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EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL INTEGRODIFFERENTIAL EQUATIONS WITH INFINITE DELAY IN BANACH SPACES
EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL INTEGRODIFFERENTIAL EQUATIONS WITH INFINITE DELAY IN BANACH SPACES
Journal of Applied Mathematics & Informatics. 2015. Jan, 33(1_2): 45-60
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : May 19, 2014
  • Accepted : August 05, 2014
  • Published : January 30, 2015
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S. CHANDRASEKARAN
S. KARUNANITHI

Abstract
This paper is concerned with the existence of mild solutions for partial neutral functional integrodifferential equations with infinite delay in Banach spaces. The results are obtained by using resolvent operators and Krasnoselski-Schaefer type fixed point theorem. An example is provided to illustrate the results. AMS Mathematics Subject Classification: 35R12, 45K.
Keywords
1. Introduction
The aim of this paper is to establish the existence results for the following neutral functional integrodifferential equations with infinite delays:
PPT Slide
Lager Image
PPT Slide
Lager Image
where A is the infinitesimal generator of a compact, analytic resolvent operator R ( t ), t ≥ 0 in a Banach space X , a : D × Bh X , g : J × Bh × X X , k : D × Bh X and f : J × Bh × X X are given functions, where Bh is a phase space defined later and D = {( t, s ) ∈ J × J : s t }.0 < t 1 < t 2 < ... < tm < b are bounded functions. B ( t ), t J is a bounded linear operator.
The histories xt : (−∞, 0] → X , xt ( s ) = x ( t + s ), s ≤ 0, belong to an abstract phase space Bh .
Neutral differential and integrodifferential equations arise in many areas of applied mathematics and for this reason these equations have been investigated extensively in the last decades. There are many contributions relative to this topic and we refer the reader to [1 , 2 , 3 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15] .
The theory of nonlinear functional differential or integrodifferential equations with resolvent operators is an important branch of differential equations, which has an extensive physical background, see for instance [16 , 17 , 18] .
Since many control systems arsing from realistic models depend heavily on histories ( that is, the effect of infinite delay on the state equations [23] ), there is real need to discuss the existence results for partial neutral functional integrodifferential equations with infinite delay. The development of the theory of functional differential equations with infinite delays depends on a choice of a phase space. In fact, various phase spaces have been considered and each different phase space has required a separate development of the theory [20] . The common space is the phase space B proposed by Hale and Kato in [19] .
The main purpose of this paper is to deal with the existence of mild solutions for the problem (1)-(2). Here, we use an abstract phase space adopted in [6 , 24] . Sufficient conditions for the existence results are derived by means of the Krasnoselski-Schaefer type fixed point theorem combined with theories of analytic resolvent operators. The results generalise the results of [4 , 6 , 7 , 21] .
2. Main results
Throughout this paper, we assume that ( X , ∥·∥) is a Banach space, the notaion L ( X , Y ) stands for the Banach space of all linear bounded operators from X into Y , and we abbreviate this notation to L ( X ) when X = Y . R ( t ), t > 0 is compact, analytic resolvent operator generated by A .
Assume that
  • (A1)Ais a densely defined, closed linear operator in a Banach space (X, ∥ · ∥) and generates aC0-semigroupT(t). HenceD(A) endowed with the graph norm |x| = ∥x∥ + ∥Ax∥ is a Banach space which will be denoted by (Y, | · |).
  • (A2) {B(t) :t∈J} is a family of continuous linear operator from (Y, | · |) into (X, ∥ · ∥) . Moreover, there is an integrable functionc: [0,b] →R+such that for anyy∈Y, the mapt→B(t)ybelongs toW1,1(J,X) and
PPT Slide
Lager Image
Definition 2.1. A family { R ( t ) : t ≥ 0} of continuous linear operators on X is called a resolvent operator for
PPT Slide
Lager Image
if:
  • (R1)R(0) =I( the identity operator onX),
  • (R2) For allx∈X, the mapt→R(t)xis continuous fromJtoX,
  • (R3) For allt∈J,R(t) is a continuous linear operator onY, and for ally∈Y, the mapt→R(t)ybelongs toC(J,Y) ∩C′(J,X) and satisfies
PPT Slide
Lager Image
Theorem 2.1 ( [21] ). Let the assumptions (A1) and (A2) be satisfied. Then there exists a constant H = H ( b ) such that
PPT Slide
Lager Image
where L ( X ) denotes the Banach space of continuous linear operators on X.
