J. Nonlinear Sci. Appl. 3 (2010), no. 3, 164–178
The Journal of Nonlinear Sciences and Applications
http://www.tjnsa.com
REDUCTION OF AN OPERATOR EQUATION IN TO AN
EQUIVALENT BIFURCATION EQUATION THROUGH
SCHAUDER’S FIXED POINT THEOREM
PALLAV KUMAR BARUAH1∗ , B V K BHARADWAJ2 AND M VENKATESULU3
This paper is dedicated to Bhagawan Sri Sathya Sai Baba.
Abstract. In this paper we deal with the Nonlinear Coupled Ordinary Differential Equations(Nonlinear CODE). A Multipoint Boundary Value Problem(MBVP) associated with these Nonlinear Equations is defined as an Operator Equation. This equation(infinite dimensional) is reduced to an Equivalent
Bifurcation Equation(finite dimensional) using Schauder’s Fixed Point Theorem. This Bifurcation Equation being on a finite dimensional space can be
easily solved by using standard approximation techniques.
1. Introduction and preliminaries
A lot of work has been done on the Differential Operator and the associated
Boundary Value Problems. Coupled Systems are very common in nature. Many
real life situations can be modeled using these coupled differential equations.
Some applications of such equations can be found in the prey-predator models
[14] where the interdependence of the dependent variables is clearly seen. So there
is a need for a unified theory for these Coupled Ordinary Differential Equations.
There are various methods to analyse differential equations and show the existence
of their solutoins, one such is through the fixed point theorems. Some papers in
literature dealing with operators and fixed point theorems are [1, 2, 13]. We
study these equations through a Functional Analytic approach. This approach
was first given by Kantorovich [8]. Cesari and Locker [9, 10, 3, 4] have adopted
Date: Received: 26, Feb., 2010.
∗
Corresponding author
c 2010 N.A.G.
°
2000 Mathematics Subject Classification. 34G20, 34L30, 34B05.
Key words and phrases. Coupled Differential Operator, Nonlinear Operator, Hilbert Space.
164
REDUCTION OF OPERATOR EQUATION
165
this approach for studying the nonlinear Differential Equations and the associated
BVPs. It is very useful to study the linear equations first before embarking on the
nonlinear equations. So we have studied the Linear Coupled Ordinary Differential
Equations and the associated BVPs in [11] and [12]. P C Das and Venkatesulu
[5],[6],[7] have studied a MBVP associated with Nonlinear Ordinary Differential
Equations using the alternative method given by Cesari [3]. We adopted the same
approach to study the Nonlinear Coupled Ordinary Differential Equations and
the associated MBVP.
First we have given few mathematical preliminaries useful in the study. We
define the Projections Pm , Qm and the right inverse H. Then we prove some
relations involving H, L, Pm and Qm . We gave an integral representation for H
and H(I − Pm ). In the succeeding sections we define the MBVP associated with
Nonlinear CODE as an operator equation. Then we reduce this operator equation
(on an infinite dimensional space) in to an equivalent Bifurcation Equation (on a
finite dimensional space) using the Schauder’s Fixed Point Theorem as in [6].
1.1. Mathematical Preliminaries. In this section we have given some preliminary definitions about the setting in which we are dealing the problem. We
have also defined the formal coupled differential operator, maximal and the minimal operator and stated some proven results, which characterize these operators.
Definition 1.1. We denote by L2 [a, b]×L2 [a, b] a real or complex Hilbert space of
ordered pairs(formed by the cartesian product) of all square integrable functions
with inner product defined as
Rb
Rb
({u1 , u2 }, {v1 , v2 }) = (u1 , v1 )1 + (u2 , v2 )1 = a u1 (t)v1 (t)dt + a u2 (t)v2 (t)dt
and norm
µ
¶1/2
2
2
k{u1 , u2 }k = ku1 (t)k1 + ku2 (t)k1
and convergence is denoted by {u1 , u1 }k → {u1 , u1 } where (., .)1 and
k . k1 denote the inner product and norm in L2 [a, b] defined as
Z b
(u1 , u2 ) =
u1 (t)u2 (t)dt
a
n
Definition 1.2. we denote by H [a, b], the subspace of L2 [a, b], consisting of
all functions u ∈ C n−1 [a, b], with u(n−1) absolutely continuous on [a,b] and with
u(n) ∈ L2 [a, b]. Also we let
H 0 [a, b] = L2 [a, b].
We denote by H n [a, b] × H m [a, b] the cartesian product of H n [a, b] and H m [a, b],
taken in that order.(i.e., It consists of all ordered pairs of the form {u, v}, where,
u ∈ H n [a, b] and v ∈ H m [a, b]) The inner product and norm are induced from the
space L2 [a, b] × L2 [a, b]
Definition 1.3. A formal Coupled Ordinary Differential Operator(CODO) of
order (n, m,
b n
b, m) on the interval [a, b] is an operator of the form:
(L1 M1 ; L2 M2 )({f, g}) = {h1 , h2 },
166
PALAV BARUAH, B V K BHARADWAJ AND M VENKATESULU
such that
L1 (f ) + M1 (g) = h1
L2 (f ) + M2 (g) = h2
n
m
where {f, g} ∈ H [a, b] × H [a, b] , {h1 , h2 } ∈ L2 [a, b] × L2 [a, b]
L1 , M1 , L2 , M2 are formal differential operators of orders n, m,
b n
b, m respectively
on the interval [a, b], and are of the form
P
P
L = n p (t)( d )i , M = m̂ q̂ (t)( d )i
1
i=0
Pn̂
i
dt
d i
i=0 p̂i (t)( dt )
1
i=0
Pm
i
dt
d i
L2 =
, M2 = i=0 qi (t)( dt
)
where all coefficients
pi (i = 0, 1, 2, ......, n), qbi (i = 0, 1, 2, ......, m),
b pbi (i = 0, 1, 2, ......, n
b)
and qi (i = 0, 1, 2, ......, m) belong to L2 [a, b] and t ∈ [a, b]
Definition 1.4. If L be a (n, n, n, n) ordered coupled ordinary differential operator it is said to be invertible iff L is one to one and on to L2 [a, b] × L2 [a, b].
