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Determinantal Representations of Solutions and Hermitian Solutions to Some System of Two-Sided Quaternion Matrix Equations.

1. Introduction

The study of matrix equations and systems of matrix equations is an active research topic in matrix theory and its applications. Research on the classical system of two-sided matrix equations

[A.sub.1]X[B.sub.1] = [C.sub.1], [A.sub.2]X[B.sub.2] = [C.sub.2] (1)

over the complex field, a principle domain, and the quaternion skew field has been actively ongoing for many years. For instance, Mitra [1,2] gave necessary and sufficient conditions of system (3) over the complex field and the expression for its general solution. Navarra et al. [3] derived a new necessary and sufficient condition for the existence and a new representation of the general solution to (3) over the complex field and used the results to give a simple representation. Ozgiiler et al. [4] gave solutions to (3) over a principle domain. Wang [5] got solvability conditions of system (3) over the quaternion skew field and represented its general solution in terms of generalized inverses.

Since quaternion matrices play an important role in quantum mechanics, signal processing, and control theory, research on quaternion matrix equations and systems of quaternion matrix equations, their general solutions, especially Hermitian solutions, has been actively developing for more recent years (see, e.g., [6-26]).

Throughout the paper, we denote the real number field by R, the set of all m x n matrices over the quaternion algebra

[mathematical expression not reproducible] (2)

by [H.sup.mxn], and by [H.sup.mxn.sub.r] its subset of matrices of a rank r. Let M(n, H) be the ring of n x n quaternion matrices. For A [member of] [H.sup.mxn], the symbol [A.sup.*] stands for the conjugate transpose (Hermitian adjoint) matrix of A. The matrix A = ([a.sub.ij]) [member of] [H.sup.nxn] is Hermitian if [A.sup.*] = A.

Motivated by the wide application of quaternion matrix equations and in order to improve the theoretical development of solutions and Hermitian solutions to quaternion matrix equations, we consider a special case of (3), more specifically,

[A.sub.1]X[A.sup.*.sub.1] = [C.sub.1], [A.sub.2]X[A.sup.*.sub.2] = [C.sub.2], (3)

Generalized inverses are useful tools used to solve matrix equations. The definition of the Moore-Penrose inverse matrix has been extended to quaternion matrices as follows.

Definition 1. The Moore-Penrose inverse of A [member of] [H.sup.mxn], denoted by [A.sup.[dagger]], is the unique matrix [A.sup.[dagger]] [member of] [H.sup.mxn] satisfying the following four equations:

(1) A[A.sup.[dagger]]A = A,

(2) [A.sup.[dagger]]A[A.sup.[dagger]] = [A.sup.[dagger]],

(3) [(A[A.sup.[dagger]]).sup.*] = A[A.sup.[dagger]],

(4) [([A.sup.[dagger]]A).sup.*] = [A.sup.[dagger]]A. (4)

The main goal of this paper is to derive determinantal representations of solutions and Hermitian solutions to system (1) over the quaternion skew field using previously obtained determinantal representations of the Moore-Penrose inverse. Evidently, determinantal representation of a solution gives a direct method of its finding analogous to the classical Cramer's rule that has important theoretical and practical significance [27].

Through the noncommutativity of the quaternion algebra when difficulties arise already in determining the quaternion determinant, the problem of the determinantal representation of generalized inverses only now can be solved due to the theory of column-row determinants introduced in [28, 29]. Within the framework of the theory of column-row determinants, determinantal representations of various kinds of generalized inverses (generalized inverses) solutions of quaternion matrix equations have been derived by the author (see, e.g., [30-39]) and by other researchers (see, e.g., [40-43]).

The paper is organized as follows. In Section 2, we start with preliminary introduction of row-column determinants, determinantal representations of the Moore-Penrose inverse previously obtained within the framework of the theory of row-column determinants, and Cramer's rules for the two-sided matrix equation and of its special cases, left- and right-sided equations. We derive some simplified expressions of general and partial solutions to (3), a solvability criterion and expressions of general and partial solutions to system (1), and determinantal representations (analogs of Cramer's rule) of its solution and Hermitian solution. A numerical example to illustrate the main results is considered in Section 4. Finally, the conclusion is drawn in Section 5.

2. Preliminaries

For A [member of] [H.sup.nxn], we define n row determinants and n column determinants. Suppose [S.sub.n] is the symmetric group on the set [I.sub.n] = {1, ..., n}.

