On a singular system of fractional nabla difference equations with boundary conditions

In this article, we study a boundary value problem of a class of linear singular systems of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient matrix singular, square with an identically zero matrix pencil and non-square, we provide necessary and sufficient conditions for the existence and uniqueness of solutions. More analytically, we study the conditions under which the boundary value problem has a unique solution, infinite solutions and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally, numerical examples are given to justify our theory.


Introduction
Difference equations of fractional order have recently proven to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, and so forth [-]. There has been a significant development in the study of fractional differential/difference equations and inclusions in recent years; see the monographs of Baleanu [], and the survey by Agarwal et al. []. For some recent contributions on fractional differential/difference equations, see [, , , -] and the references therein. In this article we provide an introductory study for a boundary value problem of a class of singular fractional nabla discrete time systems. If we define N α by N α = {α, α + , α + , . . .}, α integer, and n such that  < n <  or  < n < , then the nabla fractional operator in the case of Riemann-Liouville fractional difference of nth order for any Y k : N a → R m× is defined by, see [, -, ], where the raising power function is defined by kᾱ = (k + α) (k) . http://www.boundaryvalueproblems.com/content/2013/1/148 The following problem will then be considered. The singular fractional discrete time systems of the form F∇ n  Y k = GY k , k = , , . . . , N with known boundary conditions where F, G ∈ M(r × m; F) (i.e., the algebra of matrices with elements in the field F ) with is called the generalized eigenvalue problem. Although the generalized eigenvalue problem looks like a simple generalization of the usual eigenvalue problem, it exhibits some important differences. In the first place, it is possible for F, G to be non-square matrices. Moreover, even with F, G square it is possible (in the case F is singular) for det(sF -G) to be identically zero, independent of s. Finally, even if we assume F, G square matrices with a non-zero pencil, it is possible (when F is singular) for the problem to have infinite eigenvalues. To see this, write the generalized eigenvalue problem in the reciprocal form http://www.boundaryvalueproblems.com/content/2013/1/148

Regular case
In this section, we consider the case of the system () with a regular pencil. , where q is the algebraic multiplicity of the infinite eigenvalues. We assume that ν i= p j = p and p + q = m.
From the regularity of sF -G, there exist nonsingular matrices P, Q ∈ M m such that and The complex Weierstrass form sF w -Q w of the regular pencil sF -G is defined by where the first normal Jordan-type element is uniquely defined by the set of the finite eigenvalues of sF -G and has the form The second uniquely defined block sH q -I q corresponds to the infinite eigenvalues of sF -G and has the form The matrix H q is a nilpotent element of M q with index q * = max{q j : j = , , . . . , σ }, where H q * q =  q,q and I p j , J p j (a j ), H q j are defined as  For the regular matrix pencil of the system (), there exist nonsingular matrices P, Q ∈ M m as applied in (), (). Let where Q p ∈ M mp is a matrix with columns p linear independent (generalized) eigenvectors of the p finite eigenvalues of sF -G, and Q q ∈ M mq is a matrix with columns q linear independent (generalized) eigenvectors of the q infinite eigenvalues of sF -G.
Lemma . Consider the system () with a regular pencil. Then the system () is divided into two subsystems: Proof Consider the transformation and by substituting () into (), we obtain Whereby multiplying by P, we arrive at Moreover, let where Z p k ∈ M p , Z q k ∈ M q , and by using () and (), we obtain From the above expressions, we arrive easily at the subsystems and The proof is completed.
Definition . With F n,n (J p (k + n)n) we denote the discrete Mittag-Leffler function with two parameters defined by if and only if where · is an induced matrix norm and F n,n (J p (k + n)n) is the discrete Mittag-Leffler function with two parameters as defined by Definition ..
Proof From [-, , ] the solution of () can be calculated and given by the formula Z p k = (k + ) n- F n,n J p (k + n)n (I p -J p )Z p  http://www.boundaryvalueproblems.com/content/2013/1/148 or, equivalently, by The existence and uniqueness of the above solution depends on the convergence of the matrix power series or, equivalently, if and only if By using the property we get or, equivalently, The proof is completed.

Proposition . The subsystem () has the unique solution
Proof Let q * be the index of the nilpotent matrix H q , i.e., H q * q =  q,q . Then if we obtain the following equations: by taking the sum of the above equations and using the fact that H q * q =  q,q , we arrive easily at the solution (). The proof is completed. .

