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

Ioannis K Dassios; Dumitru I Baleanu

doi:10.1186/1687-2770-2013-148

Research
Open Access

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

Ioannis K Dassios¹Email author and
Dumitru I Baleanu^{2, 3, 4}

Boundary Value Problems20132013:148

https://doi.org/10.1186/1687-2770-2013-148

Received: 20 February 2013
Accepted: 18 May 2013
Published: 19 June 2013

Abstract

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.

Keywords

boundary conditions
singular systems
fractional calculus
nabla operator
difference equations
linear
discrete time system

1 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 [1–7]. There has been a significant development in the study of fractional differential/difference equations and inclusions in recent years; see the monographs of Baleanu et al. [1], Kaczorek [4], Klamka et al. [8], Malinowska et al. [5], Podlubny [7], and the survey by Agarwal et al. [9]. For some recent contributions on fractional differential/difference equations, see [1, 4, 5, 8–27] 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 + 1, α + 2, \dots}

, α integer, and n such that

0 < n < 1

or

1 < n < 2

, then the nabla fractional operator in the case of Riemann-Liouville fractional difference of n th order for any

Y_{k} : N_{a} \to R^{m \times 1}

is defined by, see [5, 10–12, 23],

\nabla_{α}^{n} Y_{k} = \frac{1}{Γ (- n)} \sum_{j = α}^{k} {(k - j + 1)}^{\bar{- n - 1}} Y_{j},

where the raising power function is defined by

k^{\bar{α}} = \frac{Γ (k + α)}{Γ (k)} .

The following problem will then be considered. The singular fractional discrete time systems of the form

F \nabla_{0}^{n} Y_{k} = G Y_{k}, k = 1, 2, \dots, N

(1)

with known boundary conditions

A_{1} Y_{0} = B_{1}, A_{2} Y_{N} = B_{2},

(2)

where

F, G \in M (r \times m; F)

(i.e., the algebra of matrices with elements in the field ℱ) with

Y_{k}, V_{k} \in M (m \times 1; F)

,

A_{1} \in M (r_{1} \times m; F)

,

A_{2} \in M (r_{2} \times m; F)

,

B_{1} \in M (r_{1} \times 1; F)

and

B_{2} \in M (r_{2} \times 1; F)

. For the sake of simplicity, we set

M_{m} = M (m \times m; F)

and

M_{r m} = M (r \times m; F)

. The matrices F and G can be non-square (when

r \neq m

) or square (

r = m

) and F singular (

det F = 0

). The main purpose will be to provide necessary and sufficient conditions for the existence and uniqueness of solutions of the above boundary value problem, i.e., to study the conditions under which the system has unique, infinite and no solutions and to provide a formula for the case of the unique solution (if it exists). Many authors use matrix pencil theory to study linear discrete time systems with constant matrices; see, for instance, [28–43]. A matrix pencil is a family of matrices

s F - G

, parametrized by a complex number s, see [39, 41, 44, 45]. When G is square and

F = I_{m}

, where

I_{m}

is the identity matrix, the zeros of the function

det (s F - G)

are the eigenvalues of G. Consequently, the problem of finding the nontrivial solutions of the equation

s F X = G X

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 (s F - 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

F X = s^{- 1} G X .

If F is singular with a null vector X, then $F X = 0_{m, 1}$ , so that X is an eigenvector of the reciprocal problem corresponding to eigenvalue $s^{- 1} = 0$ ; i.e., $s = \infty$ .

Definition 1.1 Given

F, G \in M_{r m}

and an arbitrary

s \in F

, the matrix pencil

s F - G

is called:

1.

Regular when $r = m$ and $det (s F - G) \neq 0$ .
2.

Singular when $r \neq m$ or $r = m$ and $det (s F - G) \equiv 0$ .

The paper is organized as follows. In Section 2, we study the existence of solutions of the system (1) when its pencil is regular. In Section 3 we study the case of the system (1) with a singular pencil, and Section 3 contains numerical examples.

2 Regular case

In this section, we consider the case of the system (1) with a regular pencil. The class of $s F - G$ is characterized by a uniquely defined element, known as complex Weierstrass canonical form, $s F_{w} - Q_{w}$ , see [39, 41, 44, 45], specified by the complete set of invariants of $s F - G$ . This is the set of elementary divisors (e.d.) obtained by factorizing the invariant polynomials into powers of homogeneous polynomials irreducible over the field ℱ. In the case where $s F - G$ is regular, we have e.d. of the following type:

e.d. of the type ${(s - a_{j})}^{p_{j}}$ are called finite elementary divisors (f.e.d.), where $a_{j}$ is a finite eigenvalue of algebraic multiplicity $p_{j}$ ;
e.d. of the type ${\hat{s}}^{q} = \frac{1}{s^{q}}$ are called infinite elementary divisors (i.e.d.), where q is the algebraic multiplicity of the infinite eigenvalues.

We assume that $\sum_{i = 1}^{ν} p_{j} = p$ and $p + q = m$ .

Definition 2.1 Let $B_{1}, B_{2}, \dots, B_{l}$ be elements of $M_{m}$ . The direct sum of them denoted by $B_{1} \oplus B_{2} \oplus \dots \oplus B_{l}$ is the $blockdiag [B_{1} B_{2} \dots B_{l}]$ .

From the regularity of

s F - G

, there exist nonsingular matrices

P, Q \in M_{m}

such that

P F Q = F_{w} = I_{p} \oplus H_{q}

(3)

and

P G Q = G_{w} = J_{p} \oplus I_{q} .

(4)

The complex Weierstrass form

s F_{w} - Q_{w}

of the regular pencil

s F - G

is defined by

s F_{w} - Q_{w} : = s I_{p} - J_{p} \oplus s H_{q} - I_{q},

where the first normal Jordan-type element is uniquely defined by the set of the finite eigenvalues of

s F - G

and has the form

s I_{p} - J_{p} : = s I_{p_{1}} - J_{p_{1}} (a_{1}) \oplus \dots \oplus s I_{p_{ν}} - J_{p_{ν}} (a_{ν}) .