Next, if the C 0 - semigroup T (·) generated by A is compact ( that is, T ( t ) is a compact operator for all t > 0 ), then the corresponding resolvent operator R (·) is also compact ( that is, R ( t ) is a compact operator for all t > 0 ) and is operator norm continuous ( or continuous in the uniform operator topology) for t > 0.
Proof . Now, we define the abstract phase space Bh as given in [24 , 7] .
Assume that h : (−∞, 0] → (0,+∞) is a continuous function with
PPT Slide
Lager Image
For any a > 0, we define
PPT Slide
Lager Image
and equip the space B with the norm
PPT Slide
Lager Image
Let us define
PPT Slide
Lager Image
If Bh is endowed with the norm
PPT Slide
Lager Image
then it is clear that ( Bh , ∥ · ∥ Bh ) is a Banach space.
Now we consider the space
PPT Slide
Lager Image
Set ∥ · ∥ b be a semi norm in B′h defined by
PPT Slide
Lager Image
Next, we introduce the basic definitions and lemmas which are used throughout this paper.
Let A : D ( A ) → X be the infinitesimal generator of a compact, analytic resolvent operator R ( t ), t ≥ 0. Let 0 ∈ ρ ( A ), then it is possible to define the fractional power (− A ) α , for 0 < α ≤ 1, as closed linear operator on its domain D (− A ) α . Further more, the subspace D (− A ) α is dense in X , and the expression
PPT Slide
Lager Image
defines a norm on D (− A ) α .
Furthermore, we have the following properties appeared in [22] . □
Lemma 2.2. The following properties hold:
(i) If 0 < β < α ≤ 1, then Xα Xβ and the imbedding is compact whenever the resolvent operator of A is compact.
(ii) For every 0 < α ≤ 1 there exists Cα > 0 such that
PPT Slide
Lager Image
Lemma 2.3 ( [5] ). Let Փ 1 2 be two operators satisfying Փ 1 is contraction and Փ 2 is completely continuous. Then either
(i) the operator equation Փ 1 x + Փ 2 x = x has a solution, or
(ii) the set
PPT Slide
Lager Image
is unbounded for λ ∈ (0, 1).
Lemma 2.4 ( [12] ). Let v (·), w (·) : [0, b ] → [0,∞) be continuous functions. If w (·) is nondecreasing and there are constants θ > 0, 0 < α < 1 such that
PPT Slide
Lager Image
then
PPT Slide
Lager Image
for every t ∈ [0, b ] and every n N such that nα > 1, and Γ(·) is the Gamma function.
Lemma 2.5 ( [6] ). Assume x B′h , then for t J, xt Bh. Moreover ,
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Definition 2.2. A function x : (−∞, b ] → X is called a mild solution of problem (1)-(2) if the following holds: x 0 = ϕ Bh on (−∞, 0]; the restriction of x (·) to the interval J is continuous, and for each s ∈ [0, t ), the function
PPT Slide
Lager Image
is integrable and the integral equation
PPT Slide
Lager Image
is satisfied.
Definition 2.3. A map f : J × Bh × X X is said to be an L 1 -Caratheodory if
(i) For each t J , the function f ( t , ·, ·) : Bh × X X is continuous.
(ii) For each ( ϕ, x ) ∈ Bh × X ; the function f (·, ϕ, x ) : J X is strongly measurable.
(iii) For every positive integer q > 0, there exists αq L 1 ( J,R + ) such that
PPT Slide
Lager Image
3. Existence Results
In this section, we shall present and prove our main result. For the proof of the main result, we will use the following hypotheses:
(H1) ([see Lemma 2.2]) A is the infinitesimal generator of a compact analytic resolvent operator R ( t ), t > 0 and 0 ∈ ρ ( A ) such that
PPT Slide
Lager Image
(H2) There exist a constant N 1 > 0 such that
PPT Slide
Lager Image
(H3) There exist constants 0 < β < 1, C 0 , c 1 , c 2 , N 2 such that g is Xβ -valued, (− A ) βg is continuous, and
  • (i)(t, x) ∈J×Bh,
  • (ii)∥(−A)βg(t,x1,y1) − (−A)βg(t,x2,y2)∥ ≤N2[∥x1−x2∥Bh+ ∥y1−y2∥] fort∈J,x1,x2∈Bh,y1,y2∈X, with
PPT Slide
Lager Image
(H4) (i) For each ( t, s ) ∈ D , the function k ( t, s , ·) : Bh X is continuous and for each x Bh , the function k (·, ·, x ) : D X is strongly measurable.
