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1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 2 Issue 4 || April. 2014 || PP-58-66
www.ijmsi.org 58 | P a g e
Controllability of Neutral Integrodifferential Equations with
Infinite Delay
Jackreece P. C.
Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria.
ABSTRACT: In this paper sufficient conditions for the controllability of nonlinear neutral integrodifferential
equations with infinite delay where established using the fixed point theorem due to Schaefer.
KEYWORDS: Controllability, Infinite delay, Neutral Integrodifferential equation, Schaefer fixed point.
I. INTRODUCTION
In this paper, we establish a controllability result to the following nonlinear neutral integrodifferential
equations with infinite delay:
              
 1.1
0,,,,,
0 
  
x
txtfdsxstgtButxtAxthtx
dt
d
t
t
st
where the state .x takes values in the Banach space X endowed with the norm . , the control function .u
is given in the Banach space of admissible control function  UJL ,2
with U as a Banach space.
    XADtA : is an infinitesimal generator of a strongly continuous semigroup of bounded linear operator
  0, ttT in X .  is a bounded linear operator from U into X , where XJJg : ,
XJf : , and XJh : are given functions. The delay   Xxt  0,: defined by
     txxt belongs to some abstract phase space  , which will be a linear space of functions mapping
 0, into X endowed with the seminorm

. in . Controllability problems of linear and nonlinear
systems represented by Ordinary differential Equations in finite dimensional space has been studies extensively
[6, 15]. Several authors extended the controllability concept to infinite dimensional systems in abstract spaces
with unbounded operators [1, 2, 8]. There are many systems that can be written as abstract neutral functional
equations with infinite delays. In recent years, the theory of neutral functional differential equations with infinite
delay in infinite dimension has received much attention [4, 7, 10]. Meanwhile, the controllability problem of
such systems was also discussed by several scholars. Meili et al. [13] studied the controllability of neutral
functional integrodifferential systems in abstract spaces using fractional power of operators and Sadovski fixed
point theorem. Balachandran and Nandha [5] established sufficient conditions for the controllability of neutral
functional integrodifferential systems with infinite delay in Banach spaces by means of Schaefer fixed point
theorem. Li et al. [12] used Hausdoff measure of noncompactness and Kakutani’s fixed point theorem to
establish sufficient conditions for the controllability of nonlinear integrodifferential systems with nonlocal
condition in separable Banach space. The purpose of this paper is to establish a set of sufficient conditions for
the controllability of neutral integrodifferential equations with infinite delay using Schaefer fixed point theorem.
II. PRELIMINARIES
In this section, we introduce notations, definitions and theorems which are used throughout this paper.
In the study of equations with infinite delay such as Eq. (1.1), we need to introduce the phase space. In this
paper we employ an axiomatic definition of the phase space first introduced by Hale and Kato [9] and widely
discussed by Hino et al [11]. The phase space  is a linear space of functions mapping ]0,( into
X endowed with the seminorm

. and assume that  satisfies the following axioms:

2. Controllability of Neutral Integrodifferential Equations with Infinite Delay
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(A1) If   ,0,,:  bXbx is continuous on  b,0 and 0x , then for every  bt ,0 the
Following conditions hold:
(i) tx is in 
(ii)   
 txHtx
(iii)        
 00:sup xtMtssxtKxt
where 0H is a constant; ),0[),0[:, MK , K is continuous and M is locally bounded, and
MKH ,, are independent of .x .
(A2) For the function .x in (A1), tx is a  -valued continuous function on ],0[ b .
(A3) The space  is complete.
We need the fixed point theorem as stated below,
Schafer’s theorem [14]: Let S be a convex subset of a normed linear space E and S0 . Let SSF : be
completely continuous operator and let
    1,0:   someforxFxSxF
Then either  F is unbounded or F has a fixed point.
We assume the following hypothesis:
(H1) A is the infinitesimal generator of a compact semigroup of bounded linear operators   0, ttT on
X and their exists 1M and 01 M such that
  MtT  and   1MtAT  .
(H2) For XJh : is completely continuous and there exists constants 21,cc such that 1
~
1 cK
and
  21, ccth  
   ,,0 bt , where   JttKK  :max
~
.
(H3) The function XJf : satisfies the following conditions
(i). For each Jt  , the function XJf : is continuous.
(ii) For each  , the function   XJf :,;  is strongly measurable.
(iii). For each positive integer k, there exists    bLk ,0. 1
 such that
     eaJttkxxtf kt .,:,sup   .
and
   xJtxxtf ,,,
where  ,0[),0[: is a continuous nondecreasing function.

