Journal of Computations & Modelling, vol.5, no.3, 2015, 99-122
ISSN: 1792-7625 (print), 1792-8850 (online)
Scienpress Ltd, 2015
On Quadratic Abstract Measure
Integro-Differential Equations
S.S. Bellale1 and S.B. Birajdar 2
Abstract
In this paper a nonlinear abstract measure quadratic integro – differential equation
is studied in Banach Algebra. The existence of solution abstract measure integro –
differential equations is proved for extremal solutions for Caratheodory as well as
discontinuous cases of the non-linearity involved in the equations.
Mathematics Subject Classification: 34G99
Keywords: Abstract measure integro-differential equation; existence theorem;
extrermal solutions and Caratheodary condition
1 Introduction
For a given closed and bounded interval J = [ 0 ,a ] in R the set of real numbers,
1
2
Department of Mathematics, Dayanand Science College, Latur, Maharashtra, India.
E-mail: sidhesh.bellale@gmail.com
Samarth Krupa, Shrdha nagar, Latur, Maharashtra, India.
E-mail: birajdarsujata1@gmail.com
Article Info: Received : March 12, 2015. Revised : May 5, 2015.
Published online : June 10, 2015.
100
On Quadratic Abstract Measure Integro-Differential Equations
consider the integro differential equation in short (IGDE)

x (t )

 f t,x (ξ ( t ) )

(
1
)




g ( t,x (δ ( t ) ) , ∫ k ( s,x (η ( s ) ds ) , a.e. t ∈ J
t
0
(1.1)
Where f : J × R → R − {0} is continuous,
g : J × R× R → R & k : J × R → R
ξ , δ ,η : J → J
and
The existence of solutions of IGDE (1.1) is proved in Dhage [ 7 ] by using a
new non-linear alternative of Leray – Schauder type developed in same paper. In this
chapter we apply a nonlinear alternative of Leray – Schauder type involving the
product of two operators in a Banach algebra under some weaker conditions than
that given in Dhage and Regan [8] to a quadratic measure differential equation
related to IGDE (1.1) for proving the existence results. The existence of exterimal
solutions is also proved using a fixed point theorem of in ordered Banach algebras.
In the first section introduction is given in section II we state the abstract
measureintegro differentiate equation to be discussed in this paper. The section III
the auxiliary results are given and the existence result is discussed in section
IV.Finally the existence results for extermal solutions for the integro differential
equations is discussed in section V.
2
Quadratic Integro differential equations
Let X be a real Banach algebra with a convenientnorm
⋅ .
Let x, y ∈ X . Then the line segment xy in X is defined by
{
}
xy = z ∈ X z = x + r ( y − x ) , 0 ≤ r ≤ 1
Let x0 ∈ X be a fixed pointand z ∈ X .
Then for any x ∈ x0 z, we define the sets S x and S x in X by
(2.1)
S. S. Bellale and S. B. Birajdar
=
Sx
S=
x
Let x1 ,x2 ∈ xy
101
{rx −∞ < r ≤ 1}
{rx −∞ < r ≤ 1}



(2.2)
be arbitrary. We say x1 < x2 if S x1 ⊂ S x2 , or equivalently
x0 x1 ⊂ x0 x2 . In this case we also write x2 > x1 let M denote the σ -algebra of all
subsets of X such that
( X ,M )
is a measureable space. Let AC' ( X ,M ) be the
space of all vector measures (real signed measures) and define a norm ⋅
on
AC' ( X ,M ) by
p = p (X )
(2.3)
where p is a total variation measure of p and is given by
p ( X ) sup
=
∞
∑ p ( Ei ) ,
Ei ⊂ X
i =1
where supremum is taken over all possible partition
(2.4)
{Ei;i ∈ N }
that AC ( X ,M ) is a Banach space with respect to the norm
of X. It is known
⋅ , given by (2.3).
For any nonempty subset S of X, let L1µ ( S ,R ) denote the space of µ -integrable
real valued functions on S which is equipped with the norm
φ
L1µ
⋅
given by
L1µ
= ∫ φ ( x) d µ
S
For φ ∈ L1µ ( S ,R ) . Let
p1 , p2 ∈ AC ( X ,M )
and
define
a
multiplication
composition in AC ( X ,M ) by
=
( p1 ∗ p2 )( E ) p1 ( E ) p2 ( E ) for all E ∈ M . Then we have
Lemma 2.1 AC ( X ,M ) is a Banach algebra.
Proof. Let p1 , p2 ∈ AC ( X ,M ) be two elements. Let σ = { E1 ,E2 .....En .....} be a
disjoint partition of X. Then by (2.3) – (2.4),
102
On Quadratic Abstract Measure Integro-Differential Equations
p1 p2 = p1 p2 ( X )
∞
= sup ∑ ( p1 ∗ p2 )( Ei )
σ
i =1
∞
= sup ∑ p1 ( E1 ) p2 ( E2 )
σ
i =1
 ∞
 ∞

