Dependence on parameters for a discrete Emden-Fowler equation

0.16 MB·English

by  Marek Galewski

Dependence on parameters for a discrete
0
1 Emden-Fowler equation
0
2

n
Marek Galewski
a
J
Faculty of Mathematics and Computer Science,

8
1 University of Lodz,

Banacha 22, 90-238 Lodz, Poland,
]
A
[email protected]
C
.
h
January 18, 2010
t
a
m
[
Abstract

1
v Weinvestigatethedependenceonparametersforthediscretebound-
3
aryvalueproblemconnectedwiththeEmden-Fowlerequation. Avari-
3
1 ational method is used in order to obtain a general scheme allowing
3
for investigation the dependence on paramaters of discrete boundary
.
1 value problems.
0
0
1 MSC Subject Classification: 34B16, 39M10
v: Keywords: variational method; discrete Emden-Fowler equation; coer-
i civity; dependence on parameters
X
r
a
1 Introduction
The discrete version of the Emden-Fowler equation received some consider-
able interest lately by the use of critical point theory, see for example [7], [8],
[9], [11]. Various variational approaches towards the existence of solutions
for this problem can be applied as being the finite dimensional counterparts
of the methods used in the continuous variational case. However, due to the
finite dimensionality of the space in which solutions are obtained, we have
1
much more tools at our disposal. In finite dimensional space the weak solu-
tion - which is a key notion in the variational approach - is always a strong
one. Also weak convergence, and thus weak lower semicontinuity coincides
with strong convergence and classical lower semicontinuity which can in turn
be obtained with no additional convexity assumption. Moreover, the coer-
civity of the action functional can be very often investigated together with
its anti-coercivity which of course involves different growth conditions on the
nonlinear term. Thus we have a lot of tools at our disposal as far as the
existence is concerned, compare with [1], [6]. Although the application of
variational methods to the discrete problems is rather a new topic, started
apparently by [3], [6] the list of our references is by no means complete since
research in this area has been very active.
In the boundary value problems for differential equations it is also im-
portant to know whether the solution, once its existence is proved, depends
continuously on a functional parameter. This question has a great impact
on future applications of any model. As it is well known difference equations
serve as mathematical models in diverse areas, such as economy, biology,
computer science, finance, see for example [2], [4], [10]. Thus it is desirable
to know whether the solution to the small deviation from the model would
return, in a continuous way, to the solution of the original model. This is
known in differential equation as stability or continuous dependence on pa-
rameter, see [12], but it has not been investigated in the area of boundary
value problems for difference equations, apart from some work done in [5].
In this paper we are going to get some general scheme for investigating the
dependence on parameter in difference equations which we illustrate with the
Emden-Fowler equation as a model example. Thus our interest lies not in the
existence of solutions, which in fact has been vastly researched, but in their
dependence on parameters. We mention that the approach of [5] required
that each boundary value problem should be treated separately, while in this
submission we provide some general result which could be further applied
for any discrete boundary value problem for which the action functional is
either coercive or anti-coercive.
2
2 Problem formulation and main results
In what follows T is a fixed natural number, T 3; [a,b], where a b
≥ ≤
are integers, stands for the discrete interval a,a+1,...,b 1,b ; M > 0 is
{ − }
fixed, L = v C([1,T],R) : v M ; v = max v(k) . We
M { ∈ k kC ≤ } k kC k∈[1,T]| |
consider the discrete equation
∆(p(k 1)∆x(k 1))+q(k)x(k)+f (k,x(k),u(k)) = g(k) (1)
− −
subject to a parameter u L and with boundary conditions
M
∈
x(0) = x(T), p(0)∆x(0) = p(T)∆x(T) (2)
known as the discrete version of the Emden-Fowler equation. We assume
that
A1 f C([1,T] R [ M,M],R), p C([0,T +1],R), q,g C([1,T],R);
∈ × × − ∈ ∈
g(k ) = 0 for certain k [1,T].
1 1
6 ∈
The growth conditions on f will be given later on. Solutions to (1)-(2)
are such functions v : [0,T +1] R that satisfy (1) as identity and further
→
v(0) = v(T), p(0)∆v(0) = p(T)∆v(T). Hence solutions to (1)-(2) are
investigated on a finite dimensional space
E = v : [0,T +1] R : v(0) = v(T), p(0)∆v(0) = p(T)∆v(T) .
{ → }
Any function from E can be identified with a vector from RT and therefore
solutions to (1)-(2) can be investigated in RT with classical Euclidean norm.
By we denote the Euclidean norm, and by , the scalar product. We
|·| h· ·i
note that with A1 any solution to (1)-(2) is in fact nontrivial in the sense
thatno function v : [0,T +1] R such that v(k) = 0for all k [1,T] would
→ ∈
satisfy (1)-(2). To reach this conclusion suppose that v : [0,T +1] R such
→
that v(k) = 0 for all k [1,T] satisfies (1)-(2). We see that at least for k it
1
∈
follows 0 = g(k ) = 0.
1
6
Variational approach towards (1)-(2) relays on investigation of critical
points to a suitable action functional. Thus problem (1)-(2), but without
parameter u and a forcing term g, can be considered either as it stands, as
3
it is done in [13], or else one may write it in a matrix form in which form we
will further investigate it. Let us denote as in [9]
p(0)+p(1) p(1) 0 ... 0 p(0)
− −
 p(1) p(1)+p(2) p(2) ... 0 0 
− −
0 p(2) p(2)+p(3) ... 0 0
M =  ... − ... ... ... ... ... 
 