Furthermore, when the boundary value problem ()-() is consistent, it has a unique solution if and only if:
In this case the unique solution is then given by where C is the unique solution of the algebraic system Proof By applying the transformation () where J p is the Jordan matrix related to the p finite eigenvalues of the pencil sF -G, which is equivalent to the fact that the finite eigenvalues of the pencil must be distinct and all lie within the unit disk |s| < . Based on these results, the solution of () can be written as or, equivalently, by using (), () The initial value Z p  of the subsystem () is not known and can be replaced by a constant vector

The above solution exists if and only if
or, equivalently, For the above algebraic system, there exists at least one solution if and only if The algebraic system () contains r  + r  equations and p unknowns. Hence the solution is unique if and only if = p, http://www.boundaryvalueproblems.com/content/2013/1/148 where C is then the unique solution of (). This can be proved as follows. If we assume that the algebraic system has two solutions C  and C  , then or, equivalently, But the matrix is left invertible since it is assumed to have p linear independent columns and r  + r  ≥ p and hence The unique solution is then given from (). The proof is completed.

Singular case
In this section, we consider the case of the system () with a singular pencil. The class of sF -G in this case is characterized by a uniquely defined element, sF K -Q K , known as the complex Kronecker canonical form, see [, , , ], specified by the complete set of invariants of the singular pencil sF -G. This is the set of the elementary divisors (e.d.) and the minimal indices (m.i.). Unlike the case of the regular pencils, where the pencil is characterized only from the e.d., the characterization of a singular matrix pencil apart from the set of the determinantal divisors requires the definition of additional sets of invariants, the minimal indices. The distinguishing feature of a singular pencil sF -G is that either r = m or r = m and det(sF -G) ≡ . Let N r , N l be the right and the left null space of a matrix respectively. Then the equations where () T is the transpose tensor, have solutions in U(s), V (s), which are vectors in the rational vector spaces N r (sF -G) and N l (sF -G) respectively. The binary vectors U(s) and V T (s) express dependence relationships among the columns or rows of sF -G respectively. Note that U(s) ∈ M m and V (s) ∈ M r are polynomial vectors. Let d = dim N r (sF -G) and t = dim N l (sF -G). It is known, see [, , , ], that N r (sF -G) and N l (sF -G) as rational vector spaces are spanned by minimal polynomial bases of minimal degrees  =  = · · · = g =  < g+ ≤ · · · ≤ d () http://www.boundaryvalueproblems.com/content/2013/1/148 and ζ  = ζ  = · · · = ζ h =  < ζ h+ ≤ · · · ≤ ζ t () respectively. The set of minimal indices  ,  , . . . , d and ζ  , ζ  , . . . , ζ t are known as column minimal indices (c.m.i.) and row minimal indices (r.m.i.) of sF -G respectively. To sum up, in the case of a singular pencil, we have invariants of the following type: • finite elementary divisors of the type (sa j ) p j ; • infinite elementary divisors of the typeŝ q =  s q ; • column minimal indices of the type  =  = · · · = g =  < g+ ≤ · · · ≤ d ; • row minimal indices of the type ζ  = ζ  = · · · = ζ h =  < ζ h+ ≤ · · · ≤ ζ t . The Kronecker canonical form, see [, , , ], is defined by where sI p -J p , sH q -I q are defined as in Section . The matrices F , G , F ζ and G ζ are defined by For algorithms about the computations of these matrices, see [, , , ]. Following the above given analysis, there exist nonsingular matrices P, Q with P ∈ M r , Q ∈ M m such that where -h) and Q g ∈ M mg . http://www.boundaryvalueproblems.com/content/2013/1/148 Lemma . The system () is divided into five subsystems: and the subsystem Proof Consider the transformation Substituting the previous expression into (), we obtain Whereby multiplying by P and using (), we arrive at Taking into account the above expressions, we arrive easily at the subsystems (), (), (), (), and (). The proof is completed.

Proposition . The subsystem () has infinite solutions and can be taken arbitrarily
by using (), (), the system () can be written as Then, for the non-zero blocks, a typical equation from () can be written as or, equivalently, () The system () is a regular-type system of difference equations with i equations and i + unknowns. It is clear from the above analysis that in every one of the dg subsystems one http://www.boundaryvalueproblems.com/content/2013/1/148 of the coordinates of the solution has to be arbitrary by assigned total. The solution of the system can be assigned arbitrarily The proof is completed.