The second uniquely defined block

s H_{q} - I_{q}

corresponds to the infinite eigenvalues of

s F - G

and has the form

s H_{q} - I_{q} : = s H_{q_{1}} - I_{q_{1}} \oplus \dots \oplus s H_{q_{σ}} - I_{q_{σ}} .

The matrix

H_{q}

is a nilpotent element of

M_{q}

with index

q^{*} = max {q_{j} : j = 1, 2, \dots, σ}

, where

H_{q}^{q^{*}} = 0_{q, q}

and

I_{p_{j}}

,

J_{p_{j}} (a_{j})

,

H_{q_{j}}

are defined as

\begin{matrix} I_{p_{j}} = [\begin{array}{ccccc} 1 & 0 & \dots & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & \dots & 0 & 1 \end{array}] \in M_{p_{j}}, \\ J_{p_{j}} (a_{j}) = [\begin{array}{ccccc} a_{j} & 1 & \dots & 0 & 0 \\ 0 & a_{j} & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & \dots & a_{j} & 1 \\ 0 & 0 & \dots & 0 & a_{j} \end{array}] \in M_{p_{j}}, \\ H_{q_{j}} = [\begin{array}{ccccc} 0 & 1 & \dots & 0 & 0 \\ 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & \dots & 0 & 1 \\ 0 & 0 & \dots & 0 & 0 \end{array}] \in M_{q_{j}} . \end{matrix}

For algorithms about the computations of the Jordan matrices, see [39, 41, 44, 45].

Definition 2.2 If for the system (1) with boundary conditions (2) there exists at least one solution, the boundary value problem (1)-(2) is said to be consistent.

For the regular matrix pencil of the system (1), there exist nonsingular matrices

P, Q \in M_{m}

as applied in (3), (4). Let

Q = [\begin{array}{cc} Q_{p} & Q_{q} \end{array}],

(5)

where $Q_{p} \in M_{m p}$ is a matrix with columns p linear independent (generalized) eigenvectors of the p finite eigenvalues of $s F - G$ , and $Q_{q} \in M_{m q}$ is a matrix with columns q linear independent (generalized) eigenvectors of the q infinite eigenvalues of $s F - G$ .

Lemma 2.1 Consider the system (1) with a regular pencil. Then the system (1) is divided into two subsystems:

\nabla_{0}^{n} Z_{k}^{p} = J_{p} Z_{k}^{p}

and

H_{q} \nabla_{0}^{n} Z_{k}^{q} = Z_{k}^{q} .

Proof Consider the transformation

Y_{k} = Q Z_{k}

(6)

and by substituting (6) into (1), we obtain

F \nabla_{0}^{n} Q Z_{k} = G Q Z_{k}

or, equivalently,

F Q \nabla_{0}^{n} Z_{k} = G Q Z_{k} .

Whereby multiplying by P, we arrive at

F_{w} \nabla_{0}^{n} Z_{k} = G_{w} Z_{k} .

Moreover, let

Z_{k} = [\begin{array}{c} Z_{k}^{p} \\ Z_{k}^{q} \end{array}],

where

Z_{k}^{p} \in M_{p 1}

,

Z_{k}^{q} \in M_{q 1}

, and by using (3) and (4), we obtain

[\begin{array}{cc} I_{p} & 0_{p, q} \\ 0_{q, p} & H_{q} \end{array}] \nabla_{α}^{n} [\begin{array}{c} Z_{k}^{p} \\ Z_{k}^{q} \end{array}] = [\begin{array}{cc} J_{p} & 0_{p, q} \\ 0_{q, p} & I_{q} \end{array}] [\begin{array}{c} Z_{k}^{p} \\ Z_{k}^{q} \end{array}] .

From the above expressions, we arrive easily at the subsystems

\nabla_{0}^{n} Z_{k}^{p} = J_{p} Z_{k}^{p}

(7)

and

H_{q} \nabla_{0}^{n} Z_{k}^{q} = Z_{k}^{q} .

(8)

The proof is completed. □

Definition 2.3 With

F_{n, n} (J_{p} {(k + n)}^{\bar{n}})

we denote the discrete Mittag-Leffler function with two parameters defined by

F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) = \sum_{i = 0}^{\infty} J_{p}^{i} \frac{{(k + n)}^{\bar{i n}}}{Γ ((i + 1) n)} .

(9)

See [10–12, 23, 46].

Proposition 2.1 The subsystem (7) has the solution

Z_{k}^{p} = {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) Z_{0}^{p}

(10)

if and only if

∥ J_{p} ∥ < 1,

(11)

where $∥ \cdot ∥$ is an induced matrix norm and $F_{n, n} (J_{p} {(k + n)}^{\bar{n}})$ is the discrete Mittag-Leffler function with two parameters as defined by Definition 2.3.

Proof From [10–12, 23, 46] the solution of (7) can be calculated and given by the formula

Z_{k}^{p} = {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) Z_{0}^{p}

or, equivalently, by

Z_{k}^{p} = {(k + 1)}^{\bar{n - 1}} (\sum_{i = 0}^{\infty} J_{p}^{i} \frac{{(k + n)}^{\bar{i n}}}{Γ ((i + 1) n)}) (I_{p} - J_{p}) Z_{0}^{p} .

The existence and uniqueness of the above solution depends on the convergence of the matrix power series

\sum_{i = 0}^{\infty} J_{p}^{i} \frac{{(k + n)}^{\bar{i n}}}{Γ ((i + 1) n)}

or, equivalently, if and only if

lim_{i \to \infty} \frac{∥ J_{p}^{(i + 1)} \frac{{(k + n)}^{\bar{(i + 1) n}}}{Γ ((i + 2) n)} ∥}{∥ J_{p}^{i} \frac{{(k + n)}^{\bar{i n}}}{Γ ((i + 1) n)} ∥} < 1

or, equivalently,

∥ J_{p} ∥ < lim_{i \to \infty} \frac{\frac{{(k + n)}^{\bar{i n}}}{Γ ((i + 1) n)}}{\frac{{(k + n)}^{\bar{(i + 1) n}}}{Γ ((i + 2) n)}} .

By using the property

Γ (z + 1) = z Γ (z),

we get

∥ J_{p} ∥ < lim_{i \to \infty} \frac{((k - 1) + (i + 1) n) \dots (1 + (i + 1) n) ((i + 1) n)}{((k - 1) + (i + 2) n) \dots (1 + (i + 2) n) ((i + 2) n)}

or, equivalently,

∥ J_{p} ∥ < 1 .