The operator (L1 M1 , L2 M2 ) is linear and closed.
The proof of the statement can be concluded from the actual definitions of the
linearity and the closedness of the operator and some of the other properties of
it.
Assumption 1:
n≥n
b, m ≥ m,
b n ≥ m,
b m≥n
b.
Theorem 1.5. Basic Existence-Uniqueness Theorem for IVP :
Let assumption 1 be true. Also, assume that there exist positive constants α and
β such that |pn (t)| ≥ α and |qm (t)| ≥ β for all t ∈ [a, b]. Let h1 , h2 belong to
L2 [a, b], let to ∈ [a, b] and let c0 , c1 ........, cn−1 , cn , ......, cn+m−1 be any arbitrary
set of (n + m) scalars. Then there exists a unique pair of functions {(f, g)} ∈
H n [a, b] × H m [a, b] such that
L1 (f ) + M1 (g) = h1 , L2 (f ) + M2 (g) = h2
(1.1)
a.e., on [a, b] and satisfying the initial conditions
f (i) (t0 ) = ci , i = 0, ....., n − 1
(1.2)
g (i) (t0 ) = cn+i , i = 0, ....., m − 1
(1.3)
and
For the next results, we assume the following:
Assumption 2:
(L1 M1 ; L2 M2 ) is a formal coupled ordinary differential equation of order (n, m,
b n
b, m)
on [a, b], such that the following conditions are satisfied.
Suppose that the coefficients, pi , (i = 0, 1, 2, ...., n), qbi , (i = 0, 1, 2, ...., m),
b pbi ,
(i = 0, 1, 2, ...., n
b), qi , (i = 0, 1, 2, ...., m) satisfy the following conditions:
pi ∈ H i [a, b],i = 0, 1, ...., n, q̂i ∈ H i [a, b], i = 0, 1, ...., m
b
i
i
p̂i ∈ H [a, b], i = 0, 1, ...., n
b , qi ∈ H [a, b], i = 0, 1, ...., m
REDUCTION OF OPERATOR EQUATION
167
Now we have a definition.
Definition 1.6. Let (L1 M1 ; L2 M2 ) be a formal coupled ordinary differential operator of order (n, m,
b n
b, m) on [a, b], such that assumptions 1 and 2 are satisfied.The
formal coupled ordinary differential operator
(L1 M1 ; L2 M2 )∗ = (L∗1 L∗2 ; M1∗ M2∗ )
is called the formal adjoint of (L1 M1 ; L2 M2 ) where
P
d j
) where p∗j ∈ H j [a, b]f orj = 0, 1, 2, ....n.
L∗1 = nj=0 p∗j (t)( dt
µ ¶
Pn
i
∗
i
with pj (t) = i=j (−1)
( d )i−j pi (t), j = 0, 1, 2, ....., n − 1.
j dt
Pm̂ ∗
d j
M1∗ = j=0
q̂j (t)( dt
) where q̂∗j ∈ H j [a, b]f orj = 0, 1, 2, .....m̂
µ ¶
Pm̂
i
∗
d i−j
i
( dt
with q̂j (t) = i=j (−1)
) (q̂i (t)), j = 0, 1, 2, ....., m̂ − 1.
j
Pn̂ ∗
d j
L∗2 = j=0
p̂j (t)( dt
) where p̂∗j ∈ H j [a, b]f orj = 0, 1, 2, .....n̂
µ ¶
Pn̂
i
∗
i
with p̂j (t) = i=j (−1)
( d )i−j (p̂i (t)), j = 0, 1, 2, ....., n̂ − 1.
j dt
P
d j
) where qj∗ ∈ H j [a, b]f orj = 0, 1, 2, ....m.
M2∗ = nj=0 qj∗ (t)( dt
µ ¶
Pm
i
∗
i
with qj (t) = i=j (−1)
( d )i−j (qi (t)), j = 0, 1, 2, ....., m − 1.
j dt
In case (L1 M1 ; L2 M2 ) = (L1 M1 ; L2 M2 )∗ , we say that (L1 M1 ; L2 M2 ) is formally
self-adjoint.
Now we define few norms which are used throughout this paper.
Definition 1.7.
k{x, x̂}kH n [a,b]×H n [a,b] =
√
b−a
n−1
X
¯©
ª¯
supt∈[a,b] ¯ xi , x̂i ¯ + k{xn , x̂n }k
i=0
¯©
ª¯
¢
µ({x, x̂}) = max maxi=0..n−2 supt∈[a,b] ¯ xi , x̂i ¯ , ess.supt∈[a,b] |{xn , x̂n }|
¡
Let τ be the formal coupled differential operator defined above.
For each j = 1, 2, ..., k we define,
Bj {x1 , x2 } =
n−1
2 X
X
¡
¢
l
l
l
α0ji
xil (a) + α1ji
xil (a1 ) + ... + αhji
xil (b)
(1.4)
l=1 i=0
where the interval [a, b] is divided as a = a0 ≤ a1 ≤ .... ≤ ah = b. We assume
that k ≤ n and all Bj s are linearly independent.
