© Hindawi Publishing Corp. A NECESSARY AND SUFFICIENT CONDITION FOR THE EXISTENCE Of A UNIQUE SOLUTION OF A DISCRETE BOUNDARY VALUE PROBLEM

A kth-order linear difference equation with constant coefficients subject to boundary conditions is considered. A necessary and sufficient condition for the existence of a unique solution for such a boundary value problem is established. The condition established answers a fundamental question for well-posedness and can be easily applied using a simple and computationally tractable algorithm that does not require finding the roots of the associated characteristic equation.


Introduction.
The present paper is motivated by the fact that, in general, the theory and the construction of solutions of boundary value problems are more difficult than those of initial value problems [4, page 629], and by the need for a simple criterion to determine whether a boundary value problem is well posed [5, page 43].
Our main objective in this research is to establish an easy-to-apply criterion for the fundamental question of existence and uniqueness of solutions of a discrete boundary value problem (DBVP).For definiteness, we consider a general kth-order linear difference equation with constant coefficients given as k j=0 a j y(n + j) = 0, n= 0, 1, 2,..., a 0 a k ≠ 0, (1.1) subject to separable boundary conditions of the form y(i) = y i , i= 0,...,k 1 − 1, y(j) = y j , j = N,...,N + k 2 − 1, ( where k 1 ,k 2 ≥ 1, k 1 + k 2 = k, and N > k 1 .We establish a criterion based upon the parameters of the given boundary value problem, namely, the coefficients, k, k 1 , and N. In the literature, in general, existence and uniqueness theorems for DBVPs provide sufficient conditions in which a Green's function plays an essential role [6, pages 243-250].Furthermore, unlike the results established in [1,2,3] and conjectured in [7] (despite their applicability), the result established here does not require finding the zeros of the corresponding characteristic polynomial which may be a difficult task for higher-degree polynomials.Similar determinants (see Theorem 2.1) were used to characterize disconjugacy of linear second-order selfadjoint difference equations in [6, page 251].This paper is organized as follows.In Section 2, the main result, Theorem 2.1, is presented along with several preliminary results that will be needed for the lengthy proof of the result.In Section 3, a simple algorithm for applying the criterion is presented.In Section 4, an illustrative example is given.Finally, in Section 5, we conclude our research with an important remark that illustrates the applicability of the main result to a more general type of boundary conditions.

Existence and uniqueness criterion.
Our main result in this research is the following theorem.

Theorem 2.1. DBVP (1.1) and (1.2) has a unique solution if and only if the following (N
where a j = 0 if j < 0 or j > k.
To prove Theorem 2.1, we need to recall the following two results which were established in [1].
and u (s) is the sth componentwise derivative of the vector u, then ) ) A j = u z j ,..., u (m j −1) z j .
(2.6) Also, we need to establish the following two lemmas.The first lemma establishes a formula for higher-order derivatives of determinants.The second one plays the major role in the proof of Theorem 2.1.
Proof.Clearly, the result is true for r = 1.Furthermore, if it holds for r , then (2.8) Hence, by the principle of mathematical induction, the result is true for all r .
Therefore, by Cramer's rule and the invariance under the transposition property of the determinants, we have that is, the result holds for m = 1.Now, if the result holds for m, then where b j = a j−1 − a j z are the coefficients of the (k + 1)-degree polynomial q(λ) = (λ − z)p(λ).But, by Leibnitz rule of differentiation, (2.17)However, (2.18) where M r s is the r s-minor of D a k ( , m+1) if r ,s ≥ 0 and 0 otherwise.Therefore, after some simplifications, we obtain (2.20) and, hence, by the principle of mathematical induction, the result holds for all m.
To complete the proof, we need to justify (2.18) and (2.19).Equation (2.18) can be easily checked out by differentiating the last column of V k+1 (z 1 ,...,z r ,z) i times, substituting z = 0 and applying the Laplace expansion of determinants through the (k + 1)th column.
As for (2.19), since b j are linear in z, by Lemma 2.4, where α i ∈ {0, 1} and Finally, using the fact that b j = −a j , the result follows.This completes the proof.
Proof of Theorem 2.1.First, without loss of generality, we assume that a k = 1.Now, it is not difficult to see that if the characteristic polynomial (1.3) has r distinct characteristic roots denoted by z 1 ,z 2 ,...,z r with corresponding multiplicities m 1 ,...,m r such that 1 ≤ r ≤ k, 1 ≤ m j ≤ k, j = 1,...,r , and m 1 + •••+m r = k, then the general solution of (1.1) is given by where q i (n) is a polynomial in n of degree m i −1.Therefore, the existence of a unique solution of the DBVP (1.1) and (1.2) is equivalent to the nonsingularity of the block matrix A defined by

.24)
where A j is a k × m j matrix given by .25) for j = 1,...,r .
Hence, by Theorem 2.3, (2.26) Therefore, by Lemma 2.5, A is nonsingular if and only if D a k (k 1 ,N − k 1 ) ≠ 0, and so the result follows.

A computational algorithm.
To determine the solvability of the DBVP (1.1) and (1.2), we propose the following easy-to-apply algorithm.
Step 1. Form the following table: Step 2. Skip k 1 columns by starting from the left.
Step 3. Form a determinant of size (N −k 1 )×(N −k 1 ) starting from the first row.
Output.If the determinant calculated in Step 3 is not zero, then DBVP (1.1) and (1.2) has a unique solution.
Remark 3.2.It is worthwhile mentioning that the determinant needed in the above algorithm corresponds to a banded matrix with constant diagonals.In fact, there will be at most k + 1 diagonals.This will definitely minimize the cost of computations as far as computer time is concerned.

Applications.
To illustrate the applicability of Theorem 2.1 (and Algorithm 3.1), we present the following example.where the missing elements are zeros.But where Therefore, one can see that D n = 0 for all N except for N = 0,k,2k,.... Hence, the DBVP (4.1) and ( 4.2) has a unique solution if and only if N = 0,k,2k,.... Otherwise, if there is a solution, it is not unique.

Conclusion.
Finally, it is important to mention that Theorem 2.1 is also applicable for a wider class of boundary conditions.Namely, the boundary conditions are described as . . .
in which B is a k × k matrix of full rank.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
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