On a Q-boundary value problem with discontinuity conditions

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 q -sampling 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:

l(y):= -1D -IDqy(x) + v(x)y(x) = vy(x), qq on the interval x e (0,T) with the boundary conditions

U(y):= D -1 y(0)-Yy(0) = 0, V(У):= D -1У(T) + Гy(T) = 0,(2)

qq together with the jump conditions at a point a e (0,T)

y(a + 0) = a1 y(a - 0), D -1 y(a + 0) = a-1 D -1 y(a - 0) + a2y(a - 0).(3)

Here v ( x ) e L q (0,T ) is a real-valued function, a i , « 2 , Y and Г are real numbers; « i >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 0 <  q <1. Let h be a real or complex valued function on A ( A is q -geometric set (see [2])). The q -difference operator D_q (the Jackson q -derivative) is defined as

_ _{w x}  h ( x ) - h ( qx )      _n

D,.h ( x ) =-------------, x * 0.

^q         x (1 - q )

When required we will replace q by q ^{- 1}. We can demonstrate the correctness of the following facts using the definition and will use often

D^ _{- 1} h ( x ) = ( D_qh )( q ^{- 1} x ),   D q h ( q ^{- 1} x ) = qD q [ D_qh ( q ^{- 1} x )] = D^ _{- 1} D_qh ( x ).

Let h and g be defined on a q -geometric set A such that the q -derivatives of h and g exist for all x g A . Then, there is a non-symmetric formula for the q -differentiation of a product

D q [ h ( x ) g ( x )] = h ( qx ) D q g ( x ) + g ( x ) D q h ( x ).                         (4)

The q -integral usually associated with the name of Jackson is defined in the interval (0,T ), as

”

f h ( x ) d q x = (1 - q ) £ h(Tq n )Tq n .

n =0

Let Lq (0,T) be the space of all complex-valued functions defined on (0,T), such that h = (Jo |h(x) I2 dqx)2 < ”.

The space L q (0, T ) is a separable Hilbert space (see [6]) with the inner product

T

{ h , g ) =   h ( x ) g ( x )d_qx.

If h and g are both q -regular at zero, there is a rule of q -integration by parts given by

TT

L g(x)Dqh(x)dqx = (hg)(T) - (hg)(0)- J. Dqg(x)h(qx)dqx"

v 0*0

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

T f Dqh(x)dqx = (h)(T) - (h)(0).(6)

Lemma 1. (see [2]) Let h (.), g (.) in L_q (0, T ) be defined on [0, q ^{- 1}T ]. Then, for x g (0, T ] we have

(Dqh )(xq-1) = Dq-1 h (x),(7)

n ^”

• <-1D hg,g) = lim h(Tqn-1)g(Tqn) -h(Tq-1)i(T) +

• 2.    Properties of the spectral characteristics

q q        n ^»

Let h ( x ) and g ( x ) be continuously differentiable functions on [0, a ] and [ a , T ]. Denote

W q ^{( h} , g ^{)( x} ) = < ^h , g ) ^:= ^{h ( x} ) ^D q g ^{( x} ) ^- g ^{( x} ) ^D q ^{h ( x} ).

Here W_q ( h , g ) is defined as the q -Wronskian of two function h and g . If h ( x ) and g ( x ) satisfy the jump conditions (3), then

< h, g )x=a+0= < h, g )x=a -0,                                     (10) i. e. the function (h, g) is continuous on [0,T]. Applying formula (4), we obtain

D q W q ( ^h , g )( x ) = D q ( h ⁽ x ) D q g ⁽ x ) - g ⁽ x ) D q h ⁽ x ) ) = h ⁽ Qx ) D q g ⁽ x ) - g ⁽ Qx ) D q ^{h (} x )•      ⁽¹¹⁾

On the other hand,

	D q W q ( h , g )( q 1 x )= h ( x ) D q g ( q 1 x ) - g ( x ) D q h ( q 1 x ) (12) = qh ( x )[ v ( x ) g ( x ) - v g ( x )] - qg ( x )[ v ( x ) h ( x ) - v h ( x )] = 0.
As a result,	,     W ( h , g )( q - 1 x ) - W ( h , g )( x ) 0 = D q W q ( h , g )( q - 1 x ) = qV 5A\  ---qV 5    ,                  (13) q x (1 - q )
so, for x Ф 0,	W q ( h , g )( x ) = W q ( h , g )( q - 1 x ),                                 (14)

• i.    e. the q -Wronskian W_q ( h , g )( x ) does not depend on x .