The proof is completed. □

Proposition 2.2 The subsystem (8) has the unique solution

Z_{k}^{q} = 0_{q, 1} .

(12)

Proof Let

q_{*}

be the index of the nilpotent matrix

H_{q}

, i.e.,

H_{q}^{q_{*}} = 0_{q, q}

. Then if we obtain the following equations:

\begin{matrix} H_{q} \nabla_{0}^{n} Z_{k}^{q} = Z_{k}^{q}, \\ H_{q}^{2} \nabla_{0}^{2 n} Z_{k}^{q} = H_{q} \nabla_{0}^{n} Z_{k}^{q}, \\ H_{q}^{3} \nabla_{0}^{3 n} Z_{k}^{q} = H_{q}^{2} \nabla_{0}^{2 n} Z_{k}^{q}, \\ H_{q}^{4} \nabla_{0}^{4 n} Z_{k}^{q} = H_{q}^{3} \nabla_{0}^{3 n} Z_{k}^{q}, \\ ⋮ \\ H_{q}^{q_{*} - 1} \nabla_{0}^{(q_{*} - 1) n} Z_{k}^{q} = H_{q}^{q_{*} - 2} \nabla_{0}^{(q_{*} - 2) n} Z_{k}^{q}, \\ H_{q}^{q_{*}} \nabla_{0}^{q_{*} n} Z_{k}^{q} = H_{q}^{q_{*} - 1} \nabla_{0}^{(q_{*} - 1) n} Z_{k}^{q} \end{matrix}

by taking the sum of the above equations and using the fact that $H_{q}^{q^{*}} = 0_{q, q}$ , we arrive easily at the solution (12). The proof is completed. □

Theorem 2.1 Consider the system (1) with a regular pencil and boundary conditions of type (2). Then the boundary value problem (1)-(2) is consistent if and only if:

1.

The pencil $s F - G$ has p distinct eigenvalues and all lie within the open disk
$| s | < 1;$
2.

$[\begin{array}{c} B_{1} \\ B_{2} \end{array}] \in colspan [\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] .$

(13)

Furthermore, when the boundary value problem (1)-(2) is consistent, it has a unique solution if and only if:

1.

$p \leq r_{1} + r_{2};$

(14)
2.

$rank [\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] = p .$

(15)

In this case the unique solution is then given by

Y_{k} = Q_{p} {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) C,

(16)

where C is the unique solution of the algebraic system

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] C = [\begin{array}{c} B_{1} \\ B_{2} \end{array}] .

(17)

Proof By applying the transformation (6) into the system (1), we get the systems (7), (8) with solutions (10), (12) respectively. Note that from Proposition 2.1 the solution (10) exists if and only if

∥ J_{p} ∥ < 1,

where

J_{p}

is the Jordan matrix related to the p finite eigenvalues of the pencil

s F - 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 | < 1

. Based on these results, the solution of (1) can be written as

Y_{k} = Q Z_{k} = [\begin{array}{cc} Q_{p} & Q_{q} \end{array}] [\begin{array}{c} Z_{k}^{p} \\ Z_{k}^{q} \end{array}]

or, equivalently,

Y_{k} = Q_{p} Z_{k}^{p} + Q_{q} Z_{k}^{q},

or, equivalently, by using (10), (12)

Y_{k} = Q_{p} {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) Z_{0}^{p} .

The initial value

Z_{0}^{p}

of the subsystem (7) is not known and can be replaced by a constant vector

C \in M_{p 1}

Y_{k} = Q_{p} {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) C .

The above solution exists if and only if

A_{1} Y_{0} = B_{1}, A_{2} Y_{N} = B_{2}

or, equivalently,

A_{1} Q_{p} C = B_{1}, A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) C = B_{2},

or, equivalently,

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] C = [\begin{array}{c} B_{1} \\ B_{2} \end{array}] .

For the above algebraic system, there exists at least one solution if and only if

[\begin{array}{c} B_{1} \\ B_{2} \end{array}] \in colspan [\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] .

The algebraic system (17) contains

r_{1} + r_{2}

equations and p unknowns. Hence the solution is unique if and only if

p \leq r_{1} + r_{2}

and

rank [\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] = p,

where C is then the unique solution of (17). This can be proved as follows. If we assume that the algebraic system has two solutions

C_{1}

and

C_{2}

, then

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] C_{1} = [\begin{array}{c} B_{1} \\ B_{2} \end{array}]

and

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] C_{2} = [\begin{array}{c} B_{1} \\ B_{2} \end{array}]

or, equivalently,

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] (C_{1} - C_{2}) = 0_{p, 1} .

But the matrix

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}]

is left invertible since it is assumed to have p linear independent columns and

r_{1} + r_{2} \geq p

and hence

C_{1} = C_{2} .

The unique solution is then given from (16). The proof is completed. □

3 Singular case

In this section, we consider the case of the system (1) with a singular pencil. The class of

s F - G

in this case is characterized by a uniquely defined element,

s F_{K} - Q_{K}

, known as the complex Kronecker canonical form, see [39, 41, 44, 45], specified by the complete set of invariants of the singular pencil

s F - 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

s F - G

is that either

r \neq m

or

r = m

and

det (s F - G) \equiv 0

. Let

N_{r}

,

N_{l}

be the right and the left null space of a matrix respectively. Then the equations

(s F - G) U (s) = 0_{r, 1},

(18)

V^{T} (s) (s F - G) = 0_{1, m},

(19)

where

{()}^{T}

is the transpose tensor, have solutions in

U (s)

,

V (s)

, which are vectors in the rational vector spaces

N_{r} (s F - G)

and

N_{l} (s F - G)

respectively. The binary vectors

U (s)

and

V^{T} (s)

express dependence relationships among the columns or rows of

s F - G

respectively. Note that

U (s) \in M_{m 1}

and

V (s) \in M_{r 1}

are polynomial vectors. Let

d = dim N_{r} (s F - G)

and

t = dim N_{l} (s F - G)

. It is known, see [39, 41, 44, 45], that

N_{r} (s F - G)

and

N_{l} (s F - G)

as rational vector spaces are spanned by minimal polynomial bases of minimal degrees

