Volume 2011, Article ID 767024,23pages doi:10.1155/2011/767024
Research Article
Green’s Function for Discrete Second-Order
Problems with Nonlocal Boundary Conditions
Svetlana Roman and Art ¯uras ˇStikonas
Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, Vilnius, LT-08663, Lithuania Correspondence should be addressed to Svetlana Roman,[email protected]
Received 1 June 2010; Revised 24 July 2010; Accepted 9 November 2010
Academic Editor: Gennaro Infante
Copyrightq2011 S. Roman and A. ˇStikonas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate a second-order discrete problem with two additional conditions which are described by a pair of linearly independent linear functionals. We have found the solution to this problem and presented a formula and the existence condition of Green’s function if the general solution of a homogeneous equation is known. We have obtained the relation between two Green’s functions of two nonhomogeneous problems. It allows us to find Green’s function for the same equation but with different additional conditions. The obtained results are applied to problems with nonlocal boundary conditions.
1. Introduction
The study of boundary-value problems for linear differential equations was initiated by many authors. The formulae of Green’s functions for many problems with classical boundary conditions are presented in 1. In this book, Green’s functions are constructed for regular and singular boundary-value problems for ODEs, the Helmholtz equation, and linear nonstationary equations. The investigation of semilinear problems with Nonlocal Boundary Conditions NBCs and the existence of their positive solutions are well founded on the investigation of Green’s function for linear problems with NBCs 2–7. In 8, Green’s function for a differential second-order problem with additional conditions, for example, NBCs, has been investigated.
In this paper, we consider a discrete difference equation
wherea2, a0/0. This equation is analogous to the linear differential equation
b2xux b1xux b0xux fx. 1.2
In order to estimate a solution of a boundary value problem for a difference equation, it is possible to use the representation of this solution by Green’s function9.
In10, Bahvalov et al. established the analogy between the finite difference equations of one discrete variable and the ordinary differential equations. Also, they constructed a Green’s function for a grid boundary-value problem in the simplest caseDirichlet BVP.
The direct method for solving difference equations and an iterative method for solving the grid equations of a general form and their application to difference equations are considered in 11, 12. Various variants of Thomas’ algorithm monotone, nonmonotone, cyclic, etc. for one-dimensional three-pointwise equations are described. Also, modern economic direct methods for solving Poisson difference equations in a rectangle with boundary conditions of various types are stated.
Chung and Yau 13 study discrete Green’s functions and their relationship with discrete Laplace equations. They discuss several methods for deriving Green’s functions. Liu et al.14give an application of the estimate to discrete Green’s function with a high accuracy analysis of the three-dimensional block finite element approximation.
In this paper, expressions of Green’s functions for1.1have been obtained using the method of variation of parameters12. The advantage of this method is that it is possible to construct the Green’s function for a nonhomogeneous equation 1.1with the variable coefficients a2, a1, a0 and various additional conditions e.g., NBCs. The main result of this paper is formulated in Theorem 4.1,Lemma 5.3, andTheorem 5.4.Theorem 4.1can be used to get the solution of an equation with a difference operator with any two linearly independent additional conditions if the general solution of a homogeneous equation is known.Theorem 5.4gives an expression for Green’s function and allows us to find Green’s function for an equation with two additional conditions if we know Green’s function for the same equation but with different additional conditions. Lemma 5.3is a partial case of this theorem if we know the special Green’s function for the problem with discreteinitial conditions. We apply these results to BVPs with NBCs: first, we construct the Green’s function for classical BCs, then we can construct Green’s function for a problem with NBCs directly Lemma 5.3or via Green’s function for a classical problemTheorem 5.4. Conditions for the existence of Green’s function were found. The results of this paper can be used for the investigation of quasilinear problems, conditions for positiveness of Green’s functions, and solutions with various BCs, for example, NBCs.
The structure of the paper is as follows. In Section 2, we review the properties of functional determinants and linear functionals. We construct a special basis of the solutions inSection 3and introduce some functions that are independent of this basis. The expression of the solution to the second-order linear difference equation with two additional conditions is obtained in Section 4. InSection 5, discrete Green’s function definitions of this problem are considered. Then a Green’s function is constructed for the second-order linear difference equation. Applications to problems with NBCs are presented inSection 6.
2. Notation
For allaij, bji ∈ ,i, j 1,2, the equality
b11 a11
b2 1 a21
b12 a11
b2 2 a21
b11 a12
b12 a22
b12 a12
b22 a22
b 1 1 b12
b2 1 b22
· a1 1 a12
a2 1 a22
2.1
is valid. The proof follows from the Laplace expansion theorem8.
