Abstract and Applied Analysis

Volume 2012 (2012), Article ID 918281, 18 pages

doi:10.1155/2012/918281

Research Article

Singular Initial Value Problem for a System of Integro-Differential Equations

Zdeněk Šmarda¹ and Yasir Khan²

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.

Abstract.

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 [8] and Schauder’s fixed point theorem [11]. 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 [17]). 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 [18]). 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 .

Lemma 1.5 (see [18]). Let the set be a subset of the set with respect to system (1.5). Then

Definition 1.6 (see [18]). 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 [18]). 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 [19]). 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

Theorem 2.1. Let assumptions (i) and (ii) hold, then for each there is one solution , of initial problem (1.1) and (1.2) such that

for , where , is a constant, and depends on , .

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 [3] 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 [2] 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.

Consider the following differential equation:

Diblík [3] proved asymptotic estimates of the solution of (3.5) which we can be formulated as follows.

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 .

Now we will show the results of Theorem 3.1. regarding only differential equation (3.5) we can apply to system of integrodifferential equations (3.1).

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:

Put

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:

Acknowledgments

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).

References

R. P. Agarwal, D. O'Regan, and O. E. Zernov, “A singular initial value problem for some functional differential equations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2004, no. 3, pp. 261–270, 2004.

V. A. Čečik, “Investigation of systems of ordinary differential equations with a singularity,” Trudy Moskovskogo Matematičeskogo Obščestva, vol. 8, pp. 155–198, 1959 (Russian).

I. Diblík, “Asymptotic behavior of solutions of a differential equation partially solved with respect to the derivative,” Siberian Mathematical Journal, vol. 23, no. 5, pp. 654–662, 1982 (Russian).

J. Diblík, “Existence of solutions of a real system of ordinary differential equations entering into a singular point,” Ukrainian Mathematical Journal, vol. 38, no. 6, pp. 588–592, 1986 (Russian).

J. Baštinec and J. Diblík, “On existence of solutions of a singular Cauchy-Nicoletti problem for a system of integro-differential equations,” Demonstratio Mathematica, vol. 30, no. 4, pp. 747–760, 1997.

J. Diblík, “On the existence of -curves of a singular system of differential equations,” Mathematische Nachrichten, vol. 122, pp. 247–258, 1985 (Russian).

J. Diblík and C. Nowak, “A nonuniqueness criterion for a singular system of two ordinary differential equations,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 64, no. 4, pp. 637–656, 2006.

J. Diblík and M. Růžičková, “Existence of positive solutions of a singular initial problem for a nonlinear system of differential equations,” The Rocky Mountain Journal of Mathematics, vol. 34, no. 3, pp. 923–944, 2004.

J. Diblík and M. R. Růžičková, “Inequalities for solutions of singular initial problems for Caratheodory systems via Ważewski’s principle,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 12, pp. 657–656, 2008.

Z. Šmarda, “On the uniqueness of solutions of the singular problem for certain class of integro-differential equations,” Demonstratio Mathematica, vol. 25, no. 4, pp. 835–841, 1992.

Z. Šmarda, “On a singular initial value problem for a system of integro-differential equations depending on a parameter,” Fasciculi Mathematici, no. 25, pp. 123–126, 1995.

Z. Šmarda, “On an initial value problem for singular integro-differential equations,” Demonstratio Mathematica, vol. 35, no. 4, pp. 803–811, 2002.

Z. Šmarda, “Implicit singular integrodifferential equations of Fredholm type,” Tatra Mountains Mathematical Publications, vol. 38, pp. 255–263, 2007.

A. E. Zernov and Yu. V. Kuzina, “Qualitative investigation of the singular Cauchy problem ,” Ukrainian Mathematical Journal, vol. 55, no. 10, pp. 1419–1424, 2003 (Russian).

A. E. Zernov and Yu. V. Kuzina, “Geometric analysis of a singular Cauchy problem,” Nonlinear Oscillations, vol. 7, no. 1, pp. 67–80, 2004 (Russian).

A. E. Zernov and O. R. Chaĭchuk, “Asymptotic behavior of solutions of a singular Cauchy problem for a functional-differential equation,” Journal of Mathematical Sciences, vol. 160, no. 1, pp. 123–135, 2009.

R. Srzednicki, “Ważewski method and Conley index,” in Handbook of Differential Equations: Ordinary Differential Equations, A. Canada, P. Drabek, and A. Fonda, Eds., vol. 1, pp. 591–684, Elsevier, Amsterdam, The Netherlands, 2004.

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964.

E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, vol. 108 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1999.