 0 0 0 ... p(T 2)+p(T 1) p(T 1) 
 − − − − 
 p(0) 0 0 ... p(T 1) p(T 1)+p(0) 
 − − − − 
and
q(1) 0 0 ... 0 0
−
 0 q(2) 0 ... 0 0 
−
0 0 q(3) ... 0 0
Q =  ... ... − ... ... ... ... .
 
 0 0 0 ... q(T 1) 0 
 − − 
 0 0 0 ... 0 q(T) 
 − 
For a fixed u L we introduce the action functional J : RT R for (1)-(2)
M
∈ →
by the formula
T T
1
J (x,u) = (M +Q)x,x F (k,x(k),u(k))+ g(k)x(k). (3)
2 h i−
X X
k=1 k=1
Calculating the Gaˆteaux derivative of J (x,u) with respect to x we relate
critical points to (3) with solutions to (1)-(2) as in [8]. In fact any critical
point to the action functional (3) is a solution to (1)-(2) and any solution
to (1)-(2) provides a critical point to J. Hence, in order to find at least one
solution (1)-(2) it suffice to find at least one critical point to (3). We denote
d
V = x RT : J (x,u) = inf J (v,u), J (x,u) = 0
u (cid:26) ∈ v∈RT dx (cid:27)
for any fixed u L . We will employ the following assumptions concerning
M
∈
the growth of the nonlinear term f.
A2 there exist constants ε > 0, ε R and r (1,2) such that
1 2
∈ ∈
f (k,y,u) ε y r−1 +ε (4)
1 2
≤ | |
4
uniformly for u [ M,M], k [1,T] and y B, where B > 0 is certain
∈ − ∈ | | ≥
(possibly large) constant.
A3 there exist constants ε > 0,ε R and r > 2 such that
1 2
∈
f (k,y,u) ε y r−1 +ε (5)
1 2
≥ | |
uniformly for u [ M,M], k [1,T] and y B, where B > 0 is certain
∈ − ∈ | | ≥
(possibly large) constant.
Our main results are as follows.
Theorem 1 (sublinear case) Assume A1, A2 and further that M+Q is
either positive or negative definite matrix. For any fixed u L there exists
M
∈
at least one non trivial solution x V to problem (1)-(2). Let u ∞
∈ u { n}n=1 ⊂
L be a sequence of parameters. For any sequence x ∞ of solutions
M { n}n=1
x V to the problem (1)-(2) corresponding to u , there exist subsequences
n ∈ un n
x ∞ RT, u ∞ L and elements x RT, u L such that
{ ni}i=1 ⊂ { ni}i=1 ⊂ M ∈ ∈ M
lim x = x, lim u = u. Moreover, x V (which means that x
i→∞ ni n→∞ ni ∈ u
satisfies (1)-(2) with u), i.e.
∆(p(k 1)∆x(k 1))+q(k)x(k)+f (k,x(k),u(k)) = g(k),
− −
x(0) = x(T), p(0)∆x(0) = p(T)∆x(T).
Theorem 2 (superlinear case) Assume A1, A3. Then the assertion of
Theorem 1 is valid.
We note that in Theorem 2 matrix M + Q could be singular and its
definiteness is not important.
Inorder toconsider thecase when r = 2we first introduce somenecessary
notation. In case when M + Q is positive definite there exists a number
a > 0 such that for all y RT
M+Q
∈
(M +Q)y,y a y 2, (6)
M+Q
h i ≥ | |
while incase when M+Qis negative definite there exists anumber b > 0
M+Q
such that for all y RT
∈
(M +Q)y,y b y 2 (7)
M+Q
h i ≤ − | |
5
Now we may formulate the assumptions in case r = 2.
