On the regularizability conditions of integral equations Текст научной статьи по специальности «Математика»

MSC 47A52

DOI: 10.14529/mmpl50309

ON THE REGULARIZABILITY CONDITIONS OF INTEGRAL EQUATIONS

L.D. Menikhes, South Ural State University, Chelyabinsk, Russian Federation, leonid .menikhes@gmail .com,

V. V. Karachik, South Ural State University, Chelyabinsk, Russian Federation, kar achik@susu .ru

Solving of integral equations of the first kind is an ill-posed problem. It is known that all problems can be divided into three disjoint classes: correct problems, ill-posed regularizable problems and ill-posed not regularizable problems. Problems of the first class are so good that no regularization method for them is needed. Problems of the third class are so bad that no one regularization method is applicable to them. A natural application field of the regularization method is the problems from the second class. But how to know that a particular integral equation belongs to the second class rather than to the third class? For this purpose a large number of sufficient regularizability conditions were constructed. In this article one infinite series of sufficient conditions for regularizability of integral equations constructed with the help of duality theory of Banach spaces is investigated. This method of constructing of sufficient conditions proved to be effective in solving of ill-posed problems. It is proved that these conditions are not pairwise equivalent even if we are restricted by the equations with the smooth symmetric kernels.

Keywords: integral equations; regularizability; smooth symmetric kernels.

To the memory of Alfredo Lorenzi.

1. Introduction

In 1963 a new method for solving of ill-posed problems called as regularization method was proposed by A.N. Tikhonov [1]. The subsequent development of the science has shown a great efficiency of the regularization method. However, soon it became clear that this method does not provide a satisfactory solution in all cases. Problems for which there is a convergence of the regularization method became known as regularizable problems. Thus all ill-posed problems can be divided into two classes: regularizable problems and non-regularizable problems. In [2] an example of the non-regularizable integral equation was constructed. So, finding conditions for the regularizability of a problem is an important and actual problem. In papers [3 8] using the duality theory of Banach spaces the regularizability conditions were investigated. In paper [9] integro-differential equations were considered and in paper [10J some multidimensional integral equations were investigated.

2. Sufficient Regularizability Conditions of Integral Equations

Let E and F be Banach spaces and A : E ^ Fbea linear continuous injective operator.

Definition 1. Mapping A-1 is called a regularizable if there exists a family of mappings Rs : F ^ E, where 5 € (0, 5o) such that

lim sup \\Rs y — x\\ =0

y:\\y-Ax\\<S

for any x € E.

In this case, a family of operators {Rs} is called a regularizer for the operator A-1. The operator equation Ax = y is called a regularizable equation, if the mapping A-1 is a regularizable mapping. In this case a family of elements {xs = Rs ys} gives a satisfactory approximate solution

Ax = y

ys of this equation is given approximately with a precision 5.

Consider the classical situation when E = C(0,1) and F = L2(0,1). Assume that the operator A is also continuous in the L2-norm. Then the operator A can be extended by continuity to the various subspaces M such that C(0,1) C M C L2(0,1).

In the papers [4, 5] the following theorems giving some sufficient conditions for the regularizability are proved.

Theorem 1. If the integral operator A : C(0,1) ^ L2(0,1) is injective and its extension to some Lp(0,1) p > 2 has a finite-dimensional kernel, then the mapping A-1 is regularizable.

Theorem 2. If the integral operator A : C(0,1) ^ L2(0,1) is injective and its extension to L<x(0,1) has a finite-dimensional kernel, then the mapping A-1 is regularizable.

Theorem 3. If the integral operator A : C(0,1) ^ L2(0,1) is injective and its extension to

P| Lp(0,1) has a finite-dimensional kernel, then the mapping A-1 is regularizable. p>2

Thus we have an infinite series of regularizability conditions for the integral equations at p > 2

integral operators with the smooth symmetric kernels are considered? In paper [6] the negative answer for all conditions from Theorem 1 is given. All these conditions are not pairwise equivalent. In paper [7] it is proved that any condition from Theorem 1 is not equivalent to the condition from Theorem 2. The question about equivalence of conditions from Theorems 1 and 3, and conditions from Theorems 2 and 3 remains open. In the next section a negative answer to this question is given.