LetX {0,1, . . . , n},X {0,1, . . . , n−2}.FX: {u|u:X → }be a linear space of realcomplexfunctions. Note thatFX∼ n1 and functionsδi,i0,1, . . . , n, such that
δij δj
i forj ∈Xδnmis a Kronecker symbol:δmn 1 ifmn, andδnm0 ifm /n, form a basis of this linear space. So, for allu∈FX, there exists a unique choice ofu1, . . . , un∈ n, such thatunk0ukδk. If we have the vector-functionu u1, u2∈F2X, then we consider the matrix functionu:X2 → M
2×2 ∼ 4 and its functional determinantDuij :X2 →
uij
u1, u2
ij :
u1i u1j u2i u2j
,
Duij detuijdetu1, u2
ij :
u1 i u1j
u2 i u2j
.
2.2
The Wronskian determinantWuiin the theory of difference equations is denoted as follows:
Wuj :
u1 j−1 u2j−1
u1j u2j
u1 j−1 u1j
u2j−1 u2j
Duj−1,j, j1, . . . , n. 2.3
LetifWuj2/0
Huij: Duj1,i
Wuj2
Duj1,i
Duj1,j2, i∈X, j−1,0,1, . . . , n−2. 2.4
We defineHi,n−1u Hinu 0,i∈X. Note thatHj1,j0,Hj2,j 1 forj∈X. IfuijP·uij, whereP pm
n∈M2×2 , then
IfWu/0 andP∈GL2 :{P∈M2×2 : detP/0}, then we getHu Hu. So, the functionHuijis invariant with respect to the basis{u1, u2}and we writeH
ij. Lemma 2.1. Ifw w1, w2∈F2X, then the equality
Dwik Dwjk Dwil Dwjl
Dwij·Dwkl, i, j, k, l∈X, 2.6
is valid.
Proof. If we takebm
1 wim,bm2 wjm,am1 wkm,am2 wlm,m 1,2, in 2.1, then we get equality2.6.
Corollary 2.2. Ifw w1, w2∈FX2, then the equality
WDw·k, Dw·li:Dwi−1,k Dwik
Dwi−1,l Dwil
Wwi·Dwkl, 2.7
k, l∈X,i1, . . . , nis valid.
We consider the space F∗X of linear functionals in the space FX, and we use the notation f, u, fk, u
k for the functional f value of the function u. Functionals δj,
j 0,1, . . . , nform a dual basis for basis{δi}n
i0. Thus,δj, u uj. Iff ∈F∗X,g ∈F∗Y, whereX{0,1, . . . , n}andY {0,1, . . . , m}, then we can define the linear functionaldirect productf·g∈F∗X×Y
fk·gl, wkl
: fk, gl, wkl
, wkl∈FX×Y. 2.8
We define the matrix
Mfw:
f, w1 g, w1
f, w2 g, w2
2.9
forf f, g,w w1, w2, and the determinant
Dfw: fk·gl, Dwkl
f, w1 g, w1
f, w2 g, w2
For example,
Df, δj
w fk·δjl, Dwkl
f, w1 w1 j
f, w2 w2 j
,
Dδi, δj
w δki ·δlj, DwklDwij,
Dfw, w0
i:Df, δi
w, w0 f·g·δi, D
w, w0
f, w1 g, w1 w1 i
f, w2 g, w2 w2 i
f, w0 g, w0 wi0
.
2.11
Let the functionsw1, w2∈FXbe linearly independent.
Lemma 2.3. Functionalsf, g are linearly independent on span{w1, w2} ⊂ FX if and only if
Dfw/0.
Proof. We can investigate the case where FX span{w1, w2}. The functionals f, g are linearly independent if the equality α1f α2g 0 is valid only for α1 α2 0. We can rewrite this equality asα1fα2g, w 0 for allw ∈span{w1, w2}. A system of functions
{w1, w2}is the basis of the span{w1, w2}, and the above-mentioned equality is equivalent to the condition below
α1
f, w1
f, w2
α2
g, w1
g, w2
α1fα2g, w1
α1fα2g, w2
0
0
. 2.12
Thus, the functionalsf,gare linearly independent if and only if the vectors
f, w1
f, w2
,
g, w1
g, w2
2.13
are linearly independent. But these vectors are linearly independent if and only if
f, w1 g, w1
f, w2 g, w2
/0. 2.14
IfffPf,wPww, wherePf,Pw∈M2×2 , then
D
fw detPw·Dfw·detPf, 2.15
Df, hw, w0detPw·Df, h
3. Special Basis in a Two-Dimensional Space of Solutions
Let us consider a homogeneous linear difference equation
Lu:a2iui2a1iui1a0iui0, i∈X, 3.1
wherea2, a0/0. LetS⊂FXa be two-dimensional linear space of solutions, and let{u1, u2}
be a fixed basis of this linear space. We investigate additional equations
L1, u0, L2, u0, u∈S, 3.2
whereL1, L2 ∈ S∗ are linearly independent linear functionals, and we use the notationL
L1, L2. We introduce new functions
v1i :Dδi, L2u, v2i :DL1, δiu. 3.3
For these functions Lm, vn δnmDLu, m, n 1,2, that is, vn ∈ KerLm for m /n. So, the functionv1 satisfies equationL2, u, and the functionv2 satisfies equationL1, u. Components of the functionsv1andv2in the basis{u1, u2}are
L2, u2
−L2, u1
,
−L1, u2
L1, u1
, 3.4
respectively. It follows that the functionsv1,v2are linearly independent if and only if
L2, u2
−L1, u2
−L2, u1 L1, u1
L1, u1 L2, u1
L1, u2 L2, u2
/0. 3.5
But this determinant is zero if and only ifDLu 0. We combine Lemma 2.3and these results in the following lemma.