A4 let M + Q be positive definite and let there exist constants ε
1
∈
(0,2a ), ε R such that
M+Q 2
∈
f (k,y,u) ε y +ε (8)
1 2
≤ | |
uniformly for u [ M,M], k [1,T] and y B, where B > 0 is certain
∈ − ∈ | | ≥
(possibly large) constant.
A5 let M + Q be negative definite and let there exist constants ε
1
∈
(0,2b ), ε Rsuch that (8) holds uniformly for u [ M,M], k [1,T]
M+Q 2
∈ ∈ − ∈
and y B, where B > 0 is certain (possibly large) constant.
| | ≥
Theorem 3 (r = 2) Assume either A1, A4 or A1, A5. Then the asser-
tion of Theorem 1 is valid.
3 Auxiliary results
We will prove the following lemmas concerning the existence (1)-(2) with
fixed u L . These lemmas improve certain existence results from [9].
M
∈
In contrast to the boundary values for ODE, compare with [14], it is the
superlinear case which is easier to be dealt with, while the sublinear one
involves more restrictive assumptions.
Lemma 4 Assume A1, A2 and that M +Q is either positive or negative
definite matrix. Then for any fixed u L there exists at least one non
M
∈
trivial solution x V to problem (1)-(2).
u
∈
Proof. Let us fix u L . We see that x J (x,u) is continuous and
M
∈ →
differentiable the sense of Gaˆteaux on RT in either case. Thus we would
have the assertion provided that x J (x,u) is coercive or anti-coercive
→
since functional x J (x,u) would have either an argument of a minimum
→
or an argument of a maximum which must be its critical point in turn.
Let M + Q be a positive definite matrix. Let us take sufficiently large
B > 0 as in A2. We denote g = T g2(k) and observe that for
| | q k=1
P
6
c1 = T1/(1−2r) it follows by H¨older’s inequality
T y(k) T y(k) 2 T 1 = √T y ,
k=1 ≤ q k=1| | q k=1 | |
P P P
T y(k)g(k) g y (9)
k=1 ≤ | || |
P
Tk=1|y(k)|r ≤ 2/qr Tk=1|y(k)|r·2r (1−2rq) Tk=11 = c1|y|r.
P P P
Now by (6), (4), (9) we have for all y B
| | ≥
J (x,u) a y 2 ε1 T y(k) r ε T y(k) y g
≥ M+Q| | − r k=1| | − 2 k=1| |−| || | ≥
P P (10)
a y 2 ε1c y r ε √T y y g .
M+Q| | − r 1| | − 2 | |−| || | |y|→→∞ ∞
NowletM+Qbeanegativedefinitematrix. Again,letustakesufficiently
large B > 0 as in A2. By (7), (4), (9) we have for all y B
| | ≥
ε
J (x,u) b y 2 + 1c y r +ε √T y + y g . (11)
≤ − M+Q| | r 1| | 2 | | | || | |y|→→∞ −∞
Lemma 5 Assume A1, A3. Then for any fixed u L there exists at least
M
∈
one non trivial solution x V to problem (1)-(2).
u
∈
Proof. Fix u L . We see that for all y RT
M
∈ ∈
(M +Q)y,y M +Q y 2, (12)
h i ≤ k k| |
where M +Q denotes the norm of a matrix M + Q. Further by (5) we
k k
have for all y B, where B is as in A3
| | ≥
ε
J (x,u) M +Q y 2 1c y r ε √T y y g .
≤ k k| | − r 1| | − 2 | |−| || | |y|→→∞ −∞
Hence the assertion follows with the same arguments as in Lemma 4.