3. Comparison of Regularizability Conditions

Consider the integral operator

Q : f (x) ^ r K(x,t)f (t) dt, (1)

J 0

acting from C(0,1) to L2(0,1). Let K(x, t) be a continuous function on the unit square [0,1] x [0,1]. Then the operator Q is also continuous according to L2-norm in C (0,1) Let us denote by Q

Q L2(0, 1) Q

investigate the regularizability of Q-1. However, the operator Q is not necessarily be injective. Moreover, in the given above theorems a connection between ker Q and regularizability of mapping Q-1 is established.

C(0, 1) L2(0, 1)

smooth symmetric kernel, extension of which to any Lp(0,1), p > 2 has an infinite-dimensional

kernel and extension to P| Lp(0,1) has a finite-dimensional kernel.

p>2 Q

and the statement of Theorem 4. This is easy to see by looking at the corresponding proof.

□

Corollary 1. The sufficient regularizability conditions from Theorems 1 and 3 are not equivalent.

Proof. Indeed, for the operator from Theorem 4 by virtue of Theorem 3 it follows the regularizability of mapping A-1, while Theorem 1 does not give an answer about its regularizability.

□

Theorem 5. There exists an injective integral operator acting from C(0,1) to L2(0,1) with the

smooth symmetric kernel, extension of which to P| Lp(0,1) has an infinite-dimensional kernel,

p>2

but its extension to L^(0,1) has a finite-dimensional kernel.

Proof. Let us introduce a sequence of the intervals

Jk =

1 1 1 - 1 -

к € N.

' 2fc+i

We denote by hk(x) the functions defined on [0,1] having the support in Jk and such that

hk(x) € p| Lp(0,1), p>2

but hk(x) € L^(0,1) for k € N. It is easy to see that such functions exist. Indeed,

ln x € p| Lp(0,1) p>2

since the integral

/ | ln x\p dx J 0

converges for any p > 2, but ln x € L^(0,1). If we now linearly mapping the interval [0,1] on the internal Jk then the function ln x transforms to a function which satisfies all conditions required from the function hk (x).

Denote by M a closure of the linear span of functions hk (x)

M = span{hk(x) : к £ N},

i.e. the smallest closed subspace from L2(0,1) containing all the functions hk(x) for k € N. As usual C0^(a, b) is a subspace of C^(a,b) consisting of functions with a compact support, i.e. infinitely differentiable and vanish in neighborhoods of the points a and b. Consider the following lemma from [2].

Lemma 1. Let h(t) € L2(a, b) and

H = { f (t) € L2(a, b) : £ f (t)h(t) dt = 0} .

Then

ffp|Co°°M) = H.

Let us verify that the functions from Co°(a,b) are dense not only in the hyperplanes, but also in N = M±, which is an orthogonal complement to M, i.e. the following equality

Nf]C^(0,1) = N (2)

holds.

Let f € N and e > 0. We show that there exists a function from C0°(0,1) R N and such that \\f — g\\ < e. Choose a number n such that

f1 e2

I f 2(t) dt < 4. (3)

1 2"+1

Since the function f (t) is orthogonal to the functions hk (t), k € N, by virtue of Lemma 1 applied to the function hk (t) there exists a family of fu nctions {fk (t), k = 1, 2,... ,n} for which the following conditions are fulfilled

1. fk(t) € C0?(Jk), k = 1, 2,... ,n;

2. J fk(t)hk(t) dt = 0, k = 1,2,... ,n; (4)

£

33 \\fk — fJ\\< k = 1, 2,...,n,

where f | jk is a restriction of the function f (t) on the interval Jk- Besides, there exists a function fo(t) € C0°(O, 2) in the interval J0 = [0,1 ] such that

£

\\fo— fJ \< WTT). (5)

Now consider the function

(t) i fk (t), fort € Jk ,k = 0,1,... ,n; ) [0, fort € [1 — 2n+, 1].

Using (4) we obtain that g € C0^(0,1) R N since for any k € N

/ g(t)hk(t) dt = f g(t)hk(t) dt = 0,

o jk

and therefore, for any m(t) € M we get

f g(t)m(t) dt = 0. o

Finally, because of (3), (4) and (5) and from the following relation

L

(g(t) — f (t))2 dt <

\\g — f \\ = \l [\g(t) — f (t))2 dt < (g(t) — f (t))2 dt + e2 <

V0 \t=0J Jk 4

n I Z n

(g(t) — f (t))2 dt + 2 <E k=0\ Jk k=0

<>.,i /torn -f m>dt + 2 < ^ 2cn+T) + i=6

it follows that ^Cg°(0,1) = N.