≤ sup  ∑ p1 ( E1 )   ∑ p2 ( E2 ) 
σ  i 1=
=
i1

∞
∞



=  sup ∑ p1 ( E1 )   sup ∑ p2 ( E2 ) 
=
 σ i 1=
 σ i1

= p1 p2
Hence AC ( X ,M ) is a Banach algebra.
Let µ be a σ finite measure on X, and let p ∈ AC ( X ,M ) . We say p is a
absolutely continuous with respect to the measures µ if µ ( E ) = 0 implies
p ( E ) = 0 for some E ∈ M . In this case we also write p << µ .
Let x0 ∈ X be fixed and let M 0 denote the σ -algebra on Sx0 let z ∈ X
be such that z > x0 and let M z denote the σ -algebra of all sets containing M.
and the sets of the form Sx,x ∈ x0 z. Given a p ∈ AC ( X ,M ) with p << µ ,
consider the abstract measure integro - differential equation (AMIGDE) of the form
( )  = g x, p S
(
( ( )  (

p Sx
d 
d µ  f x, p S x (ξ )

and
x
(δ ) ) , ∫Sx k ( t, p ( S t (η ) ) d µ
=
p(E) q(E) ,
) , a.e.[ µ ] on x z
0
E ∈ M0
( )
where q is a given known vector measure, λ S x =
(2.5)
(2.6)
( )
f ( x, p ( S ) )
p Sx
is a singed
x
measure suchthat λ << µ ,
to µ ,
dλ
is a Radon
dµ
Nikodym derivative of λ with respect
S. S. Bellale and S. B. Birajdar
103
f : S x × R → R − {0} , g : S z × R × R → R and the map
) ( (
( (
) )
x → g x, p S x (δ ) , ∫ k k , p S t (η ) d µ
Sx
is µ -integrable for each p ∈ AC ( X ,M z )
.
Definition 2.1 Given an initial real number
q
on
M 0 , a vector
p ∈ AC ( S z ,M z ) ,( z > x0 ) is said to be a solution of AMIGDE (2.5) – (2.6), if
p (E) q (E) , E ∈ M0
i) =
ii)
p << µ on x0 z , &
iii)
satisfies ( 2.5 ) a.e.[ µ ] on x0 z
Remark 2.1 The AMIGDE (2.5) (2.6) is equivalent to the abstract measure integral
equation (in short AMIE).
( (


