Dependence Of Eigenvalues Of Some Boundary Value Problems
Ekin U¼ gurlu
y, Kenan Tas
zReceived 20 Feburary 2020
Abstract
In this work we deal with a system of two …rst-order di¤erential equations containing the same eigen- value parameter. We consider some suitable separated real and complex coupled boundary conditions, and show that the eigenvalues generated by this system are continuous in an eigenvalue branch. Also we introduce the ordinary and Frechet derivatives of these eigenvalues with respect to some elements of the data.
1 Introduction
In this paper we will concern with a system of two …rst-order di¤erential equations that contains the con- tinuous analogous of the orthogonal polynomials on the unit circle. In the literature, this construction has been introduced by Atkinson in his book [1], Chap. 7. Indeed, the recurrence equations
yn+1= yn+bnzn; zn+1= bnyn+zn; wheren= 0;1;2; :::;imply the equations
8>
>>
>>
><
>>
>>
>>
:
yn+1 yn = 2i e
1 ie yn+bnzn;
zn+1 zn=
1 +ie 1 ie bnyn;
(1)
where = 1 +ie = 1 ie . Considering yn+1 yn as the discrete version of y(x)dx; bn as b(x)dx and 2eas dx;the system of equations (1) leads to the continuous version of the recurrence relations as
y0=i y+b(x)z;
z0 =b(x)y; (2)
whereb(x)is a continuous function on a given interval. However, there does not exist a detailed analysis of the system of equations (2) both in the literature and in [1]. Beside this, the system of equations that we will study is much more general than (2) as can be seen in (3). In this paper, we focus on the di¤erentiability property of the eigenvalues of some boundary value problems generated by (3) and separated, real and complex-coupled boundary conditions. Recently, such an investigation has been introduced in [2]–[4] for odd- order boundary-value problems and some background information on di¤erentiable properties of eigenvalues of even-order boundary-value problems can be found in, for example, [2].
Mathematics Sub ject Classi…cations: Primary 34B09, 34L15; Secondary 34L40
yDepartment of Mathematics, Çankaya University, Central Campus, 06790 Etimesgut, Ankara, Turkey
zDepartment of Mathematics, Çankaya University, Central Campus, 06790 Etimesgut, Ankara, Turkey
81
2 System of Equations
In this paper we consider the following system of equations
y10 =i w(x)y1+ia(x)y1+b(x)y2;
y20 = i r(x)y2+b(x)y1+ic(x)y2; (3)
where x 2[t1; t2]; a; c; r; w are real-valued, integrable functions on [t1; t2]; bis a complex-valued function, w >0and r >0for almost allxon[t1; t2]:
The system of equations (3) can also be handled as the following …rst-order equation y1
y2
0
= i w 0
0 r + ia b
b ic
y1
y2 : (4)
The assumptions on the coe¢ cients of (3) and (4) allow us to introduce the following lemma.
Lemma 1 The system of equations (3) has one and only one solution Y(x; ) = y1(x; )
y2(x; ) (5)
satisfying the initial conditions
y1(l; ) =l1; y2(l; ) =l2;
wherel2[t1; t2]andl1; l2are arbitrary complex numbers. Moreover,y1(:; )andy2(:; )are entire functions of :
Lemma 2 Lety1(x; ),y2(x; )andz1(x; ),z2(x; )be the solutions of (3) corresponding to the parameters and ; respectively. Then we have
i( )
t2
Z
t1
z1(x; ) z2(x; ) w(x) 0
0 r(x)
y1(x; ) y2(x; ) dx
= h
Y(x; ); Z(x; ) i
jtt21; (6)
where Y(x; ) and Z(x; ) are the vectors given by the rule (5) corresponding with y1(x; ); y2(x; ) and z1(x; ); z2(x; ); respectively and
[Y(x; ); Z(x; )] : = y1(x; )z1(x; ) +y2(x; )z2(x; ): (7) Proof. Using the equations in (3) we get
d
dx( y1z1+y2z2) = (i wy1+iay1+by2)z1 y1 i wz1 iaz1+bz2
+ i ry2+by1+icy2 z2+y2(i rz2+bz1 icz2)
= i( )y1z1w i( )y2z2r: (8)
Integrating both sides of (8) fromt1 tot2we obtain the result.
We shall note that the representation (7) can also be introduced as follows
[Y(x; ); Z(x; )] =Zt(x; )J Y(x; ) (9)
where the superscript denotes the transpose of the vector and
J = 1 0
0 1 :
Representation (9) will allow us to impose real and complex-coupled boundary conditions.
