On Application of the Fixed-Point Theorem to the Solution of Ordinary Differential Equations

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ISSN 2581-3463
RESEARCH ARTICLE
On Application of the Fixed-Point Theorem to the Solution of Ordinary
Differential Equations
Eziokwu C. Emmanuel1
, Akabuike Nkiruka2
, Agwu Emeka1
1
Department of Mathematics, Michael Okpara University of Agriculture, Umudike Abia State, Nigeria,
2
Department Mathematics and Statistics, Federal Polytechnic, Oko, Anambra State, Nigeria
Received: 25-12-2018; Revised: 27-01-2019; Accepted: 20-03-2019
ABSTRACT
We know that a large number of problems in differential equations can be reduced to finding the solution
x to an equation of the form Tx=y. The operator T maps a subset of a Banach space X into another Banach
space Y and y is a known element of Y. If y=0 and Tx=Ux−x, for another operator U, the equation Tx=y
is equivalent to the equation Ux=x. Naturally, to solve Ux=x, we must assume that the range R (U)
and the domain D (U) have points in common. Points x for which Ux=x are called fixed points of the
operator U. In this work, we state the main fixed-point theorems that are most widely used in the field
of differential equations. These are the Banach contraction principle, the Schauder–Tychonoff theorem,
and the Leray–Schauder theorem. We will only prove the first theorem and then proceed.
Key words: Banach spaces, compact Banach spaces, continuous functions, contraction principle,
convex spaces, point-wise convergence, Schauder–Tychonoff
2010 Mathematics Subject Classification: 46B25
INTRODUCTION
Banach contraction principle: Let X be a Banach
space and M be a nonempty closed subset of X.
Let T: M−M be an operator such that there exists a
constant k∈ (0,1) with the property.[1-5]
Tx Ty k x y for every x y M
   
‖ ‖ ‖ ‖ (1.1)
Then, T is said to have a unique fixed point in M.
This result is known as the contraction mapping
principle of the fixed-point theorem which we can
establish as follows:
Let x∈M be given with T(x0
)=x0
, define the
sequence 0
{ }∞
m
x as follows:
1, 1,2
j j
x Tx j
−
= = …(1.2)
Then, in William et al.’s study (2001), we have
‖xj+1
−xi
‖≤k‖xj
−xj−1
‖≤k2
‖Xj−1
−xj−2
‖≤∈≤kj
‖x1
−x0
‖
For every j≥1, if mn≥1, we obtain
Address for correspondence:
Eziokwu C. Emmanuel,
E-mail: okereemm@yahoo.com
( )
1 1 2
1 0
1 2
1 0 1 0
1 0
2 1
1 0
1 0
,
1
[ / (1 )]
n n m m m m
n
n
m m
n
n m n
n
x x x x x x
k x x
k x x k x x
k x x
k k k k x x
k k x x
− − −
+
− −
− −
≤ − + − +…
+ −
≤ − + − +…
+ −
≤ + + +…+ −
≤ − −
‖ ‖‖ ‖ ‖ ‖
‖ ‖
‖ ‖ ‖ ‖
‖ ‖
‖ ‖
‖ ‖
Since kn
→0 as n→∞, it follows that { }
0
m
x
m
∞
=
is
a Cauchy sequence. Since X is complete, there
exists x
̄ ∈X such that xm
→ x
̄ . Obviously, x
̄ ∈X
because M is closed; taking limits as j → ∞, we
obtain x
̄ =Tx. To show uniqueness, let y be another
fixed point of T in M. Then,
‖x
̄ −y‖=‖Tx−Ty‖≤k‖x
̄ −y‖
This completes the proof, which implies x
̄ =y an
operator T. M−X, M⊂X, satisfying (1.2) on M is
called a “contraction operator on M.”
To illustrate the above due to Kelly (1955) 1.2, let
F: R. XRn
→Rn
be is continuous such that
‖F(t,x1
)−F(t,x2
)‖≤λ(t)‖x1
−x2
‖
For every t∈R+
, x1
, x2
∈Rn
where λR+
→R+
is
continuous such that

Emmanuel, et al.: On application of the fixed-point theorem to the solution
AJMS/Apr-Jun-2019/Vol 3/Issue 2 8
( )
0
L t dt

