c
IJSRE Volume 05 Issue 06 June 2017 Page 6643
Volume||5||Issue||06||June-2017||Pages-6643-6656||ISSN(e):2321-7545 Website: http://ijsae.in
Index Copernicus Value- 56.65 DOI: http://dx.doi.org/10.18535/ijsre/v5i06.19
Almost automorphic mild solutions to Integro-differential equations with Iterated
deviating arguments
Authors
Lalit Kumar Som1 , Vikram Singh2
Department of Mathematics, Motilal Nehru College New Delhi 110021 Indian Institute of Technology Roorkee, Roorkee-247667.
ABSTRACT
In this article, we provide a set of sufficient conditions for the existence and uniqueness of compact almost automorphic mild solutions for integro functional differential equations with iterated deviating arguments. These results are carried out by means of the basic tools of semigroup theory and Banach contraction principal. In the last, an example is considered to illustrate the significance of our results.
Keywords: Almost automorphic functions, compact almost automorphic function, evolution semigroup, mild solution, differential equation with iterated deviating arguments, Banach fixed point theorem.
INTRODUCTION
The concept of almost automorphy is a generalization of almost periodicity, introduced by S. Bochner[1] in 1962. Afterward, the concept of almost automorphy was studied by Zaki for vector valued function, paving the way to many applications to differential equations. N’Gue´rekata [11] has introduced the concept of asymptotically almost automorphic functions. Veech and Zaki [21, 22] have studied extensively almost automorphic solutions together with compact almost automorphic solutions which is central issue to be discussed in this article. For more details see [1, 2, 5, 15, 17, 8, 9] and reference cited therein.
Morover, there is considerably interest on the part of mathematics in the examination of differential equations with a deviated argument, both in connection with problems in the hypothesis of control system, and because of the intrinsic richness and beauty of such equations. The abundant applications of differential equations involving deviating argument inspired us for revolutionary development of the theory of differential equations with deviating arguments and its generalization, for instance iterated deviating arguments(see, [8, 9, 19, 18]). For more study of such type equations, we refer to monograph [7].
Recently, P. Kumar et al.[16] have been studied the existence of piecewise continuous mild solution for the impulsive differential equations with iterated deviating arguments in a Banach space. Compact almost automorphic solutions of functonal delay differential equations have been studied extensively in([17],[4],[6]) and references there in, but in best of authors’ knowledge, the existence of compact almost automorphic solutions for delay differential equations with iterated deviating arguments is a subject that has not been treated in the literature.
, y = y(0)
3....m, 2, 1, = k )), t (y( I = |
y
(1.1) t
t ], T [0, = I t t))), (y(t), u(h y(t), (t, F = Ay(t) + y(t)
0 k -k tk
= t
0 1
dt d
where h1(t, y(t)) = b1(t, y(b2(t, · · · , y(bm(t, y(t))) · · · ))). The operator B(t) : D(B(t)) ⊂ E → E is closed linear operator and A(t) : D(A(t)) E → E is a family of densely defined closed linear operator on a domain D(A(t)), which is indepent of t and generate a family of exponentially stable evolution operator {U (t, s) : t ≥ s > −∞}, in the sense that there exist constant C > 0, γ > 0, such that
||U (t, s)|| ≤ Ce−γ(t−s). The function (t, s) ›→ U (t, s)y is continuous for each y E and U (t, s)
L(E, D)
for every t > s. The history yt : (−∞, 0] → E, defined as yt(τ ) : y(t + τ ) for each τ (−∞, 0],, belongs to an axiomatically defined abstract phase B . The functions f, g : R × B → E are appropriate continuous functions.
PRELIMINARIES
This section is organized for some basic definitions, facts on almost automorphic, compact almost au- tomorphic functions and some assumptions that have been made to established the main results. We denote the space of bounded linear operator by L(E, Y ) from E to Y .