  • (ii) There exist an integrable functionm:J→ [0,∞) and a constantγ> 0, such that
PPT Slide
Lager Image
  • where Ω : [0,∞) → (0,∞) is a continuous nondecreasing function. Assume that the finite bound ofisL0.
(H5) The function f : J × Bh × X X satisfies the following caratheodory conditions:
  • (i)t→f(t, x, y) is measurable for each (x, y) ∈Bh×X,
  • (ii) (x, y) →f(t, x, y) is continuous for almost allt∈J.
(H6) ∥ f ( t, x, y )∥ ≤ p ( t ) Ψ (∥ x Bh +∥ y ∥) for almost all t J and all x Bh , y X , where
p L 1 ( J,R + ) and Ψ : R + → (0,∞) is continuous and increasing with
PPT Slide
Lager Image
where
PPT Slide
Lager Image
with lM 0 N 2 (1 + N 1 ) < 1,
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
We consider the operator Փ : Bh Bh defined by
PPT Slide
Lager Image
From hypothesis ( H 1), ( H 2) and Lemma 2.3, the following inequality holds:
PPT Slide
Lager Image
Then from Bochner theorem [25] , it follows that
PPT Slide
Lager Image
is integrable on [0, t ).
For ϕ Bh , we defined by
PPT Slide
Lager Image
by
PPT Slide
Lager Image
and then
PPT Slide
Lager Image
It is easy to see that x satisfies (3) if and only if y satisfies y 0 = 0 and
PPT Slide
Lager Image
Let
PPT Slide
Lager Image
PPT Slide
Lager Image
thus
PPT Slide
Lager Image
is a Banach space. Set
PPT Slide
Lager Image
for some q ≥ 0, then
PPT Slide
Lager Image
is uniformly bounded, and for y Bq , from Lemma 2.5, we have
PPT Slide
Lager Image
Define the operator
PPT Slide
Lager Image
PPT Slide
Lager Image
Now we decompose
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Obviously the operator Փ has a fixed point is equivalent to
PPT Slide
Lager Image
has one. Now, we shall show that the operators
PPT Slide
Lager Image
satisfy all the conditions of Lemma 2.3.
Lemma 3.1. If assumptions (H1)-(H6) hold, then
PPT Slide
Lager Image
is a contraction and
PPT Slide
Lager Image
is completely continuous .
Proof . First we show that
PPT Slide
Lager Image
is a contraction on
PPT Slide
Lager Image
From (H1)-(H3) and Lemma 2.5, we have
PPT Slide
Lager Image
Since ∥ u 0 Bh = 0, ∥ v 0 Bh = 0. Taking supremum over t ,
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Thus
PPT Slide
Lager Image
is a contraction on
PPT Slide
Lager Image
Next we show that the operator
PPT Slide
Lager Image
is completely continuous. First we prove that
PPT Slide
Lager Image
maps bounded sets into bounded sets in
PPT Slide
Lager Image
Indeed, it is enough to show that there exists a positive constant Λ such that for each
PPT Slide
Lager Image
Now for each t J ,
PPT Slide
Lager Image
By (H1)-(H6) and (5), we have for t J ,
PPT Slide
Lager Image
Then for each
PPT Slide
Lager Image
Next we show that
PPT Slide
Lager Image
maps bounded sets into equicontinuous sets of
PPT Slide
Lager Image
Let 0 < r 1 < r 2 b , for each
PPT Slide
Lager Image
Let r 1 , r 2 J − { t 2 , t 2 , ..., tm }. Then we have
PPT Slide
Lager Image
The right-hand side from Theorem 2.1 of the above inequality tends to zero as r 2 r 1 and for ϵ sufficiently small. Thus the set
PPT Slide
Lager Image
is equicontinuous. Here we consider only the case 0 < r 1 < r 2 b , since the other cases r 1 < r 2 ≤ 0 or r 1 ≤ 0 ≤ r 2 b are very simple.
