Abstract and Applied Analysis

Volume 2013 (2013), Article ID 607204, 20 pages

http://dx.doi.org/10.1155/2013/607204

Research Article

Some Bounded Linear Integral Operators and Linear Fredholm Integral Equations in the Spaces and

İsmet Özdemir,¹ Ali M. Akhmedov,² and Ö. Faruk Temizer¹

¹İnönü Üniversitesi, Eğitim Fakültesi, A-Blok, Posta Kodu: 44280, Malatya, Turkey

²Baku State University, Department of Mech. & Math., Z. Khalilov Street, 23, P.O. Box 370145, Baku, Azerbaijan

Received 14 November 2012; Accepted 16 January 2013

Academic Editor: Juan Carlos Cortés López

Copyright © 2013 İsmet Özdemir et al. 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.

The spaces and were defined in ((Hüseynov (1981)), pages 271–277). Some singular integral operators on Banach spaces were examined, (Dostanic (2012)), (Dunford (1988), pages 2419–2426 and (Plamenevskiy (1965)). The solutions of some singular Fredholm integral equations were given in (Babolian (2011), Okayama (2010), and Thomas (1981)) by numerical methods. In this paper, we define the sets and by taking an arbitrary Banach space instead of , and we show that these sets which are different from the spaces given in (Dunford (1988)) and (Plamenevskiy (1965)) are Banach spaces with the norms and . Besides, the bounded linear integral operators on the spaces and , some of which are singular, are derived, and the solutions of the linear Fredholm integral equations of the form and are investigated in these spaces by analytical methods.

1. Preliminaries, Background, and Notation

The approximate solution of the singular integral equation

was obtained in [1], where is a real-valued kernel, is a given function, and is the unknown function.

High-order numerical methods for the singular Fredholm integral equations of the form:

were developed, where and are given functions, and is the unknown function. Equations of this form often arise in practical applications such as Dirichlet problems, mathematical problems of radiative equilibrium, and radiative heat transfer problems, [2].

The polar kernel of integral equations was introduced in [3, 4]. This singular kernel is in the following form:

where the first term of this kernel is weakly singular and and are bounded on the square and . With , we have the special case of the above kernel:

see [5]. One of the weakly singular integral and integrodifferential equations with this kernel was given in [6–8]. The solution of the singular integral equation of the form:

was examined in [5] by numerical methods, where and are given functions and is the unknown function to be determined.

The integral operator defined by

was studied in [9].

An integral equation of the form:

is called Fredholm integral equation of the second type. Here, is a given interval, is a function on which is unknown, and is a parameter. The kernel of the equation is a given function on the square and is a given function on .

Now, we may give some required definitions and theorems.

Definition 1 (see [10, page 41]). Let be a Banach space and be a finite measure space. Then is called measurable simple function if there exist and such that , where

Definition 2 (see [11, page 88]). The Bochner integral of a simple function given by Definition 1 with respect to on is defined by

Definition 3 (see [12, page 201]). Let be a Banach space with the norm and be a finite measure space. Then is called strongly measurable if there exists a sequence of -valued simple functions defined on such that

Theorem 4 (see [11, page 88]). All continuous functions are strongly measurable.

Theorem 5 (see [13, page 336]). Let be a Banach space with the norm and be a finite measure space. If is strongly measurable, then the scalar function

is -measurable.

Definition 6 (see [11, page 88]). Let be a finite measure space and be a Banach space, and then the Bochner integral of a strongly measurable function is the strongly limit of the Bochner integral of an approximating sequence of simple functions satisfying (10). That is,

Theorem 7 (see [12, page 202]). Let be a finite measure space, be a Banach space with the norm , and be a strongly measurable function. If exists, then

Theorem 8 (see [12, page 203]). Let be a finite measure space and be a Banach space with the norm . The Bochner integral exists if and only if is strongly measurable and .

Theorem 9 (see [14, page 82]). Let be a real or complex Banach space with the norm . Let be the real numbers satisfying and be a nonnegative constant. The set consisting of all functions fulfilling the conditions:

for all is a linear space with the algebraic operations:

and is a Banach space with the norm

where

Again, let be a real or complex Banach space with the norm . Let be the real numbers with and also be a nonnegative constant. Let be an -valued function defined on such that

for all with . By , we denote the set of all functions satisfying (18). is a linear space with the algebraic operations:

and is a Banach space with the norm:

and are called a Hölder space. The functions spaces which are similar to and were investigated in [15, pages 2419–2426], [16, pages 25–51], and [17, pages 18–33]. The class of the functions satisfying the equalities

and

was introduced in [18], where functions are increasing,

and is a natural number.