Definition 2 (see [28]). The ith row determinant of A = ([a.sub.ij]) [member of] [H.sup.nxn] is defined for all i = 1, ..., n by putting

[mathematical expression not reproducible], (5)

where [mathematical expression not reproducible] and [mathematical expression not reproducible] and s = 1, ..., q.

Definition 3 (see [28]). The jth column determinant of A = ([a.sub.ij]) [member of] [H.sup.nxn] is defined for all j = 1, ..., n by putting

[mathematical expression not reproducible], (6)

where [mathematical expression not reproducible] and [mathematical expression not reproducible] and s = 1, ..., [l.sub.t].

Since [28] for Hermitian A we have

[mathematical expression not reproducible], (7)

the determinant of a Hermitian matrix is defined by putting det A = [rdet.sub.i] A = [cdet.sub.i] A for all i = 1, ..., n.

The properties of row and column determinants are completely explored in [29]. We note the following that will be required below.

Lemma 4. Let A [member of] [H.sup.mxn]. Then

[cdet.sub.i] [A.sup.*] = [bar.[rdet.sub.i] A], [rdet.sub.i] [A.sup.*] = [cdet.sub.i] A. (8)

We shall use the following notations. Let [alpha] := {[[alpha].sub.1], ..., [[alpha].sub.k]} [subset or equal to] {1, ..., m} and [beta] := {[[beta].sub.1], ..., [[beta].sub.1]} [subset] {1, ..., n} be subsets of the order 1 [less than or equal to] k [less than or equal to] min{m, n}. Let [A.sup.[alpha].sub.[beta]] be a submatrix of A whose rows are indexed by a and columns indexed by [beta]. Similarly, let [A.sup.[alpha].sub.[alpha]] be a principal submatrix of A whose rows and columns are indexed by [alpha]. If A [member of] M(n, H) is Hermitian, then [[absolute value of A].sup.[alpha].sub.[alpha]] is the corresponding principal minor of det A. For 1 [less than or equal to] k [less than or equal to] n, the collection of strictly increasing sequences of k integers chosen from {1, ..., n} is denoted by [mathematical expression not reproducible]. For fixed i [member of] [alpha] and [mathematical expression not reproducible].

Let [a.sub..j] be the jth column and [a.sub.i.] be the ith row of A. Suppose [A.sub..j](b) denotes the matrix obtained from A by replacing its jth column with column b, and [A.sub.i.](b) denotes the matrix obtained from A by replacing its ith row with the row b. Denote by [a.sup.*.sub..j] and [a.sup.*.sub.i.] the jth column and the ith row of [A.sup.*], respectively.

Theorem 5 (see [30]). If A [member of] [H.sup.mxn.sub.r] then the Moore-Penrose inverse [A.sup.[dagger]] = ([a.sup.[dagger].sub.ij]) [member of] [H.sup.mxn] has the following determinantal representations,

[mathematical expression not reproducible], (9)

and

[mathematical expression not reproducible]. (10)

Remark 6. For an arbitrary full-rank matrix A [member of] [H.sup.mxn.sub.r], we put

[mathematical expression not reproducible], (11)

where a column vector [d.sub..j] and a row vector [d.sub.i.] have appropriate sizes.

Remark 7. First note that [([A.sup.*]).sup.[dagger]], = [([A.sup.[dagger]]).sup.*]. Because of symbol equivalence, we shall use the denotation [A.sup.[dagger],*] := [([A.sup.*]).sup.[dagger]] as well. So, by Lemma 4, for the Hermitian adjoint matrix [A.sup.*] [member of] [H.sup.nxm.sub.r] determinantal representations of its Moore-Penrose inverse

[mathematical expression not reproducible], (12)

and

[mathematical expression not reproducible]. (13)

Corollary 8. If A [member of] [H.sup.nxm.sub.r], then the projection matrix [A.sup.[dagger]] A =: [P.sub.A] = [([P.sub.ij]).sup.nxn] has the determinantal representation

[mathematical expression not reproducible], (14)

where [[??].sub..j] is the jth column of [A.sup.*] A [member of] [H.sup.nxn].

Corollary 9. If A [member of] [H.sup.mxn.sub.r], then the projection matrix A[A.sup.[dagger]] = [Q.sub.A] = [([q.sub.ij]).sub.mxm] has the determinantal representation

[mathematical expression not reproducible], (15)

where [[??].sub.i.] is the ith row of A[A.sup.*] [member of] [H.sup.mxm].