p ( E ) =  f x, p ( E1 (ξ )   ∫ g x, p S x (δ ) , ∫ k t, p S x (η ) d µ  ,


Sx
E

(
if E ∈ M z ,
)
( (
)
) )
E ⊂ x0 z
(2.7)
=
p(E) q(E)
if E ∈ M 0
(2.8)
A solution p of abstract measure AMIGDE (5.2.5) – (5.2.6) in x0 z will be denoted
(
)
by p Sx0 ,q .
Note that the above equations includes the abstract measure differential equation
considered in Dhage and Bellale [ 6] as a special case. The see this, define
f ( x, y ) = 1 for all x ∈ x0 z & y ∈ R then AMIGD (2.5) – (2.6) reduces to
( (
)
( (
) ) ( a.e.[ µ ] on x z
dp
= g x, p S x (δ ) , ∫ k t, p S t (η ) d µ
Sx
dµ
p (E) q (E) , E ∈ M0
=
0
( 2.9 ) )
( 2.10 )
Thus our AMIGDE (2.5) – (2.6) is more general and we claim that it is a new to the
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On Quadratic Abstract Measure Integro-Differential Equations
literature on measure differential equations.
Now we shall prove the existence theorem.
3
Auxiliary Results
Let X be a Banach space andlet T : X → Y . T is called compact if T ( x ) is a
compact subset of X. T is called totally bounded if for any bounded subset S of X, T
(S) is a totally bounded subset of X. T is called completely continuous if T is
continuous and totally bounded on X. Every compact operator is totally bounded,
but the converse may not be true, however two notions are equivalent on a bounded
subset of X.
An operator T : X → Y is called D – Lipschitz if these exists a continuous
and non-decreasing function ψ : R + → R + such that
Tx − Ty ≤ ψ ( x − y
)
(3.1)
for all x, y ∈ X , where ψ ( 0 ) = 0 . The function ψ is called a D – function of T on
X. In particular if ψ ( r ) =∝ r, ∝> 0 , T is called a Lipschitz with the Lipschitz
constants ∝ further if ∝< 1 ,
then T is called a contraction with contraction
constant ∝ . Again if ψ ( r ) < r for r > 0 , then T is called a non-linear contraction
on X with D – function ψ .
Theorem 3.1 Let U and U denote respectively the open and closed bounded subset
of a Banach algebra X such that 0∈U let A,B :U → X be two operators such
that
i)
A is D – Lipschitz
ii)
B is completely continuous and
iii)
M φ ( r ) < r, r > 0 where M = B U .
( )
S. S. Bellale and S. B. Birajdar
105
Then either
a)
The equation Ax Bx = x has a solution in U , as
b)
These is a point u ∈ d U such that u = λ Au Bu for some 0 < λ < 1 ,
where ∂ U is a boundary of U in X.
Corollary 3.1 Let Br ( 0 ) & Br ( 0 ) denote respectively the open and closed balls in
a Banach algebra centred at origin 0 of radius r for some real number r > 0 . Let
A,B : Br ( 0 ) → X be two operators such that
i)
A is Lipschitz with Lipschitzcontent α .
ii)
B is compact and continuous, and
iii)
α M < 1 , where M = B Br ( 0 )
(
)
then either
a)
The operator equation Ax Bx = x has a solution x in X with x ≤ r or
b)
These is an u ∈ X with u = r such that λ AuBu = u for some 0 < λ < 1 .
We define an order relation ≤ in AC ( S z ,M z ) with the help of the cone K in
AC ( S z ,M z ) given by
{
k=
p ∈ AC ( S z ,M z ) p ( E ) ≥ 0 for all E ∈ M z
}
(3.2)
Thus for any p1 , p2 ∈ AC ( Sz,M z ) are have p1 ≤ p2 if and only if
p2 − p1 ∈ k
(3.3)
or equivalently
p1 ≤ p2 ⇔ p1 ( E ) ≤ p2 ( E ) ........( 3.4 ) for all E ∈ M z .
Obviously the cone K is positive in AC ( S z ,M z ) . To see this,let p1 p2 ∈ K .
Then p1 ( E ) ≥ 0 and p2 ( E ) ≥ 0 for all E ∈ M z . By multiplication composition
p2 ) E
( p1 ∗=
AC ( S z ,M z ) .
p1 ( E ) p2 ( E ) ≥ 0 for all p1 ∗ p2 ∈ k , and so K is a positive come in
106
On Quadratic Abstract Measure Integro-Differential Equations
The following lemma follow immediately from the definition of the positive cone K
in AC ( S z ,M z )
Lemma 3.1 Dhage[3] if u1 ,u2 ,v1 ,v2 ∈ K
are such that u1 < v1 & u2 < v2 , then
u1 u2 ≤ v1v2 .
Lemma 3.2 The cone K is normalin AC ( S z ,M z ) .
Proof. To prove it is enough to show that norm
⋅
is semi monotone on K.Let
p1 , p2 ∈ K be such that p1 ≤ p2 on M z . Then we have
0 ≤ p1 ( E ) ≤ p2 ( E )
For all E ∈ M z .
Now for a countable partition
{En :n ∈ N } of
= p1 ( S z )
