Internat. J. Math. & Math. Sci.
VOL. 11 NO. 1 (1988)201-204
201
RESEARCH NOTES
A NOTE ON THE UNIQUE SOLVABILITY OF A
CLASS OF NONLINEAR EQUATIONS
RABINDRANATH SEN
Department of Applied Mathematics
University College of Science
92, Acharya Prafulla Chandra Road
Calcutta
700009
and
SULEKHA MUKHERJEE
Department of Mathematics
University of Kalyani
Kalyani, Dt.Nadia
West Bengal, India
(Received April 28, 1986 and in revised form September 17, 1986)
ABSTRACT.
The aim of the present note is to devise a simple criterion for the
existence of the unique solution of a class of nonlinear equations whose solvability
is taken for granted.
KEY WORDS AND PHRASES. Hammerstein equation, Monotonically Decomposable operators.
1980 AMS SUBJECT CLASSIFICATION CODES. PRIMARY. 66J15, 47H17.
I.
INTRODUCTION.
In equations describing physical problems presence of more than one solution may
sometimes create complications.
One is often led to a solution that may differ from
the desired solution and hence a lack of agreement of the solution with the experimen-
We are therefore motivated in devising a simple criterion by which
tal result occurs.
the uniqueness of the solution of a class of equations is guaranteed.
In what follows we take
element of
A
X
to be a complete supermatlc space
[I] and
f
to be an
X.
X
is a nonlinear mapping of
into
X
and we are interested in solving the
equation
Au + f,
u
X
f
(I.I)
by the iterates of the form
Un
+
A
Un
+ f
(Uo
prechosen)
(1.2)
Section 2 contains the convergence theorem and an example is appended in section 3.
Earlier Sen [2], Sen and Mukherjee [3] proved the unique solvability of the nonlinear
equation
Au
Pu
in the setting of a metric space.
R. SEN and S. MUKHERJEE
202
2.
CONVERGENCE.
THEOREM 2.1.
i)
Let the following conditions be fulfilled:
L
There exists a bounded linear operator
ii)
a)
0
b)
0
c)
0
O(Lu, Lv), V
(Au, Av)
0(LP+Iu, LP+Iv),
(LPAu, LPAv)
(Lmu, Lmv)
X
into
X
s.t.
(m-l)
0,I
p
X, 0 < q <
q 0(u,v), V u, v
and for fixed
m
(l-A).
belongs to the range of
f
mapping
X
u, v
u
defined by (1.2) converges to the unique solution of (I.I).
n
there exists a
condition
(ii)
By
Then the sequence
PROOF.
X
u*
u*
s.t
+
Au*
(2.1)
f
The space being supermetric and the use of the conditions a), b) and c) yields
u*)
0(Um+ I,
(Aum, Au*)
0
0(L(m-l)u o L(m-l)u *)
--<
Hence
0
If
(Unm,
v*
qn0 (u o,
u*)
(2.2)
q 0 (Lu o, Lu*)
u*)
(2.3)
(0 < q < 1)
0 as n
is another solution of the equation
0 (u*,
v*)
--<
Hence, u*
(Au* + f, Av* + f)
0
q 0(u*, v*) (0 < q < 1)
(2.4)
v*
Therefore
{u
converges uniquely to the solution of (1.1) where
n
f
belongs to
the range of (I.A).
3.
EXAMPLE.
In this section we consider the following Hammerstein equation
Ix-tl
+ I
u(x)
2
u(t)
7
u (t)
dt
(3.1)
in the setting of C(O,I).
By using the theory of Monotonically Decomposable Operator (MDO) [I] Collatz proved
[4] the existence of a solution
2(x
Let
x
2)
{u(x) ! 2(x-
X
=<
x
2)
We would show that in
f
Au
Let us choose
We take
0
(u,v)
--< 2(1
x
II
x2)
(3.2)
2
x- x )}
2(1
the equation (3. I) admits of a unique solution.
[u(t)
Ix-t[
f
u-v
with
u(x)
X
Ix-tl
Lu
]I
u(x)
u(x)
u
2
(t)
dt
lu(x)
v(x)
0 < x _-<
0<_x<l
V u(x), v (x)
(3.3)
(3.4)
u(t) dt
max
Here
C(O,I).
(3.5)
203
UNIQUE SOLVABILITY OF A CLASS OF NONLINEAR EQUATIONS
w.r.t,
induced by the metric in C(0,1) and Complete
X
Let us consider the metric in
the induced metric so that
Ix-tl
f
Av
Au
X
is a complete supermetric space.
X
(u(t)-v(t) )dr
(u() + ())
Ix-tl
u(t) dt
(u(n) + v(n))
Ix-tl
v(t) dt
(3.6)
0<<
0 < n <
u(x) +v(x)
max
oxl
(Lu,Lv)
0
lu()-v()
max
x
v u(x), v(x)
I,
2
Ix-tl
f
dt
(3.7)
(0<<I, o<<I)
(3.8)
0(Lu,Lv)
(3.9)
0=<x<l
Ix-tl
’I
v()
v(t) dt
lu()
,n
Neglecting quantities of second order in
IAu-Av
<=
Now
2
Therefore
p(Au, Av)
+ v()]
(u() + v()
3
(u() + v() +
(-2) +
v()l
0(Lu, Lv)
v())
+ 2)
2 (i
p(Lu, Lv), Vu, v
(3.11)
(3.12)
X
Simple manipulation shows that
(L2u, L2v)
(LAu, LAv)
(3.13)
Using Schwartz inequality we have
0(L2u, L2v)
<-
(.3.14)
O(u,v)
It may however be noted that both
A
and
L
map
X
X
into
where
X
is a
complete metric space.
Hence by Theorem 2.1
Un+
1-
{u
01
n
defined by
Ix-tl [Un(t)
u
2
n(t))]
dt
Converges to the unique solution of equation (3.1) in
3.1.
n
2
0, ,I
X.
In many nonlinear equations it is possible to generate
A.
L’s
which could be
prototypes of the linearized versions of
ACKNOWLEDGEMENT.
The authors are grateful to the referee for some comments.
R. SEN and S. MUKHERJEE
204
REFERENCES
I.
COLLATZ, L.
1966.
2.
SEN, R. Approximate Iterative Process in a Supermetric Space, Bull. Cal. Math.
Soc. 63(1971), 121-123.
3.
SEN, R. and MUKHERJEE, S. On Iteratlve Solutions of Nonlinear Functional Equations
in a Metric Space, International J. Math. and Math. Sci. 6(1983), 161-170.
4.
COLLATZ, L.
Functional Analysis and Numerical Mathematics, Academic Press, N.Y.,
Nonlinear Functional Analysis and Applications, (Ed.L.B.RalI),
Academic Press, 1971.