So let us define the Linear Coupled Ordinary Differential Operator L here as the
operator generated by the formal coupled ordinary differential operator and the
boundary conditions in the equation (1.4).
Then let the linear operators T1 (τ ) and T0 (τ ) be defined as follows
D(T1 (τ )) = {{x, y} ∈ H n [a, b] × H n [a, b]}
T1 (τ ) {x, y} = τ {x, y}
168
PALAV BARUAH, B V K BHARADWAJ AND M VENKATESULU
and
D(T0 (τ )) = {{x, y} ∈ H0n [a, b] × H0n [a, b]}
T0 (τ ) {x, y} = τ {x, y}
Clearly, T1 (τ ) is an extension of both T0 (τ ) and L.
Moreover, T1 (τ ) is the adjoint of the operator To (τ ∗ ) where τ ∗ represents the
formal adjoint of the operator τ .
Now let us recollect the following well known facts(Proved in [11]) about L:
D(L) is dense in L2 [a, b] × L2 [a, b].
L is a closed linear operator.
R(L) is closed in L2 [a, b] × L2 [a, b].
L2 [a, b] × L2 [a, b] = R(L) ⊕ N (L∗ ) where N (L∗ ) denotes the null space of the
adjoint of L.
We know thatnN (T1 (τ
o)) is n-dimensional with N (L) ⊆ N (T1 (τ )). Let us choose
function pairs φi , φbi , i = 1, 2, ..., n. in C ∞ [a, b] to form an orthonormal basis for
n
o
N (T1 (τ )) in such a way that φi , φbi , i = 1, 2, ..., p. forms an orthonormal basis
for N (L). we also chose {ωi , ωbi }, i = 1, 2, .., q in D(L∗ ) to form an orthonormal
basis for N (L∗ ).
We note that the operator L/D(L) ∩ N (L)⊥ is a one-to-one closed linear operator
having the same range as L. Now Let H denote the inverse of this operator.
n
o−1
H = L/D(L) ∩ N (L)⊥
By the closed graph theorem, H is a one-to-one continuous linear operator.
Clearly, D(H) = R(L), R(H) = L/D(L) ∩ N (L)⊥ .
Moreover,
LH {y1 , y2 } = {y1 , y2 }
for all {y1 , y2 } ∈ R(L)
and
HL {x1 , x2 } = {x1 , x2 } −
p ³
X
n
o´ n
o
{x1 , x2 } , φi , φbi
φi , φbi ,
i=1
for all {x1 , x2 } ∈ D(L)
Thus H is a continuous right inverse of L.
1.2. Projections Pm and Qm and their Relation with L and H. We assume
that there exist elements {ωq+1 , ωd
d
c
q+1 } , {ωq+2 , ω
q+2 } , ...., {ωm , ω
m } .... belonging to
∗
D(L ) such that the sequence of functions {ω1 , ω
c1 } , {ω2 , ω
c2 } , ....,
{ωq , ωbq } {ωq+1 , ωd
d
c
q+1 } , {ωq+2 , ω
q+2 } , ...., {ωm , ω
m } .... form a complete orthonormal set in L2 [a, b]×L2 [a, b]. Since L2 [a, b]×L2 [a, b] = R(L)⊕N (L∗ ), the elements
{ωq+1 , ωd
d
c
d
q+1 } , {ωq+2 , ω
q+2 } , ...., {ωm , ω
m } .... belong to R(L). Hence H {ωq+i , ω
q+i },
⊥
i ≥ 1 are definednand belong
to D(L) ∩ N (L) . Let So be the subspace spanned
o
b
by the elements φi , φi , i = 1, 2, ..., p. and H {ωq+i , ωd
q+i }, i = 1, 2, ..., m.
REDUCTION OF OPERATOR EQUATION
169
Now we define the sequences of projections Pm and Qm as
m
X
Pm {x, x
b} =
({x, x
b} , {ωi , ωbi }) {ωi , ωbi }
i=1
Qm {x, x
b} =
p ³
X
n
o´ n
o
{x, x
b} , φi , φbi
φi , φbi
i=1
Pm
+
i=q+1
({x, x
b} , L∗ {ωi , ωbi }) H {ωi , ωbi }
where m > q.
These operators Pm , Qm have the following properties.
Pm and Qm are continuous linear operators on all of L2 [a, b] × L2 [a, b].
R(Pm ) =< ω1 , ω2 , ....., ωm >.
R(Qm ) = S0 ⊂ D(L).
Pm2 = Pm and Q2m = Qm .
The range of (I − Pm ) is a subset of R(L) and H(I − Pm ) is a continuous linear
operator defined on all of S.
Proofs of these statements are very trivial from the definitions.
We now prove a theorem which we use in the subsequent discussions.
Theorem 1.8. The following results are valid:
(1) H(I − Pm )L {x1 , x2 } = (I − Qm ) {x1 , x2 } for all {x1 , x2 } ∈ D(L).
(2) LH(I − Pm ) {x1 , x2 } = (I − Pm ) {x1 , x2 } for all {x1 , x2 } ∈ S.
(3) LQm {x1 , x2 } = Pm L {x1 , x2 } for all {x1 , x2 } ∈ D(L).
(4) Qm H(I − Pm ) {x1 , x2 } = 0 for all {x1 , x2 } ∈ S.
Proof.
