On the Correct Solvability of the Boundary-Value Problem for One Class Operator-Differential Equations of the Fourth Order with Complex Characteristics
© A.R. Aliev and A.A. Gasymov. 2009
Received: 19 February 2009
Accepted: 26 August 2009
Published: 27 September 2009
Sufficient coefficient conditions for the correct and unique solvability of the boundary-value problem for one class of operator-differential equations of the fourth order with complex characteristics, which cover the equations arising in solving the problems of stability of plastic plates, are obtained in this paper. Exact values of the norms of operators of intermediate derivatives, which are involved in the perturbed part of the operator-differential equation under investigation, are found along with these in subspaces in relation to the norms of the operator generated by the main part of this equation. It is noted that this problem has its own mathematical interest.
It is well known that a number of problems in mechanics lead to studying the completeness of all or part of the eigenvectors and joint vectors of certain polynomial operator groups and the completeness of elementary solutions of the operator-differential equations corresponding to these groups (see, e.g., [1, 2], and their references). In this case, it is first necessary to investigate the correct solvability of Cauchy or boundary-value problems for these equations, and only after this it will be possible to proceed to the abovementioned problems. The present paper is dedicated to the problem of correct solvability of the boundary-value problem for one class of operator-differential equations of the fourth order, considered on a semiaxis.
Let be a separable Hilbert space and be a self-adjoint positively defined operator in .
If the vector function satisfies ( 1.1 ) almost everywhere in , then it is called a regular solution of ( 1.1 ).
holds, then it can be said that problem ( 1.1 ), ( 1.2 ) is regularly solvable.
It should be noted that the solvability theory for the Cauchy problem and the boundary-value problems for first- and second-order operator-differential equations have been studied in more detail elsewhere. In addition to books [6, 7], these problems have been considered also by Agmon and Nirenberg , Gasymov and Mirzoev , Kostyuchenko and Shkalikov , and in works in their bibliographies. Other papers in which issues of the solvability of various problems for operator-differential equations of higher order have been studied have appeared alongside these works, and sufficiently interesting results have been obtained. Among these papers are those by Gasymov [11, 12], Dubinskii , Mirzoev , Shakhmurov , Shkalikov , Aliev [17, 18], Agarwal et al. , Favini and Yakubov , the book by Yakubov , and other works listed in their bibliographies.
Sufficient coefficient conditions for regular solvability of the boundary-value problem stated in (1.1) and (1.2) are presented in this paper. To obtain these conditions, the main challenge is to find the exact values of the norms of operators of intermediate derivatives in subspaces , the norms of which are expressed by the main part of (1.1). This problem has its own mathematical interest (see, e.g., [21, 22], and works given in their bibliographies). Estimation of the norms of operators of intermediate derivatives, which are involved in the perturbed part of (1.1), is performed with the help of a factorization method for one class of polynomial operator groups of eighth order, depending on a real parameter. A similar approach has been presented in [9, 14], which makes it possible to formulate solvability theorems for the boundary-value problems, with conditions which can be easily checked.
It should be noted that if the main part of the equation has the operator in the form , then a biharmonic equation results, which is of mathematical interest not only theoretically, and also from a practical point of view. Many problems of elasticity theory (e.g., the theory of bending of thin elastic slabs ) can be reduced to studying the boundary-value problems for such equations. Much research has been performed to investigate the solvability of such problems, for example, that reported in . Operator-differential equations, which are studied in the present paper, include the fourth-order equations which arise when solving the stability problems of plates made of plastic material (see [25, pages 185–196]). It is very difficult to solve such problems because the differential equation must be solved in a more complete form, that is, when the main part of the equation has terms containing and . As a result, the equation has more complex characteristics, and (1.1) is of this type.
Furthermore, let us denote by the spectrum of the operator .
2. Auxiliary Results
First, let us study the main part of (1.1):
The following theorem is true.
is an isomorphism between the spaces and .