σ
=
p1
S 2 , one has
∞
= sup ∑ p1 ( Ei )
i =1
∞
≤ sup ∑ p2 ( Ei )
σ
i =1
= p2 ( S z )
= p2
As a result
⋅
is semi monotone on k andconsequently the cone K is normal in
AC ( S z ,M z ) . The proof of the lemma is complete.
An operator T : X → X is called positive if the range r (T ) of T is contained in
the cone K in X.
Theorem 3.2 Dhage [ 4] . Let [u,v ] be an order interval in the real Banach algebra
X and let A,B : [u,v ] → [u,v ] be positive and non-decreasing operators such that
S. S. Bellale and S. B. Birajdar
107
i)
A is Lipschitz with a Lipschitzconstant α ,
ii)
B is compact & continuous, and
iii)
The elements u,v ∈ X with u ≤ v .
Satisfy u ≤ AuBu and AvBv ≤ v .
Further if the cone K is normal, then the operator AxBx = x has a least and a
greatest
=
M
positive
solution
in
[u,v ]
,
whenever
αM <1 ,
where
B
=
([u,v ]) sup { Bx : x ∈ [u,v ]} .
Theorem 3.3 Dhage [ 4] let k be a positive cone in a real Banach algebra X
and let A,B : K → K be non-decreasing operators such that
i)
A is Lipschitz with the Lipschitz constant ∝
ii)
B is bounded, and
iii)
There
exist
elements
u,v ∈ k
such
that
u≤v
satisfying
u ≤ AuBu and AvBv ≤ v .
Further, if the cone k is normal then the operator equation AxBx = x has a least
and greatest positive solution in [u,v ] , whenever ∝ M < 1 where
=
M
4
B
=
([u,v ]) sup { Bx : x ∈ [u,v ]}
Existence Result
We need the following definition.
Definition 4.1 A functions β : S z × R × R → R is called Caratheodory if
i) x → β ( x, y1 , y2 ) is µ -measurable for each y1 , y2 ∈ R and
ii) The function
( y1 , y2 ) → β ( x, y1 , y2 )
on x0 z .
is continuous almost everywhere [ µ ]
108
On Quadratic Abstract Measure Integro-Differential Equations
A Caratheodary function β on S z × R × R is called L1µ Caratheodary if
iii) For each real number r > 0 there exists a functions hr ∈ L1µ ( S z Rt ) such that
β ( x, y1 , y2 ) ≤ hr ( x ) a.e.[ µ ] on x0 z
For all y1 , y2 ∈ R with y1 ≤ r & y2 ≤ r .
A function ψ : R+ → R+ is called sub multiplicative if ψ ( λ r ) ≤ λψ ( r ) for
all real number λ > 0 . Let ψ denote the class of function ψ : R+ → R+ satisfying
the following properties:
i)
ψ is continuous
ii)
ψ is non-decreasingand
iii)
ψ issub multiplicative
A member ψ ∈Ψ is called a D – function on R+ . These do exist D – function, in
fact, the function ψ : R+ → R+ defined by ψ ( λ ) = λ r , λ > 0 is a D – function on
R+ we consider the following set of assumptions:
( A0 ) for
any z > x0 , the σ -algebra M z is compact with respect to the topology
generated by the Pseudometric d defined on M z by
D ( E1 ,E2 ) =
µ ( E1 ∆ E2 ) , E1 E2 ∈ M z
( A1 )
The function x → f ( x,o ) is bounded with Fo = sup x∈S z f ( x,o )
( A2 )
The function is continuous and these exists a bounded function α : S z → R +
with bound α
such that
f ( x, y1 ) − f ( x, y2 ) ≤ α ( x ) y1 − y2 , a.e.[ µ ] ,x ∈ x0 z
For all y1 , y2 ∈ R
( H0 )
q is continuous on M z with respect to the pseudo-metric d defined in
( H1 )
The function x → k x, p S x (η )
( (
)}
is µ -integrable& satisfies
( A1 ) .
S. S. Bellale and S. B. Birajdar
109
k ( t, y ≤ γ ( x ) y , a.e.[ µ ] on x0 z
For all y ∈ R
( H2 )
The function g ( x, y1 , y2 ) is Caratheodary
( H3 )
There exists a function φ ∈ L1µ ( S z ,R + )
on x0 z and
such that φ ( x ) > 0 a.e. [ µ ]
D – function ψ : [ 0 , ∞ ] → ( 0 , ∞ ) such that
g ( x, y1 , y2 ) ≤ φ ( n )ψ ( y1 + y2 ) a.e. [ µ ] on x0 z
For all y1 , y2 ∈ R
We frequently use the following estimate of the function g in the subsequent part of
the paper. For any p ∈ AC ( S z ,M z ) , one has
( (
)
( (
))
g x, p S x (δ ) , ∫ k t, p S t (η ) d µ
(
Sx
)
)) d µ )
≤ φ ( x )ψ ( p ( S (δ ) ) + ∫ γ ( x ) p ( S (η ) ) d µ )
(
( (
≤ φ ( x )ψ p S x (δ ) + ∫ k t, p S t (η )
Sx
z
≤ φ ( x )ψ
( p +∫
Sz
Sz
γ ( x) p d µ
( p+γ p)
≤ φ ( x ) (1 + γ )ψ ( p )
≤ φ ( x )ψ
z
)
L1µ
L1µ
Theorem 4.1 Suppose that the assumptions
( A0 ) − ( A2 ) & ( H1 ) − ( H 3 )
holds.
Suppose that there exists a real number r > 0 such that
(
)
F0  q + φ L1 1 + γ L1 ψ ( r ) 
µ
µ


