Math. Si.
ltrnat. J. Math.
(1983) 161-170
Vol. 6 No.
161
ON ITERATIVE SOLUTION OF NONLINEAR FUNCTIONAL
EQUATIONS IN A METRIC SPACE
RABINDRANATH SEN
Department of Applied Mathematics
University College of Science
92 Acharya Prafulla Chandra Road
Calcutta-700009
INDIA
SULEKHA MUKHERJEE
Department of Mathematics
University of Kalyanl
Kalyani, Dt. Nadia
West Bengal, INDIA
(Received September 4, 1981)
ABSTRACT.
Given that
metric space
A and P as nonlinear onto and into self-mapplngs of a complete
R, we offer here a constructive proof of the existence of the unique solu-
tion of the operator equation Au
quence
AUn+1
Pu
n
(u
Pu, where
u E
).
0,1,2
prechosen, n
R, by considering the iteratlve seWe use
Kannan’s
criterion [i] for
the existence of a unique fixed point of an operator instead of the contraction mapping
principle as employed in [2].
Operator equations of the form
Anu
Pmu,
where u
E
R,
n and m positive integers, are also treated.
Kannan’s xcd point theorem, Nonlinear Integral Equatn.
1980 THEMATICS SUBJECT CLASSIFICATION CODES. Primay: 65J15, 47H17; Sendy:
KEY WORDS 4D PHES.
47HI0.
INTRODUCTION.
R is a complete metric space.
R.
A is an operator possibly nonlinear mapping R onto
P is a nonllnear operator mapping R into R.
We investigate the unique solution of
the equation
Au
by considering the iterates of the form
Pu,
u E R
(i.i)
162
R. SEN AND S. MUKHERJEE
AUn+1
where u
o
Pu
(1.2)
n
--0,1,2,3,
is prechosen and n
Using the contraction mapping principle, we have proved in [2] the convergence of
(1.2).
By considering the sequence (m a positive integer, u
[3] proved the convergence of {u
n
o
prechosen), Chatterjee
to the unique solution of (i.i).
By arguing along
the lines of [2], Chakravorty has proved the solvability of the equation
Anu pmu
where u e R, n and m are positive integers, as well as the system of simultaneous equations Au
AlU PI v,
Pv,
u,v
R.
In this paper we are using Kannan’s [i] criterion for the existence of the unique
fixed point of an operator to build up a sequence of sufficient conditions which will
guarantee the convergence of the sequence (1.2).
{u
pmu
n
i
given by A
Ui+l
Conditions for the convergence of
(n,m positive integers
i
u
o
pmu
prechosen
...) under
0 1,2
i
suitable conditions to the unique solution of
Anu
Section 2 contains the convergence theorems.
Section 3 contains a nonlinear integral
where u e, R are also formulated.
equation where our method can be effectively applied to ensure the existence and uniqueness of the solution of the equation.
2.
CONVERGENCE.
We first state
"If T is
O[T(x), T(y)]
Kannan’s
a map of a complete metric space E into itself and if
<
{O[x, T(x)] + O[Y, T(y)]},
unique fixed point in
THEOREM 2.1.
(i)
(ii)
(iii)
theorem as follows:
where x,y e E and 0 < e
,
< 1
E".
Let the following conditions be fulfilled for all u,v
O(u,v) -> 0(Au,Av)
o(APu, Pu)
then T has a
<
>
s0(u,v),
B
R.
> e > i
y0(Au,Pu)
28y < (- i)
Then the sequence {u
n
defined by (i 2), will converge to the unique solution of
the equation (I.i).
The error estimate is given by
(un,u*)
PROOF.
_<
q(
q
n-1
-q
The existence of A-1
O(uo
,A-ipuo
q
By
e(- i)
(2 i)
its boundedness and continuity follow from (i).
SOLUTION OF NONLINEAR FUNCTIONAL EQUATIONS
un
Thus, the sequence
where u
A-ipun
n
1,2,
n
i’
and u
163
prechosen, is well-
o
defined.
P,
It then follows from (i) and (iii) that, for all u,v
0(A-IPu,A-1pv)
or
O
(A-Ipu, A-IPv)
<
_<
i/a o(Pu,Pv)
_<
1/a
-
[0(A-ieu, A-Ipv)
[0(A-Ipu’Pu)
i
1
+
+
0(A-Ipu,Pu)
+
0(A-Ipv,Pv)]
0(A-Ipv’Pv)]
i
(- i) [0(APu,Pu) + O(APv,Pv)]
-<
_<
By condition (iii)
q
(a
Y
a(Y
a(a- l)
Y
n
-<
q[O(Un_ I,
qO(Un_l,
-< q(
Since 0 < q <
.
given equation as n
+
O(v,A-ipv)]
Therefore
by
Kannan’s
criterion
(2.2)
A-Ip
will
A-Ipu*)
A-ipun_l
A-ipun_l
n-i
q
’i
1/2, 0 <---q---<
i,
i- q
[0(u,A-ipu)
To find the error estimates, we note that
o(A-ipun_ 1
u*)
[0(Au,Pu) + 0(Av,Pv)]
1/2
<
have a unique fixed point u* (say).