Because , then from the theorem on trace [3–5], it follows that all elements of the vector belong to . Continuing this process, it is apparent that the operator matrix is boundedly invertible in . Therefore, all . Consequently, . In the same way, it can be established that the equation has only a trivial solution.
are used in the expression .
As a result, is bounded and acts mutually and uniquely from the space to the space . Then, taking into account the Banach theorem on the inverse operator, it can be established that the operator carries out the isomorphism from the space to . Thus, the theorem is proved.
the following statement results.
Let be bounded operators in . Then the operator is a bounded operator from to .
Thus, the lemma is proved.
Now certain properties of polynomial operator groups will be investigated, which will have in the future a special role.
The following can then be established.
where and the numbers satisfy the following systems of equations:
and moreover , and the numbers satisfy the systems of equations shown in Lemma 2.3, which are obtained from (2.29) in the process of comparing the coefficients for the same degrees. Then, from the spectral decomposition of operator , the proof of the lemma can be obtained from (2.29). Thus, the lemma is proved.
The next step is to prove the theorem, which will play an important role in future investigations and will show the special importance of the spectral properties of the polynomial operator groups and .
Substituting (2.35) into (2.34), from Lemma 2.3, (2.31) can be obtained. Thus, the theorem is proved.
From Theorem 2.4, it follows that:
3. Norms of the Operators of Intermediate Derivatives
First, let us calculate .
It holds that
because in this case, for sufficiently small , the smallest value is negative for some . Then there exists an interval such that for .
Consequently, , and the lemma is proved.
is true, where is obtained from by removing the first three rows and columns, . The correctness of (3.8) follows directly from Theorem 2.4.
The following statement indicates when the numbers , can be equal to .
To establish the condition , it is necessary and sufficient that be positive for any .
where are uniquely determined from the conditions at zero in (3.12) and (3.13). As a result, writing inequality (3.9) for the vector function , for . Necessity is thereby proved.
As this expression goes to the limit as , it can be observed that , and from this, . Sufficiency is thereby proved, and thus the lemma is completely proved.
It is interesting that for some , it may occur that .
It holds that if and only if has a solution in the interval ; moreover, this root is equal to .
Thus, there exists a vector such that for , . Because is a continuous function of the argument then , and this means that has a root in the interval .
Inversely, if has a root in the interval , then this means that for any the number cannot be positive. This is why, from Lemma 3.2, . Denoting the root of by , it can be seen that , because from the proof of the lemma, it was obtained that for , is positive. Moreover, because , it can be determined that . The lemma is thereby proved.
By generalizing the last two lemmas, the following theorem can be derived.
In the same way, it is possible to determine the results for boundary-value problems of the form ( 1.1 ), ( 1.2 ) for having any three values from the collection .
By considering concretely the cases , the following statement results.
Taking into account the abovementioned procedure for finding the numbers , it is necessary to solve the systems from the proof of Lemma 2.3 together with the equation .
In the case , it can be determined that . This is why . To find the number , it is necessary to solve the system from Lemma 2.3 for together with the equation . In this case, , and consequently and . As a result, . In the case , it is found that . Then, from the corresponding system, it can be obtained that and or . It is clear that . From the other side, if in the equation , it is assumed that , then the result is that , which has only one real root, . Therefore, , and the theorem is proved.
4. Solvability Conditions for the Boundary-Value Problem (1.1), (1.2)
The results obtained make it possible to determine sufficient coefficient conditions of regular solvability for the boundary-value problem (1.1), (1.2). In particular, the following main theorem is true.
is satisfied, where the numbers , are as defined in Theorem 3.6. Then the boundary-value problem (1.1), (1.2) is regularly solvable.
Thus, the theorem is proved.
The conditions of regular solvability obtained here for the boundary-value problem (1.1), (1.2) are not improvable in terms of the operator coefficients of (1.1).
the given problem has a unique solution in the space .
The authors have dedicated this paper, in gratitude for useful consultation in their work, as a sign of deep respect to the memory of the Academician of the Azerbaijan National Academy of Sciences, Professor M. G. Gasymov.
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