(1) Let {x1 , x2 } ∈ D(L). ThenP
(I − Pm ) {x1 , x2 } = L {x1 , x2 } − m
i=1 (L {x1 , x2 } , {ωi , ω̂i }) {ωi , ω̂i }
m
X
= L {x1 , x2 } −
({x1 , x2 } , L∗ {ωi , ω̂i }) {ωi , ω̂i }
i=1
m
X
= L {x1 , x2 } −
({x1 , x2 } , L∗ {ωi , ω̂i }) {ωi , ω̂i }
i=q+1
Therefore
H(I − Pm )L {x1 , x2 } = HL {x1 , x2 }
−
m
X
({x1 , x2 } , L∗ {ωi , ω̂i }) H {ωi , ω̂i }
i=q+1
= {x1 , x2 } −
p ³
X
n
o´ n
o
{x1 , x2 } , φi , φ̂i
φi , φ̂i
i=1
−
m
X
({x1 , x2 } , L∗ {ωi , ω̂i }) {ωi , ω̂i }
i=q+1
= (I − Qm ) {x1 , x2 }.
170
PALAV BARUAH, B V K BHARADWAJ AND M VENKATESULU
(2) Since (I − Pm ) {x1 , x2 } ∈ R(L) for all {x1 , x2 } ∈ L2 [a, b] × L2 [a, b], we
have
LH(I − Pm ) {x1 , x2 } = (I − Pm ) {x1 , x2 }
(3) Let {x1 , x2 } ∈ D(L). Then
Qm {x1 , x2 } =
p ³
X
i=1
m
X
n
o´ n
o
{x1 , x2 } , φi , φ̂i
φi , φ̂i +
({x1 , x2 } , L∗ {ωi , ω̂i }) H {ωi , ω̂i }
i=q+1
Therefore
LQm {x1 , x2 } =
p ³
X
n
o´ n
o
{x1 , x2 } , φi , φ̂i L φi , φ̂i +
i=1
m
X
({x1 , x2 } , L∗ {ωi , ω̂i }) LH {ωi , ω̂i }
i=q+1
=
m
X
({x1 , x2 } , L∗ {ωi , ω̂i }) LH {ωi , ω̂i }
i=q+1
Thus
LQm {x1 , x2 } =
m
X
({x1 , x2 } , L∗ {ωi , ω̂i }) LH {ωi , ω̂i }
i=1
=
m
X
(L {x1 , x2 } , {ωi , ω̂i }) {ωi , ω̂i }
i=1
= Pm L {x1 , x2 }
(4) Let {x1 , x2 } ∈ L [a, b] × L2 [a, b]. Then
Ã
!
m
X
H(I − Pm ) {x1 , x2 } = H {x1 , x2 } −
({x1 , x2 } , {ωj , ω̂j }) {ωj , ω̂j }
2
j=1
= H {x1 , x2 } −
m
X
({x1 , x2 } , {ωj , ω̂j }) H {ωj , ω̂j }
j=1
Therefore
Qm H(I − Pm ) {x1 , x2 } ³
´
Pm
({x
,
x
}
,
{ω
,
ω̂
})
H
{ω
,
ω̂
}
= Qm H {x1 , x2 } − Qm
1
2
j
j
j
j
j=1
n
o´ n
o P
Pp ³
m
φi , φ̂i + i=q+1 (H {x1 , x2 } , L∗ {ωi , ω̂i }) H {ωi , ω̂i }
= i=1 H {x1 , x2 } , φi , φ̂i
n
o´ n
o
³
Pm
Pp
φi , φ̂i
− i=1
j=1 ({x1 , x2 } , {ωj , ω̂j }) H {ωj , ω̂j } , φi , φ̂i
´
³P
Pm
m
∗
({x
,
x
}
,
{ω
,
ω̂
})
H
{ω
,
ω̂
}
,
L
{ω
,
ω̂
}
H {ωi , ω̂i }
− i=q+1
1
2
j
j
j
j
i
i
j=1
Pm
= i=q+1 (H {x1 , x2 } , L∗ {ωi , ω̂i }) H {ωi , ω̂i }
REDUCTION OF OPERATOR EQUATION
171
³P
´
P
m
− m
({x
,
x
}
,
{ω
,
ω̂
})
{ω
,
ω̂
}
,
{ω
,
ω̂
}
H {ωi , ω̂i }
1
2
j
j
j
j
i
i
i=q+1
j=1
Thus
Qm H(I − Pm ) {x1 , x2 } = 0(i.e {0, 0})
¤
Now we give an integral representation to H and H(I − Pm ).
The right inverse operator H, being a one-to-one continous linear operator has
an integral representation given by
Z
b
(H {y1 , y2 }) (t) =
K(t, s) {y1 , y2 }T
(1.5)
a
where K(t, s) is called the Generalise Green’s function.
Similarly H(I − Pm ) is also given the following representation
Z
b
(H(I − Pm ) {x1 , x2 }) (t) =
Km (t, s) {x1 (s), x2 (s)}T ds
(1.6)
a
where Km (t, s) be the function defined by
Km (t, s) =
K(t, s) −
n µZ
X
i=1
b
¶T
T
K(t, ξ) {ωi (ξ), ω̂i (ξ)} dξ
{ωi (s), ω̂i (s)} , a ≤ t, s ≤ b (1.7)
a
2. Main results
2.1. The MBVP, Assumptions and Inequalities. In this section we follow
the notations of the previous one. We deal with the the Nonlinear Multipoint
Boundary Value Problem(MBVP) L {x, x̂} = N {x, x̂}, where N is defined subsequently. Indeed we reduce the equation L {x, x̂} = N {x, x̂} to an equivalent
bifurcation equation by making use of the Schauder’s fixed point theorems.
We reduce the equation L {x, x̂} = N {x, x̂} under the following assumption
We assume the following for the rest of our study.