Theorem 10 (see [14, page 16]). Let be a bounded linear operator mapping a Banach space into itself with and denote the identity operator. Then has a bounded inverse operator on which is given by the Neumann series:

which satisfies

The iterated operators are defined by and . The series in the right of (24) is convergent in the norm on .

2. The Main Results

2.1. The Spaces and

In this section, we determine essentially the spaces and .

Theorem 11. Let be a real or complex Banach space with the norm , be real numbers with , and be a nonnegative constant. Then the set of all functions

satisfying the inequalities

for all with is a linear space with the usual algebraic operations addition and scalar multiplication defined by

and is a Banach space with the norm

where and are defined by

Proof. Let be a Banach space with the norm . It is known that the set is a linear space with operations in . Also, it is obvious that . On the other hand, since

is a linear space.

Furthermore, is a normed space with the norm . Indeed, consider the following.

 (N1) It is clear that for all . (N2) If and , then for all , since

So . On the other hand, if , it is found that by (30), (31) and (32). Hence, the proposition “ if and only if ” is true.(N3) Let . Since

by (31) and (32),

 (N4) If ,

by (31) and

by (32). Hence,

From the inequality

which holds for all real numbers , we obtain .

As a result, is a normed space with the norm .

The space is a Banach space with respect to . To see this we consider an arbitrary Cauchy sequence in , and we show that converges to a function . Since is Cauchy, for every , there exists such that

So from (41),

where

By (42), while or , we have

for all . We see by (44) that is Cauchy in . Since is complete, there exists a unique such that

The limit depends on the choice of . This defines a function:

where

Now, we want to show that and . By (48), the continuity of the norm gives together with (44) and (45) that

for all with such that or . Since for all by (49) and (50), we derive . Furthermore, since is bounded, there exists the nonnegative constant such that which yields

for all . Thus, it is obtained by (51) that

for all with and . By (52) and (49),

By (53) and (50),

for all and . Therefore, we obtain from (54) and (55) that there exists the nonnegative constant such that

Hence, . This step completes the proof.

Theorem 12. The inclusion

and the inequality

hold, where is defined by (16).

Proof. If , then by (16) and (17) there exists the constant satisfying the inequalities:

for all with . By taking

it is obtained by (59) that and . That is,

as desired.

Example 13. Let us take instead of and define the function as

Then, we have

for all . Thus, we obtain by (63) that and . From Theorem 12, we conclude that and

Theorem 14. The function is continuous with respect to the Euclidean metric on .

Proof. Let and be usual metric on . Then, we wish to show that the function is continuous. It is clear that

Besides, since

we have by (65) that and . Thus, since the equalities:

hold, there exists the nonnegative constant by (28) that the inequalities:

hold from (67) and (68), respectively. By (69), we have

Hence, the function is continuous at the arbitrary point which means that it is continuous on .

Theorem 15. Let be a real or complex Banach space with the norm , be real numbers such that , and be a nonnegative constant. The set of all functions satisfying

for all with is a linear space with the usual algebraic operations addition and scalar multiplication

and is a Banach space with the norm:

where .

Theorem 16. The function is continuous with respect to the Euclidean metric on .

Theorem 17. The inclusion

and the inequality

hold, where is the norm in the space .

Since the proofs of Theorems 15–17 are completely similar to that of Theorems 11–14, we leave them to the reader.

Lemma 18. Let be a real or complex Banach algebra with the norm and define by

where

Then, if

then and

Proof. We use the induction method. If , then

Thus, Lemma is true for .

Assume that if

then and

Now, let and . Then, we must show that and

Then,

(1) If , then we have from that

Furthermore,

for all . Since for all , . So, we have

from (87). Additionally,

By (89) and (83), we get

where . Thus, we conclude that

from (86) and (90), where .

(2) If , then , and since for all , we have . That is, (88) is obtained from (87). Besides,

(i) if , then (91) is derived from (86) and (90).(ii) If , since , we get

by taking instead of in (86). Let

Since , (91) is derived from (90). By and , we show that (88) and (91) hold. Therefore, and

So, Lemma 18 holds for all .