Lemma 3.1. Let {u1, u2} be the basis of the linear space S. Then the following propositions are equivalent:
1the functionalsL1,L2are linearly independent;
2the functionsv1,v2are linearly independent;
3DLu/0.
If we takebm
1 umi ,b2mumj ,amn Ln, um,m, n1,2, in formula2.1, then we get
Dδi, L1u D
δj, L1
u
Dδi, L2u D
δj, L2
u
The left-hand side of this equality is equal to
Dδi, L2u D
δj, L2
u
DL1, δiu D
L1, δj
u
v
1 i v1j
v2i v2j
. 3.7
Finally, we havesee3.3
Dv Du·DLu. 3.8
Similarly we obtain
Wv Wu·DLu. 3.9
Lemma 3.2. Let {u1, u2} be a fundamental system of homogeneous equation3.1. Then equality
3.9is valid, and
Wv/0⇐⇒DLu/0. 3.10
Propositions inLemma 3.1are equivalent to the conditionWv/0.
Corollary 3.3. If functionalsL1,L2are linearly independent, that is,DLu/0, and
vi1: Dδi, L2u
DLu , v
2 i :
DL1, δiu
DLu , 3.11
that is,vv/DL, then the two bases{v1, v2}and{L
1, L2}are biorthogonal:
Lm, vnδmn, m, n1,2, 3.12
Dv Du
DL, Wv
Wu
DL, Hv Hu. 3.13
Remark 3.4. Propositions inLemma 3.1are valid if we take{v1, v2}instead of{v1, v2}.
Remark 3.5. If{u1, u2}is another fundamental system anduPu, whereP∈GL2 , then
Dδi, L2u
DLu
Dδi, L2u
DLu ,
DL1, δiu
DLu
DL1, δiu
DLu 3.14
see2.15. So, the definition ofv : v1, v2is invariant with respect to the basis{u1, u2}:
4. Discrete Difference Equation with Two Additional Conditions
Let{u1, u2}be the solutions of a homogeneous equation
Lu:a2iui2a1iui1ai0ui0, a2i, ai0/0, i∈X. 4.1
ThenDui·is the solution of4.1, that is,
a2iDui2,ja1iDui1,ja0iDuij0, i∈X, j ∈X. 4.2
For j i1, this equality shows that −a2
iWui2 a0iWui1 0, and we arrive at the conclusion that Wui ≡ 0 the case where {u1, u2} are linearly dependent solutions or
Wui/0 for alli1, . . . , nthe case of the fundamental system.
In this section, we consider a nonhomogeneous difference equation
Lu:a2iui2a1iui1a0iuifi, i∈X, 4.3
with two additional conditions
L1, ug1∈ , L2, ug2∈ , 4.4
whereL1,L2are linearly independent functionals.
4.1. The Solution to a Nonhomogeneous Problem with Additional
Homogeneous Conditions
A general solution of4.1isuC1u1C2u2, whereC1,C2are arbitrary constants and{u1, u2} is the fundamental system of this homogeneous equation. We replace the constantsC1,C2by the functionsc1, c2 ∈FX Method of Variation of Parameters12, respectively. Then, by substituting
uf;ic1;iu1i c2;iu2i, i∈X, 4.5
into4.3and denotingdki
2
l1cl;ik−cl;iulik,k −1,0,1,2,imax0,−k, . . . ,minn−
k, n 12, we obtain
fi 2
k0
akiuf;ik 2
k0
aki
2
l1
cl;ikulik 2
k0
akidki 2
k0
aki
2
l1
cl;iulik
2
k0
akidki 2
l1
cl;i
2
k0
akiulik
.
The functionsu1andu2are solutions of the homogeneous equation4.1. Consequently,
fi 2
k0
akidki, fori∈X. 4.7
Denoteblicl;i1−cl;i,l1,2. We derivek0,1,2
dki−dk−1,i1 2
l1
cl;ik−cl;iulik− 2
l1
cl;ik−cl;i1ulik 2
l1
bliulik,
2
k0
akidki−dk−1,i1 2
l1
bli 2
k0
akiulik0.