∞
= +∞
∫
Assume further that
( )
0
,0
F t dt
∞
+∞
∫‖ ‖
Then, the operator T with
( )( ) ( )
( )
, , .
t
TX t F s x s ds t R
∞
= ∈
∫
mapsthespaceCn
(R+
)intoitselfandisacontraction
operator on Cn
(R+
) if L1.
In fact x, y, C R
n
n
+
( )be given. Then, due to Bielicki,
1956, we have[6]
( )
( )
( ) ( )
0
0 0
0 0
0 0
( , )
, ,0 ( ,0)
( ) ( ) ( ,0)
( ) ( ,0)
Tx F t x t dt
F t x t F t dt F t dt
t x t dt F t dt
t dtx F t


∞
∞
∞ ∞
∞ ∞
∞ ∞
∞
≤
≤ − +
≤ +
≤ +
∫
∫ ∫
∫ ∫
∫ ∫
Which shows that TC R C R
n
n
n
n
+ +
( )⊂ ( ). We also
have
( )
( )
( )
0
0
, ( , ( )
( ) ( )
∞
∞
∞
∞
− ≤ −
≤ −
≤ −
∫
∫
Tx Ty F t x t F t y t dt
t x t y t dt
x y

‖ ‖ ‖ ‖
‖ ‖
‖ ‖
It follows that if L1, the equation Tx=x has a
unique solution x
̄ in C R
n
n
+
( ), thus a unique
x C R
n
n
�∈ ( )
+ such that
x t F s x s ds t R
t
( ) = ( )
( ) ∈
∞
+
∫ , ,� �
It is easily seen that under the above assumptions
on F and L (Banach, 1922), the equation is as
follows as:
( ) ( ) ( )
( , )
t
x t f t F s x s ds
∞
= + ∫
Furthermore, it is a unique solution in C R
n
n
+
( ) if
f is a fixed function in C R
n
n
+
( ). This solution
belongs to Cn
if fϵCn
.
THE SCHAUDER–TYCHONOOFF
FIXED-POINT THEOREM
Before we state the Schauder–Tychonoff theorem,
we characterize the compact subsets of Cn
[a,b]. [7]
This characterization, which is contained
in theorem 2 and 5, allows us to detect the relative
compactness of the range of an operator defined on
a subset of Cn
[a,b] and has values in Cn
[a,b]. We
define below the concept of a relatively compact,
and a compact, set in a Banach space.[11-16]
Definition 2.1[8]
Let X be a Banach space. Then, a subset M of X is
said to be “compact” if every sequence x
n
n
{ }
∞
=1
in M contains a subsequence which converges,
i.e., x
n
n
{ }
∞
=1
from M contains a subsequence
which converges to a vector in X.
It is obvious from this definition that is relatively
compact if and only if M (the closure of M in the
norm of X) is compact. The following theorem
characterizes the compact subsets of Cn
[a,b].
Theorem 2.1 (H-gham and Taylor, 2003)
Let M be a subset of Cn
[a,b]. Then, M is relatively
compact if and only if
(i) There exists a constant K such that ‖f‖∞≤K, f∈M
(ii) The set M is “equicontinuous” that is for every
ϵ0 there exists δ(ϵ)0 depending only on (ϵ)
such that ‖f (t1
)−f(t2
)‖ϵ for all t1
, t2
∈ [a,b] with
|t1
−t2
|δ(ϵ) and all f∈M
The proof is based on Lemma 2.3. We start with
definition 2.2.
Definition 2.2[9]
Let M be a subset of the Banach space X and let ϵ0
be given. Then, the set M1
⊂X is said to be an “ϵ-net
of M” if for every point x∈M1
such that ‖x−y‖ϵ.
Lemma 2.2 (Andrzej and Dugundji, 2003)
Let M be a subset of a Banach space X. Then, M is
relatively compact if and only if for every ϵ0 and
there exists a finite ϵ net of M in X.