Definition 2.1. A continuous function f : R → E is almost automorphic if for each sequence of real
numbers {'n}, there exists a subsequence {τn} of {'n}, such that for each t R, g(t) = limn→∞ f (t + τn) is well defined and f (t) = limn→∞ g(t − τn).
The set of all almost automorphic functions f : R → E is denoted by AA(E).
Remark 2.1.[10] The set AA(E) is a Banach space with the supremum norm given by ||f||∞ = sup ||f(t)||.
Definition 2.2. A continuous function f : R × E → E is said to be almost automorphic if every
sequence of real numbers {'n} contains a subsequence {τn} such that for each t R and for every y E g(t, y) = limn→∞ f (t + τn, y) is well defined and f (t, y) = limn→∞ g(t − τn, y).
The space of all such functions is denoted by AA(R × E, E).
Remark 2.2. The range of almost automorphic functions is relatively compact on E, therefore it is bounded.
Definition 2.3. A continuous function f : R → E is a compact almost automorphic if for each sequence
of real numbers {τr }, there exists a subsequence {τn} of {τr } such that for each t R, g(t) = limn→∞ f (t+ n n τn), and f (t) = limn→∞ g(t − τn). converge uniformly on a compact subset of R. The set of all compact almost automorphic functions f : R → E is denoted by AAc(E).
Definition 2.4. A continuous function f : R × E → E is said to be compact almost automorphic if every
sequence of real numbers {'n}contains a subsequence {τn} such that for each t R and for every y E g(t, y) = limn→∞ f (t + τn, y) and f (t, y) = limn→∞ g(t − τn, y), converges uniformly on a compact subset of R. The set of all compact almost automorphic functions is denoted by AAc(R × E, E).
In this work we will define a phase space B axiomatically with the similar ideas developed in [12]. The phase space B denote the vector space of functions yt : (−∞, 0] → E defined as yt(τ ) = (t + τ ) for τ (−∞, 0] with the seminorm||.|| B and the following axioms hold.
If yt : (−∞, c + b) → E is continuous on [c, c + b) for b > 0 and yc B, then for each t [c, c + b) the following conditions hold :
yt is in B.
||yt||B ≤ K(t − c)sup{||y(s)|| : c ≤ s ≤ t} + M(t − c) ||yc|| B ,
where the functions M, K : [0, ∞) → [1, ∞) are locally bounded and continuous respectively and independent of t. Here H > 0 is constant and independent of t.
If y(.) is a function defined in (A), then yt B on [c, c + b). The space B is complete.
If the sequence (ψn)n N of bounded continuous functions defined from (−∞, 0] to E formed by the functions with compact support such that (ψn) → ψ uniformly on a compact subset, then ψ is in
B and ||(ψn) → ψ|| → 0 as n → ∞.
Remark 2.3. Throughout in this paper, by axiom (D) for every ψ B C((−∞, 0]; E) there exists a constant L
> 0 such that ||ψ|| B ≤ L sup ||ψ|| θ , see proposition 7.1.1 of [10]. θ≤0
Definition 2.5. Let S (t) : B → B be a C0 semigroup defined by ψ(0) if θ [−t, 0];
S (t)ψ(θ) =
ψ (t + θ) if θ
(−∞, − t]. (2.1)
The phase space B is said to be a fading memory if "S(t)ψ"B → 0 as t → ∞ with ψ (0) = 0 for each ψ B.
Remark 2.4. We will assume throughout in this work that there exists a positive R such that max{K(t), M(t)} ≤ R
for each t ≥ 0. We investigate that this condition is verified, for example, if B is a fading memory space, for more details see [[12], proposition 7.1.5].
Theorem 2.1 ([10], Theorem 2.2.6, p.22). Let f: R × E → E is an compact almost automorphic function in tR, for each y E. Suppose that f satisfies the Lipschitz condition in y uniformly in t R. If υ :R → E
is a compact almost automorphic function, then the function F : R → E given by F (t) = f (t, υ(t)) is in AAc(E).