(1) L satisfies
all the assumptions of the previousosection.
n
(2) Let X(t, x0 , x1 , ..., xn−1 ), X̂(t, x̂0 , x̂1 , ..., x̂n−1 ) be a pair of nonlinear real
valued function defined on the interval [a, b].and |{xi , x̂i }| ≤ Ri , , i =
0,
n 1, ..., n − 1.
o
(3) X(t, x0 , x1 , ..., xn−1 ), X̂(t, x̂0 , x̂1 , ..., x̂n−1 ) ∈ L2 [a, b] × L2 [a, b] for each
fixed {xi , x̂i } satisfying |{xi , x̂i }| ≤ Ri , i = 0, 1, ..., n − 1.
(4) There exists a real number k0 ≥ 0n such othat for |{xi , x̂i }| ≤ Ri and
|{yi , ŷi }| ≤ Ri , the pair of functions X, X̂ satisfies the following
172
PALAV BARUAH, B V K BHARADWAJ AND M VENKATESULU
n
o
k X(t, x0 , x1 , ..., xn−1 ), X̂(t, x̂0 , x̂1 , ..., x̂n−1 )
n−1
n
o
X
− X(t, y0 , y1 , ..., yn−1 ), X̂(t, ŷ0 , ŷ1 , ..., ŷn−1 ) k ≤ k0 (
|{xi , x̂i } − {yi , ŷi }|)
i=0
(2.1)
, t ∈ [a, b].
We now define the operator N as following
Definition 2.1. D(N ) = {{x, x̂} ∈ H n−1 [a, b] × H n−1 [a, b] : supt∈[a,b] |{xi , x̂i }| ≤
Ri ,
n−1
i = 0, .., n − 2, ess.sup
, x̂n−1 }| ≤ Rn−1 }
t∈[a,b] |{x
n
o
N {x(t), x̂(t)} = X(t, x(t), x1 (t), ..., xn−1 (t)), X̂(t, x̂0 , x̂1 , ..., x̂n−1 )
for all t ∈
[a, b] for which |{xn−1 , x̂n−1 }| ≤ Rn−1
Clearly we can see that N {x, x̂} ∈ L2 [a, b] × L2 [a, b] for {x, x̂} ∈ D(N ).
Now we develop an existential theory for the Nonlinear multipoint boundary value
problem(NL MBVP)
L {x, x̂} = N {x, x̂}
(2.2)
Now we prove some inequalities useful in our analysis.
kN
n {x, x̂} − N {y, ŷ}k =
o
k X(., x(.), x1 (.), ..., xn−1 (.)), X̂(., x̂(.), x̂1 (.), ..., x̂n−1 (.))
n
o
1
n−1
1
n−1
− X(., y(.), y (.), ..., y (.)), X̂(., ŷ(.), ŷ (.), ..., ŷ (.)) k
°P
°
≤ k0 °( n−1 |{xi , x̂i } − {y i , ŷ i }|)°
i=0
≤ k0
¡Pn−1
i=0
¢
k{xi , x̂i } − {y i , ŷ i }k
√
P
i
i
i i
≤ k0 { b − a{ n−2
i=0 supt∈[a,b] |{x , x̂ } − {y , ŷ }|}+
k{xn−1 , x̂n−1 } − {y n−1 , ŷ n−1 }k}
≤ k0 k{x, x̂} − {y, ŷ}kH n [a,b]×H n [a,b]
where k.kH n [a,b]×H n [a,b] is the norm also on H n−1 [a, b] × H n−1 [a, b].
Thus for {x, x̂} , {y, ŷ} ∈ D(N ) we have
Let
kN {x, x̂} − N {y, ŷ}k ≤ k0 k{x, x̂} − {y, ŷ}kH n [a,b]×H n [a,b]
(2.3)
¶¯
Z b ¯µ i
¯ ∂ Km (t, s) ¯2
i
¯ ds) 21 , i = 0, 1, ..., n − 2
¯
ρm = (supt∈[a,b]
¯
¯
i
∂t
(2.4)
a
and
Z
ρn−1
m
=(
a
¶¯
Z b ¯µ n−1
¯ ∂ Km (t, s) ¯2
¯
¯ dsdt) 12
b
¯
¯
n−1
∂t
a
(2.5)
REDUCTION OF OPERATOR EQUATION
173
R b ¯¯³ i m (t,s) ´¯¯2
1
Noting that a ¯ ∂ K∂t
¯ ds) 2 is a monotone decreasing sequence for every t,
i
Dini’s theorem assures uniform convergence of this sequence to zeros m → ∞ for
→ 0 as
i = 0, 1, ..., n − 1. Thus ρim → 0 as m → ∞, i = 0, 1, ..., n − 1 and ρn−1
m
m → ∞.
So define
θm =
√
n−2
X
b − a(
ρim ) + ρn−1
m
(2.6)
i=0
θm = maxi=0,1,..,n−1 ρim
(2.7)
We observe that both θm , θm tend to zero as m → ∞. let X be the vector with
the
°R pair of functions° {x, x̂}. Then
° b
°
=
° a Km (., s)X(s)ds° n
n [a,b]
H [a,b]×H
¯
¯´ °R ³ n−1
°
³
³
´
´
√
Pn−2
° b ∂ Km (t,s)
¯
°
¯R b ∂ i Km (t,s)
X(s)ds
+
X(s)ds
b−a
sup
¯
°
°
¯
t∈[a,b]
i−0
∂ti
∂tn−1
a
a
Ã
!