ϵ_{1} = ϵ_{2} = \dots = ϵ_{g} = 0 < ϵ_{g + 1} \leq \dots \leq ϵ_{d}

(20)

and

ζ_{1} = ζ_{2} = \dots = ζ_{h} = 0 < ζ_{h + 1} \leq \dots \leq ζ_{t}

(21)

respectively. The set of minimal indices $ϵ_{1}, ϵ_{2}, \dots, ϵ_{d}$ and $ζ_{1}, ζ_{2}, \dots, ζ_{t}$ are known as column minimal indices (c.m.i.) and row minimal indices (r.m.i.) of $s F - G$ respectively. To sum up, in the case of a singular pencil, we have invariants of the following type:

finite elementary divisors of the type ${(s - a_{j})}^{p_{j}}$ ;
infinite elementary divisors of the type ${\hat{s}}^{q} = \frac{1}{s^{q}}$ ;
column minimal indices of the type $ϵ_{1} = ϵ_{2} = \dots = ϵ_{g} = 0 < ϵ_{g + 1} \leq \dots \leq ϵ_{d}$ ;
row minimal indices of the type $ζ_{1} = ζ_{2} = \dots = ζ_{h} = 0 < ζ_{h + 1} \leq \dots \leq ζ_{t}$ .

The Kronecker canonical form, see [39, 41, 44, 45], is defined by

s F_{K} - G_{K} : = s I_{p} - J_{p} \oplus s H_{q} - I_{q} \oplus s F_{ϵ} - G_{ϵ} \oplus s F_{ζ} - G_{ζ} \oplus 0_{h, g},

(22)

where

s I_{p} - J_{p}

,

s H_{q} - I_{q}

are defined as in Section 2. The matrices

F_{ϵ}

,

G_{ϵ}

,

F_{ζ}

and

G_{ζ}

are defined by

F_{ϵ} = blockdiag {L_{ϵ_{g + 1}}, L_{ϵ_{g + 2}}, \dots, L_{ϵ_{d}}},

(23)

where

L_{ϵ} = [\begin{array}{c} I_{ϵ} & ⋮ & 0_{ϵ, 1} \end{array}]

for

ϵ = ϵ_{g + 1}, \dots, ϵ_{d}

G_{ϵ} = blockdiag {{\bar{L}}_{ϵ_{g + 1}}, {\bar{L}}_{ϵ_{g + 2}}, \dots, {\bar{L}}_{ϵ_{d}}},

(24)

where

{\bar{L}}_{ϵ} = [\begin{array}{c} 0_{ϵ, 1} & ⋮ & I_{ϵ} \end{array}]

for

ϵ = ϵ_{g + 1}, \dots, ϵ_{d}

. The matrices

F_{ζ}

,

G_{ζ}

are defined as

F_{ζ} = blockdiag {L_{ζ_{h + 1}}, L_{ζ_{h + 2}}, \dots, L_{ζ_{t}}},

(25)

where

L_{ζ} = [\begin{array}{c} I_{ζ} \\ 0_{1, ζ} \end{array}]

for

ζ = ζ_{h + 1}, \dots, ζ_{t}

G_{ζ} = blockdiag {{\bar{L}}_{ζ_{h + 1}}, {\bar{L}}_{ζ_{h + 2}}, \dots, {\bar{L}}_{ζ_{t}}},

(26)

where ${\bar{L}}_{ζ} = [\begin{array}{c} 0_{1, ζ} \\ I_{ζ} \end{array}]$ for $ζ = ζ_{h + 1}, \dots, ζ_{t}$ .

For algorithms about the computations of these matrices, see [39, 41, 44, 45].

Following the above given analysis, there exist nonsingular matrices P, Q with

P \in M_{r}

,

Q \in M_{m}

such that

\begin{aligned} P F Q = F_{K}, \\ P G Q = G_{K} . \end{aligned}

(27)

Let

Q = [\begin{array}{ccccc} Q_{p} & Q_{q} & Q_{ϵ} & Q_{ζ} & Q_{g} \end{array}],

(28)

where $Q_{p} \in M_{m p}$ , $Q_{q} \in M_{m q}$ , $Q_{ϵ} \in M_{m (d - g)}$ , $Q_{ζ} \in M_{m (t - h)}$ and $Q_{g} \in M_{m g}$ .

Lemma 3.1 The system (1) is divided into five subsystems:

\nabla_{0}^{n} Z_{k}^{p} = J_{p} Z_{k}^{p},

(29)

the subsystem

H_{q} \nabla_{0}^{n} Z_{k}^{q} = Z_{k}^{q},

(30)

the subsystem

F_{ϵ} \nabla_{0}^{n} Z_{k}^{ϵ} = G_{ϵ} Z_{k}^{ϵ},

(31)

the subsystem

F_{ζ} \nabla_{0}^{n} Z_{k}^{ζ} = G_{ζ} Z_{k}^{ζ},

(32)

and the subsystem

0_{h, g} \cdot \nabla_{0}^{n} Z_{k + 1}^{g} = 0_{h, g} \cdot Z_{k}^{g} .

(33)

Proof Consider the transformation

Y_{k} = Q Z_{k} .

(34)

Substituting the previous expression into (1), we obtain

F Q \nabla_{0}^{n} Z_{k} = G Q Z_{k} .

Whereby multiplying by P and using (27), we arrive at

F_{K} \nabla_{0}^{n} Z_{k} = G_{K} Z_{k} .

(35)

Moreover, let

Z_{k} = [\begin{array}{c} Z_{k}^{p} \\ Z_{k}^{q} \\ Z_{k}^{ϵ} \\ Z_{k}^{ζ} \\ Z_{k}^{g} \end{array}],

where $Z_{p}^{k} \in M_{p 1}$ , $Z_{q}^{k} \in M_{q 1}$ , $Z_{ϵ}^{k} \in M_{(d - g) 1}$ , $Z_{ζ}^{k} \in M_{(t - h) 1}$ and $Z_{g}^{k} \in M_{g 1}$ . Taking into account the above expressions, we arrive easily at the subsystems (29), (30), (31), (32), and (33). The proof is completed. □

Solving the system (1) is equivalent to solving subsystems (29), (30), (31), (32) and (33). The solutions of the systems (29), (30) are given by (10) and (12) respectively; see Propositions 2.1 and 2.2.