To establish the results, the following assumptions are considered on the basis of the data for the system (1.1) -(1.2).
(N1) The nonlinear function F: R × E × E → E is a almost automorphic function in t for each t
R
and y, z E such that
||F (t, y1, z1) − F (t, y2, z2)|| ≤ LF (||y1 − y2|| + ||z1 − z2||), (2.2)
for each yi, zi E, i = 1, 2 and t R. Here, LF is a positive constant known as Lipschitz constant. (N2) The functions bi : R × E → R, (i = 1, 2 · · · , m) are in AAc(R × E, E) and there exist positive
constants Lbi such that
||bi(t, y) − bi(t, z)|| ≤ Lbi ||y – z||, for all y, z E, t R. (2.3) (N3) The operator U (r, s) and A(t) commute that is,
(N4) The operator U (t, s) bAAc(E) i.e. there exists a subsequence (an) for every sequence (a ) of real
numbers such that
||U (t + an, s + an) − U (t, s)|| < s.
(N5) The functions f, g : R × B → E are almost automorphic in t and Lipschitz continuous in second argument in the sense there exists bounded functions Kf , Kg : R → (0, ∞) respectively such that
||f (t, y) − f (t, z)|| ≤ Kf (t)||(y − z)|| ||g(t, y) − g(t, z)|| ≤ Kg (t)||(y − z)||.
(N6) There exists a non-increasing function H: [0, ∞) → [0, ∞) and γ > 0 with the condition
e−γsH(s) L1([0, ∞) for the strongly measurable function s → A(t)U (t, s), define from (−∞, t) into L(E) such that
||A(t)U (t, s) || ≤ e−γ(t−s) H(t − s).
(N7) For t ≥ 0, there exists a function κ(.) L1(R) such that ||B(t)y|| ≤ κ(t)||y|| and B(·)y is strongly measurable on (R) for each y D(A(t)).
Definition 2.6. A continuous function y : [c, c + b) → E, b > 0, is a mild solution for the system (1.1) on [c, c +
b), if ys B for every s R and the function s → A(s)U (t, s)f (s, ys) is integrable on [c, t) for every c < t < c
+ b, and
1 ( ) ( , )( ( ) ( , )) ( , )
( , ) ( ) ( , ) ( , ) ( , )
( , ) ( ) ( , ) ( , ) ( , ( ), ( ( , ( ))) .
t
t t
s s
c c
t s t
c c
y t U t c y c f c f t y
U t s A s f s y ds U t s g t y ds
U t s B s f y d U t s F s y s y h s y s ds
Remark 2.5. Generally the operator function A(s)U (t, s) is not integrable over (−∞, t). If L and R are constant defined in Remark 2.3 and 2.4 respectively and f satisfies (N5), then the following estimate and Bochner’s criteria of integrability of functions
( )
( )
|| ( ) ( , ) ( , ) ( ) ( , ) || ( ) || ( , ) ||
( ) ( )( ( ) || || ( ) || || ) || ( , 0) ||
( ) ( )(1 ) || || ) || ( , 0)
s s
t s
f s B
t s
f
A s U t s f s y A s U t s L X f s y X
e H t s K s M s y K s y f s
e H t s K s L R y f s
||
follows that the function s → A(s)U (t, s)f (t, ys) is integrable on (−∞, 0) for each t > 0.
From the above estimation we observe that the mild solution is well defined over the interval (−∞, 0).
Thus mild solution is defined on (−∞, t), t > 0, ∈R.
MAIN RESULTS
This section is devoted to establish the existence and uniqueness results for the system ( 1.1) -(1.2). First we will mention some appropriate lemmas on the basis on our assumptions that are required for the proof of existence and uniqueness results.
Lemma 3.1. [4] The function defined as s → ys belongs to AAc(X), for y AAc(B).