Proof
Necessity. Assume that M is relatively compact
and the condition in the statement of the lemma
is not satisfied. Then, there exists some ϵ0
0, for
which there is no finite ϵ0
net of M, Choose x1
∈M,
then, {x1
} is not an ϵ0
net of M. Consequently,
‖x2
−x1
‖≥ϵ0
for some x2
∈M. Now consider the set
{x2
−x1
}. Since this set is not an ϵ0
net of M, there
exists x3
, ∈, with ‖x3
−x1
‖≥ϵ0
for i=1, 2. Continuing
the same way, we construct as certain any Cauchy
sequence, and it follows that no convergent
subsequence can be extracted from {xn
}. This is
a contradiction to the compactness of M; thus, for
any ϵ0, there exists a finite ϵ net for M
Sufficiency
Suppose that for every ϵ0, there exists a finite ϵ net
for M and consider a strictly decreasing sequence,
n=1, 2… of positive constants such that limn→∞
ϵn
=0.
Then, for each n=1, 2…, there exists a finite ϵ net
of M; if we construct open balls with centers at the
points of the ϵ1
net and radii equal to ϵ1
, then every
point of M belongs to one of these balls.
Now, let { }
x
n
n
∞
=1
be a sequence in M. Applying
the above argument, there exists a subsequence of
{ } 1,2 { }
1
j
n n
x n say x
j
∞
= …
= which belongs to
one of these ϵn
- balls. Let B (y1
) be the ball with
center y1
. Now, we consider the ϵ2
net of M. The
sequence{x’} has now a subsequence
{ }
''
, 1,2
n
x n
= … which is contained in some
ϵ2
- ball. Let us call this ball B (y2
) with center at y2
.
Continuing the same way, we obtain a sequence of
balls { }
y
n
n
∞
=1
with centers at yn+1
, radii −ϵn
, and
with the following property: The intersection of
any finite number of such balls contains a
subsequence of {xn
}. Consequently, choose a
subsequence { } { }
1
k
n n
x of x
k
∞
=
as follows:
( ) ( ) ( ) ( )
1 2
1 2 1
1
1
, ... ,
with
j
n n i
i
j j i
x B y x B y B y x B y
n n n
=
− −
∈ ∈ ∩ ∈
…

Since xnj
, xnk
∈B(y2
) for j≤k, we must have
2
j k j k k
n n n k k n
x x x y y x
     
‖ ‖‖ ‖ ‖ ò
Thus, {xnj
} is a Cauchy sequence, and since X
is complete, it converges to a point in X. This
completes the proof.
Proof of Theorem 2.1
Necessity suffices to give the proof for n=1. We
assume that M is relatively compact. Lemma 2.1
implies now the existence of a finite ϵ-net of M
for any ϵ0. Let x1
, x2
(t)… xn
(t), t∈ [a.b] be the
function of such an ϵ-net. Then, for every f∈M,
there exists xk
(t) for which ‖f−xk
‖∞ϵ consequently,

( )
( ) ( ) ( ) .
k
k k
k
f t x t f t x t
x f x
x 
∞ ∞
∞
≤ + −
≤ −
+
‖ ‖‖ ‖
‖ ‖
(2.1)
Choose, now, K=max‖xk
‖∞+ϵ. Since each function
xk
(t) is uniformly continuous on [a,b,], there exists
δk
(ϵ)0, k=1,2… such that
|xk
(t1
)|∞ϵ for |t1
−t2
|δk
(ϵ)
δ=min {δ1
,δ2
,…δn
} suppose that x is a function in
M and let xj
be a function of the ϵ-net for which
‖x−xj
‖∞ϵ. Then,
     
 
1 2 1 1 1 2
1 2
( ) ( ) ( )
( )

     
    
   
j j j
j j j
x t x t x t x t x t x t
x x x t x t

  
‖ ‖
(2.2)
For all t1
, t2
∈ [a,b] with |t1
−t2
|δ(ϵ). Consequently,
M is equicontinuous. The boundedness of M is as
follows.
Sufficiency
Fix ϵ0 and pick δ=δ(ϵ)0 from the condition
of equicontinuity. We are going to show the
existence of a finite ϵ-net for M Divide [a,b] into
subintervals [tk−1
, tk
], k=1,2,…n with t0
=a, tn
=b,
and tk−
tn−1
δ. Now, define a family P of polygons
on [a,b] as follows: the function f: [a,b]→[−K,K]
belongs to P if and only if f is a line segment on
[tk−1
, tk
] for k=1,2…and f is continuous. Thus, if
f∈P, its vertices (endpoints of its line segments)
can appear only at the points (tk
, f(tk
), k=0,1…n.
It is easy to see that P is a compact set in C1
[a,b]. We show that P is a compact ϵ net of M. To
this end, let t ∈ [a,b]. Then, t ∈ [tj−1
, tj
] for some
j=1,2…n. If Mj and mj
denote the maximum and
the minimum of tj−1
, tj
, respectively, then
mj
≤x(t)≤Mj
mj
≤x
̄ (t)≤Mj