4.8
Then we rewrite equality4.7asd0i0 by definition
fi 2
k0
akidki 2
k0
akidk−1,i1a2id1,i1a0id−1,i1. 4.9
We can taked−1,i1 0,i 0, . . . , n−1. Thend1,i1 fi/a2i for alli∈ X, and we obtain the following systems:
b1,i1u1i1b2,i1u2i10,
b1,i1u1i2b2,i1u2i2
fi
a2 i
, i∈
X. 4.10
Sinceu1,u2are linearly independent, the determinantWuis not equal to zero and system 4.10has a unique solution
b1,i1c1;i2−c1;i1−
u2 i1fi
a2
iWui2
, b2,i1 c2;i2−c2;i1
u1 i1fi
a2
iWui2
. 4.11
Then
c1;i− i−2
j0
u2j1fj
a2
jWuj2
c1;1, c2;i i−2
j0
u1j1fj
a2
jWuj2
c2;1, i2, . . . , n, 4.12
and the formula for solution of nonhomogeneous equationwith the conditionsu0u1 0 is
ui i−2
j0
fj
a2
jWuj2
u1 j1 u1i
u2 j1 u2i
i−2
j0
Duj1,i Wuj2
fj
a2 j
i−2
j0
Hij
a2 j
fori2, . . . , n. We introduce a functionHθ∈FX×X:
Hijθ: θi−jHij
a2 j
, θi:
⎧ ⎨ ⎩
1 i >0,
0 i≤0.
4.14
Then we rewrite4.13and the conditionsu00,u10 as follows:
ui n−2
j0
Hijθfj
Hijθ, fj
X
Hi,θ·, f
X, i∈X, 4.15
wherew, gX wl, glX :
n−2
l0 wlgl,w, g ∈FX. So, we derive a formula for the general solutionui Hi,θ·, fXC1ui1C2u2i. We use this formula for the special basis{v1, v2}see
3.11. In this case, we have
ui
Hi,θ·, f
XC1v 1
i C2vi2, i∈X. 4.16
Let there be homogeneous conditions
L1, u0, L2, u0. 4.17
So, by substituting general solution4.16into homogeneous additional conditions, we find see3.12
C1− Lk1,
Hk,θ·, f
X
− Lk1, Hk,θ·, f
X,
C2− Lk2,
Hk,θ·, f
X
− Lk2, Hk,θ·, f
X.
4.18
Next we obtain a formula for solution in the case of difference equation with two additional homogeneous conditions
uf;i
Hi,θ·, f
X−v 1
i Lk1, Hk,θ·
, f
X−v 2
i Lk2, Hk,θ·
, f
X
δik−Lkvi, Hk,θ·
, f
X,
4.19
wherev1i Dδi, L2/DL,vi2DL1, δi/DL,vi v1i, v2i,Lk L1k, Lk2,i, k∈X,Lkvi :
Lk
1v1i Lk2vi2.
4.2. A Homogeneous Equation with Additional Conditions
Let us consider the homogeneous equation4.1with the additional conditions4.4
We can find the solution
u0;ig1·v1i g2·v2i, i∈X, 4.21
to this problem if the general solution is inserted into the additional conditions.
The solution of nonhomogeneous problems is of the formuiuf;iu0;isee4.19and
4.21. Thus, we get a simple formula for solving problem4.3-4.4.
Theorem 4.1. The solution of problem4.3-4.4can be expressed by the formula
ui δik−Lkvi, Hk,θ·
, f
Xg1·v 1
i g2·v2i, i∈X. 4.22
Formula4.22can be effectively employed to get the solutions to the linear difference equation, with variousa0,a1,a2, any right-hand side functionf, and any functionalsL
1,L2 and anyg1,g2, provided that the general solution of the homogeneous equation is known. In this paper, we also use4.22to get formulae for Green’s function.
4.3. Relation between Two Solutions
Next, let us consider two problems with the same nonhomogeneous difference equation with a difference operator as in the previous subsection
Luf, Lvf,
lm, ufm, m1,2, Lm, vFm, m1,2,
4.23
andDL/0. The differencewv−usatisfies the problem
Lw0,
Lm, wFm− Lm, u, m1,2.
4.24
Thus, it follows from formula4.21that
wi F1− L1, uv1i F2− L2, uv2i, i∈X, 4.25
or
viui F1− L1, u
Dδi, L2
DL F2− L2, u
DL1, δi
DL , i∈X, 4.26
Corollary 4.2. The relation
vi 1
DLu
L1, u1
L2, u1
u1i
L1, u2
L2, u2
u2i
L1, u −F1 L2, u −F2 ui
, i∈X, 4.27
between the two solutions of problems4.23is valid.