Proposition 3.1 The subsystem (31) has infinite solutions and can be taken arbitrarily

Z_{k}^{ϵ} = C_{k, 1} .

(36)

Proof If we set

Z_{k}^{ϵ} = [\begin{array}{c} Z_{k}^{ϵ_{g + 1}} \\ Z_{k}^{ϵ_{g + 2}} \\ ⋮ \\ Z_{k}^{ϵ_{d}} \end{array}]

by using (23), (24), the system (31) can be written as

blockdiag {L_{ϵ_{g + 1}}, \dots, L_{ϵ_{d}}} [\begin{array}{c} \nabla_{0}^{n} Z_{k}^{ϵ_{g + 1}} \\ \nabla_{0}^{n} Z_{k}^{ϵ_{g + 2}} \\ ⋮ \\ \nabla_{0}^{n} Z_{k}^{ϵ_{d}} \end{array}] = blockdiag {{\bar{L}}_{ϵ_{g + 1}}, \dots, {\bar{L}}_{ϵ_{d}}} [\begin{array}{c} Z_{k}^{ϵ_{g + 1}} \\ Z_{k}^{ϵ_{g + 2}} \\ ⋮ \\ Z_{k}^{ϵ_{d}} \end{array}] .

(37)

Then, for the non-zero blocks, a typical equation from (37) can be written as

L_{ϵ_{i}} \nabla_{0}^{n} Z_{k}^{ϵ_{i}} = {\bar{L}}_{ϵ_{i}} Z_{k}^{ϵ_{i}}, i = g + 1, g + 2, \dots, d

(38)

or, equivalently,

[\begin{array}{ccc} I_{ϵ_{i}} & ⋮ & 0_{ϵ_{i}, 1} \end{array}] \nabla_{0}^{n} Z_{k}^{ϵ_{i}} = [\begin{array}{ccc} 0_{ϵ_{i}, 1} & ⋮ & I_{ϵ_{i}} \end{array}] Z_{k}^{ϵ_{i}},

or, equivalently,

[\begin{array}{ccccc} 1 & 0 & \dots & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & \dots & ⋮ & ⋮ \\ 0 & 0 & \dots & 1 & 0 \end{array}] [\begin{array}{c} \nabla_{0}^{n} z_{k}^{ϵ_{i}, 1} \\ \nabla_{0}^{n} z_{k}^{ϵ_{i}, 2} \\ ⋮ \\ \nabla_{0}^{n} z_{k}^{ϵ_{i}, ϵ_{i}} \\ \nabla_{0}^{n} z_{k}^{ϵ_{i}, ϵ_{i} + 1} \end{array}] = [\begin{array}{ccccc} 0 & 1 & \dots & 0 & 0 \\ 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & \dots & ⋮ & ⋮ \\ 0 & 0 & \dots & 0 & 1 \end{array}] [\begin{array}{c} z_{k}^{ϵ_{i}, 1} \\ z_{k}^{ϵ_{i}, 2} \\ ⋮ \\ z_{k}^{ϵ_{i}, ϵ_{i}} \\ z_{k}^{ϵ_{i}, ϵ_{i} + 1} \end{array}],

or, equivalently,

\begin{aligned} \nabla_{0}^{n} z_{k}^{ϵ_{i}, 1} = z_{k}^{ϵ_{i}, 2} + v_{k}^{ϵ_{i}, 1}, \\ \nabla_{0}^{n} z_{k}^{ϵ_{i}, 2} = z_{k}^{ϵ_{i}, 3} + v_{k}^{ϵ_{i}, 2}, \\ ⋮ \\ \nabla_{0}^{n} z_{k}^{ϵ_{i}, ϵ_{i}} = \nabla_{0}^{n} z_{k}^{ϵ_{i}, ϵ_{i} + 1} . \end{aligned}

(39)

The system (39) is a regular-type system of difference equations with

ϵ_{i}

equations and

ϵ_{i} + 1

unknowns. It is clear from the above analysis that in every one of the

d - g

subsystems one of the coordinates of the solution has to be arbitrary by assigned total. The solution of the system can be assigned arbitrarily

Z_{k}^{ϵ} = C_{k, 1} .

The proof is completed. □

Proposition 3.2 The solution of the system (32) is unique and is the zero solution

Z_{k}^{ζ} = 0_{t - h, 1} .

(40)

Proof From (25), (26) the subsystem (32) can be written as

blockdiag {L_{ζ_{h + 1}}, \dots, L_{ζ_{t}}} [\begin{array}{c} \nabla_{0}^{n} Z_{k}^{ζ_{h + 1}} \\ \nabla_{0}^{n} Z_{k}^{ζ_{h + 2}} \\ ⋮ \\ \nabla_{0}^{n} Z_{k}^{ζ_{t}} \end{array}] = blockdiag {{\bar{L}}_{ζ_{h + 1}}, \dots, {\bar{L}}_{ζ_{t}}} [\begin{array}{c} Z_{k}^{ζ_{h + 1}} \\ Z_{k}^{ζ_{h + 2}} \\ ⋮ \\ Z_{k}^{ζ_{t}} \end{array}] .