B(R, E)
such that y(s + τn) → x uniformly on a compact subset of R. Since the phase space B satisfies the axiom (C2) in proposition [14], we deduce that ys+τn → xs B for each s R. Let l > 0 such that K [−l, l] for arbitrary compact subset K R. For s > 0, fix Ns,l N such that
|| ( y s n) ( ) || , x s s s [ , ],l l whenever n ≥ Ns,l. We have from the above estimation
[ , ]
|| t n || t ( ) || l n l||B ( ). sup sup || ( n) ( ) ||
l l
y x M l t y x K l t y x
2 R .s
where R is constant defined in remark (2.4). Thus yt+τn converges to xt uniformly on the compact subset
K. Similarly, it can be shown that xt−τn − yt converges uniformly to 0 on K.
Lemma 3.2. The function defined by ( , ) ( ) ,
t
y U t s y s ds
is in AA (E) for y AA (E).Proof. We first examine that Ψy is bounded. Since y AAc(E), so is bounded i.e. ||y|| ≤ ∞. Then || y || ||y|| C .
(3.1)
Thus yis bounded. Now our object to show that (Ψy) AAc(E). Since y AAc(E), so there exists
subsequence { }n nN of a sequence { ' } n nN of real numbers and x BC (R, E) such that y t( n) ( ) as x t n and (x tn) ( ) as y t n ,
converge uniformly on the compact subset K R. We have form the following estimation ( )
||U t s y t( , ) ( ) ||Ce t s || y||
and Bochner’s criterion of integrability of the functions that the function s → U (t, s)y(s) is integrable over (−∞, t), t R. Furthermore, we have form (3.2) and Lebesgue dominated convergence theorem that (Φy)(t + τn) converges to
( )( ) ( , ) ( )
t
x t U t s x s ds
for all t R. Now the remaining task consists of showing that the convergence is uniform on the compact subset K R. Taking l ≥ 0, Ns N such that K [−l/2, l/2], with the conditions
( )
1/ 2
|| ( n) ( ) || , s, [ , ]
s
y s x s s n N s l l
and
e ds s
For all t K, we have
|| ( )( ) ( )( ) || || ( , ) ( ) ( , ) ( ) ||
t
y t n x t U l n s n y s n t s x s ds
U|| ( , ) |||| ( ) ( ) ||
t
n n n
U t
s
y s
x s ds
|| ( , ) |||| ( ) ( ) ||
t
n n n
l
U t s y s x s ds
|| ( , ) ( , ) |||| ( ) ||
t
n n
U t s U t s x s ds
( ) ( ) 0
2 || || || ||
t
s s
l t
y Ce ds Ce ds x ds
( ) ( )
1/2 0
2 || || || ||
t
s s
y Ce ds Ce ds x ds
2 || || || || .
t
C
y C x ds
Thus, we have (y)( t n) ( x)( ) t uniformly on K as n → ∞. Similarly, one can show that (x)(tn) ( y)( ) t converges uniformly to 0 on K as n → ∞.
Lemma 3.3. The function defined by ( ) ( , ) ( ) ,
t
y A s U t s y s ds
is in AA (E) for y AA ( E).