2.2. Some Bounded Linear Integral Operators on the Spaces and

Hereafter, by , we mean the constant defined by (79), and, by the integral of the Banach valued functions, we mean the Bochner integral unless stated otherwise.

Theorem 19. Let be a real or complex Banach algebra with the norm and .

(i) Then, the integral operator

defined by

is bounded with

that is, .

(ii) The operator

defined by

is bounded with

that is, .

(iii) Suppose that and . Then, the operator

defined by

is bounded with

that is, , where the constants , , and are given by

for all the real numbers such that .

Proof. (i) We have

for all and with . By (107), it is obtained that

such that

Thus, it is concluded from (108) that and

Besides, the norm of holds the inequality

from (110). Also, since is linear, it is an element of the space .

On the other hand:

(ii) The inequalities

hold for all    with  .

So, by (112),

and by (113),

where . By usage of (114) and (115), we get that and

Thus, is bounded with

from (116). Furthermore is linear, and, hence, .

(iii) It is clear that

for all   with  .

By (118),

and by taking

Thus, we have by (119) and (121) that and

So, by (122), is a bounded linear operator and also

Therefore, .

Hereafter, by , and for all the real numbers with , we denote the constants given by (104), (105), and (106), respectively.

Theorem 20. Let be a real or complex Banach algebra with the norm and such that . Then, given by

is a linear operator from into , and, also, is bounded with

where

Proof. The function given by

is continuous for each . Therefore, the function is strongly measurable by Theorem 4, and, hence, the function is Lebesgue measurable from Theorem 5. Since

Lebesgue integral exists, and, so, Bochner integral in (124) exists by Theorem 8. Now, by taking , we get

from (128). Let define

for all and . Since

we have

So, we obtain which yields

where . Since , it is found by (129) and (135) that

Hence, by (136), and

That is, is bounded by (137) and

Clearly, is linear. So, .

2.3. The Solutions of the Linear Fredholm Integral Equations in the Spaces and

We have the following.

Theorem 21. Let be a real or complex Banach algebra and define the function by

such that

(i) Then, the linear Fredholm integral equation of the form

has a unique solution in the space with

where and is a real or complex parameter satisfying the inequality:

(ii) The solution of the equation

uniquely exists in the space . Besides,

where , is a real or complex parameter satisfying the inequality:

(iii) Suppose that

Then, (141) has a unique solution in the space such that

where ,

and is a real or complex parameter satisfying

Proof. (ii) Equation (144) may be rewritten as

or

which yields

where is defined by

and is an identity operator on .

By Lemma 18, we have that and

Also, we derive by Theorem 19 that and

from (117), and so

Since by (146), the inverse operator

exists and the norm of satisfies the inequality:

from Theorem 10. So, (144) has a unique solution in the space such that

and this also completes the proof of the second part.

The proof of the first and third parts of Theorem can be completed by the similar way to that of the second part, using Lemma 18, Theorem 19, (111), Lemma 18, Theorem 19, and (123), respectively.

Theorem 22. Let be a real or complex Banach algebra. Then Fredholm integral equation of the form

has a unique solution in with

where , is the constant given in Theorem 20, that is,

and is a real or complex parameter fulfilling

Proof. Equation (161) can be expressed as

or

which implies

where is defined by

and is an identity operator on .

By Theorem 20, we have that is in the space , and

Since by (164), the inverse operator

exists and the norm of satisfies the inequality:

from Theorem 10. So, (161) has a unique solution in the space such that

which completes the proof.

2.4. The Bounded Singular Integral Operators on the Space

Lemma 23. Let , , and with . Then the improper integral given by

exists, and there exists the nonnegative constant depending only on , , and such that

hold for all , where is defined by (73).

Hereafter, by , we mean the interval with .

Proof. We can write

for all . Here, is defined for each by and . Since the functions and are continuous for each , the function defined by is continuous, and, so, it is strongly measurable by Theorem 4, where and .

Since

and for all , we get . So,

By (176),

On the other hand, since , we get . Thus,

Since , we obtain , and . So, we have by (28) that

By (180) and

we have

Similarly, since for all , we get

from (28). Also, by

and (183), we derive

Thus, by taking ,

from (182) and (185).