O(u
)
q
O(u o
+ O(u*,
A-Ipu*)]
(2.3)
A-Ieuo)
(2.4)
so that u converges to the unique solution of the
n
The above inequality gives the apriori error estimate.
We next consider the equation
tegers (n > m).
(ii)
where u
R and n and m are positive in-
A and P are the same as prescribed earlier.
THEOREM 2.2.
(i)
Anu pmu,
Let the following conditions be fulfilled for all u,v
80(u,v) -> 0(Au, Av) -> a0(u,v),
0(APu, Pu)
_<
70(Au, Pu);
(iii)
A and P commute;
(iv)
2By < (a- i).
Then the sequence
8 > a > i;
{u.} defined by
1
AnUi+l
pmu i
R,
164
R. SEN AND S. MUKHERJEE
where u
o
prechosen, n and m positive integers and n > m, i
to the unique solution u* of the equation
,
<
0(ui’u
Let n
PROOF.
m
+ p,
pmu.
Anu
im-i
q
c.p
0
q
1
0,i,2,...
will converge
The error estimate is given by
(Uo ,A-ipuo
where p is a positive integer.
(2.5)
Hence, sequence (u i} is
expressed by
ui+1
pmu.
Ui+l
A-ipmu i
n
A
1
or
An-I
Hence,
-1) n pmu i
(A
ui+I
(An)-ipmu.1
-I
Since A
exists and A commutes with
P,
we have
(2.6)
A-Ip
PA
-I,
so that A
-I commutes
with P.
There fo re,
(A-I) P (A-I) mpm
(An)-ipm
I
(A-I)P (A-Ip)m’
(2.7)
(A-IF) TM,
Hen ce,
(A-l)p (A-IF) m
Ui+l
u.
l
Now proceeding as in the previous theorem, we prove that
A-1p
will have a unique fixed
point u* (say).
u*
Thus
A-Ip
(A-Ip) TM
and so
Therefore, u* is also a fixed point of
fixed point of
(A-Ip) TM
u*
u*
(A-Ip) TM.
O((A-Ip) m
-<
To prove that u* is the unique
we proceed as follows
If possible, let v* be another fixed point of
O(u*,v*)
(2.8)
u
u*,
(A-Ip) m
(A-Ip) m
such that v* # u*.
v*)
q[0((A-Ip)mu*, (A-ip)m-lu*)
+
o((A-Ip) m
v*,
(A-Ip) m-I
v*)]
Then,
SOLUTION OF NONLINEAR FUNCTIONAL EQUATIONS
q
I
0((A-Ip) m
q
<- q(
m-i
q
m-i
q
O
<q(
((A-Ip) m
2m-i
q
q(l-q
Since A is an onto mapping,
(A-Ip) ln+l
v*
(A-Ip) v *
0(v*,
(A-Ip)v*),
,
0, i
v*)
O(v*
im-i
q
l-q
(A-Ip) m-I
A-Ipv*)
0(v*
i- q
q(1-q
<
v*,
165
0 <
q
q
i
v*)
1,2,3
i
(2.9)
< 1
A-I exists and is continuous and is also an onto mapping.
(A-I) p
Furthermore, it follows from (i) that
is a contraction mapping and hence
has a unique fixed point in R.
Since
it follows
(A-I) p and (A-Ip) m commute and since each of
m
p
that (A-I) (A-Ip) has a unique fixed point
Now,
0(ui,u*
-
0((A-I)P(A-Ip) m
1
0 ((A
-<--q--(
alp
1
THEORI 2.3.
(i)
(ii)
(iii)
0
q
B0(u,v) -> o(Au,Av) -> O(u,v),
((A-npm)lu,@) -< k0(u,O)
A-n pm
u*)
u
(2.10)
o
Let the following conditions exis=
> 0 for all
u,v
R;
for all positive integers %;
is continuous at its fixed point;
(iv)
A and P commute;
(v)
P is compact and
P
(vi)
o((Anf u, (pmf u)
>
{u..}
is closed for all finite positive integers
O(Anu,pmu)
difined by
tive integers and n < m, i
for all finite positive integers
Anui+1
Pmu
i
0,1,2,...) will converge
pmU.
PROOF.
(Uo A-Ip
Let R be a metric linear space.