Proof. Similarly, as in Lemma3.2 it is easy to show that Ψy is bounded. Now we show that Ψy AAc(E). Since y AAc(E), so there exists subsequence { }n nN of a sequence { ' } n nNof real numbers
and x BC (R, E) such that
y t( n) ( ) as x t n and ( x t n) ( ) as y t n ,
converge uniformly on the compact subset K R. We have form the following estimation || ( ) ( , ) ||A s U t s e(t s)H t( s) || y|| (3.4)
and Bochner’s criterion of integrability of the functions that the function s → A(s)U (t, s) y(s) is integrable over (−∞, t), t R. Furthermore, we have form (3.3) and Lebesgue dominated convergence theorem that (y)( t n) converges to
/ 2
( ) ( ) ( , ) ( )
x l
t A s U t s x s ds
for all t R. Now we our task is to show that the convergence in uniform on the compact subset
Lalit Kumar Som, Vikram SinghIJSRE Volume 05 Issue 06 June 2017 Page 6649 Taking l ≥ 0, and Ns N, such that K [−l/2, l/2], with the condition
( )
/ 2
|| ( ) ( ) || , [ / 2, / 2] ( )
n s
s
l
y s x s s n N s l l
e H s ds s
For all t K, we have
( ) ( ) ( ) || ( )( ) ( )( ) || ( ) ( , ) ( ) ( ) ( , ) ( ) ||
2 ( )(|| ( ) ( ) || || || )
2 ( )(|| ( ) ( ) || || || )
6 ( ) || ||
t
y n n n n n
t l t s n l t s n s l t
t x t A s U t s y s ds
A s U t s x s ds
e H t s y s x s y ds
e H t s y s x s y ds
e H s y
( ) 0 ( ) ( )/ 2 0
( )
0
) 2 ( )( || ||
8 || || ( ) 2 ( )
2 4 || || ( )
l t s
s s
l
s
ds e H s y ds
y e H s ds e H s ds
y e H s ds
Thus the convergence is uniform on K. Similarly one can show that (x)( t n) ( y)( ) t converges uniformly to 0 on K as n → ∞.
emma 3.4. The function defined by
( , ) ( ) ( )
t s
y U t s B s y d
Is in AAc(E) for y AAc(E)
Proof. Proceeding as previous Lemma3.2 it is sufficient to prove that Ψy(t) AAc(E) defined by
( ) ( ) ( )
t
y t B t y d
(3.5)
Since y AAc(E), so there exists subsequence { }n nN of a sequence { ' } n nN of real
numbers and x BC(R, E) such that
( n) ( ) as and ( n) ( )as ,
y t x t n x t y t n
converge uniformly on the compact subset K R. We have form the following estimation
||B(t − s)y(t)|| = ||B(t − s)||L(R,E)||y(t)|| ≤ κ(t − s)||y(t)||, (3.5)
Lalit Kumar Som, Vikram SinghIJSRE Volume 05 Issue 06 June 2017 Page 6650 convergence theorem that (Ψy)(t + τn) converges to
( )( ) ( ) ( , ) ( )
t
x t A s U t s x s ds
For
for all t R. Now we have to show that the convergence is uniform on the compact subset K R.
taking l0and NsN such that K [ l/ 2, / 2]l with condition || (y sn)x s( ) || nNs, s [ l/ 2, / 2]l
/ 2 ( )
l
k s ds
For all t K, we have
|| ( )( ) ( )( ) || || || ( ) || ( , ) || ( ) ( ) ||
t
y s n x t B t s L R E y t n x t ds
( )(|| ( ) ( ) ||)
t
n
k l s y s x t ds
( )(|| ( ) ( ) ||)
t
n l
k t s y s x t ds
/ 2 0
2 || || ( ) ( )
l
y k s ds k s ds
0
2 ||y|| k s ds( )
This implies that the convergence is uniform on K. Similarly one can show that (x)( t n) ( y t)( ) converges uniformly to 0 on K as n → ∞.
Lemma 3.5. The function defined by
1
( )( ) ( , ) ( , ( ), ( ( , ( ))))
t
y t U t s F s y s y h s y s ds
Is in A Ac(E )for yA Ac(E ).
Proof. Proceeding as previous Lemma3.2 it is sufficient to prove that F (s) AAc(E) defined by
F (s) = F (s, y(s), y(h1(s, y(s)))),
where h1(t, y(t)) = b1(t, y(b2(t, · · · , y(bm(t, y(t))) · · · ))). Since bm AAc(R × E, E) and we have y(t)
AAc(E), using the assumption (N 2) and the Lemma2.1 in [4] we obtain bm(t, y(t)) is in
AAc(E). Since y is continuous in bm, so y(bm) is compact almost automorphic. Continuing this process for (m − 1), (m − 2), ..2, 1, we obtain β(t) = y{b1(t, y(b2(t, · · · , y(bm(t, y(t))) · · · )))} is in
AAc(E).