Proof. If we expand the determinant in4.27according to the last row, then we get formula 4.26.
Remark 4.3. The determinant in formula4.27is equal to
L1, u1 L2, u1
u1 i
L1, u2 L2, u2
u2 i
L1, u L2, u ui
−
L1, u1 L2, u1
u1 i
L1, u2 L2, u2
u2 i
F1 F2 0
. 4.28
In this way, we can rewrite4.27as
vi DL, δiu, u
DLu
F1Dδi, L2u F2DL1, δiu
DLu , i∈X. 4.29
Note that in this formula the functionuis in the first term only andviis invariant with regard to the basis{u1, u2}.
5. Green’s Functions
5.1. Definitions of Discrete Green’s Functions
We propose a definition of Green’s functionsee9,12. In this section, we suppose that andXn : X {0,1, . . . , n}. LetA : FXn → FXn−m ImAbe a linear operator,
0 ≤ m ≤ n. Consider an operator equation Au f, whereu ∈ FXn is unknown and
f∈FXn−mis given. This operator equation, in a discrete case, is equivalent to the system of linear equations
n
i0
ajiuifj, j 0,1, . . . , n−m, 5.1
that is,Auf, whereu∈
n1,f∈
n−m1,A a
ji∈Mn1×n−m1, rankAn−m1.
We have dim KerAm. In the casem >0, we must add additional conditions if we want to get a unique solution. Let us addM−nmhomogeneous linear equations
n
i0
whereB bji∈Mn1×M−nm, rankBM−nm, and denote
aji:
⎧ ⎨ ⎩
aji, j0,1, . . . , n−m,
bj−nm,i, jn−m1, . . . , M,
i∈Xn,
fj:
⎧ ⎨ ⎩
fj, j0,1, . . . , n−m,
0, jn−m1, . . . , M.
5.3
We have a system of linear equations Au f, where f fj ∈ Mn1×1, A
aji∈Mn1×M1. The necessary condition for a unique solution isM ≥n. Additional
equations 5.2 define the linear operator B : FXn → FXM−nm and the additional operator equationBu0, and we have the following problem:
Auf, Bu0. 5.4
If solution of5.4allows the following representation:
ui n−m
j0
Gijfj, i∈Xn, 5.5
thenG∈FXn×Xn−mis calledGreen’s functionof operatorAwith the additional condition
Bu0. Green’s function exists if KerA∩KerB{0}. This condition is equivalent to detA/0 forMn. In this case, we can easily get an expression for Green’s function in representation 5.5from the Kramer formula or from the formula foru A−1f. IfA−1 gij, thenGij
gij fori ∈ Xn,j ∈ Xn−m andAG E,BG O, where G Gij ∈ Mn1×n−m1 or
n
k0aikGkj δji,i∈Xn−m,
n
k0bikGkj 0,i∈Xm,j ∈ Xn−m. So,G0j, . . . , Gnjis a unique solution of problem5.4withfj δ0j, . . . , δnj,j ∈Xn−m.
Example 5.1. In the casem2, formula5.5can be written as
ui n−2
j0
Gijfj
Gi,·, f
X, i∈Xn. 5.6
The functionHθ∈FX×Xis an example of Green’s function for4.3with discreteinitial conditionsu0u10. In the casem2, formula5.6is the same as4.15,X Xn−2.
Remark 5.2. Let us consider the casem 2. Iffi fi1, where the functionf is defined on
X:{1,2, . . . , n−1}, then we use the shifted Green’s functionG∈FX×X
ui n−1
j1
For finite-difference schemes, discrete functions are defined in pointsxi ∈ 0, Land
fifxi. In this paper, we introduce meshes
ωh{0x0< x1<· · ·< xnL},
ωhωh\ {x0, xn}, ωhωh\ {xn−1, xn}
5.8
with the step sizeshixi−xi−1, 1≤i≤n,h0hn10, and a semi-integer mesh
ω1/2h xi1/2|xi1/2
xixi1
2 , 0≤i≤n−1
5.9
with the step sizeshi1/2 hihi1/2, 0≤i≤n. We define the inner product
U, Vωh : n
i0
UiVihi1/2, 5.10
whereU, V ∈Fωh, and the following mesh operators:
δZi1/2 Zi1−Zi
hi1 , Z∈F
ωh, δZi Zi1/2−Zi−1/2
hi1/2 , Z∈F
ωh1/2. 5.11
IfA:Fωh → Fωandf∈Fω, whereωωh, ωh,ωh, then we define the Green’s functionG∈Fωh×ω
ui
j:xj∈ω
Gijfj, i∈Xn. 5.12
For many applications another discrete Green’s functionGhis used9,11
ui n
j0
Ghijfjhj1/2
Ghi,·, f
ωh, i∈Xn, 5.13
wherefj0 forxj∈ωh\ω. The relations between these functions are
Ghij Gij
hj1/2 forj:xj∈ω, G h
ij0 forj:xj∈ωh\ω. 5.14
Note that the Wronskian determinant can be defined by the following formulasee 10:
Whuj
u
1
j−1 u2j−1
δu1j−1/2 δu2j−1/2
u1j−1 u2j−1 u1j−u1j−1
hj
u2j−u2j−1 hj
Wuj hj
, j 1, . . . , n. 5.15
5.2. Green’s Functions for a Linear Difference Equation with Additional
Conditions
Let us consider the nonhomogeneous equation 4.3with the operator: L : U → FX, where additional homogeneous conditions define the subspaceU {u ∈ FX : L1, u 0,L2, u0}.