Corollary 3 If y1(x; ),y2(x; )andz1(x; ),z2(x; )are the solutions of (3) corresponding with the same parameter ;then [Y(:; ); Z(:; )] is independent fromxand depends only on :
3 Boundary Conditions
In this section we will share some suitable separated, real-coupled and complex-coupled boundary conditions for the solutions of (3).
Firstly, we can impose the separated boundary conditions as
(i+ tan )y1(t1) (1 +itan )y2(t1) = 0;
(i+ tan )y1(t2) (1 +itan )y2(t2) = 0; (10) where and are some real numbers.
Secondly, the real-coupled boundary conditions can be given as y1(t2)
y2(t2) =M y1(t1)
y2(t1) ; (11)
whereM 2E2(C);i.e.
M = m11 m12
m21 m22 ; mij 2C; (12)
such that
M J M =J; (13)
whereM =Mt:
Finally, the complex-coupled boundary conditions can be introduced as y1(t2)
y2(t2) =ei M y1(t1)
y2(t1) ; (14)
where is a real number andM 2E2(C):
Before passing to the details on the boundary-value problems we shall pay a special attention on the equation (13) together with the matrix (12). The equation (13) can also be written as
jm11j2+jm21j2 m11m12+m21m22
m11m12+m21m22 jm12j2+jm22j2 = 1 0
0 1
which implies that 8
<
:
jm11j2+jm21j2= 1;
m11m12=m21m22; jm12j2+jm22j2= 1:
(15)
It is possible to consider a matrixM with entriesmij satisfying (15). Indeed, for the matrix
M = 2 64
p2 i+ tan 1 +itan
1 itan i+ tan i+ tan
1 +itan
p21 itan i+ tan
3 75
equations in (15) are satis…ed, where and are some real numbers.
Finally, we should note that all the conditions (10), (11) and (14) can be embedded into the following boundary conditions
y1(t1)
y2(t1) =A:v; y1(t2)
y2(t2) =B:v (16)
whereAandB are some2 2complex matrices and vis a 2 1 vector.
Now we can introduce the following theorem.
Theorem 4 Each boundary-value problem generated by (3) and (10), (11), (14) has only real and discrete eigenvalues with possible accumulation points at positive and negative in…nity.
Proof. LetY(x; )be the corresponding vector given by (5) and generated byy1(x; )andy2(x; )of the solutions of (3). For each problem we will use the following equation
2 Im 0
@
t2
Z
t1
jy1(x; )j2w(x)dx+
t2
Z
t1
jy2(x; )j2r(x)dx 1 A=h
Y(x; ); Y(x; )i
jtt21 : (17)
Separated boundary conditions give that
y1(t2; )y1(t2; ) +y2(t2; )y2(t2; ) +y1(t1; )y1(t1; ) y2(t1; )y2(t1; )
= 1 +itan i+ tan
1 itan
i+ tan + 1 jy2(t2; )j2+ 1 +itan i+ tan
1 itan
i+ tan 1 jy2(t1; )j2
= 0: (18)
Therefore (17) and (18) show thatIm = 0. This completes the proof for the problem (3), (10).
Real-coupled boundary conditions implies that
Y (t2; )J Y(t2; ) Y (t1; )J Y(t1; )
= Y (t2; )M J M Y(t2; ) Y (t1; )J Y(t1; ) = 0: (19) Therefore (17) and (19) prove the result for the problem (3), (11).
The proof for the problem (3), (14) can now be given similarly. Therefore the reality of the eigenvalues of each problem has been proved.
For the other assertion we shall use the conditions in (16).
LetY(x; )be a fundamental matrix solution of (4) satisfyingY(t1; ) =I;whereI is the2 2 identity matrix. For any other solutionY(x; )of (4) one may write the following
Y(x; ) =Y(x; )Y(t1; ):
Considering the conditions (16) we get
fB Y(t2; )Ag:v= 0 and forv6= 0we obtain that
( ) := detfB Y(t2; )Ag= 0:
Clearly, the zeros of ( )coincide with the eigenvalues of the problem (4), (16) and vice versa. Finally using Lemma 2.1 we complete the proof.
Moreover, for the eigenvalues of the corresponding problems we can introduce the following theorem.
Theorem 5 If we denote the eigenvalues of the problem (3), (16) by 0; 1; 2; ::: then the series X
r6=0
j rj 1 is convergent for each >0:
Proof. The proof follows from Theorem4, representation (4) and Gronwall’s inequality.
