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.
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.
2. Auxiliary Results
First, let us study the main part of (1.1):
The following theorem is true.
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.
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.
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.
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
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 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.
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 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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