3. Controllability of Neutral Integrodifferential Equations with Infinite Delay
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(H4) XJJg : is continuous and there exists 0N , such that
  Nstg ,,
For every .0 bts 
(H5) for each  ,   

0
,,lim)(
a
a dssstgtq  exists and it is continuous. Further there exists
01 N such that   .1Ntq 
(H6) there exists a compact set XV  , such that             ,,,,, sgtTsButTsftT and
    VsqtT  for all , , and .0 bts 
(H7) The linear operator   XUJLW ,: 2
, defined by
    
b
dssBusbTWu
0
has an inverse operator
1
W , which takes values in   WUJL ker/,2
and there exist positive constants
0, 43 MM such that 4
1
3 MWandMB  
(H8)  
  



b
s ss
ds
dssm
0
~
where
    
dsxMMNbbMN
dscxcMcxcccMxMMN
b
s
b
st




0
1
0
21121211433 0 
   MNbMNMNbcMcccMKM
cK
c 

 1321121
1
0
~~
~
1
1

  JttMM  :max
~
 








11
11
~
1
~
,~
1
~
max~
cK
MK
cK
cMK
tm
Definition 2.1: The system (1.1) is said to be controllable on the interval J iff, for every Xxx b ,0 , there
exists a control  UJLu ,2
 such that the mild solution  tx of (1.1) satisfies   00 xx  and   bxbx  .
Definition 2.2: A function   Xbx  ,: is called a mild solution of the integrodifferential equation (1.1)
on [0, b] iff ,
              
             1.2,0,,,
,,,00
0 0
0
btdsxsfdxsgsqsBustT
dsxshstATxthhtTtx
t
s
t
s
t
tt




 

 



is satisfied. See [3,10]

4. Controllability of Neutral Integrodifferential Equations with Infinite Delay
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III. MAIN RESULT
Theorem 3.1: Assume that (H1)-(H8) holds. Then the system (1.1) is controllable.
Proof: Consider the map defined by
              
           )1.3(,,,
,,,00
0 0
0
 





 

t
s
t
s
t
st
JtdsxsfdxsgsqsBustT
dsxshstATxthhttx


and define the control function  tu as
              
              )2.3(,,,
,,,00
0 0 0 0
0
1
   


 
b b s b
s
b
sbb
tdsxsfsbTdsdxsgsbTdssqsbT
dsxshsbATxbhhbTxWtu



First we show that there is a priori bound 0K such that JtKxt  , , where K depends only on b and
on the function . . Then we show that the operator has a fixed point, which is then a solution to the system
(1.1). Obviously,    1xbx  , which means that the control u steers the system from the initial function 
to 1x in time b, provided that the nonlinear operator has a fixed point.
Substituting (3.2) into (2.1) we obtain
               
t
st dsxshstATxthhtTtx
0
,,,00 
                 
t b
sb dsxshsbATxbhhbTxBWtT
0 0
1
1
,,,00 
                
b b s b
s ddsxsfsbTdsdxsgsbTdssqsbT
0 0 0 0
,,,  
         


 
t
s
s
dsxsfdxsgsqstT
0 0
,,,  
Then, we have
       MNbMNMNdscxcMcxcccMtx
t
st   13
0
21121210 
dsxM
t
s 
0
From which using Axiom (A1)(iii), it follows that
      tMtssxtKxt  0:sup
   MtssxK
~
0:sup
~