Lalit Kumar Som, Vikram SinghIJSRE Volume 05 Issue 06 June 2017 Page 6651 Since
F (s) AAc(R × E × E, E), so there exists subsequence{ }n nN of a sequence { ' } n nN of real numbers and G BC(R × E × E, E) such that
F (t + τn, u, v) → G(t, u, v) and G(t − τn, u, v) → F (t, u, v), for u, v E,
converge uniformly as n → ∞ for u, v E on a arbitrary compact subset K R. We have y, β AAc(E),
so every sequence of real numbers { ' } n nN contains a subsequence { }n nN such that y, β
BC(R, E) are associate with y, β as in Definition2.1. Since R(y) = {y(t):t R} and R(β)={β(t): t
R} are relatively compact sets, so there exists points pi , qi in R, i = 1, 2, 3, ...n, one can find
i(t) {1,
2...n} for each t R such that
1 2
|| ( ) y t pi|| s and||( )t qi||s
Consider a natural number Ns s u c h that
3 ||F s ( n, , ) p qi i ( , , ) || G s p qi i ,s
for all s K and i = 1, 2, 3, ...n, whenever n ≥ Ns. Now we have for s K and n ≥ N
||F (t + τn, y(t + τn), β(t + τn)) − G(t, y(t), β(t))|| ≤||F (t + τn, y(t + τn), β(t + τn)) − F (t + τn, y(t), β(t))||
+ ||F (t + τn, y(t), β(t)) − F (t + τn, pi(t), qi(t)) ||
+ ||F (t + τn, pi(t), qi(t)) − G(t, pi(t), qi(t))||
+ ||G(t, pi(t), qi(t)) − G(t, y(t), β(t)||
≤LF {||y(t + τn) − y(t)|| + ||β(t + τn) − β(t)||} + 2 LF ||y(t) − pi(t)||+ ||β(t) − qi(t)||} + s3
≤ 2 LF {s1 + s2} + s3
this implies that convergence is uniform on the compact subset K. Arguing as perviously one can show
that G (t − τn, y(t − τn), β(t − τn)) − F (t, y(t), β(t)) converges uniformly to 0 on the compact subset K. This completes the proof.
Theorem 3.6. Let the assumptions (N 1) -(N 6) hold, then there exists a unique compact almost auto- morphic mild solution of the equations (1.1) -(1.2) in the Banach space AAcL(E) = {y AAcL(E) : "y(t1) −
Lalit Kumar Som, Vikram SinghIJSRE Volume 05 Issue 06 June 2017 Page 6652
( ) ( )
( ) 1
0 0, 0,
sup ( ) sup ( )
sup (2 )) 1
sup || ( ) ||, sup || ( , ) ||, sup || ( , ) ||,
sup || ( ) |
t t
t t
t s t s
f H f g
t R t R
t
t s
f b f b
t R
H g t f t
t t y B t y B
f f
t R
K e M K s ds C e K s ds
C
C e M M L ds L LL
where M H t M g t y M f t y
K K t
1|, MbL sup s k s( )d .