(41)

Then for the non-zero blocks, a typical equation from (41) can be written as

L_{ζ_{j}} \nabla_{0}^{n} Z_{k}^{ζ_{j}} = {\bar{L}}_{ζ_{j}} Z_{k}^{ζ_{j}}, j = h + 1, h + 2, \dots, t

(42)

or, equivalently,

[\begin{array}{c} I_{ζ_{j}} \\ \dots \\ 0_{1, ζ_{j}} \end{array}] \nabla_{0}^{n} Z_{k}^{ζ_{j}} = [\begin{array}{c} 0_{1, ζ_{j}} \\ \dots \\ I_{ζ_{j}} \end{array}] Z_{k}^{ζ_{j}},

or, equivalently,

[\begin{array}{cccc} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & \dots & ⋮ \\ 0 & 0 & \dots & 1 \\ 0 & 0 & \dots & 0 \end{array}] [\begin{array}{c} \nabla_{0}^{n} z_{k}^{ζ_{j}, 1} \\ \nabla_{0}^{n} z_{k}^{ζ_{j}, 2} \\ ⋮ \\ \nabla_{0}^{n} z_{k}^{ζ_{j}, ζ_{j}} \end{array}] = [\begin{array}{cccc} 0 & 0 & \dots & 0 \\ 1 & 0 & \dots & 0 \\ ⋮ & ⋮ & \dots & ⋮ \\ 0 & 0 & \dots & 0 \\ 0 & 0 & \dots & 1 \end{array}] [\begin{array}{c} z_{k}^{ζ_{j}, 1} \\ z_{k}^{ζ_{j}, 2} \\ ⋮ \\ z_{k}^{ζ_{j}, ζ_{j}} \end{array}],

or, equivalently,

\begin{aligned} \nabla_{0}^{n} z_{k}^{ζ_{j}, 1} = 0, \\ \nabla_{0}^{n} z_{k}^{ζ_{j}, 2} = z_{k}^{ζ_{j}, 1}, \\ ⋮ \\ \nabla_{0}^{n} z_{k}^{ζ_{j}, ζ_{j} - 1} = z_{k}^{ζ_{j}, ζ_{j} - 2}, \\ \nabla_{0}^{n} z_{k}^{ζ_{j}, ζ_{j}} = z_{k}^{ζ_{j}, ζ_{j} - 1}, \\ 0 = z_{k}^{ζ_{j}, ζ_{j}} . \end{aligned}

(43)

We have a system of

ζ_{j}

+1 difference equations and

ζ_{j}

unknowns. Starting from the last equation, we get the solutions

\begin{matrix} z_{k}^{ζ_{j}, ζ_{j}} = 0, \\ z_{k}^{ζ_{j}, ζ_{j} - 1} = 0, \\ z_{k}^{ζ_{j}, ζ_{j} - 2} = 0, \\ ⋮ \\ z_{k}^{ζ_{j}, 1} = 0, \end{matrix}

which means that the solution of the system (32) is unique and is the zero solution. The proof is completed. □

Proposition 3.3 The subsystem (33) has an infinite number of solutions that can be taken arbitrarily

Z_{k}^{g} = C_{k, 2} .

(44)

Proof It is easy to observe that the subsystem

0_{h, g} \cdot \nabla_{0}^{n} Z_{k + 1}^{g} = 0_{h, g} \cdot Z_{k}^{g}

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 3.1 Consider the system (1) with a singular pencil and known boundary conditions of type (2). Then the boundary value problem (1)-(2) is consistent if and only if:

1.

$∥ J_{p} ∥ < 1;$
2.

the column minimal indices are zero, i.e.,
$dim N_{r} (s F - G) = 0;$

(45)
3.

$[\begin{array}{c} B_{1} \\ B_{2} \end{array}] \in colspan [\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] .$

(46)

Furthermore, when the boundary value problem (1)-(2) is consistent, it has a unique solution if and only if

1.

$p \leq r_{1} + r_{2};$

(47)
2.

$rank [\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] = p .$

(48)

In this case the unique solution is given by the formula

Y_{k} = Q_{p} {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) C,

(49)

where C is the unique solution of the algebraic system

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] C = [\begin{array}{c} B_{1} \\ B_{2} \end{array}] .

(50)

In any other case the system has infinite solutions.

Proof First we consider that the system has non-zero column minimal indices and non-zero row minimal indices. By using the transformation (34), the solutions of the subsystems (29), (30), (31), (32) and (33) are given by (10), (12), (36), (40) and (44) respectively. Note that from Proposition 2.1 the solution (10) exists if and only if

∥ J_{p} ∥ < 1 .

Furthermore, if

Z_{k} = [\begin{array}{c} Z_{k}^{p} \\ Z_{k}^{q} \\ Z_{k}^{ϵ} \\ Z_{k}^{ζ} \\ Z_{k}^{g} \end{array}] = [\begin{array}{c} {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) Z_{0}^{p} \\ 0_{q, 1} \\ C_{k, 1} \\ 0_{t - h, 1} \\ C_{k, 2} \end{array}] .

Since

Z_{0}^{p}

is unknown, it can be replaced with the unknown vector C. Then

Y_{k} = Q Z_{k} = [\begin{array}{ccccc} Q_{p} & Q_{q} & Q_{ϵ} & Q_{ζ} & Q_{g} \end{array}] [\begin{array}{c} {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) C \\ 0_{q, 1} \\ C_{k, 1} \\ 0_{t - h, 1} \\ C_{k, 2} \end{array}]

or, equivalently,

Y_{k} = Q_{p} {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) C + Q_{ϵ} C_{k, 1} + Q_{g} C_{k, 2} .

Since

C_{k, 1}

and

C_{k, 2}

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 (31) and consequently (33) exist. These systems as shown in Propositions 3.1 and 3.3 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} (s F - G) = 0 .

In this case the Kronecker canonical form of the pencil

s F - G

has the following form:

s F_{K} - G_{K} : = s I_{p} - J_{p} \oplus s H_{q} - I_{q} \oplus s F_{ζ} - G_{ζ} .

(51)

Then the system (1) is divided into three subsystems (29), (30), (32) with solutions (10), (12), (40) respectively. Thus

Y_{k} = Q Z_{k} = [\begin{array}{ccc} Q_{p} & Q_{q} & Q_{ζ} \end{array}] [\begin{array}{c} {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) C \\ 0_{q, 1} \\ 0_{t - h, 1} \end{array}]

or, equivalently,

Y_{k} = Q_{p} {(k + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(k + n)}^{\bar{n}}) (I_{p} - J_{p}) C .

The solution exists if and only if

A_{1} Y_{0} = B_{1}, A_{2} Y_{N} = B_{2}

or, equivalently,

A_{1} Q_{p} C = B_{1}, A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) C = B_{2},

or, equivalently,

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] C = [\begin{array}{c} B_{1} \\ B_{2} \end{array}] .

For the above algebraic system, there exists at least one solution if and only if

[\begin{array}{c} B_{1} \\ B_{2} \end{array}] \in colspan [\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] .

