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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

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

Ioannis K Dassios1* and Dumitru I Baleanu234

Author 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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Boundary Value Problems 2013, 2013:148  doi:10.1186/1687-2770-2013-148

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/148


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

© 2013 Dassios and Baleanu; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M1">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M2">View MathML</a>, α integer, and n such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M3">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M4">View MathML</a>, then the nabla fractional operator in the case of Riemann-Liouville fractional difference of nth order for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M5">View MathML</a> is defined by, see [5,10-12,23],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M6">View MathML</a>

where the raising power function is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M7">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M8">View MathML</a>

(1)

with known boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M9">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M10">View MathML</a> (i.e., the algebra of matrices with elements in the field ℱ) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M15">View MathML</a>. For the sake of simplicity, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M17">View MathML</a>. The matrices F and G can be non-square (when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M18">View MathML</a>) or square (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M19">View MathML</a>) and F singular (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M20">View MathML</a>). 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a>, parametrized by a complex number s, see [39,41,44,45]. When G is square and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M22">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M23">View MathML</a> is the identity matrix, the zeros of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M24">View MathML</a> are the eigenvalues of G. Consequently, the problem of finding the nontrivial solutions of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M25">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M24">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M27">View MathML</a>

If F is singular with a null vector X, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M28">View MathML</a>, so that X is an eigenvector of the reciprocal problem corresponding to eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M29">View MathML</a>; i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M30">View MathML</a>.

Definition 1.1 Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M31">View MathML</a> and an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M32">View MathML</a>, the matrix pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> is called:

1. Regular when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M35">View MathML</a>.

2. Singular when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M18">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M38">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> is characterized by a uniquely defined element, known as complex Weierstrass canonical form, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M40">View MathML</a>, see [39,41,44,45], specified by the complete set of invariants of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> is regular, we have e.d. of the following type:

• e.d. of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M43">View MathML</a>are called finite elementary divisors (f.e.d.), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M44">View MathML</a> is a finite eigenvalue of algebraic multiplicity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M45">View MathML</a>;

• e.d. of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M46">View MathML</a> are called infinite elementary divisors (i.e.d.), where q is the algebraic multiplicity of the infinite eigenvalues.

We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M48">View MathML</a>.

Definition 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M49">View MathML</a> be elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M50">View MathML</a>. The direct sum of them denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M51">View MathML</a> is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M52">View MathML</a>.

From the regularity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a>, there exist nonsingular matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M54">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M55">View MathML</a>

(3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M56">View MathML</a>

(4)

The complex Weierstrass form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M57">View MathML</a> of the regular pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M59">View MathML</a>

where the first normal Jordan-type element is uniquely defined by the set of the finite eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> and has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M61">View MathML</a>

The second uniquely defined block <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M62">View MathML</a> corresponds to the infinite eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> and has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M64">View MathML</a>

The matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M65">View MathML</a> is a nilpotent element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M66">View MathML</a> with index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M67">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M68">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M71">View MathML</a> are defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M72">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M73">View MathML</a> as applied in (3), (4). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M74">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M75">View MathML</a> is a matrix with columns p linear independent (generalized) eigenvectors of the p finite eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M77">View MathML</a> is a matrix with columns q linear independent (generalized) eigenvectors of the q infinite eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M79">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M80">View MathML</a>

Proof Consider the transformation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M81">View MathML</a>

(6)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M82">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M83">View MathML</a>

Whereby multiplying by P, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M84">View MathML</a>

Moreover, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M85">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M87">View MathML</a>, and by using (3) and (4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M88">View MathML</a>

From the above expressions, we arrive easily at the subsystems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M89">View MathML</a>

(7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M90">View MathML</a>

(8)

The proof is completed. □

Definition 2.3 With <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M91">View MathML</a> we denote the discrete Mittag-Leffler function with two parameters defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M92">View MathML</a>

(9)

See [10-12,23,46].