Proof. For y AAc L(E), we define a map Q:AAcL(E) → AAc L(E) by
Q ty( ) f t y( , t) t U t s A s f s y ds( , ) ( ) ( , s) t U t s g s y ds( , ) ( , s)
1
( , ) ( ) ( , ) ( , ) ( , ( ), ( ( , ( )))) (3.8)
t
s t
U t s B s f y d ds U t s F s y s y h s y s ds
It is verified by previous Lemmas that if y AAc(E), then Qy AAc(E). Therefore, the map Q is well defined. Further, we will show that Qy AAcL(E) for each y AAcL(E)
|| ( y)( ) ( y)( ) || || ( , t) ( , z) || s|| ( , ) ( ) ( , s) ||
z
Q t Q z f t y f t y
U t s A s f s y dst|| ( , ) ( , s) || t|| ( , ) || s || ( ) ( , ) ||
z U t s g s y ds z U t s B s f y d ds
1
|| ( , ) ( , ( ), ( ( , ( )))) ||
t
z U t s F s y s y h s y s ds
( ) ( )
( ) || || t t s || ( ) ( , ) || t t s || ( , ) ||
f t z s s
z z
K t y y e H t s f s y ds e g s y ds
( ) ( )
( ) || ( , ) || || ||
t s t
t s t s
zCe k s f y d ds zCe F ds
1 || || [ ]
f H f g f bL
K L M M M M CM C F t z
Thus (Q) is a mapping from AAcL(E) to AAcL(E). Next, we will show (Q) is a contraction mapping. If
y, w AAcL(E), then we obtain
( )
|| (Qy)( ) (t Qw)( ) ||z Kf||yt wt) || t e t s H t( s K) f( ) ||s ys ws||ds
( ) ( ) || || t t sg s s
Ce K s y w ds
( )
0 ( ) ( ) || ||
t t s s
f
Ce b s K s y w d ds
( )
1 1
|| ) || t t s || ( , ( ), ( ( , ( )))) ( , ( ), ( ( ( ), ))) ||
f t t
K y w Ce F s y s y h s y s F s w s w h w s s ds
Now we estimate
1 1
|| ( , ( ), ( ( , ( )))) F t y t y h t y t ( , ( ), ( ( , ( )))) ||F t w t w h t w t
1 1
[|| ( ) ( ) || || ( ( , ( ))) ( ( , ( ))) ||]
f
L y t w t y h t y t w h t w t
1 1
[|| ( ) ( ) || || ( ( , ( ))) ( ( , ( ))) ||
f
L y t w t y h t w t w h t w t
|| ( ( , ( )))y h t y t1 y h t w t( ( , ( ))) ||]1
Let (h t y tj( , ( )))b t y bj( , ( j1( ,... ( ,t y t b t y tm( , ( )))...))), j1, 2,... ,m yAAcL( ),E
1( , ( )) [[17] .2183].
m
with h t u t t p Thus we obtain
1 1 1 2 1 2
|h t y t( , ( )h t w t( , ( ) | | ( , ( ( , ( )))b t y h t y t b t w h t w t( , ( ( , ( )))) |
1|| ( ( , ( )))2 ( ( , ( )))) ||2
b
L y h t y t w h t w t
Lb1[|| ( ( , ( )))y h t y t2 y h t w t( ( , ( )))) ||2
2 2
|| ( ( , ( )))y h t w t w h t w t( ( , ( )))) ||]
Lb1[|L b t y h t y t| 2( , ( ( , ( )))3 b t w h t w t2( , ( ( , ( ))) |3
|| ( )y l w l( ) ||]
1 1 1 2 1
1 2
1
[ ... ... ... ] || ||
m m
m m
b b b b b b b
L L L L L L LL L L y w
1 1 1 2 1
1 2
1
[ ... ... ... ]|| || 0
m m
m m
b b b b b b b
L L L L L L LL L L y w
Now from (3.10) and (3.9) we have
( )
|| ( )( ) ( )( ) || | | ||
sup ( ) || ||
y w
t t s H f t R
Q t Q t L y w
e M K s y w ds
sup t (t s) g( ) || ||
t R
Ce K s L y w ds
1
( )
sup t t s bL || ||
t R
Ce M L y w ds
( )
(2 ) || ||
t
t s
f b
Ce L LL L y w ds
1 ( ) ( ) ( )sup ( )
sup ( )
sup
(2 ) || ||
|| || (3.11)
t t s
f H f
t R
t t s
g t R
t t s
f bL t R
f b
K e M K s ds
Ce K s ds
Ce M M ds
C
L LL L y w
y w
Where L is a constant appearing in the Remark2.3. Thus the mapping Qy is a contraction on the Banach space AAcL(E). Then by the virtue of Banach contraction principle, there exists a unique y ∈ AAcL(E) such that (Qy) = y. This completes the proof.