The orthogonal projectors [L.sub.A] := I - [A.sup.[dagger]]A and [R.sub.A] := I - A[A.sup.[dagger]] induced by A will be used below.

Theorem 10 (see [44]). Let A [member of] [H.sup.mxn], B [member of] [H.sup.rxs], C [member of] [H.sup.mxs] be known and X [member of] [H.sup.nxr] be unknown. Then the matrix equation

AXB = C (16)

is consistent if and only if A[A.sup.[dagger]]C[B.sup.[dagger]]B = C. In this case, its general solution can be expressed as

X = [A.sup.[dagger]]C[B.sup.[dagger]] + [L.sub.A]V + W[R.sub.B], (17)

where V, W are arbitrary matrices over H with appropriate dimensions.

Theorem 11 (see [31]). Let [mathematical expression not reproducible]. Then the partial solution [X.sub.0] = [A.sup.[dagger]]C[B.sup.[dagger]] = ([x.sup.0.sub.ij]) [member of] [H.sup.nxr] to (16) has determinantal representations,

[mathematical expression not reproducible], (18)

or

[mathematical expression not reproducible], (19)

where

[mathematical expression not reproducible], (20)

are the column vector and the row vector, respectively. [[??].sub.i.] and [[??].sub..j] are the ith row and the jth column of [??] = [A.sup.*]C[B.sup.*].

Corollary 12. Let A [member of] [H.sup.mxn.sub.k], C [member of] [H.sup.mxs] be known and X [member of] [H.sup.nxs] be unknown. Then the matrix equation AX = C is consistent if and only if A[A.sup.[dagger]]C = C. In this case, its general solution can be expressed as X = A[A.sup.[dagger]] + [L.sub.A]V, where V is an arbitrary matrix over H with appropriate dimensions. The partial solution [X.sup.0] = A[A.sup.[dagger]]C has the following determinantal representation,

[mathematical expression not reproducible]. (21)

where [[??].sub..j] is the jth column of [??] = [A.sup.*]C.

Corollary 13. Let B [member of] [H.sup.rxs.sub.k], C [member of] [H.sup.nxs] be given and X [member of] [H.sup.nxr] be unknown. Then the equation XB = C is solvable if and only

if C = C[B.sup.[dagger]]B and its general solution is X = C[B.sup.[dagger]] + W[R.sub.B], where W is any matrix with conformable dimension. Moreover, its partial solution X = C[B.sup.[dagger]] has the determinantal representation,

[mathematical expression not reproducible]. (22)

where [[??].sub.i.] is the ith row of [??] = C[B.sup.*].

3. Cramer's Rules for the Solution and Hermitian Solution to System (3)

First, consider the general system (1).

Lemma 14 (see [5]). Let [mathematical expression not reproducible] be given and X [member of] [H.sup.nxr] is to be determined. Put [mathematical expression not reproducible]. Then system (1) is consistent if and only if

[A.sub.i][A.sup.[dagger].sub.i][C.sub.i][B.sup.[dagger].sub.i][B.sub.i] = [C.sub.i], I = 1, 2; (23)

[mathematical expression not reproducible]. (24)

In that case, the general solution to (1) can be expressed as the following:

[mathematical expression not reproducible], (25)

where Z and W are arbitrary matrices over H with compatible dimensions.

Some simplification of (25) can be derived due to the quaternionic analogue of the following proposition.

Lemma 15 (see [45]). If A e [H.sup.nxn] is Hermitian and idempotent, then the following equation holds for any matrix B [member of] [H.sup.mxn],

A [(BA).sup.[dagger]] = [(BA).sup.[dagger]]. (26)

It is evident that if A [member of] [H.sup.nxn] is Hermitian and idempotent, then the following equation is true as well:

[(AB).sup.[dagger]] A = [(AB).sup.[dagger]]. (27)

Since [mathematical expression not reproducible], and [R.sub.H] are projectors, then by (26) and (27), we have, respectively,

[mathematical expression not reproducible]. (28)

Using (28) and (23), we obtain the following expressions of (25):

[mathematical expression not reproducible]. (29)

By putting Z, W as zero-matrices in (29), we obtain the following partial solution of (25):

[mathematical expression not reproducible]. (30)

Now consider system (1). Since

[mathematical expression not reproducible], (31)

so [mathematical expression not reproducible], and [mathematical expression not reproducible]. Moreover, substituting [B.sub.i] = [A.sup.*.sub.i], we have [mathematical expression not reproducible]; similarly, [mathematical expression not reproducible]. Due to the above, we obtain the following analog of Lemma 14.

