is the solution of (2.35).
Denote , . Substituting into (2.35), we obtain Let where If (2.37) had only the trivial solution lying in, then would be only one solution of (2.37) and from here, by (2.35), would be only one solution of (1.1) satisfying (1.2) for .
We will suppose that there exists nontrivial solution of (2.37) lying in . Substituting instead of , into (2.37), we obtain the following differential equation:
Calculating the derivative along the trajectories of (2.40) 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 (2.40). The proof is complete.
3. Asymptotic Expansions of Solutions
Diblík  investigated a singular initial problem for implicit ordinary differential equations and constructed asymptotic expansions of solutions in a right-hand neighbourhood of a singular point. Some results about asymptotic expansions of solutions for integrodifferential equations with separable kernels are given in [3, 10, 12].
The aim of this section is to show that results of paper  for ordinary differential equations are possible to extend on certain classes systems integrodifferential equations with a separable kernel in the following form: where , , , , ,
We will construct the solution of (3.1) in the form of one parametric asymptotic expansions as where is the general solution of the differential equation so that , , , are unknown functions, is a constant.
Theorem 3.1. Assume that (I)Let be a constant, , , , , as , , is any positive number. (II), , , , as , , , .
Then (3.5) has a unique solution on , satisfying asymptotic estimates where .
Substituting (3.3) into (3.1) and comparing the terms with the same powers of , we obtain the following system of recurrence equations: and Denote then it is obvious that the recurrence equations , have the same form as (3.5) with the constant . Hence we can apply Theorem 3.1, after the modification of assumption (II) of Theorem 3.1 for indices , , to recurrence (3.10) which we will demonstrate with the following example.
Example 3.2. Consider the following system of integrodifferential equations:
System (3.11) has the form of system (3.1) for
We will construct a solution of system (3.11) in the following form:
where is the general solution of the equation . We will demonstrate the calculation of coefficients for . Substituting (3.13) in (3.11) and comparing the terms with the same powers of , we obtain the following system of recurrence equations:
Differentiating both equations (3.17), we obtain the following differential equations:
Equation (3.18) satisfies assumptions of Theorem 3.1. with following functions and coefficients:
Hence we can choose a constant and similarly . By Theorem 3.1., we have Second equation (3.19) is different from (3.18) only in the constant . Thus Substituting solutions (3.21) and (3.22) into (3.15) instead of integral terms, we obtain for unknown coefficients the following differential equations: For (3.23), we can put
Then we can choose a constant . By Theorem 3.1., we get Similarly for (3.24), we can put , , , , ,
Then we can choose a constant . By Theorem 3.1., we have Substituting coefficients , into (3.16) and using the same method as in the calculation of coefficients , , we have Thus the solution of system (3.11) has for the following asymptotic expansions:
The first author is supported by Grant FEKT-S-11-2-921 of the Faculty of Electrical Engineering and Communication, Brno University of Technology and Grant P201/11/0768 of the Czech Grant Agency (Prague).
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