r>
1 − α  q + φ L1 1 + γ L1 ψ ( r ) 


µ
µ
where α  q + φ

L1µ
(
)
(1 + γ )ψ ( r ) < 1
L1µ
& F0 = sup f ( x,o ) . Then the AMIGDE (2.5) – (2.6) has a solution on x0 z .
x ∈ Sz
( 4.1)
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On Quadratic Abstract Measure Integro-Differential Equations
Proof :- Consider on open ball B r ( 0 ) in AC ( S z ,M z ) centered at the origin 0and
of radius r. Where r satisfies the inequalities in (5.4.1). Define two operators.
A,B : Br ( 0 ) → AC ( S z ,M z )
by
1

Ap ( E ) = 
 f x, p ( E (ξ )
(
)
if E ∈ M 0
if E ∈ M z E ⊂ x0 z
( 4.2 )
&

q(E)

Bp ( E ) = 
g x, p S x (δ ) ,
∫
E
( (
)
if E ∈ M 0
∫
Sx
( (
) )
k t, p S t (η ) d µ d µ
( 4.3)
if E ∈ M z ,E ⊂ x0 z
We show that the operators A and B satisfy all the condition of Corollary 3.1 on
Br ( 0 ) .
Step – I
First, we show that A is a Lipschitz on Br ( 0 ) . Let p1 , p2 ∈ Br ( 0 ) be
arbitrary, then by assumption
( A2 ) ,
Ap1 ( E ) − Ap2=
(E)
(
(
f x, p1 ( E (ξ ) ) − f x, p2 ( E (ξ ) )
≤ α ( x ) p1 ( E (ξ ) ) − p2 ( E (ξ ) )
≤ α x p1 − p2 ( E )
for all E ∈ M z . Hence by definition of the norm in AC ( S z ,M z ) one has
Ap1 − Ap2 ≤ α
p1 − p2
For all p1 , p2 ∈ AC ( S z ,M z ) . As a result. A is a Lipschitz operator Br ( 0 ) with the
Lipschitzconstant α .
Step – IIWe show that B is continuous on Br ( 0 ) . Let
{ pn }
be sequence of vector
measure in Br ( 0 ) converging to a vector measure. Then by dominated convergence
theorem,
S. S. Bellale and S. B. Birajdar
111
( (
( (
)
))
lim Bpn ( E ) = lim ∫ g x, pn S x (δ ) , ∫ k t, p S t (η ) d µ
Sx
h→∞
h→∞ E
( (
) (t, p ( S (η )) d µ ) d µ
= ∫ g x, p S x (δ ) , ∫
E
t
Sx
= Bp ( E )
for all E ∈ M z , E ⊂ x0 z . Similarly if E ∈ M 0 then
lim Bpn=
( E ) q=
( E ) Bp ( E ) and so B is a continuous operator on Br ( 0 ) .
n→∞
Step – III Next, we show that B is a totally bounded operator on Br ( 0 ) . Let
be a sequence an Br ( 0 ) . Then
{Bpn :n ∈ N }
the set
we have
{ pn }
pn ≤ r for all n ∈ N . We show that
is uniformly bounded andequicontinuous set in AC ( S z ,M z ) .
{Bpn }
In this step, we first show that
is uniformly bounded.
Let E ∈ M z . Then there exists two subsets F ∈ M 0 & G ∈ M z , G ⊂ x0 z such that
E = F ∪G & F ∧G =φ
Hence ,
( (
)
( (
)
Bpn ( E (σ ) ) ≤ q ( F ) + ∫ g x, pn S x (δ ) , ∫ k t, pn S t (η ) d µ d µ
G
)ψ ( p
(
≤ q + ∫ φ ( x ) (1 + γ )ψ ( p
=+
q φ (1 + γ ψ ( p )
G
L1µ
n
)dµ
E
L1µ
n
)dµ
≤ q + ∫ φ ( x) 1+ γ
L1µ
L1µ
Sx
n
For all E ∈ M z .
from (3.3) it follows that
∞
sup ∑ Bpn ( Ei ) =
Bpn =
Bpn ( S z ) =
q +φ
σ
=+
q φ
L1
i =1
L1µ
(1 + γ )ψ ( p )
L1µ
(1 + γ )ψ ( r )
L1µ
For all n ∈ N . Hence the sequence
{Bpn }
(
is uniformly bonded in B Br ( 0 )
)
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On Quadratic Abstract Measure Integro-Differential Equations
{Bpn : n∈ N }
Step – IV Next we show that
is equicontinuous set in AC ( S z ,M z ) .
Let E1 ,E2 ∈ M z , then there exist
F1 ,F2 ∈ M 0 & G1 ,G2 ∈ M z , G1 ⊂ x0 z
and G2 ⊂ x0 z
φ ) and E=
such that E=
F1 ∪ G1 with ( F1 ∩ G1 =
F2 ∪ G2 with
1
2
F2 ∩ G2 =
φ.
We know the identities
G1 = ( G1 − G2 ) ∪ ( G1 ∩ G2 ) 

G2 = ( G2 − G1 ) ∪ ( G1 ∩ G2 ) 
( 4.4 )
There fore, we have
Bpn ( E1 ) − Bpn ( E2 ) ≤ q ( E1 ) − q ( F2 )
+∫
G1 −G2
+∫
G2 −G1
( ( ) ( ( )) )
g ( x, p ( S (δ ) ) , ∫ k ( t, p ( S (η ) ) ) d µ ) d µ
g x, pn S x (δ ) , ∫ k t, pn S t (η ) d µ d µ
Sx
x
n
n
Sx
t
( H3 )
Since g is Caratheodory and satisfies
We have that
Bpn ( E1 ) − Bpn ( E2 ) ≤ q ( F1 ) − q ( F2 )
+∫
G1∆G2
(
)
(
( (
))
g x, pn S (δ ) , ∫ k t, pn S t (η ) d µ d µ
≤ q ( F1 ) − q ( F2 ) + ∫
G1∆G2
Sx
(
φ ( x) 1+ γ
L1µ
)ψ ( p
n
)dµ
Assume that d ( E1 ,E2 )= µ ( E1∆E2 ) → 0 .
Then we have E1 → E2 . As a result F1 → F2 and µ ( G1∆G2 ) → 0 . As q is
continuous on compact M z , it is uniformly continuous and so
Bpn ( E1 ) − Bp2 ( E2 ) ≤ q ( F1 ) − q ( F2 ) + ∫
G1∆G2
as
(
φ ( x) 1+ γ
L1µ
)ψ ( p
n
)dµ →0
E1 → E2
This shows that
{Bpn :n ∈ N }
is a equicontinuous set in AC ( S z M z ) . Now an
S. S. Bellale and S. B. Birajdar
113
application of the Arzela – Ascolli theorem yields that B is a totally bounded
operator on Br ( 0 ) . Now B is continuous and totally bounded operator on Br ( 0 ) ,
it is completely continuous operator on Br ( 0 ) .
Step V Finally we show that hypothesis (iii) of Corollary 3.1. The Lipschitz constant
of A is α . Here the number M in the hypothesis (iii) is given by
( )
sup { Bp : p ∈ Br ( 0 )}
sup { Bp ( S ) : p ∈ Br ( 0 )}
M = B Br ( 0 )
=
=
( 4.5)
z
Now let E ∈ M z . Then there are sets F ∈ M 0 and G ∈ M z , G ⊂ x0 z such that
E = F ∪ G and F ∩ G = φ
From the definition of B is follows that
( (
)
( (
)
)
Bp=
( E ) q ( F ) + ∫ g x, p S x (δ ) , ∫ k t, p S t (η ) d µ d µ
G
Sx
Therefore,
( (
)
( (
) )
Bp ( E ) ≤ q ( F ) + ∫ g x, p S x (δ ) , ∫ k t, p S t (η ) d µ d µ
(
G
≤ q + ∫ φ ( x) 1+ γ
G
(
L1µ
≤ q + ∫ φ ( x) 1+ γ
x0 z
q +φ
=
L1µ
)ψ ( p ) d µ
)ψ ( p ) d µ
Sx
L1µ
(1 + γ )ψ ( p )
L1µ
Hence from (4.6) it follows that
Bp ≤ q + φ
For all p ∈ Br ( 0 ) . As a result are have
L1µ
(1 + γ )ψ ( p )
L1gm
114
On Quadratic Abstract Measure Integro-Differential Equations
(
)
=
M B Br ( 0 ) ≤ q + φ
Now α M ≤ α ⋅  q + φ