The algebraic system (50) contains

r_{1} + r_{2}

equations and p unknowns. Hence the solution is unique if and only if

p \leq r_{1} + r_{2}

and

rank [\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] = p,

where C is then the unique solution of (50). The uniqueness of C can be proved as follows. If we assume that the algebraic system has two solutions

C_{1}

and

C_{2}

, then

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] C_{1} = [\begin{array}{c} B_{1} \\ B_{2} \end{array}]

and

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] C_{2} = [\begin{array}{c} B_{1} \\ B_{2} \end{array}]

or, equivalently,

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}] (C_{1} - C_{2}) = 0_{p, 1} .

But the matrix

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(N + 1)}^{\bar{n - 1}} F_{n, n} (J_{p} {(N + n)}^{\bar{n}}) (I_{p} - J_{p}) \end{array}]

is left invertible since it is assumed to have p linear independent columns and

r_{1} + r_{2} \geq p

and hence

C_{1} = C_{2} .

The unique solution is then given from (49). The proof is completed. □

4 Numerical examples

Example 1

Assume the system (1) for

k = 0, 1, 2, 3

and

n = \frac{3}{2}

. Let

F = [\begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1 \\ 1 & - 1 & 0 & 1 & - 1 \\ 1 & - 1 & 0 & 2 & - 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}]

(52)

and

G = [\begin{array}{ccccc} 0 & 0 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 0 & - \frac{1}{4} \\ \frac{1}{2} & - \frac{1}{2} & 0 & - \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & - \frac{1}{2} & 1 & - \frac{1}{2} & \frac{3}{4} \\ 0 & 0 & 0 & 1 & - 1 \end{array}] .

(53)

Then

det (s F - G) = (s - \frac{1}{2}) s (s - \frac{1}{4})

and the pencil is regular. We assume the boundary conditions (2) with

\begin{matrix} A_{1} = [\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}], \\ A_{2} = [\begin{array}{ccccc} 1 & 0 & 0 & 0 & 1 \end{array}], \\ B_{1} = [\begin{array}{c} 0 \\ 0 \\ 36 \\ 36 \end{array}] \end{matrix}

and

B_{2} = 24 .

The three finite eigenvalues (

p = 3

) of the pencil are

\frac{1}{2}

, 0,

\frac{1}{4}

, and the Jordan matrix

J_{p}

has the form

J_{p} = [\begin{array}{ccc} \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{1}{4} \end{array}] .

It is easy to observe that

∥ J_{p} ∥ < 1 .

By calculating the eigenvectors of the finite eigenvalues, we get the matrix

Q_{p}

Q_{p} = [\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{array}] .

Moreover,

A_{1} Q_{p} = [\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{array}] .

(54)

From (9) we get the Mittag-Leffler function

F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) = \sum_{i = 0}^{\infty} J_{p}^{i} \frac{{(3 + \frac{3}{2})}^{\bar{i \frac{3}{2}}}}{Γ ((i + 1) \frac{3}{2})}

or, equivalently,

F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) = \sum_{i = 0}^{\infty} J_{p}^{i} \frac{Γ (3 + \frac{3}{2} (i + 1))}{Γ (3 + \frac{3}{2}) Γ ((i + 1) \frac{3}{2})},

or, equivalently,

F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) = \sum_{i = 0}^{\infty} J_{p}^{i} \frac{(2 + \frac{3}{2} (i + 1)) (1 + \frac{3}{2} (i + 1)) (\frac{3}{2} (i + 1)) Γ (\frac{3}{2} (i + 1))}{Γ (3 + \frac{3}{2}) Γ ((i + 1) \frac{3}{2})},

or, equivalently,

F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) = \frac{1}{Γ (3 + \frac{3}{2})} \sum_{i = 0}^{\infty} J_{p}^{i} (2 + \frac{3}{2} (i + 1)) (1 + \frac{3}{2} (i + 1)) (\frac{3}{2} (i + 1))

and since

∥ J_{p} ∥ < 1

, by using the sum

\sum_{i = 0}^{\infty} x = {(1 - x)}^{- 1}

for

∥ x ∥ < 1

, we calculate the sum of the matrix power series

\sum_{i = 0}^{\infty} J_{p}^{i} (2 + \frac{3}{2} (i + 1)) (1 + \frac{3}{2} (i + 1)) (\frac{3}{2} (i + 1))

, and we get

F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) = \frac{1}{Γ (3 + \frac{3}{2})} (- \frac{1}{9} J_{p}^{2} + \frac{20}{9} J_{p} + \frac{35}{9} I_{p}) {(I_{p} - J_{p})}^{- 4} .

And since

A_{2} Q_{p} = [\begin{array}{ccc} 1 & 1 & 1 \end{array}],

by using the above expression

A_{2} Q_{p} {(4)}^{\bar{\frac{1}{2}}} F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) (I_{p} - J_{p}) = \frac{1}{36} [\begin{array}{ccc} 1 & 1 & 1 \end{array}] [\begin{array}{ccc} 358 & 0 & 0 \\ 0 & 35 & 0 \\ 0 & 0 & 24 \end{array}]

or, equivalently,

A_{2} Q_{p} {(4)}^{\bar{\frac{1}{2}}} F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) (I_{p} - J_{p}) = \frac{1}{36} [\begin{array}{ccc} 358 & 35 & 24 \end{array}],

(55)

it is easy to observe that the conditions (13), (14) and (15) are satisfied. Thus from Theorem 2.1 the unique solution of the boundary value problem (1)-(2) is given by

Y_{k} = Q_{p} {(k + 1)}^{\bar{\frac{3}{2} - 1}} F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(k + \frac{3}{2})}^{\bar{\frac{3}{2}}}) (I_{p} - J_{p}) C

or, equivalently, by

Y_{k} = \frac{1}{Γ (k + 1)} \sum_{i = 0}^{\infty} \frac{Γ (k + \frac{3}{2} (i + 1))}{Γ (\frac{3}{2} (i + 1))} [\begin{array}{ccc} \frac{1}{2^{i + 1}} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{3}{4^{i + 1}} \\ 0 & 0 & \frac{3}{4^{i + 1}} \end{array}] C,

where C is the unique solution of the algebraic system

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(4)}^{\bar{\frac{1}{2}}} F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) (I_{p} - J_{p}) \end{array}] C = [\begin{array}{c} B_{1} \\ B_{2} \end{array}]

or, equivalently,

C = [\begin{array}{c} 0 \\ 0 \\ 36 \end{array}],

and thus the unique solution of the boundary value problem is

Y_{k} = \frac{27}{Γ (k + 1)} \sum_{i = 0}^{\infty} \frac{Γ (k + \frac{3}{2} (i + 1))}{Γ (\frac{3}{2} (i + 1))} [\begin{array}{c} 0 \\ 0 \\ 0 \\ \frac{1}{4^{i}} \\ \frac{1}{4^{i}} \end{array}] .