Let n(x, v) and ^(x,v) be the solution of equation (1) under the boundary conditions n(0,v) = ^(T,v) = 1, D -1n(0,v) = y, D -^(T,v) = -Г.(15)

qq and under the jump conditions (3). Then

U (n) = V (^) = 0.(16)

Since the q -Wronskian is independent of x , we can evaluate

Wq Ш>) := Wq (v) = -V (n) = U (£)•(17)

W_q ( v ) is called the characteristic function of L .

Lemma 2. The eigenvalues { v _n } _n > 0 of the boundary value problem L coincide with zeros of the characteristic function. The functions n ( x , v n ) and ^ (x , v n ) are eigenfunctions and

£(x,vn) = впП(x,vn), вп * 0.(18)

Denote

T an = f n(x,vn ) dqx •

The set { v _n, a _n } _n > 0 is called the spectral date of L .

Lemma 3. The following relation holds

Pn«n = Wq (vn),(20)

where W q ( v ) = D_qW_q ( v ) (respect to v ).

The proof of Lemma 2 and Lemma 3 can be done similar to [12].

Theorem 1. The q -Sturm–Liouville eigenvalue problem (1)–(3) is self-adjoint on

CqQ (0) П L q (0, T ) .

Proof. We first prove that h (.), g (.) in L q (0, T ), we have the following q -Lagrange's identity f, ( l ( h ( x ) ) g ( x ) - h ( x ) l ⁽ g ( x ) ) ) d q x = [ h , g ]( T ) - lim [ h , g ]( Tq n ), ^{J 0}                                         n ^^

where

[ h , g ]( x ):= h ( x ) D - 1 g ( x ) - D - 1 h ( x ) g ( x )•

q

q

Applying (9) with h ( x ) = D_qh ( x ) and g ( x ) = g ( x ) , we obtain

< ^{- 1} D - 1 D q h ⁽ x ), g ⁽ x )> q^q

= lim ( D q h ) TTq^{n - 1}) g(Tqⁿ ) - ( D q h\lq ^{- 1}) g ( T ) + <  D q h , D q g >  n →∞

= lim D - J h(Tqⁿ ) g(Tqⁿ ) - D - - h ( T ) g ( T ) + <  D q h , D q g >. n →∞ ^{q                 q}

Applying (8) with h ( x ) = h ( x ), g ( x ) = D_qg ( x ), we obtain

< D q h , D q g >= h(T ) D q g(Tq ^{- 1}) - lim h(Tqⁿ ) D q g(Tq^{n - 1}) + <  h , - ¹ D - 1 D q g >  n →∞                        q ^q

= h(T ) D - 1 g ( T ) - lim h(Tqⁿ ) D - 1 g(Tqⁿ ) + <  h , - - D

q

n →∞

q

qq

,-1 D q g >.

Therefore,

< - ¹ D - J D_qh ( x ), g ( x )>=[ h , g ]( T ) - lim[ h , g](Tqⁿ ) + <  h , - ¹ D - - D q g >. q ^q                          n →∞                 q ^q

Lagrange's identity (21) results from (25) and the reality of v ( x ). Letting h (.), g (.) in assuming the that they satisfy (2)–(3), we obtain

D₄h (0) - yh (0) = 0, D₄g (0) - y? (0) = 0.

qq

C_q ²(0) and

The continuity of h (.), g (.) at zero implies that li_{m n} ^^ [ h , g](Tqⁿ ) = [ h , g ](0). Then (25) will be

• < - ¹ D - 1 D q h , g >=[ h , g ]( T ) - [ h , g ](0) + <  h , - ¹ D - 1 D q g >. q^q                                        q^q

From (26), we have

[ h , g ](0)= h (0) D - 1 g (0 ) - D - 1 h (0) g (0) = 0. qq

Similarly, from (2) we obtain

[ h , g ]( T ) = h ( T ) D gg ( T ) - D ₄h ( T ) KT) = 0. qq

Since v ( x ) is real-valued function, then

< l( h ), g > = <  - ¹ D - 1 D q h ( x ) + v ( x ) h ( x ), g ( x ) > = <  - ¹ D - 1 D q h ( x ), g ( x ) >  + <  v ( x ) h ( x ), g ( x ) >  q^qq            q^qq

= <  h , - ¹ D - 1 D q g q q ^q

+ < h (x), v (x) g (x) > = < h ,l( g) >, i.e. l is a self-adjoint operator.