(1 + γ )ψ ( p )
(1 + γ )ψ ( r )
L1µ
L1µ
L1µ
L1µ
<1
and so, hypothesis (iii) of corollary 3.1 is satisfied.
Now an application of corollary 3.1 yields that either the operator Ax Bx = x has a
solution, or there exist u ∈ AC ( S z ,M z ) such that u = r satisfying u = λ AxBx
for some 0 < λ < 1 . We show that this letter assertion does not hold. Assume the
contrary. Then we have
( (
)
(
) )
( (
)
λ  f x,u G  g x,u S δ , k t,u S η d µ d µ ,
x( )
t ( )
( ) ) ∫
∫S x
  (

u(E)= 
if E ∈ M z , E ⊂ x0 z.


λ q ( E ) , if E ∈ M 0
For some 0 < λ < 1 .
If
E ∈Mz ,
then
these
sets
F ∈ M 0 and G ∈ Mz, G ⊂ x0 z
such
that
E = F ∪ G and F ∩ G = φ . Then we have
u ( E ) = λ Au ( E ) , Bu ( E )
(
( (
)
( (
)
λ  f ( x,u ( G ) )  q ( F ) + ∫ g x,u S x (δ ) , ∫ k t,u S t (η ) d µ
G
(
Sx
( (
)
( (
)
) ))
=λ  f ( x,u ( G ) ) − f ( x, 0 )  q ( F ) + ∫ g x,u S x (δ ) d µ , ∫ k t,u S t (η ) d µ
G
Sx
( (
( (
) )
+ λ f ( x, 0 )  q ( F ) + ∫ g x,u S x (δ ) , ∫ k t,u S t (η ) d µ d µ 
G
Sx


Hence
S. S. Bellale and S. B. Birajdar
(
115
u ( E ) ≤ 1 + ( x,u ( G ) − f ( x, 0 )
( (
)
( (
)
) )
. q ( F ) + ∫ g x,u S x (δ ) d µ , ∫ k t,u S t (η ) d µ d µ 
G
Sx


+ f ( x, 0 )  q ( F ) + ∫ g x,u S x (δ ) , ∫ k t,u S t (η ) d µ d µ 
G
Sx


( (
(
≤ λ α ( x ) u ( G ) + F0
( (
)
) ( q + ∫ φ ( x ) (1 + γ
G
L1µ
) )
)ψ ( u ) d µ )


≤  α u ( E ) + F0   q + ∫ φ ( x ) 1 + γ L1 ψ ( u ) d µ 
µ


x0 z


≤  α u + F0   q + φ L1 1 + γ L1 ψ ( u ) 

µ
µ

)
(
)
(
which further implies that
(
u ≤  α u  q + φ L1 1 + γ L1

µ
gm

F0  q + φ L1 1 + γ L1 ψ ( u )
µ
µ

≤
1 − α  q + φ L1 1 + γ L1 ψ u

µ
µ
(
)
(
)
)ψ ( u )  + F  q + φ (1 + γ )ψ ( u )
L1µ
0
L1µ
)


Substituting u = r in the above inequality yields
(
)
)
F0  q + φ L1 1 + r L1 ψ ( γ )