Where, x
̄ 0
:[a,b]−[K,K] is a polygon in P such that
x
̄ 0
(tk
)=tk
k=1,2…n. It follows that
|x(t)−x
̄ 0
(t)|=≤Mj
−mjϵ
Thus, P is a compact ϵ net for M. The reader can
now easily check that since P has a finite ϵ-net, say
N, the same N will be a finite 2ϵ-net for M. This
completes the proof.
The following two examples give relatively
compact subsets of functions in Cn
[a,b].
Example 2.3: let M C a b
n
∈ 1
[ , ] be such that there
exists positive constants K and L with the property:
i. ‖x(t)‖≤K, t∈ [a,b] (2.3)
ii. ‖x’(t)‖≤L, t∈ [a,b]
For every x∈M, M is a relatively compact subset
of C a b
n
1
[ , ]; in fact, the equicontinuity of M
follows from the Mean Value Theorem for
scalar-valued functions.
Proof for Equicontinuity
Consider the operator T of example 1.3.2. Let
M⊂Cn
[a,b] be such that there exists L0 with the
property:
‖x(t)‖≤L for all x∈M
Then, the set S={Tu: u∈M} is a relatively compact
subset of Cn
[a,b]. In fact, if
N K t s ds
t a b a
b
=
∈
∫
sup ( , )
[ , ]
　　
1 ,
‖f‖≤LN for any f∈S. Moreover, for f=Tx, we have
　　　
f t f t K t s K t s x s ds
L K t s K
a
b
a
b
1 2 1 2
1
( )− = ( )− ( )

 
 ( )
≤ ( )−
∫
∫
( ) , ,
, t
t s ds
2 ,
( )
This proves the equicontinuity of S
Definition 2.3 (Banach, 1922)
Let X be a Banach space. Let M be a subset of X.
Then, M is called “convex” if λx+(1−λ)y∈M for
any number λ∈ [0,1] and any x,y∈M.
Theorem 2.2 (Schauder–Tychonoff)
Let M be a closed, convex subset of a Banach
space X. Let T: M→M be a continuous operator
such that TM is a relatively compact subset of X.
Then, T has a fixed point in M.
It should be noted here that the fixed point of T in
the above theorem is not necessarily unique. In the
proof of the contraction mapping principle, we
saw that the unique fixed point of a contraction
operator T can be approximated by terms of a
sequence { } 1
with 1,2,
0
n j j
x x Tx j
n −
∞
= = …
=
Unfortunately, general approximation methods
are known for fixed points of operators as in
theorem 2.2. which suggests the following
definition of a compact operator.
Definition 2.4[10]
Let X, Y be two Banach spaces and M a subset of
X. An operator T: M→Y is called “compact” if it
is continuous and maps bounded subsets of into
relatively compact subsets of Y.
The example 2.14 below is an application of the
Schauder–Tychonoff theorem.
( ) ( )+ ( ) ( )
∫
Tx t F T K t s x s ds
a
b
,
Where, f∈Cn
[a,b] is fixed and K: [a,b]x[a,b]→Mn
is continuous. It is easy to show as in examples
1, 3 and 2, 9, that T is continuous on Cn
[a,b] and
that every bounded set M⊂Cn
[a,b] mapped by T
onto the set TM is relatively compact. Thus, T is
compact. Now let
M={u∈Cn
[a,b]; ‖u‖∞≤L}
Where L is a positive constant. Moreover, let
K+LN≤L where,
1
[ , ]
, sup ( , )
b
t a b a
K f N K t s ds
∞
∈
= = ∫
‖ ‖ ‖
Then, M is a closed, convex, and bounded
subset of Cn
[a,b] such that TM⊂M. By the
Schauder–Tychonoff theorem, there exists at least
one x0
∈Cn
[a,b] such that x0
=Tx0
. For this x0
, we
have
( ) ( ) ( ) ( )
0 0
, , [ , ]
b
a
x t f t K t s x s ds t a b
=
+ ∈
∫
Corollary 2.4 (Brouwer’s Theorem: Let
S={u∈Rn
;‖u‖≤r}
where r is a positive constant. Let T: S→S be
continuous. Then, T has a fixed point in S
Proof
This is a trivial consequence of theorem 2.2
because every continuous function f: S→Rn
is
compact.