Lemma 6 AssumeA1 and either A4 or A5. Then for any fixed u L
M
∈
there exists at least one non trivial solution x V to problem (1)-(2).
u
∈
Proof. Fix u L . With A4 we see by (10) that x J (x,u) is
M
∈ →
coercive while with A5 we see by (11) it is anti-coercive. Hence the assertion
follows with the same arguments as in the above lemmas.
7
4 Dependence on parameters
In order to derive the results concerning the dependence on parameters for
problem (1)-(2), we employ the following general principle which we could
further apply in a finite dimensional setting.
Let E be finite dimensional Euclidean space with inner product , and
h· ·i
with norm . Let C be a finite dimensional complete normed space with
k·k
norm . Let us consider a family of action functionals x J (x,u), where
k·kC →
x E and where u C is a parameter.
∈ ∈
Theorem 7 Assume that there exist constants µ,a,β > 0, b R such that
∈
J (x,u) a x µ +b for all x E with x β and all u C. (13)
≥ k k ∈ k k ≥ ∈
Assume also that x J (x,u) continuous and differentiable in the sense of
→
Gˆateaux in the first variable for any u C. Then for any u C there exists
∈ ∈
at least one solution x to problem
u
d
J (x,u) = 0. (14)
dx
Assume further that there exists a constant α > 0 such that
J (0,u) α for all u E. (15)
≤ ∈
Let u ∞ C be a convergent sequence of parameters, where lim u =
{ n}n=1 ⊂ n→∞ n
u C and let us assume that either J is continuous on E C or the Gˆateaux
∈ ×
derivative of J with respect to x, d J (x,u), is bounded on bounded sets in
dx
E C and J is continuous with respect to u on C for any x E. Then for
× ∈
any sequence x ∞ of solutions x E to the problem (14) corresponding
{ n}n=1 n ∈
to u for n = 1,2,... there exist a subsequence x ∞ E and an element
n { ni}i=1 ⊂
x E such that lim x = x and
∈ i→∞ ni
J (x,u) = inf J (y,u).
y∈E
Moreover, x satisfies (14) with u = u, i.e.
d
J (x,u) = 0. (16)
dx
8
Proof. Let us fix u C. Assumption (13) leads to the coercivity of
∈
x J (x,u). Since x J (x,u) is also continuous and since E is finite
→ →
dimensionalitfollowsthatJ hasanargumentofaminimumx whichsatisfies
u
(14). Next, let us take a sequence u ∞ X converging to some u X
{ n}n=1 ⊂ ∈
and let x ∞ be a sequence of solutions to (14) corresponding to the
{ n}n=1
relevant elements of the sequence u ∞ . By (13) and by (15) we see that
{ n}n=1
for all n N we have
∈
a x µ +b J (x ,u ) α.
n n n
k k ≤ ≤
Hence a sequence x ∞ is norm bounded, say by some c > 0, and again
{ n}n=1
sinceE isfinitedimensional, itcontainstheconvergentsubsequence, x ∞ ,
{ ni}i=1
such that lim x = x, where x E.
i→∞ ni ∈
We will prove that (16) holds. We observe that there exists x E such
0
∈