Lemma 16. Let [A.sub.1] [member of] [H.sup.mxn], [A.sub.2] [member of] [H.sup.kxn], [C.sub.1] [member of] [H.sup.mxm], [C.sub.2] [member of] [H.sup.kxk] be given and X [member of] [H.sup.nxn] is to be determined. Then system (3) is consistent if and only if

[mathematical expression not reproducible]. (32)

In that case, the general solution to (3) can be expressed as follows:

[mathematical expression not reproducible]. (33)

Z and W are arbitrary matrices over H with compatible dimensions.

By putting Z, W as zero-matrices, the partial solution of (3) is

[mathematical expression not reproducible]. (34)

Further we give determinantal representations of (34).

Let [mathematical expression not reproducible].

Consider each term of (34) separately.

(i) Denote [C.sub.1]1 := [A.sub.2][C.sub.1][A.sub.1]. For the first term of (34) [X.sub.1] = [A.sup.[dagger].sub.1][C.sub.1] [([A.sub.1.sup.*]).sup.[dagger]] = ([x.sup.(1).sub.ij], we have

[mathematical expression not reproducible]. (35)

By using determinantal representations (9) and (13) of the Moore-Penrose inverses [A.sup.[dagger].sub.1] and [([A.sub.1.sup.*]).sup.[dagger]], respectively, we obtain

[mathematical expression not reproducible] (36)

Suppose [e.sub.l.] and [e.sub..l] are the unit row vector and the unit column vector, respectively, such that all their components are 0, except the lth components, which are 1. Since [mathematical expression not reproducible], then

[mathematical expression not reproducible] (37)

If we denote by

[mathematical expression not reproducible] (38)

the sth component of a row vector [v.sup.(1).sub.i.] = [[v.sup.(1).sub.i1], ..., [v.sup.(1).sub.in]],then

[mathematical expression not reproducible]. (39)

Further, it is evident that [mathematical expression not reproducible], so the first term of (34) has the determinantal representation

[mathematical expression not reproducible], (40)

where

[mathematical expression not reproducible]. (41)

If we denote by

[mathematical expression not reproducible] (42)

the fth component of a column vector [v.sup.(2).sub..j] = [[v.sup.(2).sub.1j], ..., [v.sup.(2).sub.nj]], then

[mathematical expression not reproducible]. (43)

So, another determinantal representation of the first term of (34) is

[mathematical expression not reproducible], (44)

where

[mathematical expression not reproducible], (45)

are the column vector.

(ii) Similarly above, for the second term [X.sub.2][H.sup.[dagger]][C.sub.2] [([A.sup.*.sub.2]).sup.[dagger]] = ([x.sup.(2).sub.ij]) of (34), we have

[mathematical expression not reproducible], (46)

or

[mathematical expression not reproducible], (47)

where

[mathematical expression not reproducible], (48)

are the column vector and the row vector, respectively. [c.sup.(21).sub.q.] and [c.sup.(21).sub.l.] are the qth row and the lth column of [C.sub.21] = [H.sup.*][C.sub.2][A.sub.2]. Note that [mathematical expression not reproducible].

(iii) The third term of (34) can be obtained similarly as well. So,

[mathematical expression not reproducible], (49)

or

[mathematical expression not reproducible], (50)

where

[mathematical expression not reproducible], (51)

are the column vector and the row vector, respectively. [c.sup.(22).sub.q.] are the qth row, and [c.sup.(22).sub..l] are the lth column of [C.sub.22] = [T.sup.*][C.sub.2]H. The following expressions give some simplifications in computing. Since [mathematical expression not reproducible] and [mathematical expression not reproducible], then [mathematical expression not reproducible].

(iv) Using (9) for determinantal representations of [H.sup.[dagger]] and [T.sup.[dagger]] in the fourth term of (34), we obtain

[mathematical expression not reproducible], (52)

where [a.sup.(2,H).sub..i], [a.sup.(2,T).sub..i] are the Zth columns of the matrices [H.sup.*][A.sub.2] and [T.sup.*][A.sub.2], respectively; [x.sup.(1).sub.zf] is the first term; [p.sub.fj] is the (fj)th element of [mathematical expression not reproducible] with determinantal representation by (14) as

[mathematical expression not reproducible], (53)

where [[??].sup.(2).sub..j] is the jth column of [A.sup.*.sub.2][A.sub.2]. Note that [mathematical expression not reproducible].