From the inequality which is satisfied for all , we get , and, thus, . Hence,

From

and (187), it is obtained that

By (178), (186), and (189), we have

such that

Furthermore, since

we can write

Thus,

since and for all . Besides, by

and (194), we get

Since for all , the inequality holds, and, so,

By

and (195), we obtain

from (197). Then,

Also, we have

since . Therefore, we get by (195) that

since which implies that for all . So,

Since and which yields for all , we obtain

from (195). Hence,

By (192),

Since and which imply that and for all , it is found that

By

and (207),

Since

and , which yields for all , we get

By

and (211),

Since for all , we derive

By

and (214),

So, we derive

from (213) and (216).

Since and , which implies for all ,

Since for all ,

By

equations (219) and (220), it is obtained that

By (222),

Since and which yield and ,

By

and (224),

Thus, by taking

we obtain

from (203), (205), (209), (217), (223), (226). By (196), (199), and (228), we find

such that

By (190) and (229),

with

which terminates the proof.

Hereafter, we assume unless stated otherwise that is defined by (232).

Theorem 24. The operator defined by

is the singular integral operator in the space , and satisfies the inequality .

Proof. If , we get by Lemma 23 that

hold for all , . Furthermore, if , we obtain by (234) and

that

which holds for all , where . Thus, by (235) and (237),

and by (234),

Since and

from (238) and (239), we get

Also, is linear. So, . This also concludes the proof.

Theorem 25. The operator defined by

is the singular integral operator in the space such that

where and is given by

Proof. Equation (242) may be rewritten as

From (239) and (245),

for all . Thus, we have by (246) that

So, it is obvious by (247) that the inequality

holds. From (248), we obtain and since is linear, , and this completes the proof of theorem.

Acknowledgment

The authors would like to thank the referee for his/her careful reading and helpful comments that have improved the quality of the paper.

References

K. S. Thomas, “Galerkin methods for singular integral equations,” Mathematics of Computation, vol. 36, no. 153, pp. 193–205, 1981.

T. Okayama, T. Matsuo, and M. Sugihara, “Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind,” Journal of Computational and Applied Mathematics, vol. 234, no. 4, pp. 1211–1227, 2010.

A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998.

P. K. Kythe and P. Puri, Computational Methods for Linear Integral Equations, Birkhäauser, Boston, Mass, USA, 2002.

E. Babolian and A. A. Hajikandi, “The approximate solution of a class of Fredholm integral equations with a weakly singular kernel,” Journal of Computational and Applied Mathematics, vol. 235, no. 5, pp. 1148–1159, 2011.

M. A. Abdou and A. A. Nasr, “On the numerical treatment of the singular integral equation of the second kind,” Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 373–380, 2003.

A. A. Badr, “Integro-differential equation with Cauchy kernel,” Journal of Computational and Applied Mathematics, vol. 134, no. 1-2, pp. 191–199, 2001.

S. Bhattacharya and B. N. Mandal, “Numerical solution of a singular integro-differential equation,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 346–350, 2008.

M. R. Dostanić, “The asymptotic behavior of the singular values of the convolution operators with kernels whose Fourier transforms are rational functions,” Journal of Mathematical Analysis and Applications, vol. 395, no. 2, pp. 496–500, 2012.

J. Diestel and J. J. Uhl, Jr., Vector Measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, RI, USA, 1979.

R. F. Curtain and A. J. Pritchard, Functional Analysis in Modern Applied Mathematics, Academic Press, New York, NY, USA, 1977.

B. D. Craven, Lebesgue measure & integral, Pitman, Toronto, Canada, 1982.

C. Swartz, An Introduction to Functional Analysis, vol. 157 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992.

R. Kress, Linear Integral Equations, vol. 82, Springer, Berlin, Germany, 1989.

N. Dunford and J. T. Schwartz, Linear Operators Part III, Spectral Operators, John Wiley & Sons, New York, NY, USA, 1988.

F. D. Gakhov and F. D., Kraevıe Zadaçi, 1963.

N. L. Muskhelişvili, Singulyarnie Integralnie Uravneniya, Nauka, Moscow, Russia, 1968.

T. G. Gegeliya, “Nekotorie Spesialnie Klassi Funksiy i ih Svoystva,” Tr. Tbilissk. Matem. in-Ta An Gruz SSR, vol. 32, pp. 94–139, 1966.