THE LERAY–SCHAUDER THEOREM
Theorem 3.1 (Leray–Schauder)[8]
Let X be a Banach space and consider the equation.
S(x,μ)−x=0(3.1)
where:
i) SX x [0,1]→X is compact in its first variable
for each μ∈ [0,1]. Furthermore, if M is a
bounded subset of X, S (u,μ) is continuous
in μ uniformly with respect to u∈M; that is
for every ϵ0, there exists sδ(ϵ)0 with the
property: ‖S(u,μ1
)−S(u,μ2
‖ϵ for every μ1
, μ2
∈
[0,1] with |μ1
−μ2
|δ(ϵ) and every u∈M
ii) S(x,μ0
)=0 for some μ0
∈ [0,1] and every x∈X
iii) Ifthereareanysolutionsxμ
oftheequation(2.4),
they belong to some ball of X independently of
μ∈ [0,1].
Then, there exists a solution of (3.1) for every
μ∈ [0.1],
The main difficulty in applying the above theorem
lies in the verification of the uniform boundedness
of the solutions (condition (iii). There are no
general methods that may be applied to check
condition.
As an application of theorem 3.1, we provide
example 3.1
Example 3.1 (Arthanasius, 1973)
Let F: Rn
→Rn
be continuous and such that for
some r0
F(x), x≤‖x‖2
whenever ‖x‖r
Then, F(x) has at least one fixed point in the ball
Sr
=[u∈Rn
: ‖u‖r]
Proof
Consider the equation
μF(x)−(1+ϵ)x=0
With constants μ∈ [0,1], ϵ0. Since every
continuous function F: Rn
→Rn
is compact, the
assumptions of theorem 3.1 will be satisfied for
(3.1) with S(x,μ)=[μ⁄((1+ϵ)]F(x)) if we show that
all possible solutions of (3.1) are in the ball Sr
. In
fact, let x
̄ be a solution of (3.1) such that ‖x
̄ ‖r.
Then, we have
μF(x
̄ )−(1+ϵ) x
̄ , x
̄ =0
Or
μF(x
̄ ) x
̄ =(1+ϵ)x
̄ , x
̄ =(1+ϵ)‖x
̄ ‖2
This implies that
F(x
̄ )x
̄ )≥(1+ϵ)‖x
̄ ‖2
For some x
̄ ∈Rn
with ‖x
̄ ‖r, which is a contradiction
to (3.1).
It follows by theorem 3.1 that for every ϵ0, the
equation (3.1) has a solution xϵ
for μ=1 such that
‖xϵ
‖≤r. Let ϵm
=1⁄(m,m=1,2…) and let x(∈m
)=xm
since the sequence x
m
n
{ }
∞
=1
is bounded, it
contains a convergent subsequence x
k
n
{ }
∞
=1
� let
xmk
→x0
as k→∞ then x0
∈Sr
and
( ) (1 (1/ )) 0, 1,2 .
k k
m k m
F x m x k
− + = = …
Taking the limit of the left hand side of the above
equation as k→∞ and using the continuity of F,
we obtain
F (x0
)=x0
Thus, x0
is a fixed point of F in Sr
.
CONCLUSION
The problem of fixed point is the problem of
finding the solution to the equation y=Tx
=0. It is
important that the domain of T and the range of
T have points in common, and in this case, such
points of x for which Tx
=x are regarded as the fined
points of the operator T; also, the work reveals that
the contraction mapping principle must be satisfied
for a fixed point to exist as other basic results
center on the need for the relative compactness of
a subset, M of Cn
[a,b] if there must be a fixed
point of T in M.
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On Application of the Fixed-Point Theorem to the Solution of Ordinary Differential Equations