Lemma 5.3. Green’s function for problem 4.3 with the homogeneous additional conditions
L1, u0,L2, u0, where functionalsL1andL2are linearly independent, is equal to
Gij
DL, δi
u, Hθ ·,j
DLu , i∈X, j∈X. 5.16
Proof. In the previous section, we derived a formula of the solution see Theorem 4.1 for
g1, g20
ui δki −Lkvi, Hk,θ·
, f
X, i∈X, 5.17
wherevi1Dδi, L2/DL,v2i DL1, δi/DL. So, Green’s function is equal to
Gij δki −Lkvi, Hkjθ
Hθ
ij− Lk1, Hkjθ
Dδ
i, L2
DL − L
k 2, Hkjθ
DL
1, δi
DL . 5.18
We have
DL, δi
u, H·θ,j
L1, u1
L2, u1
u1 i
L1, u2
L2, u2
u2 i
Lk 1, Hkjθ
Lk 2, Hkjθ
Hθ ij
, 5.19
too. If we expand this determinant according to the last row and divide byDLu, then we get the right-hand side of5.18. The lemma is proved.
IfuPu, whereP∈GL2, then we get that Green’s functionGij Gu
For the theoretical investigation of problems with NBCs, the next result about the relations between Green’s functionsGuijandGvijof two nonhomogeneous problems
Luf, Lvf,
lm, u0, m1,2, Lm, v0, m1,2,
5.20
with the samef, is useful.
Theorem 5.4. If Green’s functionGuexists and the functionalsL
1andL2are linearly independent, then
Gvij
DL, δi
u, Gu ·,j
DLu , i∈X, j∈X. 5.21
Proof. We have equality4.26 the caseF1, F20
vu− L1, uv1− L2, uv2. 5.22
Ifui Gui,·, fX, then
viui− n
k0
ukLk1v1i − n
k0
ukLk2vi2
Gui,·−
n
k0
Guk,·Lk1v1i −
n
k0
Guk,·Lk2vi2, f
X
. 5.23
So, Green’s functionGvis equal to
Gv
ijGuij− n
k0
Gu kjLk1v1i −
n
k0
Gu kjLk2vi2
δk
i −Lk1vi1−Lk2v2i, Gukj
δk
i −Lkvi, Gukj
.
5.24
A further proof of this theorem repeats the proof ofLemma 5.3we haveGuinstead ofHθ.
Remark 5.5. Instead of formula5.18, we have
Gvij Guij− Lk1, GukjDδi, L2 DL − L
k 2, Gukj
DL1, δi
DL . 5.25
We can write the determinant in formula5.21in the explicit way
Gvij DL, δi
u, Gu·,j
DLu
1
DLu
L1, u1
L2, u1
u1i
L1, u2
L2, u2
u2 i
Lk1, Gukj Lk2, Gukj Guij
Formulaes5.25and5.26easily allow us to find Green’s function for an equation with two additional conditions if we know Green’s function for the same equation, but with other additional conditions. The formula
ui
Gi,·, f
Xg1v 1
i g2v2i, i∈X 5.27
can be used to get the solutions of the equations with a difference operator with any two linear additionalinitial or boundary or nonlocal boundaryconditions if the general solution of a homogeneous equation is known.
6. Applications to Problems with NBC
Let us investigate Green’s function for the problem with nonlocal boundary conditions
Lu:a2iui2a1iui1a0iuifi, i∈X, 6.1
L1, u:κ0, u −γ00, u0, 6.2
L2, u:κ1, u −γ11, u0. 6.3
We can write many problems with nonlocal boundary conditionsNBCin this form, where
κm, u:κim, ui,m0,1, is a classical part andm, u:
i
m, ui,m0,1, is a nonlocal part of boundary conditions.