4 Banach Space
In this section we will share some suitable Banach spaces to introduce the ordinary and Frechet derivatives of the corresponding boundary-value problems.
We shall denote byBthe Banach space consisting of all vectors such that
= (t1; t2; A1; A2; f1; g1; h1; m1; n1) with the norm
k k=jr1j+jr2j+kA1k+kB1k+
et2
Z
et1
(jf1j+jg1j+jh1j+jm1j+jn1j);
where
B = R R M2;2(C) M2;2(C) L1(et1;et2) L1(et1;et2) L1(et1;et2) L1(et1;et2) L1(et1;et2);
M2;2(C)is the set of all complex2 2 matrices and(et1;et2) [t1; t2]: Here the functionsf1; g1; h1; m1; n1 are de…ned asf; g; h; m; n2L1(t1; t2)on[t1; t2]and zero otherwise.
LetX1be a subset ofBconsisting of all vectors
!1= (t1; t2; A; B;ea;eb;ec;w;e er):
There is no confusion with identifying the subsetX1 withX that is a subset ofBwith the vectors
!1= (t1; t2; A; B; a; b; c; w; r):
Therefore we will identifyX withX1 to inherit the norm fromBand convergence in X:
Lemma 6 The solutionsy1(x; ); y2(x; )of (4), (16) satisfying the initial conditions y1( ; ) = 1; y2( ; ) = 2;
where 2(et1;et2) and 1; 2 are complex numbers, are continuous functions of the variables ; 1; 2; a; b;
c; w; r:
Proof. The proof follows from (4) and Theorem 2.7 in [5].
Lemma 7 The eigenvalue = (!0) of (4), (16) is continuous at !02X:
Proof. Consider the function
(!; ) = detfB Y(t1; t2; a; b; c; w; r; )Ag:
The zeros of coincide with the eigenvalues of (4), (16). As we have discussed earlier that is an entire function and therefore it is not constant in :
Now let!0 2X and (!0; ) = 0:Then for the values of satisfyingj j>0 we have (!0; )6= 0 and applying the well-known result on continuity of roots of an equation of [6] we complete the proof.
Remark 1 In Lemma7, we, in fact, prove that there exists an eigenvalue branch such that it is continuous at!0: Therefore from now on we will only consider such continuous eigenvalue branches.
Finally, using Lemma 6, Lemma 7and the results in [5] we can introduce the following results.
Lemma 8 Suppose that = (!0),!02X;is an eigenvalue of (4), (16) of order one or two. There exist normalized eigenfunctions u1(x; !); u2(x; !) of of order one or uj1(x; !); uj2(x; !); j = 1;2; of of order two such that uk(x; !) converges uniformly to uk(x; !0) and ujk(x; !) converges uniformly to ujk(x; !0) on any compact subintervals of (et1;et2); wherek= 1;2:
We recall that an operatorT acting on Banach spacesK1; K2asT:K1!K2is said to be di¤erentiable atf 2K1 if
kT(f +h) T f T0(f)k=o(h)ash!0;
whereT0:K1!K2 is a bounded operator and it is said to be the Frechet derivative ofT:
Finally in the next theorem we will …x all the other variables except the variable that the derivative of the eigenvalue is being taken with respect to it.
Theorem 9 Let = (!); !2X;be the eigenvalue of (3), (16). Then with the normalized eigenfunctions u1(x; ); u2(x; )we have the following equations:
(i) 0( ) = 2ju1(t1)j2= 2ju2(t1)j2; (ii) 0( ) = 2ju1(t2)j2= 2ju2(t2)j2;
(iii) 0(M) = i u1(t1) u2(t1) H J M u1(t1)
u2(t1) ; H2E2(C);
(iv) 0( ) =ju1(t1)j2 ju2(t1)j2; (v) 0(a) =
t2
R
t1
ju1j2h; h2L1(t1; t2);
(vi) 0(b) =
t2
R
t1
2 Im(u1u2)h; h2L1(t1; t2);
(vii) 0(c) =
t2
R
t1
ju2j2h; h2L1(t1; t2);
(viii) 0(w) =
t2
R
t1
ju1j2h; h2L1(t1; t2);
(ix) 0(r) =
t2
R
t1
ju2j2h; h2L1(t1; t2):
Proof. To prove the assertions we will use the equation (6).