1
¯ ³
¯
µ
¶
¯R b ∂ i Km (t,s) ´2 ¯ 2
√
Pn−2
≤ b−a
supt∈[a,b] ¯¯ a
ds¯¯ kXk
i=0
∂ti
µ
¶ 12
R b R b ³ ∂ n−1 Km (t,s) ´2
+ a a
dsdt kXk
∂tn−1
(by¡√
Schwartz
¡Pinequality)
¢
n−2 i ¢
n−1
b−a
=
kXk
i=0 ρm + ρm
= θm kXk
Hence for all X ∈ L2 [a, b] × L2 [a, b] we have
°Z b
°
°
°
°
Km (., s)X(s)ds°
°
° n
a
≤ θm kXk
(2.8)
H [a,b]×H n [a,b]
Also R
¯R i
¯
¯ b ∂ Km (t,s)
¯
b
µ( a Km (., s)X(s)ds) = maxi=0,1..n−1 supt∈[a,b] ¯ a
X(s)ds¯
∂ti
R b ¯¯ i m (t,s) ¯¯2
1
≤ maxi=0,1..n−1 (supt∈[a,b] a ¯ ∂ K∂t
¯ ds) 2 kXk
i
(by Schwartz inequality)
= (maxi=0,1,..,n−1 ρim ) kXk
(by it’s definition)
= θm kXk
Therefore for X ∈ L2 [a, b] × L2 [a, b] we have
Z b
µ(
Km (., s)X(s)ds) ≤ θm kXk
(2.9)
a
2.2. Construction of Sets V and S0 . Let us consider the Banach Space H n−1 [a, b]×
e n−1 [a, b] × H
e n−1 [a, b] is a linear manifold
H n−1 [a, b]. We observe that the set H
n−1
n−1
e n−1 [a, b] ×
of H [a, b] × H [a, b]. We remember the function µ defined on H
e n−1 [a, b].
H
We consider the p + m − q dimensional space S0 . Clearly S0 ⊂ D(L) ⊂ H n [a, b] ×
174
PALAV BARUAH, B V K BHARADWAJ AND M VENKATESULU
H n [a, b]. We chose {x0 , x̂0 } ∈ S0 such that β = µ({x0 , x̂0 }) < R where R =
mini=0,..,n−1 Ri . Here Ri s are the constants defined previously in the assumptions. Let {z0 , ẑ0 } = H(I − Pm )N {x0 , x̂0 }, and let e and e be real numbers such
that
k{z0 , ẑ0 }kH n [a,b]×H n [a,b] ≤ e, µ({z0 , ẑ0 }) ≤ e
(2.10)
Let c, d, r and R be real numbers such that
0 < c < d, 0 < r < R, c + e < d, R + β ≤ R, r + e < R
(2.11)
e n−1 [a, b] × H
e n−1 [a, b] are defined as follows:
The sets V and Se0 in H
Definition 2.2. V = {{x, x̂} ∈
S0 : k{x, x̂} − {x0 , x̂0 }kH n [a,b]×H n [a,b] ≤ c and µ({x, x̂} − {x0 , x̂0 }) ≤ r}
(2.12)
n
n
e
e
e
and S0 = {{x, x̂} ∈ H [a, b] × H [a, b] :
k{x, x̂} − {x0 , x̂0 }kH n [a,b]×H n [a,b] ≤ d and µ({x, x̂} − {x0 , x̂0 }) ≤ R}
(2.13)
We observe that V is a closed and bounded subset of L2 [a, b] × L2 [a, b]. Indeed, let {{yk , ŷk }} be any sequence contained in V and let {{yk , ŷk }} converge
to {y, ŷ} in the topology of L2 [a, b] × L2 [a, b].
Firstly, we notice that {{yk , ŷk }} ∈ H n [a, b] × H n [a, b]. Since S0 is finite dimensional, the element {y, ŷ} ∈ S0 . From the fact that the linear operators on a finite
dimensional space are bounded, it readily follows that {L {yk , ŷk }} converges to
L {y, ŷ} in the topology of S(i.e.L2 [a, b]×L2 [a, b]). Hence the sequence {{yk , ŷk }}
converges to {y, ŷ} in the topology of H n [a, b]×H n [a, b], which implies that the sequence {{yk , ŷk }} converges to {y, ŷ} in the topology of H n−1 [a, b]×H n−1 [a, b] and
µ ({yk , ŷk } − {y, ŷk }) → 0 as k → ∞. Hence k{y, ŷ} − {x0 , x̂0 }kH n [a,b]×H n [a,b] ≤ c
and
µ ({y, ŷ} − {x0 , x̂0 }) ≤ r. Thus {y, ŷ} ∈ V . Obviously V is a bounded subset of
L2 [a, b] × L2 [a, b]. Thus V is a closed and bounded subset of L2 [a, b] × L2 [a, b].
2.3. Operator T and sets A ({x∗ , x̂∗ }) and A. For each {x∗ , x̂∗ } ∈ V , let T be
the operator on Se0 defined by
Definition 2.3.
T {x, x̂} = {x∗ , x̂∗ } + H(I − Pm )N {x, x̂}
(2.14)
for {x, x̂} ∈ Se0 .
We observe that T is well defined on Se0 .
For each {x∗ , x̂∗ } ∈ V , the set A ({x∗ , x̂∗ }) is defined by
Definition 2.4.
n
o
A ({x∗ , x̂∗ }) = {x, x̂} ∈ Se0 : {x, x̂} = T {x, x̂}
(2.15)
REDUCTION OF OPERATOR EQUATION
175
We denote by A = ∪{x∗ ,x̂∗ }∈V A ({x∗ , x̂∗ }).