(56)

Example 2

We assume the system (1) as in Example 1 but with different boundary conditions. Let

\begin{matrix} A_{1} = [\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}], \\ A_{2} = [\begin{array}{ccccc} 1 & 0 & 0 & 0 & 1 \end{array}], \\ B_{1} = [\begin{array}{c} 0 \\ 0 \\ 0 \\ 36 \end{array}] \end{matrix}

and

B_{2} = 24 .

It is easy to observe that

[\begin{array}{c} B_{1} \\ B_{2} \end{array}] \notin colspan [\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(4)}^{\bar{\frac{1}{2}}} F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) (I_{p} - J_{p}) \end{array}]

since

[\begin{array}{c} 0 \\ 0 \\ 0 \\ 36 \\ 24 \end{array}] \notin colspan [\begin{array}{ccc} 36 & 36 & 0 \\ 0 & 36 & 0 \\ 0 & 0 & 36 \\ 0 & 0 & 36 \\ 358 & 35 & 24 \end{array}]

and thus from Theorem 2.1, and since (13) does not hold, the boundary value problem is not consistent.

Example 3

Consider the system (1) and let

F = [\begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 \end{array}], G = [\begin{array}{ccccc} 1 & 2 & 2 & 1 & 2 \\ 0 & 2 & 2 & 0 & 2 \\ 1 & 2 & 2 & 2 & 3 \\ 0 & 2 & 3 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 \end{array}] .

Since the matrices F, G are non-square, the matrix pencil

s F - G

is singular and has invariants such as the finite elementary divisors

s - 2

,

s - 1

, an infinite elementary divisor of degree 1 and the row minimal indices

ζ_{1} = 0

,

ζ_{2} = 1

. Since the Jordan matrix has the form

J_{p} = [\begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array}]

with

∥ J_{p} ∥ > 1

for every induced matrix norm, from Theorem 3.1 the boundary value problem (1)-(2) is non-consistent.

Example 4

Consider the system (1) for

k = 0, 1, 2, 3

and

n = \frac{3}{2}

. Let

F = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]

(57)

and

G = [\begin{array}{cc} \frac{3}{4} & 0 \\ 0 & 0 \\ 0 & 1 \end{array}] .

(58)

Since the matrices F, G are non-square, the matrix pencil

s F - G

is singular and has invariants such as a finite elementary divisor

s - \frac{3}{4}

and the row minimal indices

ζ_{1} = 0

,

ζ_{2} = 1

. We assume the boundary conditions (2) with

A_{1} = [\begin{array}{cc} 1 & 1 \end{array}], A_{2} = [\begin{array}{cc} 0 & 1 \\ 0 & 2 \end{array}], B_{1} = 2

and

B_{2} = [\begin{array}{c} 0 \\ 0 \end{array}] .

The Jordan matrix is

J_{p} = \frac{3}{4}

with

∥ J_{p} ∥ < 1

for every induced matrix norm. By calculating the matrix

Q_{p}

, we get

Q_{p} = [\begin{array}{c} 1 \\ 0 \end{array}] .

Moreover,

A_{1} Q_{p} = 1,

(59)

and since

A_{2} Q_{p} = [\begin{array}{c} 0 \\ 0 \end{array}],

we get

A_{2} Q_{p} {(4)}^{\bar{\frac{1}{2}}} F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) (I_{p} - J_{p}) = [\begin{array}{c} 0 \\ 0 \end{array}] .

(60)

By using (59), (60), it is easy to observe that the conditions (45), (46), (47) and (48) are satisfied and thus from Theorem 3.1 the unique solution of the boundary value problem (1)-(2) is given by

Y_{k} = Q_{p} {(k + 1)}^{\bar{\frac{1}{2}}} F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(k + \frac{3}{2})}^{\bar{\frac{3}{2}}}) (I_{p} - J_{p}) C

or, equivalently, by

Y_{k} = \frac{1}{Γ (k + 1)} \sum_{i = 0}^{\infty} \frac{Γ (k + \frac{3}{2} (i + 1))}{Γ (\frac{3}{2} (i + 1))} [\begin{array}{c} \frac{3^{i}}{4^{i + 1}} \\ 0 \end{array}] C,

where C is the unique solution of the algebraic system

[\begin{array}{c} A_{1} Q_{p} \\ A_{2} Q_{p} {(4)}^{\bar{\frac{1}{2}}} F_{\frac{3}{2}, \frac{3}{2}} (J_{p} {(3 + \frac{3}{2})}^{\bar{\frac{3}{2}}}) (I_{p} - J_{p}) \end{array}] C = [\begin{array}{c} B_{1} \\ B_{2} \end{array}]

or, equivalently,

C = 2,

and thus the unique solution of the boundary value problem is

Y_{k} = \frac{2}{Γ (k + 1)} \sum_{i = 0}^{\infty} \frac{Γ (k + \frac{3}{2} (i + 1))}{Γ (\frac{3}{2} (i + 1))} [\begin{array}{c} \frac{3^{i}}{4^{i + 1}} \\ 0 \end{array}] .

(61)

5 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.

Declarations

Acknowledgements

We would like to express our sincere gratitude to Professor GI Kalogeropoulos for his helpful and fruitful discussions that clearly improved this article. Moreover, we are very grateful to the anonymous referees for their valuable suggestions that improved the article.

Authors’ Affiliations

(1)

School of Mathematics and Maxwell Institute, The University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom

(2)

Department of Mathematics and Computer Sciences, Cankaya University, Ankara, Turkey

(3)

Institute of Space Sciences, Magurele, Bucharest, Romania

(4)

Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

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