If γ0, γ1 0, then problem 6.1–6.3 becomes classical. Suppose that there exists Green’s function Gclij for the classical case. Then Green’s function exists for problem6.1– 6.3ifϑDLu/0. ForLmκm−γmm,m0,1, we derive
ϑDκ0·κ1u−γ0D0·κ1u−γ1Dκ0·1u γ0γ1D0·1u. 6.4
Sinceκkm, Gclkj0,m0,1, we can rewrite formula5.26as
Gij Gclijγ0vi1
k 1, Gclkj
γ1v2i
k 2, Gclkj
Gclijγ0
k 1, Gclkj
Dδ
i, L2
ϑ γ1
k 2, Gclkj
DL
1, δi
ϑ
1
ϑ
L1, u1
L2, u1
u1 i
L1, u2
L2, u2
u2i
−γ0
k 1, Gclkj
−γ1
k 2, Gclkj
Gcl ij
.
6.5
Example 6.1. Let us consider the differential equation with two nonlocal boundary conditions
−ufx, x∈0,1,
u0 γ0uξ0, u1 γ1uξ1, 0< ξ0, ξ1 <1.
We introduce a meshωh see5.8. Denoteui uxi,fi fxiforxi ∈ ωh. Then problem6.6can be approximated by a finite-difference problemscheme
−δ2uifi, xi∈ωh, 6.7
u0γ0us0, unγ1us1. 6.8
We suppose that the pointsξ0,ξ1are coincident with the grid points, that is,ξ0xs0,ξ1xs1.
We rewrite6.7in the following form:
a2iui2a1iui1a0iui fi1, i∈X, 6.9
where
a2i − 1
hi2hi3/2, a 1 i
2
hi2hi1, a 0 i −
1
hi1hi3/2, i∈
X. 6.10
We can take the following fundamental system:u1
i 1,u2i xi. Then
Duij u
1 i u1j
u2 i u2j
1 1
xi xj
xj−xi, i, j∈X, Wjhj, j1, . . . , n,
Hij
xi−xj1
hj2
, j−1,0,1, . . . , n−2, Hi,n−1Hin0, i∈X.
6.11
As a result, we obtain
Hijθ θi−jHij a2
j
θi−j
xj1−xi
hj3/2. 6.12
For a problem with the boundary conditionsu0un0 we haveDLu 1,
DL, δi
u, H·θ,jHijθ−xiHnjθ
θi−j
xj1−xi
hj3/2−θn−jxi
xj1−1
hj3/2,
6.13
and we express Green’s functionGclof the Dirichlet problem via Green’s functionHθof the initial problem
We derive expressions for “classical” Green’s function
Gclij hj3/2
θi−j
xj1−xi
θn−jxi
1−xj1
hj3/2
⎧ ⎨ ⎩
xi
1−xj1
, i≤j1,
xj11−xi, i≥j1,
i∈X, j∈X
6.15
orsee5.7and5.13
Gclij hj1/2
⎧ ⎨ ⎩
xi
1−xj
, xi ≤xj,
xj1−xi, xi≥xj,
i∈X, j∈X
Gcl,hij
⎧ ⎨ ⎩
xi
1−xj
, 0≤xi≤xj ≤1,
xj1−xi, 0≤xj≤xi≤1,
i, j∈X.
6.16
Remark 6.2. Note that the index offon the right-hand side of6.9is shiftedcf.6.1.
Green’s functionGcl,his the same as in10, and it is equal to Green’s function
Gclx, y
⎧ ⎨ ⎩
x1−y, 0≤x≤y≤1,
y1−x, 0≤y≤x≤1
6.17
for differential problem6.6at grid points in the caseγ0γ10.
For a “nonlocal” problem with the boundary conditionsu0γ0us0,unγ1us1,
ϑ:DLu L1,1 L2,1
L1, x L2, x
1−γ0·1 1−γ1·1
x0−γ0xs0 xn−γ1xs1
1−γ0 1−γ1
−γ0ξ0 1−γ1ξ1
1−γ01−ξ0−γ1ξ1γ0γ1ξ1−ξ0.
6.18
It follows from6.5that
Ghij Gcl,hij γ0
1−xiγ1xi−ξ1
ϑ G
cl,h s0j γ1
xi−γ0xi−ξ0
ϑ G
cl,h
ifϑ /0. Green’s function does not exist forθ0. By substituting Green’s functionGcl,hfor the problem with the classical boundary conditions into the above equation, we obtain Green’s function for the problem with nonlocal boundary conditions
Ghij
⎧ ⎨ ⎩
xi
1−xj
, xi ≤xj,
xj1−xi, xi≥xj,
γ0
1−xiγ1xi−ξ1 1−γ01−ξ0−γ1ξ1γ0γ1ξ1−ξ0
⎧ ⎨ ⎩
ξ0
1−xj
, ξ0≤xj,
xj1−ξ0, ξ0≥xj,
γ1
xi−γ0xi−ξ0
1−γ01−ξ0−γ1ξ1γ0γ1ξ1−ξ0
⎧ ⎨ ⎩
ξ1
1−xj
, ξ1≤xj,
xj1−ξ1, ξ1≥xj.