Letu1=u1(:; ( )), u2 =u2(:; ( ))andv1=u1(:; ( +h)), v2=u2(:; ( +h))forh >0: Then we obtain that
i( ( ) ( +h)) 0
@
t2
Z
t1
u1v1wdx+
t2
Z
t1
u2v2rdx 1 A
= u1(t1)v1(t1) u2(t1)v2(t1)
= 1 +itan i+ tan
1 itan( +h)
i+ tan( +h) 1 u2(t1)v2(t1): (20) Dividing the equation (20) byhletting h!0we obtain the result and this proves(i):
(ii)can be proved similarly.
For the proof of (iii) we let u1 = u1(:; (M)), u2 = u2(:; (M)) and v1 = u1(:; (M +H)), v2 = u2(:; (M +H))forH 2E2(C):Then we have
i( (M) (M +H)) 0
@
t2
Z
t1
u1v1wdx+
t2
Z
t1
u2v2rdx 1 A
= (V J U)(t2) (V J U)(t1)
= V (t1)(M +H) J M U(t1) V (t1)J U(t1)
= V (t1)H J M U(t1): (21)
Therefore (21) completes the proof for (iii):
Now we letu1=u1(:; ( )),u2=u2(:; ( ))andv1=u1(:; ( +h)),v2=u2(:; ( +h))forh >0:Then we obtain that
i( ( ) ( +h)) 0
@
t2
Z
t1
u1v1wdx+
t2
Z
t1
u2v2rdx 1 A
= (V J U)(t2) (V J U)(t1)
= (e ih 1)V (t1)J U(t1): (22)
Then we divide the equation (22) byhwe leth!0 to obtain the result(iv):
For u1 =u1(:; (a)), u2 =u2(:; (a))and v1 =u1(:; (a+h)), v2 =u2(:; (a+h)), h2 L1(t1; t2); we have
i( (a) (a+h)) 0
@
t2
Z
t1
u1v1wdx+
t2
Z
t1
u2v2rdx 1 A= i
t2
Z
t1
u1v1hdx (23)
and (23) completes the proof of(v):
For the rest we can apply a similar way and this completes the proof.
5 Conclusion and Remarks
In this paper we have considered a system of …rst-order di¤erential equations containing the same eigenvalue parameter together with some suitable separated and coupled boundary conditions. As we have stressed in the Introduction the motivation of this system has been introduced by Atkinson [1]. The special form of (3) coincides with (2). In fact,w 1; r=a=c= 0on the given interval[t1; t2]gives the system of equations (2). However, in this case some modi…cations should be done. For example (6) should be given as
( )
t2
Z
t1
y1(x; )z1(x; )dx=h
Y(x; ); Z(x; )i jtt21
and this form has been given in [1].
We need to pay attention to some issues here. In the equation (3), we considered a …rst-order di¤erential system corresponding to some boundary conditions and such …rst-order systems have been investigated in the literature, e.g. orthogonality of eigenvectors, self adjointness property (but not continuity property).
However, all these resources contain abstract boundary conditions such as
A EA=B EA; (24)
where A and B are some suitable abstract matrices corresponding to some boundary conditions and E is some special matrix.
It is an ordinary way to obtain that the boundary value problem constructed by some matrices satisfying (24) is selfadjoint (symmetric) and say that all eigenvalues are real. However, it is not possible to investigate the continuity dependence of eigenvalues on data A; B: In other words, it is not clear how the continuity depends if one replacesA by some matricesA+H:For this reason, the papers cited as [2]–[5], all consider separated and coupled boundary conditions to investigate the dependence of eigenvalues.
On the other side, Eq. (3) is not a standart Hamiltonian system. This can be seen easily from the system (3) and from the boundary conditions.
In this work, we have achieved to impose suitable, well-de…ned boundary conditions in such an unusual way for the solutions of the equation. Paper therefore includes new types of boundary conditions that can not be embedded into the known ones.
References
[1] FV. Atkinson, Discrete and Continuos Boundary Problems. New York, USA: Academic Press, 1964.
[2] E. U¼gurlu, Regular third-order boundary value problems, Appl. Math. Comput., 343(2019), 247–257.
[3] E. U¼gurlu, Third-order boundary value transmission problems, Turkish J. Math., 43(2019), 1518–1532.
[4] E. U¼gurlu, Regular …fth-order boundary value problems, Bull. Malays. Math. Sci. Soc., 43(2020), 2105–
2121.
[5] Q. Kong and A. Zettl, Linear Ordinary Equations in Inequalities and Applications, Singapore: World Scienti…c, 1994.
[6] J. Dieudonne, Foundations and Modern Analysis, New York, USA, Academic Press, 1969.