   MNbMNMNbcMcccMKM  13212210
~~

  



 
t t
sss dsxMKdsxcMKxcK
0 0
111
~~
sup
~

5. Controllability of Neutral Integrodifferential Equations with Infinite Delay
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Let    tsxt s  0:sup , then the function  t is continuous and nondecreasing and from Axiom
(A2) we have
     MNbMNMNbcMcccMKMt  13211210
~~

      
tt
dsMKdsscMKtcK
00
111
~~~

From which it follows that
       




t t
dss
Kc
MK
dss
cK
cMK
ct
0 0
11
11
1
~
~
1
~

Denoting the right hand side of the above inequality as  t , we have   c0 ,     Jttt  , and
     s
cK
MK
s
cK
cMK
t  




11
11
~
1
~
~
1
~
   s
cK
MK
s
cK
cMK
 




11
11
~
1
~
~
1
~
      sstm  

which implies that
 
 
  
 
Jt
ss
ds
dssm
ss
dst t



  

 0 0 0
~
This inequality implies that there is a constant K such that   JtKt  , and hence,
    KtKttxt  , , where K only depends on b and on the function . .
We now rewrite the initial value problem (1.1) as follows:
For  , define ˆ by
 
   
      





atifhtT
tiftht
t
0,00
0,ˆ



If        atttytxandy ,,ˆ   , then it is easy to see that x satisfies (2.1) if and only if
y satisfies,
  0,00  tyty
           
               




b
ssbb
t
t
sstt
dsyshsbATybhhbTxBWtT
dsyshstATythhtTty
00
1
1
0
ˆ,ˆ,,00
ˆ,ˆ,,0


    Jts
cM
M
scMK
cK








 
11
11
1
~
~
1
1

6. Controllability of Neutral Integrodifferential Equations with Infinite Delay
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               ddsysfsbTdsdysgsbTdssqsbT
b s b
ss
b
   
0 0 00
ˆ,ˆ,,
        JtdsysfdysgsqstT
t s
ss 



    ,ˆ,ˆ,,
0 0
 
We define the operator  0:,: 0000  yy by
            
               dsyshsbATybhhbTxBWtT
dsyshstATythhtTty
t b
ssbb
t
sstt
 




0 0
1
1
0
ˆ,ˆ,,00
ˆ,ˆ,,0


            

 ddsysfsbTdsdysgsbTdssqsbT
b b b
ss    
0 0 0 0
ˆ,ˆ,,
         



 
t s
ss JtdsysfdysgsqstT
0 0
,ˆ,ˆ,,  
From the definition of an operator  defined on equation (3.1), it can be noted that the equation (2.1) can be
written as
     3.310,   tyty
Now, we prove that  is completely continuous. For any kBy  , let btt  210 , then
     21 tyty 
         2211
ˆ,ˆ,,0 2121 tttt ythythhtTtT  
           
2
1
1
ˆ,ˆ, 2
0
21
t
t
ss
t
ss dsyshstATdsyshstTstTA 
               1
0
1
1
21
ˆ,,00
t
bbybhhbTxBWtTtT 
       dssqsbTyshsbAT
b b
ss  
0 0
ˆ, 
        

 ddsysfsbTdsdysgsbT
b b
ss   
0 0 0
ˆ,ˆ,,
            2
1
ˆ,,001
1
2
t
t
bbybhhbTxBWtT 
         
b b
ss dssqsbTdsyshsbAT
0 0
ˆ, 
         

 ddsysfsbTdsdysgsbT
b
ss
b
  
00 0
ˆ,ˆ,,
            


 
1
0 0
21
ˆ,ˆ,,
t s
ss dsysfdysgsqstTstT  
         


 
2
1 0
2
ˆ,ˆ,,
t
t
s
ss dsysfdysgsqstT  

7. Controllability of Neutral Integrodifferential Equations with Infinite Delay
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         2211
ˆ,ˆ,,0 2121 tttt ythythhtTtT  
          
1 2
10
2122121
ˆˆt t
t
ssss dscycstATdscycstTstTA  (3.4)
         
1
0
12114321