Then the sequence
Anu
1
(A-I)P(A-Ip) m
m
Ui_l, (A-IP) u*)
mi-i
q
u* (say).
oo.
u* as i
which shows that u.
-Ie) m
Ui_l,
them has unique fixed points,
The sequence {u.
1
expressed by
(u
;
.
prechosen, n and m are posi-
to a solution u* of the equation
166
R. SEN AND S. MUKHERJEE
u
A-npm
I"
A-npTM
Let us denote
is well-defined.
(A-npm) i Uo,
ui-i
(2.11)
0,1,2,
i
by G.
Since R is a metric linear space, the null element also belongs to R.
By condition (ii)
0((A-npm) i Uo,@)
0(u i,@)
which implies that
{u.}
Thus,
k0(Uo,
is bounded.
1
{u.}
Since P is compact and
bounded.
-<
{Pu.}
is bounded,
1
is sequentially compact and is hence
1
p2
is again compact, so that
(P(Pui))
is compact.
In general,
pm
is
compact with m a positive integer.
-I are
both bounded,
{u
Since
and A
l
pm {(An)-ipmu
i
-I commutes with
pact and A
fined by u
Gui_l,
i
Let u.
i
Now u.
i
Thus
+
P
i
-
u* as p
Gku
{(An)-lu
0,1,2,
1
is also bounded
0 1,2,
is compact, i
contains a convergent
Since
pm
Thus {u
subsequence{Uip}
is com-
i}
de-
(Say).
oo.
for some integer k.
(p-l)
p
Gku.
oo, for finite k.
u* as p
(p-l)
-I and P
commute,
Since A
(A-ip) k
Gkum
u
(p-l)
-I is
Since A
continuous,
lira
p+
pU
-i k
(A
i(p_l
(A-l)ku i
(A-I) k
(p-l)
u*.
p
being closed x for all finite positive integers U, we obtain from above that
u*
Thus
u* is a solution of
pk(A-i)k
nNA..__u
Gku *
u*
Now, G being continuous
lim
1
Anu pmu.
at its fixed points,
p
Therefore {u
(2.12)
pink U.
By virtue of condition (vi), u* is also a solution of
3.
u
converges to u*
u
ip+I
Gu i
Gu*
u*
p
a solution of the given equation
AN EXAMPLE.
Let
u (x)
lim
p
c (0, i)
SOLUTION OF NONLINEAR FUNCTIONAL EQUATIONS
Au
u
2
(x) + 2(x + 15)u(x)
1.5;
0.05 -< u(x) -< 1.5
D(A);
(3.1)
l
Pu
167
Ix-
7
0
u
[u(t)-
t
2
(t)
]dt
8
(3.2)
0.06 -< u(x) < 0.13
D(P):
Pu.
We are interested in the solvability of the integral equation Au
We have
Pu
7
I
1
0
i
-> 7
u
Ix- I:I
[u(t)
Ix-
[0.06-
t
2
(t) ]dt
8
(0.13)2
2x + 2x
0.13311
Since
Min
[i
2x
+ 2x2]
dt
8
0
2]
(3.3)
1/2
0<x_<l
Pu
>_
0.067
for
D(P).
u
(3.4)
Again
Pu
J0
<
2
Ix-
7
(0.06)2
7(0 13
2
for
and complete
D(A)
L2(0,1)
+
2x
Ix- tldt
2x
2)
D(P)
u
(3.5)
D(P) and hence D(A)
Thus we have 0.067 -< Pu < 1.365 for all u
We now introduce in D(A) the
f
0
(I
]dt
8
8
x 0.13
< 7
1.365
u (t)
[u(t)
t
norm
with respect to the above
L2(0,1)
we can introduce the scalar product
llul12
(u,v).
II II.
f0 u2dx for all
lul 1,2
(.e.
Since
(u,v)
(P).
i
D(A) being
uv dx, u,v
u
D(A))
now a subspace of
I)(A)
and
0
On the choioe of the metric O(u,v)
I]u- vll
for all u,v
complete metric space.
Since A is a continuous operator,
(A) is closed.
We further assume that
0.093 -<
Now for all u
D(A)
l]u(x)ll
-< 0.13
for all
u e
D(P).
V(A), O(A) becomes
a
R. SEN AND S. MUKHERJEE
168
(Au- Av, u- v)
v_2dx +
(x + 15)(u
2
(u + v)(u
-> 30
v)2dx +
(u
[u(x) + v(x) l(u
v)2dx +
(u-
30
[u() + v()]
(u-
v_2dx)
where 0 <
< i
0
0
30.ii lu
v
2
lu vl
-> (30 + 2 x 0.05)
a
v)2dx
0
0
Thus we have
v)2dx
0
0
2
(3.6)
30.1.