N

{^n(t)} belonging to Cg°(0,1). It is true because we are able to choose a full linearly independent sequence of functions from N f] C0,o(0,1), orthogonalization of which gives us the necessary system of functions {^n(t)}.

Now consider the integral operator Q, given by formula (1) with the kernel of the form

<x

K(x,t) = ^ (lnVn{x)^n{t), (6)

n=1

where {(¿>n(x)} is an arbitrary orthonormal system of infinitely differentiable functions on [0,1], i^n(t)} is the constructed above system of functions and

=-L Usup .....sup

n \ же[0,1] же[0,1] у

х max( sup | фп(г)| ,..., sup | | )

V ie [0,1] te [0,1] у

Then it is clear that the series from (6) converges uniformly and the series obtained from it by any-times termwise differentiation are also converges uniformly. Therefore the function K(x,t) is an infinitely differentiate function on the unit square [0,1] x [0,1]. We show that ker Q = M. In fact, if

„1 ,, 1 ^ K(x,t)f (t) dt = У^апФи(х)фи(Ь)/(t) dt = 0, (7)

' j0 n=1

then

<x

anbn

у] апЬпфп(х) = 0, (8)

n=1

where

bn = I f (t)Mt) dt 0

is the ^^^h ^^^fcient of function f (t) by the system of functions {фn(t)}. According to

the Lebesgue theorem the series from (7) can be integrated term by term. From (8), since the system {^n(x)} is orthonormal and an = 0, it follows that bn = 0 for n € N, i.e. f (t) € N± = M. It is obvious that if f (t) € M then f (t) € ker Q.

So, we proved that ker Q = M. If we take фп = Фп, then the kernel of the оperator Q is

Q

Q

C(0,1) since all the functions in M, except the identical zero, are discontinuous. Its extension to

P| Lp(0,1) has an infinite-dimensional kernel because it contains all the functions hk(t), к € N. p>2

But extension of the operator Q to L^(0,1) has a null kernel because all the functions in M, except identical zero, are not bounded. Theorem is proved.

□

Corollary 2. Sufficient regularizability conditions from Theorems 2 and 3 are not equivalent.

Q

regularizability of Q-1, while Theorem 3 does not give an answer about regularizability of Q-1.

□

References

1. Tikhonov A.N. [The Solution of Incorrectly Formulated Problems and the Regularization Method]. Dokl. AN SSSR, 1963, vol. 151, no. 3, pp. 501-504. (in Russian)

2. Menikhes L.D. [Regularizability of Mappings of Inverse to Integral Operators]. Dokl. AN SSSR, 1978, vol. 241, no. 2, pp. 282-285. (in Russian)

3. Vinokurov V.A., Menikhes L.D. [Necessary and Sufficient Condition for the Linear Regularizability], Dokl. AN SSSR, 1976, vol. 229, no. 6, pp. 1292-1294. (in Russian)

4. Menikhes L.D. Regularizability of Some Classes of Mappings That are Inverses of Integral Operators. Mathematical Notes, 1999, vol. 65, no. 1-2, pp. 181-187. DOI: 10.1007/BF02679815

5. Menikhes L.D. On a Sufficient Condition for Regularizability of Linear Inverse Problems. Mathematical Notes, 2007, vol. 82, no. 1-2, pp. 242-246. DOI: 10.1134/S0001434607070267

6. Menikhes L.D., Kondrat'eva O.A. On Comparison of the Conditions for Regularizability of Integral Equations. Izvestiya Chelyabinskogo Nauchnogo Centra, 2009, vol. 1 (43), pp. 11-15. (in Russian)

7. Menikhes L.D. On Connection Between Sufficient Conditions of Regularizability of Integral Equations. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2013, vol. 5, no. 1, pp. 50-54. (in Russian)

8. Menikhes L.D. Linear Regularizability of Mappings Inverse to Linear Operators. Russian Mathematics, 1979, no. 12, pp. 35-38. (in Russian)

9. Favini A., Lorenzi A., Tanabe H. Singular Evolution Integro-Differential Equations with Kernels Defined on Bounded Intervals. Applicable Analysis, 2005, vol. 84, no. 5, pp. 463-497. DOI: 10.1080/00036810410001724418