Application
Let E = (L2[0, π], ||.||2) with || f || (2 0 | f ( ) |x 2 dx) 1/2
. To finish this work, we will apply ourabstract results to the following neutral differential equation to establish the significance of our result
2
( )
2 2 0
( , ) t ( ) ( ) ( , ) ( , ) ( ) ( , ) t t s ( , )
z t x t t s z s x ds z t x a t z t x e z t x ds
t x
1 1( ) ( ) ( , ) ( , ( ), ( ( , ( )))), (4.1)
t
t t s z s x ds F t x t x h t x t
( , 0) ( , ) 0, ( , ) ( , ) (0, ), [0, ), 0;
z t z t z x t
0 c( ). ( ) ( , ).
Lalit Kumar Som, Vikram SinghIJSRE Volume 05 Issue 06 June 2017 Page 6654 1
0
( ( )) { [0, ]: '' , (0) ( ) 0} ( ) '' ( )
D A t yC y E y y and A t y y a t y
The evolution family generated by A(t) is given by
0( )
{ ( , ) ( )
t sa d
t s
U t s yT ts e y for yE
Here
2 2
1
( ) ( ) n t , n n( )
n
T t y e y e L e
( ) 2 / sin , 1, 2,... || ( ) || t
n
e nt n and T t e
Thus U (t, s) satisfies the assumptions (N 3), (N 4) and (N 6). To reduce the above system into abstract form (1.1)-(1.2), let B = Cr × Lp(µ, E), r ≥ 0, 1 ≤ p < ∞ and let for a positive Lebesgue integrable function µ : (−∞, −r) → R there exists a nonnegative locally bounded function χ on (−∞, 0] such that
µ(ξ + θ) ≤ χ(ξ)µ(θ) for ξ ≤ 0 and θ (−∞, −r) \ Nξ where Nξ (−∞, −r) is a set of zero Lebesgue measure . The space Cr × Lp(µ, E) is a collection of all function ϕ : (−∞, 0] → E, such that
ϕ(·) is a continuous on [−r, 0], (Lebesgue) measurable and || ||pp is (Lebesgue) integrable on (−∞, −r). The seminorm in B is defined by
1/2 2
[ ,0]
|| || sup || || ( ) || ( ) ||
p
r p
B
r
d
We consider that µ(.) is a continuous function that fulfill the assumptions (g − 5) − (g − 7), Theorem
1.3.8 in [13]. The phase space B satisfies the axioms (A), (B), (C) and (D). Now we fix r = 0 and p = 2 moreover define ψ(θ)(x) = ψ(θ, x) B . We suppose that the function σ
AAc(E) and σ1 σ2 : [0, ∞) → R are continuous functions with
1/2 1/2
2 2
0 ( ) 0 ( )
|| || || ||
( ) ( )
f g
s s
L ds and L ds
s s
Now define the functions f, g : R × B → E by
0
1 0
1
( , )( ) ( ) ( ) ( , ) ,
g ( , )( ) ( ) ( ) ( , ) ,
f t x t s s x ds
t x t s s x ds
Now we are able to transform the system (4.1) into the abstract form (1.1). Moreover f, g are bounded operators satisfying (N 5) with ||f|| ≤ Lf , ||g|| ≤ Lg . Now consider
F t y t ( , ( ), ( ( , ( ))))y h t y t1
1/10sin ( ) 1/10cos 1 (1/10cos 2 (cosy t y y tcos 3 ))t
_________________________________________________________
(sintsin 5t5)
Now we obtain that F and bi satisfies (N 1)-(N 2). Thus all assumption are fulfilled, Now applying the Theorem3.6 , there exists a unique solution to the system (1.1)-(1.2).
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