(v) Similarly to the previous case,

[mathematical expression not reproducible], (54)

(vi) Consider the sixth term by analogy to the fourth term. So,

[mathematical expression not reproducible], (55)

where

[mathematical expression not reproducible], (56)

and

[mathematical expression not reproducible], (57)

are the column vector and the row vector, respectively. [c.sup.(23).sub.q.] and [c.sup.(23).sub..l] are the qth row and the Ith column of [C.sub.23] = [T.sup.*] [C.sub.2][A.sub.2].

(vii) Using (9) for determinantal representations of [T.sup.[dagger]] and (12) for [H.sup.*,[dagger]] in the seventh term of (34), we obtain

[mathematical expression not reproducible], (58)

where [a.sup.(2,T).sub..q], [a.sup.(2,H,*).sub..f] are the ^th column of [T.sup.*][A.sub.2] and the fth row of [mathematical expression not reproducible], respectively.

Hence, we prove the following theorem.

Theorem 17. Let [mathematical expression not reproducible]. Then for the partial solution (34) to system (3), we have

[mathematical expression not reproducible], (59)

where the term [x.sup.(1).sub.ij] has the determinantal representations (40) and (44); [x.sup.(2).sub.ij], (46) and (47); [x.sup.(3).sub.ij], (49) and (50); [x.sup.(4).sub.ij], (52); [x.sup.(5).sub.ij], (54); [x.sup.(6).sub.ij], (55); and [x.sup.(7).sub.ij], (58).

Due to Khatri and Mitra [46], the next lemma can be generalized to H.

Lemma 18. Let A [member of] [H.sup.mxn] and B [member of] [H.sup.mxm] and B = [B.sup.*] be known and X [member of] [H.sup.nxn] be unknown. Then the matrix equation

AX[A.sup.*] = B (60)

has a Hermitian solution if and only if [Q.sub.A]B = B. In that case, the general Hermitian solution of (60) is

X = [A.sup.[dagger]]B[A.sup.[dagger],*] + [L.sub.A]V + [V.sup.*][L.sub.A], (61)

where V [member of] [H.sup.nxn] is any matrix.

As it follows from the above, if rank A = r, then the determinantal representation of the partial Hermitian solution X = [A.sup.[dagger]]B[A.sup.[dagger],*] = ([x.sub.ij]) is

[mathematical expression not reproducible], (62)

where

[mathematical expression not reproducible], (63)

are the row vector and the column vector and [b.sup.(1).sub..s] and [b.sup.(1).sub.f.] are the sth column and the fth row of [B.sub.1] = [A.sup.*]BA, respectively.

The general Hermitian solution to system (3) can be expressed as Y = (1/2)(X + [X.sup.*]), where X is an arbitrary solution of (3). Since by Lemma 18, the existence of Hermitian solutions (1) needs [C.sub.1] and [C.sub.2] to be Hermitian, then

[mathematical expression not reproducible]. (64)

So, the determinantal representation of the partial Hermitian solution Y = ([y.sub.ij]) can be obtained as [y.sub.ij] = (1/2)([x.sub.ij] + [bar.[x.sub.ji]]) for all (I, j = 1, n, where [x.sub.ij] is determined by Theorem 17 and [bar.[x.sub.ji]] = [[summation].sup.[delta].sub.ji] so that

[mathematical expression not reproducible], (65)

where

[mathematical expression not reproducible], (66)

are the row vector and the column vector, respectively. [c.sup.(21,*).sub..q] and [c.sup.(21,*).sub.l.] are the qth column and the lth row of [C.sup.*.sub.21] = [A.sup.*.sub.2][C.sub.2] H;

(iii) [mathematical expression not reproducible], (67)

where

[mathematical expression not reproducible], (68)

are the row vector and the column vector, respectively. [c.sup.(22,*).sub..q] are the qth column and [c.sup.(22,*).sub.l.] are the lth row of [C.sup.*.sub.22] = [H.sup.*][C.sub.2]T;