Lemma 4. The eigenvalues {vn} of the boundary value problem (1)-(3) are real. Eigenfunctions related to different eigenvalues are orthogonal in L2q (0,T). All zeros of Wq (v) are simple, i. e.

W q ( V n ) ^ 0 .

Proof. Let v 0 be an eigenvalue with an eigenfunction т у ₀(.). Then,

< ^{1 (} П 0 ), П 0 >=< П 0 , ^{1 (} П 0 ) >.                                    ⁽²⁷⁾

Since l ( n ₀) = v ₀ n ₀ , then

^{( v} 0 ^{- V} c ) f I П 0 ^{( x} )|² d q x.                                   ⁽²⁸⁾

Since П 0 (.) is non-trivial then v 0 = V 0 . So the eigenvalues are real.

Let v , д be two distinct eigenvalues with corresponding eigenfunctions n (.), ^ (.), respectively. Then,

( v - ^ )[ n ( x Ж x)d_qx = 0.

Since v ^ д , then n (.) and Ж (.) are orthogonal.

Since n ( x , v _n ) and Ж ( x , v ) are solutions of the boundary value problem (1)-(3), we obtain

^D q ^{n ( x , v} n ^{), Ж ( x , v} ) ) = ^{( v} n ^{- v} ) П ^{( x , v} n ^{) Ж ( x , v} ).                        ⁽²⁹⁾

Integrating equation (29) from 0 to T and using the conditions (2), we obtain

Г t                    W q ( v n ) - W_q ( v )

• £ П ⁽ x , ^v n Шx , ^{v )} d q X =--------------•

• ^{J 0}                            v n - v

• 3.    Completeness of Eigenfunctions

Since § ( x,v _n ) = P _n n (x v_n ) as v >v _n , we obtain W q ( v n ) = £ n « n •

Thus it follows that W ( v ) ^ 0 .

Theorem 2. The system of eigenfunctions {n(x,vn)}n>0 of the boundary value problem (1)-(3) is complete in Lq (0,T).

Proof. Consider the function

Y ( x , v ) =

W q ( v ) L

Г x                   T

^ (x , v )   n (t, v )h ( t ) d q t + П x , v )   £ ( t , v ) h ( t ) d q t .

J 0                  ¹               J x                  ¹

It is easy to verify that

-1D XDY(x) + {-v + v(x)}Y(x) = h(x),  x e [0,T], v e С, qq    q

Furthermore, taking into account (19), from (18) and (20) we get

Res_v=v Y ( x , v ) ^{v  v} n

W q ( v n ) L

£ ( x , ^v n ) J ^ C t , ^v n ) h ⁽ t ) d q t ⁺ n ⁽ x , ^v n ) £ £ ( t , ^v n ) h ⁽ t ) d q t

в          C T                  1          Ti

= W77\ n ^{( x} , ^v n ) I П t ^{, v} n ) ^{h (} t ) d q t =--- n ^{( x} , ^v n ) Ji tt , ^v n ) h ⁽ t ) d q t • W q ( v )        ^J0                  » n        ^J0

Let the function h ( x ) e L q (0,T ) be such that

T

£ n ( t , v n ) h ( t ) d q t = 0, n = 0,1,2,...

Then in view of (31), Res _v = _v Y ( x , v ) = 0 and consequently for each fixed x e [0, T ], the function n

• Y ( x , v ) is entire in v . Furthermore, for p e G § = { p :| p - p _n | > J } and | p | > p where v = p ², p _n are the zeros of the function

Wq (p) = p(b1 sin pT - b2 sin p(2a - T)), where b1 = ^L+^L,  b2 = ^1—^L-, 5 is a fixed positive number, p* is rather large, the inequality

• ^{1    W} q ^{v )|} > C 5 ¹ p ¹ e ^W ,   p^e C 5 ,   p = ^ ⁺ i T

and consequently the inequality

• | Y ( x , v )| < C J ,   pe G ₅ ,

• ¹    p ^|

• 4.    The q -sampling theory

is obtained (see [12]). Using the maximum principle and Liouville's theorem we conclude that Y ( x , v ) = 0. From this and (30) it follows that h ( x ) = 0 a. e. on (0, T ). Thus the theorem is proved.