µ
µ
r≤
1 − α  q + φ L1 1 + r L1 ψ γ 
µ
µ


(
)
(4.6)
Which is a contradiction to the first inequality in (4.4). In consequence, the operator
(
equation p ( E ) = Ap ( E ) Bp ( E ) has a solution u Sx0 ,q
)
in AC ( S z ,M z ) with
u ≤ r . This further implies that the AMIGDE (2.5) – (2.6) has a solution on x0 z .
This completes the proof.
5
Existence of Extremal Solutions
In this section we shall prove the existence of a minimal and a maximal
116
On Quadratic Abstract Measure Integro-Differential Equations
solutions for the AMIGDE (2.5) – (2.6) on x0 z under Carathedory as well as
discontinuous case of non-lineality g involved in it.
5.1
Carathedory Case
We need following definitions
Definition 5.1 A vector measure u ∈ AC ( S z ,M z ) is called a lower solution of the
AMIGDE (2.5) – (2.6) if
( )  ≤ g x,u S
( (
( ( ) 

u Sx
d 
d µ  f x,u S x (ξ )

& u(E) ≤q(E) ,
x
(δ ) ) , ∫S
x
( (
) )
k t,u S t (η ) d µ a.e. [ µ ] on x0 z
E ∈ M0
Similarly a vector measure v ∈ AC ( S z ,M z ) is called an upper solution to
AMIGDE (2.5) – (2.6) if
( )
( ( )


v Sx
d 
 ≥ g x,v S x (δ ) , k t,v S t (η ) d µ
∫S x
d µ  f x,v S x (ξ ) 


and v ( E ) ≥ q ( E ) , E ∈ M 0
( (
)
( (
) )
a.e. [ µ ] on x0 z
A vector measure p ∈ A ( S z ,M z ) is a solution to AMIGDE (2.5) – (2.6) it is upper
as well as lower solution to AMIGDE (2.5) – (2.6) on x0 z .
Definition 5.2 A solution PM is called as maximal solution to AMIGDE (2.5) –
(
(2.6) if for any other solution p Sx0 ,q
)
for the AMIGDE (2.5) – (2.6) we have
that
p ( E ) ≤ pM ( E ) , ∀E ∈ M z
S. S. Bellale and S. B. Birajdar
117
(
Similarly a minimal solution pm Sx0 ,q
)
of AIGDE (2.5) – (2.6) is defined on x0 z .
We consider the following assumptions :
( C0 )
f & g define the functions
F : x0 z × R → R + − {0} and g : x0 z + R × R → R +
( C1 )
The functions f ( x, y1 ) ,k ( x, y1 ) and g ( x, y1 , y2 ) are non-decreasing
in y1 , y2 for each x ∈ x0 z .
( C2 )
The AMIGDE (2.6) – (2.6) has a lower solution u and an upper
solution v such that u ≤ v on M z .
( C3 ) The function
g ( x, y1 , y2 ) is L1µ Caratheodary.
Theorem 5.1 Suppose that the assumptions
( A0 ) − ( A2 ) , ( B0 ) − ( B2 ) , & ( C0 ) − ( C3 ) holds. Further suppose that
α
r
where =
( q + h ) <1
(5.1)
r L1µ
u + v . Then the AMIGDE (2.5) - (2.6) has a minimal and maximal
solution defined on x0 z .
Proof. AMIGDE (2.5) - (2.6) is equivalent to the abstract measures integral equation
( 5.7 ) & ( 2.8)
.
Define
the
operators
A,B : AC ( S z ,M z ) → AC ( S z ,M z )
by
(4.2)and(4.3) respectively. Then the AMIGDE (2.5) – (2.6) is equivalent to the
operator equation
=
p ( E ) Ap ( E ) Bp ( E ) , E ∈ M z
(5.2)
We shall show that the operator A and B satisfy all the conditions of theorem 3.2 on
AC ( S z ,M z ) since µ is a positive measure, from assumption
that A and B on positive operators on
( C0 )
if follows
AC ( S z ,M z ) . To show this let
118
On Quadratic Abstract Measure Integro-Differential Equations
p1 , p2 ∈ AC ( S z ,M z ) be such that p1 ≤ p2 on M z . From ( C2 ) it follows that
Ap1 ( E ) = f ( x, p1 ( E ) ) ≤ f ( x, p2 ( E ) ) = Ap2 ( E )
For all E ∈ M z , E ⊂ x0 z and Ap1 ( E ) = 1= Ap2 ( E )
For E ⊂ M 0 . Hence A is non-decreasing on AC ( S z ,M z )
Similarly, we have
(
)
(
) ( ( ))
= ∫ g ( x, p ( S (δ ) ) , ∫ k ( t, p ( S (η ) ) ) d µ ) d µ
Bp ( E ) = ∫ g x, p1 S x (δ ) , ∫ k t, p1 S t (η ) d µ d µ
E
2
E
Sx
x
2
Sx
t
= Bp2 ( E )
For all E ∈ M z , E ⊂ x0 z
Again if E ∈ M 0 , then
Bp1=
( E ) q=
( E ) Bp2 ( E )
Therefore the operator B is also non-decreasing on AC ( S z ,M z ) . Now it can be
shown that as the proof of theorem 3.1 that A is Lipschitz operator on [u,v ] with
the Lipschitz constant α . Since the cone K is normal in X, the order interval
[u,v ]
is norm-bounded.
r for all x ∈ [u,v ] . As
Hence there is a real number r > 0 such that x ≤ u + v =
g
is
L1 Caratheodary,
µ
g ( x, y1 , y2 ) ≤ hr ( x ) on
arguments
=
Br ( 0 )
as
=
[u,v
] , γ (u )
in
there
is
a
x0 z for all
the
function
(
hr : L1 S z ,R +
µ
)
such
that
y1 , y2 ∈ R . Now proceeding with the
proof
of
theorem
4.1
with
hr ( x ) andψ ( r ) = 1 , it can be proved that B is compact and
continuous operator on [u,v ] . Since u is lower solution of AMIGDE (2.5) – (2.6)
we have
S. S. Bellale and S. B. Birajdar
119