D(A):
0.05 < u(x) -< 1.5
u2(x)
Au
+ 2(x + 15)u(x)
2
1.5
+ 2.15 X 0.05
1.5
+ 2(x + 15)u(x)
1.5
< (1.5) 2 + 2(1 + 15)1.5
1.5
>
(0.05)
0. 0025
Aiso
u2(x)
Au
48.75.
0.0025 -< Au < 48.75.
(A):
Hence
(A)
Thus
By (3.6), A has a bounded inverse for all u
Since
(P)
c_
D(A)
c__
(A),
A-Ipu
l]Au- Avll
l[(u
D(A).
E
is well defined and the sequence
0,1,2,... is also well-defined.
n
(A).
__c
Un+I
A-iPUn
Moreover,
v) [2(x + 15) + (u +
for all
u,v
D(A).
(3.7)
Now,
(x +
15)2dx
240.333
and hence
Ix+ ll
IAu- Avll
(3.8)
: 5.503
<
(2
x
15.503
+
1.5
34.0061 lu vll
34.006.
so that we take
Now for all u
E
P(P),
+
1.5)llu- vll
(3.9)
SOLUTION OF NONLINEAR FUNCTIONAL EQUATIONS
)_u2_x_
(Au,u)
+ 2(x + 15) u(x)
169
1.5] u(x) dx
0
(x) dx + 2
u
(x + 15)
(x) dx
0
u2(x)dx
> 2 x 15
u(x) dx
1.5
0
0
+ 0.06
u
(x)dx- 1.5
d
0
0
Hence,
llAull llull
_>
30
llul12 + 0.0611ull 2
lul I, u (P),
lAull > 30.06 lull
Using the lower bound for
1.5
>- 30.06 x 0.093- 1.5
1.2956
(3.10)
1.30
Again
lPul 12
49
I]. ii Ix
0
<
I{Pull
2
Ii II Ix
06)2 )2
(0
49(0.13 +
8
6
(t))
dt]
8
0
0.83939 x
Hence,
u
(u(t)
t
0
t
2
Idt] 2
dx
0
0.097288
< 0.311.
Using the above result, we have for u
lAu Pull
leull
lAull
>
(P) that
E
>
0.311)
(1.30
0.989.
Now
APu- Puo
(Pu)
(Pu)
2
+ 2(x + 15)Pu- Pu- 1.5
2
+ (2x + 29)Pu- 1.5
Hence,
I{ APu
Pull
<
llPull 2
< 0.097
+
ll2x + 29{{ {lPull
+
(_4x
2
+ 4
x 29x
+
+ 1.5
292_dx_)] 1/2
x 0.311
+ 1.5
0
0.097 +
4
4 x 29
2
2
92
1/2
x 0.311
+ 1.5
10.930.
On choosing y
ii.i, we have
II APu Pull -< YI
Now
Au
Pull
for all
u E
(P)
(3.11)
170
R. SEN AND S. MUKHERJEE
2y
34.006
2
Ii.I
754.9332
Again
(- i)
29.1
30.1
875.91
Thus,
25y
(- i)
754.9332
=0.862 < 1
875.91
Thus, all the assumptions of Theorem 2.1 are fulfilled so that the equation
u
(x) + 2(x + 15)u(x)
Ix-
7
1.5
t
u
[u(t)
2
0
admits of a unique solution in the interval 0.06 -< u(x)
Starting from u
u
2
0.06e
o
+ 2(x +
n+l(X)
x,
15)Un+l(X)
1.5
7
0
(3.12)
-< 0.13 for 0 -< x -< I.
the sequence of iterates {u
n
fl
(t) ]dt
8
is given by
u
Ix- tl [Un(t)
2
n
(t)
]dt,
8
0,1,2
n
and the convergence of the sequence to the unique solution of the equation in
0.06 < u(x) -< 0 is assured.
For computational advantage, we can take {u’} as follows:
n
1
u’n (x)
[1.5
2(x + 15)
u’(x)
O
0.06e
{u’ (x)}
n
u
,2
n-i
(x) + 7
jrl
u
lx-tl
0
[u’
,2
(t)
n-i
(t)
]dt].
n-i
x
will converge to the unique solution of the equation Au
Pu in the
interval 0.06 < u(x) < 0.13, 0 -< x < i.
REFERENCES
i.
KANNAN, R.
Some results on fixed points, II, Amer. Math. Monthly 76 (1969),
405-408.
2.
SEN, R. Approximate iterative process in a supermetric space, Bull. Cal. Math.
Soc. 63 (197!), 121-123.
3.
CHATTERJEE, S.K. On a nonlinear functional equation, Mathematica Balkanica 2
(1972), 3-5.
4.
CHAKRAVORTY, M. On solutions of certain functional equations, Bull. Cal. Math.
Soc. 71 (1978), 7-11.