[mathematical expression not reproducible], (69)

where [a.sup.(2,T,*).sub.z.] is the z row of [A.sup.*.sub.2]T and [a.sup.(2,H,*).sub.q.] is the qth row of [A.sup.*.sub.2]H, respectively; [x.sup.(1).sub.zf] can be obtained by (40) or (44); evidently, [bar.[p.sub.if]] = [p.sub.if] and [p.sub.if] is the (if)th element of [mathematical expression not reproducible] with determinantal representation by (14) as

[mathematical expression not reproducible], (70)

where [[??].sup.(2).sub..f] is the fth column of [A.sup.*.sub.2][A.sub.2];

[mathematical expression not reproducible], (70)

where [[??].sup.(2).sub..f] is the fth column of [A.sup.*.sub.2][A.sub.2];

[mathematical expression not reproducible], (71)

where

[mathematical expression not reproducible], (72)

and

[mathematical expression not reproducible], (73)

are the row vector and the column vector, respectively. [c.sup.(23),*.sub..q] and [c.sup.(23),*.sub.l.] are the qth column and the lth row of [C.sup.*.sub.23] = [A.sub.2][C.sub.2][T.sup.*];

[mathematical expression not reproducible], (74)

where [a.sup.(2,T,*).sub.q.], [a.sup.(2,H).sub..f] are the qth row of [A.sup.*.sub.2]T and the fth column of [H.sup.*][A.sub.2], respectively.

4. An Example

In this section, we give an example to illustrate our results. Let us consider the system of matrix equations

[mathematical expression not reproducible] (75)

where

[mathematical expression not reproducible]. (76)

Since

[mathematical expression not reproducible], (77)

then rank [A.sub.1] = 2, rank [A.sub.2] = 2. By Theorem 5, one can find

[mathematical expression not reproducible]. (78)

Since [mathematical expression not reproducible], then, by Lemma 16, system (75) is consistent.

First, we can find the solution of (75) by direct calculation. Since X = [[summation].sub.[delta]][X.sub.[delta]], where

[mathematical expression not reproducible], (79)

then

[mathematical expression not reproducible]. (80)

Now, we find the solution of (75) by its determinantal representation by Theorem 17. Since

[mathematical expression not reproducible], (81)

then by (41)

[mathematical expression not reproducible]. (82)

Similarly, [v.sup.(1).sub.12] = 0.375, [v.sup.(1).sub.13] = 0.375j. So, [v.sup.(1).sub.1.] = [0.5k 0.375 0.375j]. Further, by (40),

[mathematical expression not reproducible]. (83)

Since

[mathematical expression not reproducible], (84)

and rank H = rank T = 1, then

[mathematical expression not reproducible]. (85)

So, by (47) and by (49), [x.sup.(2).sub.11] = 0 and [x.sup.(3).sub.11] = 0, respectively. Moreover, [x.sup.([delta]).sub.11] = 0 for all [delta] = 4, ..., 7. So, [x.sub.11] = [x.sup.(1).sub.11] = 0.5k.

So, [x.sub.11] obtained by Cramer's rule and the matrix method (80) are equal.

Similarly, we can obtain for all the remainder solutions. Note that we used Maple with the package CLIFFORD in the calculations.

5. Conclusions

Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we have derived determinantal representations (analogs of Cramer's rule) of the general and Hermitian solutions to the system of two-sided quaternion matrix equations [A.sub.1]X[A.sup.*.sub.1] = [C.sub.1] and [A.sub.2]X[A.sub.2.sup.*] = [C.sub.2]. Since the Hermitian solution is Y = (1/2)(X + [X.sup.*]), where X is an arbitrary solution, the determinantal representation of [X.sup.*] is derived as well. To accomplish that goal, we have used the determinantal representations of the Moore-Penrose matrix inverse which were previously introduced by the author.

Data Availability

No data were used to support this study.

https://doi.org/10.1155/2018/6294672

Conflicts of Interest

The author declares that there are no conflicts of interest.

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Ivan I. Kyrchei (iD)

Pidstrygach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Lviv, Ukraine

Correspondence should be addressed to Ivan I. Kyrchei; st260664@gmail.com

Received 5 April 2018; Accepted 27 September 2018; Published 1 November 2018

Academic Editor: Frank Uhlig
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Title Annotation:Research Article
Author:Kyrchei, Ivan I.
Publication:Journal of Mathematics
Geographic Code:7IRAN
Date:Jan 1, 2018
Words:5609
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