 

u ( E ) ≤ f ( x,u ( E ) )  ∫ g  x,u S x (δ ) , ∫ k t,u S t (η ) d µ  d µ 


 
Sx
E 
 
(
( (
)
))
E ∈ M z , E ⊂ x0 z
and
u ( E ) ≤ q ( E ) , if
E ∈M 0
From the above inequality in the gives
u ( E ) ≤ Au ( E ) Bu ( E ) , if E ∈ M z
and so u ≤ AuBu . Similarly, since v ∈ AC ( S z ,M z ) is an upper solution of
AMIGDE (2.5) – (2.6) it can be proved that Av ( E ) Bv ( E ) ≤ v ( E ) for all
E ∈ M z and consequently AvBv ≤ v on M z . Thus hypothesis (iii) of theorem 3.2
is satisfied. Now definition of norm, it follows that
M = B ([u,v ])
= sup { Bp : p ∈[u,v ]}
= sup { Bp ( S z ) : p ∈[u,v ]}
 sup ∞

B p ( Ei ) 
= sup

∑
p ∈ [u,v ]  σ i =1

For
any
=
σ
partition
{Ei :i ∈ N }
of
Sz
such
that
∪i∞=1 Ei, Ei ∩ Ej =∀
Sz =
φ , i , j∈ N
Let E ∈ M z , E ∩ x0 z . Then, for any p ∈ [u,v ] one has
(
∞
(
) (t,v ( S (η ))) d µ ) d µ
Bp ( E ) ≤ sup ∑ ∫ g x,v S x (δ ) , ∫
σ
i =1 Ei
Sx
t
∞
≤ sup ∑ ∫ hr ( x ) d µ
σ
=
i =1
Ei
hr ( u ) d µ
∫=
E
hr
L1µ
Therefore for any E ∈ M z , there are sets E ∈ M 0 and G ∈ x0 z
E = F ∪ G, F ∩ G = φ .
such that
120
On Quadratic Abstract Measure Integro-Differential Equations
Hence, we obtain
=
M
B ([u,v ]) ≤ q + hr
L1gm
( q + hr ) < 1 . Thus the operator A and B satisfy all the conditions
As α M ≤ α
L1µ
of theorem 3.2 and so an application of it yields that the operator equation
ApBp = p has a maximal and a minimal solution in [u,v ] . Thus further implies
that AMIGDE (2.5) – (2.6) has a maximal & a minimal solution on x0 z . This
completes the proof.
5.2
Discontinuous Case
In the following we obtain an existence result for external solution for the
AMIGDE (2.5) – (2.6) when the nonlinearityg is a discontinuous function in all its
three variables.
We consider the following assumptions :
( C4 )
The function h: x0 z → R + defined by
( (
)
( (
))
h ( x ) = g x,v S x (δ ) , ∫ k t,v S t (η ) d µ
Sx
)
is µ -integrableon x0 z .
Remark 5.1 Assume that the hypothesis
( (
)
( C2 ) and ( C3 )
( (
))
hold. Then
)
g x,v S x (σ ) , ∫ k t,v S x (η ) d µ ≤ h ( x )
Sx
All p ∈[u,v ] .
Theorem 5.2
( C0 ) − ( C2 ) ,( C4 )
Suppose that the assumptions
hold. Further suppose that
( A0 ) − ( A2 ) ,( B0 ) − ( B2 )
and
S. S. Bellale and S. B. Birajdar
121
α
( q + h )<1
(5.3)
L1µ
Then the AMIGDE (2.5) – (2.6) has a minimal and a maximal solution defined on
x0 z
Proof. The proof is similar to Theorem 5.2 with appropriate modifications. Here, the
function h plays the role of hr [u,v ] . Now the desired conclusion follows by an
application of theorem 5.3.Notice that we do not need any type of continuity of the
nonlinear function g in above theorem 5.2 for guaranteeing the existence of extermal
solutions for the AMIGDE (2.5) - (2.6) on x0 z instead we assumed the
monotonicity condition on it.
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