Proposition 2.1The subsystem (7) has the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M93">View MathML</a>

(10)

if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M94">View MathML</a>

(11)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M95">View MathML</a>is an induced matrix norm and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M91">View MathML</a>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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M97">View MathML</a>

or, equivalently, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M98">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M99">View MathML</a>

or, equivalently, if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M100">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M101">View MathML</a>

By using the property

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M102">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M103">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M104">View MathML</a>

The proof is completed. □

Proposition 2.2The subsystem (8) has the unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M105">View MathML</a>

(12)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M106">View MathML</a> be the index of the nilpotent matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M65">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M108">View MathML</a>. Then if we obtain the following equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M109">View MathML</a>

by taking the sum of the above equations and using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M110">View MathML</a>, we arrive easily at the solution (12). The proof is completed. □

Theorem 2.1Consider 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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a>haspdistinct eigenvalues and all lie within the open disk

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M112">View MathML</a>

2.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M113">View MathML</a>

(13)

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

1.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M114">View MathML</a>

(14)

2.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M115">View MathML</a>

(15)

In this case the unique solution is then given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M116">View MathML</a>

(16)

whereCis the unique solution of the algebraic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M117">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M118">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M119">View MathML</a> is the Jordan matrix related to the p finite eigenvalues of the pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a>, which is equivalent to the fact that the finite eigenvalues of the pencil must be distinct and all lie within the unit disk <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M121">View MathML</a>. Based on these results, the solution of (1) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M122">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M123">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M124">View MathML</a>

The initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M125">View MathML</a> of the subsystem (7) is not known and can be replaced by a constant vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M126">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M127">View MathML</a>

The above solution exists if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M128">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M129">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M130">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M131">View MathML</a>

The algebraic system (17) contains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M132">View MathML</a> equations and p unknowns. Hence the solution is unique if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M133">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M134">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M136">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M137">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M138">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M139">View MathML</a>

But the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M140">View MathML</a> is left invertible since it is assumed to have p linear independent columns and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M141">View MathML</a> and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M142">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> in this case is characterized by a uniquely defined element, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M144">View MathML</a>, known as the complex Kronecker canonical form, see [39,41,44,45], specified by the complete set of invariants of the singular pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> is that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M18">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M38">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M151">View MathML</a> be the right and the left null space of a matrix respectively. Then the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M152">View MathML</a>

(18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M153">View MathML</a>

(19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M154">View MathML</a> is the transpose tensor, have solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M156">View MathML</a>, which are vectors in the rational vector spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M158">View MathML</a> respectively. The binary vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M160">View MathML</a> express dependence relationships among the columns or rows of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> respectively. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M163">View MathML</a> are polynomial vectors. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M164">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M165">View MathML</a>. It is known, see [39,41,44,45], that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M158">View MathML</a> as rational vector spaces are spanned by minimal polynomial bases of minimal degrees

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M168">View MathML</a>

(20)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M169">View MathML</a>

(21)

respectively. The set of minimal indices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M171">View MathML</a> are known as column minimal indices (c.m.i.) and row minimal indices (r.m.i.) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> respectively. To sum up, in the case of a singular pencil, we have invariants of the following type:

finite elementary divisors of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M43">View MathML</a>;

infinite elementary divisors of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M46">View MathML</a>;

column minimal indices of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M175">View MathML</a>;

row minimal indices of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M176">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M177">View MathML</a>

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M62">View MathML</a> are defined as in Section 2. The matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M183">View MathML</a> are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M184">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M185">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M186">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M187">View MathML</a>

(24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M188">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M186">View MathML</a>. The matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M183">View MathML</a> are defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M192">View MathML</a>

(25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M193">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M194">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M195">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M196">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M194">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M199">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M200">View MathML</a>

(27)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M201">View MathML</a>

(28)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M206">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M207">View MathML</a>

(29)

the subsystem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M208">View MathML</a>

(30)

the subsystem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M209">View MathML</a>

(31)

the subsystem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M210">View MathML</a>

(32)

and the subsystem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M211">View MathML</a>

(33)

Proof Consider the transformation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M212">View MathML</a>

(34)

Substituting the previous expression into (1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M213">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M214">View MathML</a>

(35)

Moreover, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M215">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M220">View MathML</a>. 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 3.1The subsystem (31) has infinite solutions and can be taken arbitrarily

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M221">View MathML</a>

(36)

Proof If we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M222">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M223">View MathML</a>

(37)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M224">View MathML</a>

(38)

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M225">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M226">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M227">View MathML</a>

(39)

The system (39) is a regular-type system of difference equations with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M228">View MathML</a> equations and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M229">View MathML</a> unknowns. It is clear from the above analysis that in every one of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M230">View MathML</a> subsystems one of the coordinates of the solution has to be arbitrary by assigned total. The solution of the system can be assigned arbitrarily