10. Karachik V.V. Normalized System of Functions with Respect to the Laplace Operator and Its Applications. Journal of Mathematical Analysis and Applications, 2003, vol. 287, no. 2, pp. 577-592. DOI: 10.1016/S0022-247X(03)00583-3

Received May 15, 2015

УДК 517.948 Б01: 10.1152!) шшрШШШ

ОБ УСЛОВИЯХ РЕГУЛЯРИЗУЕМОСТИ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ

Л.Д. Менихес, В. В. Карачик

Решение интегральных уравнений первого рода представляет собой некорректную задачу. Как известно, все задачи можно разбить на три непересекающихся класса: корректные задачи, некорректные регуляризуемые задачи, некорректные нерегуляризуе-мые задачи. Задачи из первого класса настолько хороши, что метод регуляризации для них не нужен. Задачи третьего класса настолько плохи, что метод регуляризации к ним не применим. Естественным полем применения метода регуляризации являются

задачи второго класса. Но как узнать, что данное интегральное уравнение принадлежит ко второму, а не к третьему классу. Для этого было построено большое количество достаточных условий регуляризуемости. В данной статье исследуется одна бесконечная серия достаточных условий регуляризуемости интегральных уравнений, построенных с помощью теории двойственности банаховых пространств. Этот метод построения достаточных условий показал свою эффективность при решении некорректных задач. Доказано, что эти условия являются попарно не эквивалентными, даже если ограничиться уравнениями с гладкими симметричными ядрами.

Ключевые слова: интегральные уравнения; регуляризуемость; гладкие симметричные ядра.

Литература

1. Тихонов, А.Н. О решении некорректно поставленных задач и методе регуляризации /

A.Н. Тихонов // Докл. АН СССР. - 1963. - Т.[151, № 3. - С. 501-504.

2. Менихес, Л.Д. О регуляризуемости отображений, обратных к интегральным операторам / Л.Д. Менихес // Докл. АН СССР. - 1978. - Т. 241, № 2. - С. 282-285.

3. Винокуров, В.А. Необходимое и достаточное условие линейной регуляризуемости /

B.А. Винокуров, Л.Д. Менихес // Докл. АН СССР. - 1976. - Т. 229, № 6. - С. 1292-1294.

4. Менихес, Л.Д. О регуляризуемости некоторых классов отображений, обратных к интегральным операторам / Л.Д. Менихес // Математические заметки. - 1999. - Т. 65, № 2. - С. 222-229.

5. Менихес, Л.Д. Об одном достаточном условии регуляризуемости линейных обратных задач / Л.Д. Менихес // Математические заметки. - 2007. - Т. 82, № 2. - С. 242-247.

6. Менихес, Л.Д. О сравнении условий регуляризуемости интегральных уравнений / Л.Д. Менихес, O.A. Кондратьева // Известия Челябинского научного центра. - 2009. -Вып. 1 (43). - С. 11-15.

7. Менихес, Л.Д. О связи достаточных условий регуляризуемости интегральных уравнений / Л.Д. Менихес // Вестник Южно-Уральского государственного университета. Серия: Математика. Механика. Физика. - 2013. - Т. 5, № 1. - С. 50-54.

8. Менихес, Л.Д. Линейная регуляризуемость отображений, обратных к линейным операторам / Л.Д. Менихес // Известия ВЫСШИХ учебных заведений. Математика. - 1979. -№ 12. - С. 35-38.

9. Favini, A. Singular Evolution Integro-Differential Equations with Kernels Defined on Bounded Intervals / A. Favini, A. Lorenzi, H. Tanabe // Applicable Analysis. - 2005. - V. 84, № 5. -P. 463-497.

10. Karachik, V.V. Normalized System of Functions with Respect to the Laplace Operator and Its Applications / V.V. Karachik // Journal of Mathematical Analysis and Applications. -2003. - V. 287, № 2. - P. 577-592.

Леонид Давидович Менихес, доктор физико-математических наук, заведующий кафедрой, кафедра «Математический и функциональный анализ:», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].

Валерий Валентинович Карачик, доктор физико-математических наук, профессор, кафедра «Математический и функциональный анализ», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].

Поступила в редакцию 15 мая 2015 г.