ON A q-BOUNDARY VALUE PROBLEM WITH DISCONTINUITY CONDITIONS

In this paper, we studied q -analogue of Sturm–Liouville boundary value problem on a finite interval having a discontinuity in an interior point. We proved that the q -Sturm–Liouville problem is self-adjoint in a modified Hilbert space. We investigated spectral properties of the eigenvalues and the eigenfunctions of q -Sturm–Liouville boundary value problem. We shown that eigenfunctions of q -Sturm–Liouville boundary value problem are in the form of a complete system. Finally, we proved a sampling theorem for integral transforms whose kernels are basic functions and the integral is of Jackson’s type.


Introduction
Boundary value problems with discontinuity conditions on the interval often appear in mathematics and other branches of sciences. Quantum calculus was initiated at the beginning of the 19th century and in recent years, many papers subject to the boundary value problems consisting a q -Jackson derivative in the classical Sturm-Lioville problem have occured [1]. In [2,3], q -Sturm-Liouville problems are investigated and a space of boundary values of the minimal operator and describe all maximal dissipative, self-adjoint, maximal accretive and other extensions of q -Sturm-Liouville operators in terms of boundary conditions are raised. A theorem on completeness of the system of eigenfunctions and associated functions of dissipative operators are proved by using the Lidskii's theorem.
Also, there are a lot of physical models involving q -difference and their related problems in [4,5]. In [6], the construction of expansions in q -Fourier series was followed by the derivation of the qsampling theorems. In [7], a q -version of the sampling theorem was derived using the q -Hankel transform. The sampling theory associated with q -type of Sturm-Liouville equations is conceived (see [8,9]).
In [10], it is proved that the regular symmetric q -Sturm-Liouville operator is semi-bounded and investigated the continuous spectrum of this operator. In [11], authors established a Parseval equality and an expansion formula in eigenfunctions for a singular q -Sturm-Liouville operator. In this paper, q -analogue of Sturm-Liouville boundary value problems with discontinuity conditions in an interior point ( [12]) are discussed.
Let us consider the boundary value problem L for the equation: together with the jump conditions at a point ) Here is a real-valued function, 1  , 2  ,  and  are real numbers; 0 > 1  .

Preliminaries on q -calculus
In this section, we give some of the q -notations and we will use these q -notations throughout the paper. These standard notations are founded in [13].
Let q be a positive number with 1 < < 0 q . Let h be a real or complex valued function on A ( A is q -geometric set (see [2])). The q -difference operator q D (the Jackson q -derivative) is defined as When required we will replace q by 1  q . We can demonstrate the correctness of the following facts using the definition and will use often ).
h and g be defined on a q -geometric set A such that the q -derivatives of h and g exist for all A x  . Then, there is a non-symmetric formula for the q -differentiation of a product ).
The q -integral usually associated with the name of Jackson is defined in the interval ) (0,T , as is a separable Hilbert space (see [6]) with the inner product h and g are both q -regular at zero, there is a rule of q -integration by parts given by The q appearing in the argument of h in the right-hand side integrand is another manifestation of the symmetry that is everywhere present in q -calculus. As an important special case, we have

Properties of the spectral characteristics
Let On the other hand, As a result, i. e. the q -Wronskian ) )( , be the solution of equation (1) under the boundary conditions and under the jump conditions (3). Then is called the characteristic function of L. where The proof of Lemma 2 and Lemma 3 can be done similar to [12].
Proof. We first prove that Lagrange's identity (21) results from (25) and the reality of assuming the that they satisfy (2)-(3), we obtain 0.
is real-valued function, then Integrating equation (29) from 0 to T and using the conditions (2), we obtain . Proof. Consider the function

Completeness of Eigenfunctions
Furthermore, taking into account (19), from (18) and (20) ,  is a fixed positive number, *  is rather large, the inequality and consequently the inequality is obtained (see [12] From the rule for the q -differentiation of product (4), we can write If we apply a q -integration by means of (6) we obtain   We can therefore apply Kramer's lemma (see [14]) and write an integral transform of the form (33) as