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M231">View MathML</a>

The proof is completed. □

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M232">View MathML</a>

(40)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M233">View MathML</a>

(41)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M234">View MathML</a>

(42)

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M235">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M236">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M237">View MathML</a>

(43)

We have a system of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M238">View MathML</a>+1 difference equations and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M238">View MathML</a> unknowns. Starting from the last equation, we get the solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M240">View MathML</a>

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

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M241">View MathML</a>

(44)

Proof It is easy to observe that the subsystem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M242">View MathML</a>

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 3.1Consider 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.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M243">View MathML</a>

2. the column minimal indices are zero, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M244">View MathML</a>

(45)

3.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M245">View MathML</a>

(46)

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

1.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M246">View MathML</a>

(47)

2.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M247">View MathML</a>

(48)

In this case the unique solution is given by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M248">View MathML</a>

(49)

whereCis the unique solution of the algebraic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M249">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M250">View MathML</a>

Furthermore, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M251">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M252">View MathML</a> is unknown, it can be replaced with the unknown vector C. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M253">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M254">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M255">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M256">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M257">View MathML</a>

In this case the Kronecker canonical form of the pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> has the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M259">View MathML</a>

(51)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M260">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M261">View MathML</a>

The solution exists if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M262">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M263">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M264">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M265">View MathML</a>

The algebraic system (50) contains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M266">View MathML</a> equations and p unknowns. Hence the solution is unique if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M267">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M268">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M136">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M271">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M272">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M273">View MathML</a>

But the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M274">View MathML</a> is left invertible since it is assumed to have p linear independent columns and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M141">View MathML</a> and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M276">View MathML</a>

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

4 Numerical examples

Example 1

Assume the system (1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M278">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M279">View MathML</a>

(52)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M280">View MathML</a>

(53)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M281">View MathML</a> and the pencil is regular. We assume the boundary conditions (2) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M282">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M283">View MathML</a>

The three finite eigenvalues (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M284">View MathML</a>) of the pencil are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M285">View MathML</a>, 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M286">View MathML</a>, and the Jordan matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M119">View MathML</a> has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M288">View MathML</a>

It is easy to observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M289">View MathML</a>

By calculating the eigenvectors of the finite eigenvalues, we get the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M290">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M291">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M292">View MathML</a>

(54)

From (9) we get the Mittag-Leffler function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M293">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M294">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M295">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M296">View MathML</a>

and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M297">View MathML</a>, by using the sum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M298">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M299">View MathML</a>, we calculate the sum of the matrix power series <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M300">View MathML</a>, and we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M301">View MathML</a>

And since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M302">View MathML</a>

by using the above expression

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M303">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M304">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M305">View MathML</a>

or, equivalently, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M306">View MathML</a>

where C is the unique solution of the algebraic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M307">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M308">View MathML</a>

and thus the unique solution of the boundary value problem is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M309">View MathML</a>

(56)

Example 2

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M310">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M311">View MathML</a>

It is easy to observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M312">View MathML</a>

since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M313">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M314">View MathML</a>

Since the matrices F, G are non-square, the matrix pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> is singular and has invariants such as the finite elementary divisors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M316">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M317">View MathML</a>, an infinite elementary divisor of degree 1 and the row minimal indices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M319">View MathML</a>. Since the Jordan matrix has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M320">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M321">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M278">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M324">View MathML</a>

(57)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M325">View MathML</a>

(58)

Since the matrices F, G are non-square, the matrix pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M21">View MathML</a> is singular and has invariants such as a finite elementary divisor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M327">View MathML</a> and the row minimal indices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M319">View MathML</a>. We assume the boundary conditions (2) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M330">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M331">View MathML</a>

The Jordan matrix is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M332">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M297">View MathML</a> for every induced matrix norm. By calculating the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M290">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M335">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M336">View MathML</a>

(59)

and since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M337">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M338">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M339">View MathML</a>

or, equivalently, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M340">View MathML</a>

where C is the unique solution of the algebraic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M341">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M342">View MathML</a>

and thus the unique solution of the boundary value problem is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/148/mathml/M343">View MathML</a>

(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.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

IKD wrote the first draft of the manuscript and DB correct it and prepared the final version of it. All authors read and approved the final manuscript.

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.

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