Suppose A ({x∗ , x̂∗ }) is non-empty. Then {x, x̂} = T {x, x̂} = {x∗ , x̂∗ } + H(I −
Pm )N {x, x̂} for some X ∈ Se0 .
Clearly {x, x̂} ∈ D(L) and by Theorem 1.8 we have Qm {x, x̂} = {x∗ , x̂∗ }
Thus
L {x, x̂} = LQm {x, x̂} + LH(I − Pm )N {x, x̂}
using parts of Theorem 1.8 we get
L {x, x̂} − N {x, x̂} = Pm (L {x, x̂} − N {x, x̂})
Hence {x, x̂} ∈ Se0 is a solution of the nonlinear BVP if it satisfies th equation
Pm (L {x, x̂} − N {x, x̂})
(2.16)
This equation is called the bifurcation equation of order m.
We notice that L {x, x̂} − N {x, x̂} = Pm (L {x, x̂} − N {x, x̂}) on the set A provided A is non-empty. In the following sections we show that A ({x∗ , x̂∗ }) is
non-empty. Thus the original MBVP will be reduced to the equivalent bifurcation equation (2.16) on the set A.
2.4. Reduction of the Original MBVP in to an Equivalent Bifurcation Equation using the Schauder’s Fixed Point Theorem. The following
lemma is needed for our discussions in this section.
Lemma 2.5. Consider the Banach Spaces H n [a, b] × H n [a, b] and H n−1 [a, b] ×
H n−1 [a, b]. Let {xm , x̂m } be a bounded sequence in the space H n [a, b]×H n [a, b].Then
this sequence has a subsequence which converges in the topology of H n−1 [a, b] ×
H n−1 [a, b].
Lemma 2.6. Let all the assumptions stated at the starting of this section and
conditions (2.10), (2.11) be satisfied. Then the operator T : Se0 → H n [a, b] ×
H n [a, b] is continuous.
Proof. From the aboveR Lemma we have
b
T {x, x̂} = {x∗ , x̂∗ } + a Km (., s) (N {x, x̂} (s)) ds for {x, x̂} ∈ D(N ).
Suppose x and y ∈ Se0R. Then
b
T {x, x̂} − T {y, ŷ} = a Km (., s) (N {x, x̂} (s) − N {y, ŷ} (s)) ds
So
¯
¯´
³P
√
¯
¯
n−1
i
n
b−a
sup
(T
{x,
x̂}
−
T
{y,
ŷ})
(t)
¯
¯ + k(T {x, x̂} − T {y, ŷ}) k
t∈[a,b]
i=0
¯
¯
´
³
√
Pn−1
¯R b ∂ i Km (t,s)
¯
sup
(N {x, x̂} (s) − N {y, ŷ} (s)) ds¯
= b−a
¯ a
t∈[a,b]
i=0
∂ti
°R n
°
° b
°
+ ° a ∂ K∂tmn(.,s) (N {x, x̂} (s) − N {y, ŷ} (s)) ds°
Ã
µ
¶ 12 !
√
R b ³ ∂ i Km (t,s) ´2
Pn−1
supt∈[a,b] a
≤ b−a
ds
kN {x, x̂} (s) − N {y, ŷ}k +
i=0
∂ti
µ
R b R b ³ ∂ n Km (t,s) ´2
a
a
∂tn
dsdt
¶ 21
kN {x, x̂} (s) − N {y, ŷ}k
This
from the ¢Schwartz’s
inequality
¢
¡√comes¡P
n−1 i
n
=
b−a
i=0 ρm + ρm kN {x, x̂} (s) − N {y, ŷ}k
176
PALAV BARUAH, B V K BHARADWAJ AND M VENKATESULU
whereµ
¶ 12
R b R b ³ ∂ n Km (t,s) ´2
n
ρm = a a
dsdt
∂tn
We clearly notice that ρnm → 0 as m → ∞.
Define¡√
¡Pn−1 i ¢
¢
n
γm =
b−a
ρ
+
ρ
m
m
i=0
Then γm → 0 as m → ∞.
Therefore
¯ inequality we get that ¯´
³Pfrom the above
√
¯
¯
n−1
i
b−a
i=0 supt∈[a,b] ¯(T {x, x̂} − T {y, ŷ}) (t)¯
+ kT {x, x̂} − T {y, ŷ}k
≤ γm kN {x, x̂} (s) − N {y, ŷ}k
≤ γm k0 k{x, x̂} (s) − {y, ŷ}kH n [a,b]×H n [a,b]
Hence the map T is continuous.
¤
Corollary 2.7. with all the assumptions in the previous theorem satisfied, the
map T : Se0 → H n−1 [a, b] × H n−1 [a, b] is continuous.
Proof. The proof of the corollary follows
¯ result and the fact that
¯´
³P from the above
√
¯
¯
n−1
i
kT {x, x̂}kH n−1 [a,b]×H n−1 [a,b] ≤ b − a
sup
(T
{x,
x̂}
−
T
{y,
ŷ})
(t)
¯
¯ +
t∈[a,b]
i=0
k(T {x, x̂} − T {y, ŷ})n k
¤
In addition to the previous assumptions, only for the present case we assume
that |{x, x̂}| ≤ K0 . Thus for {x, x̂} ∈ Se0 we have
|N {x, x̂}| ≤ k0 and hence
√
kN {x, x̂}k ≤ k0 b − a
(2.17)
Lemma 2.8. Suppose all the assumptions of this section and (2.10), (2.11),(2.17)
are satisfied. Then the set T (Se0 ) is relatively compact in H n [a, b] × H n [a, b].
Proof. Firstly we observe that T (Se0 ) is bounded in H n [a, b] × H n [a, b].