Theorem 3. Let n ( x, v ) and ^ ( x, v ) be the solutions of (1) selected as above. Then every function h of the form

^{h ( v )} = I T v ⁽ x ^{n (} x ^{, v )} d q x ,   v ^e L q ^(0, T ),                          ⁽3³⁾

can be written as the Lagrange-type sampling expansion

A        W q ( v )

h ( v ) = V h ( V ) ~-----------,

^X      q v q ( V n )( v -V n )

where W q ( v ) is the q -Wronskian of the functions n ( x, v ) and ^ Cx, v ) .

Proof. We multiply equation (1) with n (x , v _n ) . Then we consider again equation (1), but replace V by v n and multiply this last equation by n (x , v ) . Subtracting the two results yields

^{( v - V} n ) n ⁽ x , ^V Ш x , ^V n ) = D q n ⁽ q ^{- 1} x , ^V n ) n ⁽ x , ^V ) ^- D q n ⁽ q ^{- 1} x , ^v Ш x , v ).

From the rule for the q -differentiation of product (4), we can write

^{( v - V} n Ш x , ^VШ x , ^V n ) = D q ^[ D q n ⁽q ^{- 1} x , ^V n M x , ^V ) - D q П ⁽q ^{- 1} x , ^VШ x , ^V n ^{) ]}

If we apply a q -integration by means of (6) we obtain

^{( v -V} n )£ ^{n ( x , V ) n x, v} n ) d q x = J_o D q [ D q n ⁽ q ^{- 1 x, V} n ^{n ( x, v} ) ^- D q n ⁽ q ^{- 1 x, V} M ^{x, v} n ) ] d q x

= D q n ⁽ q - 1 ⁰, ^V n ⁾ n ^(o, ^V ) ^- D q n ⁽ q - 1 ⁰, ^V W, v ) ^- ( D q n ⁽ q ^{- 1} o, ^v „ ) n ⁽o, ^v ) ^- D q n ⁽ q ^{- 1} o, ^{v )} u ⁽ ° , ^V n ) )

From the condition (2), we have

D q n ⁽q ^{- 10}, ^V n ) п ⁽⁰ ,^V ) ^- D q n ⁽q ^{- 10}, ^V ) п ⁽⁰ , ^V n ) = D - 1 ^ (0, ^V n ) n ^{(0, V} ) ^- D - 1 ^ (0, ^V ) п ⁽⁰ , ^V n ) qq

= D - 1 П ⁽⁰ V n ) ^- Yn ^{(0, v} n ) = U⁽ n ) = ⁰. ^q

Multiply (17) by n ( O, v _n ) to obtain

W q ^{( V )} П ⁽⁰, ^V n )  = ^- D - 1 n ⁽⁰, ^V ) П ⁽⁰, ^V n ) ^-Г n ⁽⁰, ^{V )} n ⁽⁰, ^V n )

= - Dq - 1 ^'T, ^V )^> П ^<°-⁰, ^V n ) + D_q - 1^^ ^T, ^V n M T, ^V ).

Then, we get

^{( V} -^V n ) | ° П ⁽ x , ^V ) П⁽. x , ^V n ) d q x =   W q ^{( V} ) П ⁽⁰ , ^V n ).

* 0

as a result,

W q ( V П (0 , V n )

^V-V n

T

J n( x,v )n(x,vn ) dqx = and taking the limit as v ^vn gives

J ⁰ I П ^(x , ^V n ) I² d q x = ^W q ^{( V} n ) П ⁽⁰ , ^V n ).

We can therefore apply Kramer's lemma (see [14]) and write an integral transform of the form (33) as

W_q ( v )

h ^{( v} ) = X h ^{( v} n             ;

.

n =0      W q ^{( v} n ^{)( v -V} n )