that d J (x ,u) = 0 and J (x ,u) = inf J (y,u). We see that there are
dx 0 0 y∈E
two possibilities: either J (x ,u) < J (x,u) or J (x ,u) = J (x,u). If we have
0 0
J (x ,u) = J (x,u), then by the Fermat’s rule we have (16). Let us suppose
0
that J (x ,u) < J (x,u), so there exists δ > 0 such that
0
J (x,u) J (x ,u) > δ > 0. (17)
0
−
We investigate the inequality
δ < (J (x ,u ) J (x ,u)) (J (x ,u ) J (x,u ))
ni ni − 0 − ni ni − ni
(18)
(J (x,u ) J (x,u))
− ni −
which is equivalent to (17). In case J is jointly continuous we have
(J (x ,u ) J (x,u )) (J (x,u ) J (x,u)) =
− ni ni − ni − ni −
(19)
J (x ,u )+J (x,u) 0.
− ni ni i→→∞
In case the Gaˆteaux derivative is bounded on bounded sets recalling that
x c for all i N we have
k nik ≤ ∈
d
J (x ,u ) J (x,u ) sup J (v,u ) x x 0. (20)
| ni ni − ni | ≤ kvk≤c(cid:13)(cid:13)dx ni (cid:13)(cid:13)k ni − k i→→∞
(cid:13) (cid:13)
(cid:13) (cid:13)
9
Indeed, we fix n and introduce the auxiliary function g : [0,1] R
i
→
g (t) = J (tx +(1 t)x,u ) = J (x+t(x x),u )
ni ni − ni ni − ni
(see that g (0) = J (x,u ), g (1) = J (x x,u )) and we have by the
ni ni ni ni ni
Mean Value Theorem that there exists some ξ (0,1) such that
∈
d d
g (0) g (1) = g (ξ) = J (ξx +(1 ξ)x,u ),x x
| ni − ni | (cid:12)dt ni (cid:12) (cid:12)(cid:28)dx ni − ni ni − (cid:29)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
since tx +(1 t)x c and thus (20) holds. By (20) and by continuity
k ni − k ≤
of J with respect to u for any x E we see that
∈
lim (J (x,u ) J (x,u)) = 0 and lim (J (x ,u ) J (x,u )) = 0.
i→∞ ni − i→∞ ni ni − ni
(21)
Finally, since x minimizes x J (x,u ) over E we get J (x ,u )
ni → ni ni ni ≤
J (x ,u ) and next
0 ni
lim (J (x ,u ) J (x ,u)) lim (J (x ,u ) J (x ,u)) = 0. (22)
i→∞ ni ni − 0 ≤ i→∞ 0 ni − 0
So putting (19), (22) into (18) in case J is jointly continuous and (21),
(22) in the other case, we see that δ 0, which is a contradiction. Thus
≤
J (x,u) = inf J (y,u) and thus (16) holds.
y∈E
5 Proofs of main results and some corollaries
Proof of Theorem 1. InordertoproveTheorem1weneedtodemonstrate
that allassumptions of Theorem 7 aresatisfied. Firstly, let M+Qbepositive
definite. We see that (13) is satisfied by (10). We see that F (k,0,u(y)) = 0
T
0
so that J (0,u) = f (k,t,u(k))dt = 0 and (15) holds. By A1 we
0
Xk=1R
see that J is jointly continuous in (x,u). Next, we chose a subsequence
u ∞ L from a sequence u ∞ L such that lim u = u.
{ ni}i=1 ⊂ M { n}n=1 ⊂ M i→∞ ni
Such a subsequence necessarily exists since C([1,T],R) is a finite dimen-
sional space. Next, we chose a corresponding sequence x ∞ E and
{ ni}i=1 ⊂
rename both sequences as u ∞ and x ∞ . Thus all assumptions of
{ n}n=1 { n}n=1
Theorem 7 are satisfied.
10