Indeed, let {x, x̂} ∈ Se0 . Then we have
Rb
T {x, x̂} = {x∗ , x̂∗ } + a Km (., s) [N {x, x̂}] ds
Therefore
¯
¯´
³P
√
¯
¯
n−1
i
n
b−a
sup
(T
{x,
x̂})
(t)
¯
¯ + k(T {x, x̂}) k
t∈[a,b]
i=0
¯
¯
³
´
√
Pn−1
¯ ∗ ∗ i ¯
∗
∗ n
≤ b−a
i=0 supt∈[a,b] ¯({x , x̂ }) (t)¯ + k({x , x̂ }) k +
¯R i
¯´
³P
√
¯ b ∂ Km (t,s)
¯
n−1
b−a
sup
[N
{x,
x̂}
(s)]
ds
¯
¯ +
t∈[a,b]
i=0
∂ti
a
°R n
°
° b ∂ Km (t,s)
°
[N {x, x̂} (s)] ds°
° a
∂tn
then, by a simple calculation as in the previous proof, we get
¯´
¯
³P
√
¯
¯
n
i
n−1
(T
{x,
x̂})
(t)
sup
b−a
¯ + k(T {x, x̂}) k
t∈[a,b] ¯
i=0
¯´
¯
³P
√
¯ ∗ ∗ i ¯
n
n−1
≤ b−a
({x
,
x̂
})
(t)
sup
¯ + k({x∗ , x̂∗ }) k + γm kN {x, x̂}k
¯
t∈[a,b]
i=0
¯
¯
´
³
√
√
Pn−1
¯ ∗ ∗ i ¯
∗
∗ n
+
k({x
,
x̂
})
k
+
γ
k
≤ b−a
({x
,
x̂
})
(t)
sup
b−a
¯
¯
m
0
t∈[a,b]
i=0
We notice that the right hand side of the above inequality is independent of
REDUCTION OF OPERATOR EQUATION
177
{x, x̂}. Therefore the set T (Se0 ) is bounded in H n [a, b] × H n [a, b]. Hence from the
Lemma 2.5, the set T (Se0 ) is relatively compact in H n−1 [a, b] × H n−1 [a, b]. This
completes the proof.
¤
We now present a theorem which reduces the original MBVP to an equivalent
bifurcation equation by making use of the Schauder’s fixed point theorem.
Theorem 2.9. Let all the assumptions in this section and conditions (2.10),
(2.11) and (2.17) be true. Let m be sufficiently large such that
√
√
c ≤ d − θm k0 b − a, r ≤ R − θm k0 b − a
(2.18)
∗
∗
∗
∗
Then for each {x , x̂ } ∈ V the set A({x , x̂ }) is non-empty. Moreover L {x, x̂}−
N {x, x̂} = Pm (L {x, x̂} − N {x, x̂}) on the set A.
Proof. By (2.15), it is enough to show that the map T corresponding to {x∗ , x̂∗ } ∈
V has a fixed point in Se0 . We have seen in the previous theorems that the map
T is continuous and the set T (Se0 ) is relatively compact in H n−1 [a, b] × H n−1 [a, b].
We now prove that T (Se0 ) ⊂ Se0
Let {x, x̂} ∈ Se0 , then T {x, x̂} ∈ H n [a, b] × H n [a, b] and
Rb
T {x, x̂} = {x∗ , x̂∗ } + a Km (., s) [N {x, x̂} (s)] ds
Therefore
Rb
T {x, x̂} − {x0 , x̂0 } = {x∗ , x̂∗ } − {x0 , x̂0 } + a Km (., s) [N {x, x̂} (s)] ds
Hence
kT {x, x̂} − {x0 , x̂0 }kH n [a,b]×H n [a,b] = k{x∗ , x̂∗ } − {x0 , x̂0 }kH n [a,b]×H n [a,b]
°R
°
° b
°
+ ° a Km (., s) [N {x, x̂} (s)] ds°
n
n
H [a,b]×H [a,b]
≤ c + θm kN
√{x, x̂}k
≤ c + θm k0 b − a
≤d
Also
Rb
µ(T {x, x̂} − {x0 , x̂0 }) ≤ µ({x∗ , x̂∗ } − {x0 , x̂0 }) + µ( a Km (., s) [N {x, x̂}] ds)
≤ r + θm kN
√{x, x̂}k
≤ r + θm k0 b − a
≤R
Thus T (Se0 ) ⊂ Se0
We also observed that Se0 is a closed, bounded and convex subset of H n−1 [a, b] ×
H n−1 [a, b]. Hence the application of Schauder’s fixed point theorem to the pair Se0
and T yields that the map T has a fixed point in Se0 . Moreover, since A({x∗ , x̂∗ })
is non-empty. So, thus we have the desired reduction on the non-empty set A.
So, We notice that L {x, x̂} − N {x, x̂} = Pm (L {x, x̂} − N {x, x̂}) on the set
A since A is non-empty.
¤
Acknowledgements
This study is funded under the Research Project SR/S4/MS:279/05, by DST,
Ministry of Science and Technology, Govt. of INDIA.
178
PALAV BARUAH, B V K BHARADWAJ AND M VENKATESULU
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1,2
Department of Mathematics and Computer Science, Sri Sathya Sai University, Prashanthi Nilayam, Puttaparthy, INDIA.
E-mail address: baruahpk@sssu.edu.in, bvkbharadwaj@sssu.edu.in
3
Department of Mathematics and Computer Applications, Kalasalingam University, Krishnankoil, Tamilnadu, INDIA
E-mail address: venkatesulum 2000@yahoo.co.in