Proposition . The solution of the system () is unique and is the zero solution
Proof From (), () the subsystem () can be written as Then for the non-zero blocks, a typical equation from () can be written as or, equivalently, or, equivalently, or, equivalently, () http://www.boundaryvalueproblems.com/content/2013/1/148 We have a system of ζ j + difference equations and ζ j unknowns. Starting from the last equation, we get the solutions which means that the solution of the system () is unique and is the zero solution. The proof is completed.

Proposition . The subsystem () has an infinite number of solutions that can be taken arbitrarily
Proof It is easy to observe that the subsystem does not provide any non-zero equations. Hence all its solutions can be taken arbitrarily. The proof is completed.
We can now state the following theorem.

Theorem . Consider the system () with a singular pencil and known boundary conditions of type (). Then the boundary value problem ()-() is consistent if and only if:
.

J p < ;
. the column minimal indices are zero, i.e., In this case the unique solution is given by the formula where C is the unique solution of the algebraic system In any other case the system has infinite solutions.
Proof First we consider that the system has non-zero column minimal indices and nonzero row minimal indices. Furthermore, if Since Z p  is unknown, it can be replaced with the unknown vector C. Then or, equivalently, Y k = Q p (k + ) n- F n,n J p (k + n)n (I p -J p )C + Q C k, + Q g C k, . http://www.boundaryvalueproblems.com/content/2013/1/148 Since C k, and C k, can be taken arbitrarily, it is clear that the general singular discrete time system for every suitable defined boundary condition has an infinite number of solutions. It is clear that the existence of the column minimal indices is the reason that the systems () and consequently () exist. These systems as shown in Propositions . and . have always infinite solutions. Thus a necessary condition for the system to have a unique solution is not to have any column minimal indices which are equal to dim N r (sF -G) = .
In this case the Kronecker canonical form of the pencil sF -G has the following form: Then the system () is divided into three subsystems (), (), () with solutions (), (), () respectively. Thus or, equivalently, Y k = Q p (k + ) n- F n,n J p (k + n)n (I p -J p )C.

The solution exists if and only if
or, equivalently, or, equivalently, For the above algebraic system, there exists at least one solution if and only if The algebraic system () contains r  + r  equations and p unknowns. Hence the solution is unique if and only if p ≤ r  + r  http://www.boundaryvalueproblems.com/content/2013/1/148 and rank where C is then the unique solution of (). The uniqueness of C can be proved as follows.
If we assume that the algebraic system has two solutions C  and C  , then But the matrix is left invertible since it is assumed to have p linear independent columns and r  + r  ≥ p and hence The unique solution is then given from (). The proof is completed.

Numerical examples Example 1
Assume the system () for k = , , ,  and n =   . Let (   ) http://www.boundaryvalueproblems.com/content/2013/1/148 Then det(sF -G) = (s -  )s(s -  ) and the pencil is regular. We assume the boundary conditions () with The three finite eigenvalues (p = ) of the pencil are   , ,   , and the Jordan matrix J p has the form It is easy to observe that J p < .
By calculating the eigenvectors of the finite eigenvalues, we get the matrix Q p

Moreover,
From () we get the Mittag-Leffler function , or, equivalently, , or, equivalently, and since J p < , by using the sum ∞ i= x = (x) - for x < , we calculate the sum of the matrix power series ∞ i= J i p ( +   (i + ))( +   (i + ))(   (i + )), and we get And since by using the above expression or, equivalently, it is easy to observe that the conditions (), () and () are satisfied. Thus from Theorem . the unique solution of the boundary value problem ()-() is given by or, equivalently, by where C is the unique solution of the algebraic system and thus the unique solution of the boundary value problem is

Example 2
We assume the system () as in Example  but with different boundary conditions. Let It is easy to observe that

Example 3
Consider the system () and let Since the matrices F, G are non-square, the matrix pencil sF -G is singular and has invariants such as the finite elementary divisors s -, s -, an infinite elementary divisor of degree  and the row minimal indices ζ  = , ζ  = . Since the Jordan matrix has the form J p =     with J p >  for every induced matrix norm, from Theorem . the boundary value problem ()-() is non-consistent.

Example 4
Consider the system () for k = , , ,  and n =   . Let The Jordan matrix is J p =   with J p <  for every induced matrix norm. By calculating the matrix Q p , we get Moreover, and since we get where C is the unique solution of the algebraic system

Conclusions
In this article, we study the boundary value problem of a class of a singular system of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient singular, square with an identically zero matrix pencil and non-square, we study the conditions under which the boundary value problem has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. As a further extension of this article, one can study the stability, the behavior under perturbation and possible applications in economics and engineering of singular matrix difference/differential equations of fractional order. For all this, there is already some research in progress.