ˆ,00
t
bb bMNcychMxMMstTstT 
          
b b t
t
ss hMxMMstTdsdsyMdsNM
0 0
1432
2
1
,00ˆ 
        
b b b
bbbb dsdssMdsNdMbMNdscycMcyc
0 0 0 0
121121
ˆˆ 

             
1 2
10
12121
t t
t
dssNNstTdssNNstTstT 
The right-hand side of equation (3.4) is independent of ky  and tends to zero as 012  tt , since g is
completely continuous and the compactness of  tT for 0t implies the continuity in the uniform operator
topology. Thus  maps k into an equicontinuous family of functions.
            

 

 
t
sstt dsyshstATythhtTty
0
ˆ,ˆ,,0
          





t
bbybhhbTxBWstT
0
1
1 ˆ,,00
              
b b b
ssbs dsysfsbTdssqsbTdsyshsbAT
0 0 0
ˆ,ˆ, 
               


 
b t s
ss
s
dsdysgysfsqstTdsdsdysgsbT
0 0 00
ˆ,,ˆ,ˆ,,  
           

 

 
t
sstt yshtATTythhtT
0
ˆ,ˆ,,0
            





t
bbybhhbTxBWstTT
0
1
1 ˆ,,00
             
b b
ss
b
bb dsysfsbTdssqsbTdsyshsbAT
0 00
ˆ,ˆ, 
      
b s
dsdsdysgsbT
0 0
ˆ,,  
          dsdysgysfsqstTT
t s
ss 




 

 
0 0
ˆ,,ˆ,
Since  tT is a compact operator, the set      kytytY  : is precompact in X for every ,

8. Controllability of Neutral Integrodifferential Equations with Infinite Delay
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t 0 . Moreover, for every ky  we have
             
 
t
t
sstttt dsyshstATythythtyty
  ˆ,ˆ,ˆ,
       

t
t
bbybhhMxMMstT

 ˆ,,00143
             
b
ss
b b
ss dsysfstTdssqstTdsyshstAT
00 0
ˆ,ˆ, 
    
        dsdsysgysfsqstT
dsdsdysgstT
t
t
s
ss
b s
 
 
 


 

 



0
0 0
ˆ,,ˆ,
ˆ,,
Clearly       0 tyty  as

 0 . Therefore, there is a family of precompact sets which are
arbitrarily close to the set    kyty  : . Hence, the set    kyty  : is precompact in X .
Next we prove that the set
00
: bb  is continuous. Let   0
1 bnny  with yyn  in
0
b . We have
                
t
sssntttnn dsyshyshstATythythtyty st 0
ˆ,ˆ,ˆ,ˆ, 
       
t t
sssn dsyshyshMMMstT s0 0
143
ˆ,ˆ. 
           
b s b
sssnn dsdsysfysfMdsysgysgM s0 0 0
ˆ,ˆ,ˆ,,ˆ,,  
           


 
t s
nsssn dsysgysgysfysfstT s0 0
ˆ,,ˆ,,ˆ,ˆ,   
From the assumption (H1) and the Lebesgue Dominated Convergence theorem we conclude that yyn  as
n in
0
b . Thus . is continuous, which completes the proof that . is completely continuous.
Hence there exists a unique fixed point  tx for bon  . Obviously .x is a mild solution of the system
(1.1) satisfying   1xbx  .
IV. CONCLUTION
Sufficient condition for the controllability of the nonlinear neutral integrodiffential equation was established
using Schaefer’s fixed point theorem. First we show that there is a priori bound 0K such
that JtKxt  , . Then we show that the operator has a fixed point, which is then a solution to the system
(1.1). Obviously,    1xbx  , which means that the control u steers the system from the initial function 
to 1x in time b, provided that the nonlinear operator has a fixed point.

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