6.20
This formula corresponds to the formula of Green’s function for differential problem6.6
see4
Gx, y
⎧ ⎨ ⎩
x1−y, x≤y,
xj1−x, x≥y,
γ0
1−xγ1x−ξ1
1−γ01−ξ0−γ1ξ1γ0γ1ξ1−ξ0
⎧ ⎨ ⎩
ξ0
1−y, ξ0 ≤y
xj1−ξ0, ξ0 ≥y,
γ1
x−γ0x−ξ0
1−γ01−ξ0−γ1ξ1γ0γ1ξ1−ξ0
⎧ ⎨ ⎩
ξ1
1−y, ξ1 ≤y,
xj1−ξ1, ξ1 ≥y.
6.21
Example 6.3. Let us consider the problem
−ufx, x∈0,1,
u0 γ0
1
0
α0xuxdx, u1 γ1
1
0
α1xuxdx,
6.22
whereα0, α1 ∈L 10,1.
Problem6.22can be approximated by the difference problem
−δ2uifi, xi∈ωh,
u0 γ0
A0, u
K, unγ1
A1, u
K,
6.23
The expression of Green’s function for the problem with the classical boundary conditionsγ0 γ1 0,u1i 1,u2i xiis described inExample 6.1. The existence condition of Green’s function for problem6.23isϑ /0, where
ϑDLu
1−γ0
A0,1
K 1−γ1
A1,1 K
−γ0
A0, x
K 1−γ1
A1, x K
1−γ0
A0,1−x
K−γ1
A1, x
Kγ0γ1
A0,1−x K
A0, x K
A1,1−x K
A1, x K
6.24
such a condition was obtained for problem6.23in15,16and Green’s function is equal toseeTheorem 5.4
GijGclij
γ0
1−xiγ1
xi
A1,1 K−
A1, x K
A0, Gcl ·,j K ϑ γ1
xi−γ0
xi
A0,1 K−
A0, x K
A1, Gcl ·,j
K
ϑ ,
6.25
whereGclijis defined by6.15.
Green’s function for differential problem6.22was derived in8. For this problem
ϑ1−γ0
1
0
α0x1−xdx−γ1
1
0
α1xx dx
−γ0γ1
1
0
α0xα1
yx−ydx dy,
Gx, yGclx, y γ0
1−xγ1
1
0α1tx−tdt
ϑ ·
1
0
α0tG cl
t, ydt
γ1
x−γ0
1
0α0tx−tdt
ϑ ·
1
0
α1tG cl
t, ydt
6.26
ifϑ /0, whereGclx, yis defined by formula6.17.
Remark 6.4. We could substitute6.15into6.25and obtain an explicit expression of Green’s function. However, it would be quite complicated, and we will not write it out. Note that, ifA0, uK us0,A
1, u
K us1, then discrete problem6.23is the same as6.7-6.8. For
example, it happens if a trapezoidal formula is used for the approximationαl,l0,1 and we takeAl
i δ sl
i /hsl1/2. It is easy to see that we could obtain the same expression for Green’s
Example 6.5. Let us consider a difference problem
−δ2ui fi, xi∈ωh,
u0α0u1γ0un−1, unα1u1γ1un−1.
6.27
A condition for the existence of the Green’s functionfundamental system{1−x, x}is
ϑ:DLu 1−α01−h1−γ0hn −α11−h1−γ1hn
−α0h1−γ01−hn 1−hn−α1h1−γ11−hn
1−α0 γ0
α1 1−γ1
h1
α0 1−γ0
α1 1−γ1
hn
1−α0 γ0
1−α1 γ1
/0.
6.28
We consider three typesα0 γ1 0, γ0 α1 1; α0 α1 1h1/hn−1, γ0 γ1
1hn/h1−1;α00,γ01,α1hn/h1,γ1 1−hn/h1of discrete boundary conditions
u0 un−1, u1un,
u0un, δu1/2δun−1/2,
u0un−1, δu1/2δun−1/2.
6.29
All the cases yield ϑ 0. Consequently, Green’s function for the three problems does not exist.
7. Conclusions
Green’s function for problems with additional conditions is related with Green’s function of a similar problem, and this relation is expressed by formulae5.26. Green’s function exists if
ϑDLu/0. If we know Green’s function for the problem with additional conditions and the fundamental basis of a homogeneous difference equation, then we can obtain Green’s function for a problem with the same equation but with other additional conditions. It is shown by a few examples for problems with NBCs that but formulae5.26can be applied to a very wide class of problems with various boundary conditions as well as additional conditions.
All the results of this paper can be easily generalized to then-order difference equation withnadditional functional conditions. The obtained results are similar to a differential case 8,17.
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