Next, we show that
PPT Slide
Lager Image
is continuous.
Let
PPT Slide
Lager Image
Then there is a number q > 0 such that | y (n) ( t )| ≤ q for all n and a.e. t J , so y (n) Bq and y Bq . In view of (5), we have
PPT Slide
Lager Image
By ( H 3), ( H 5) and Definition 2.2,
PPT Slide
Lager Image
We have by the dominated convergence theorem that
PPT Slide
Lager Image
Thus
PPT Slide
Lager Image
is continuous.
Next we show that
PPT Slide
Lager Image
maps Bq into a precompact in X . Let 0 < t b be fixed and ϵ be a real number satisfying 0 < ϵ < t . For y Bq , we define the operators
PPT Slide
Lager Image
From Theorem 2.1 and the compactness of the operator R ( ϵ ), the set
PPT Slide
Lager Image
is precompact in X , for every ϵ , 0 < ϵ < t . Moreover, by Theorem 2.1 and for each y Bq , we have
PPT Slide
Lager Image
So the set
PPT Slide
Lager Image
is precompact in X by using the total boundedness. Applying this idea again and observing that
PPT Slide
Lager Image
Therefore,
PPT Slide
Lager Image
and there are precompact sets arbitrarily close to the set
PPT Slide
Lager Image
. Thus the set
PPT Slide
Lager Image
is precompact in X .
Therefore from Arzela-Ascoli theorem, we can conclude that the operator
PPT Slide
Lager Image
is completely continuous. In order to study the existence results for the problem (1)-(2), we introduce a parameter λ ∈ (0, 1) and consider the following nonlinear operator equation
PPT Slide
Lager Image
where Փ is already defined. The following lemma proves that an a priori bound exists for the solution of the above equation.
Lemma 3.2. If hypotheses (H1)-(H6) are satisfied, then there exists an a priori bound K > 0 such that xt Bh K, t J, where K depends only on b and on the functions
PPT Slide
Lager Image
Proof . From the equation (6), we have
PPT Slide
Lager Image
Thus from this proof and Lemma 2.4 it follows that
PPT Slide
Lager Image
Let μ ( t ) = sup{∥ xs Bh : 0 ≤ s t }, then the function μ ( t ) is nondecreasing in J , and we have
PPT Slide
Lager Image
By using lemma 2.5, we have
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Let us take the right hand side of the above inequality as v ( t ). Then v (0) = B 0 K 1 , μ ( t ) ≤ v ( t ), 0 ≤ t b and
PPT Slide
Lager Image
Since Ψ and Ω are nondecreasing.
PPT Slide
Lager Image
Let
PPT Slide
Lager Image
Then w (0) = v (0) and v ( t ) ≤ w ( t ).
PPT Slide
Lager Image
This implies that
PPT Slide
Lager Image
This implies that v ( t ) < ∞. So there is a constant K such that v ( t ) ≤ K, t J . So ∥ xt Bh μ ( t ) ≤ v ( t ) ≤ K, t J , where K depends only on b and on the functions
PPT Slide
Lager Image
Theorem 3.3. Assume that the hypotheses (H1)-(H6) hold. Then the problem (1)-(2) has at least one mild solution on J.
Proof . Let us take the set
PPT Slide
Lager Image
Then for any
PPT Slide
Lager Image
we have by Lemma 3.2 that ∥ xt Bh K, t J , and we have
PPT Slide
Lager Image
which implies that the set G is bounded on J .
Consequently, by Krasnoselski-Schaefer type fixed point theorem and Lemma 3.2 the operator
PPT Slide
Lager Image
has a fixed point
PPT Slide
Lager Image
Then x is a fixed point of the operator Փ which is a mild solution of the problem (1)-(2). □
4. Example
Consider the following partial neutral integrodifferential equation of the form
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
where ϕ Bh . We take X = L 2 [0, π ] with the norm | · | L2 and define A : X X by Aw = w′′ with the domain
D ( A ) = { w X : w,w′ are absolutely continuous, w′′ X, w (0) = w ( π ) = 0}.Then
PPT Slide
Lager Image
where
PPT Slide
Lager Image
n = 1, 2, . . . .. is the orthogonal set of eigen vectors of A . It is well known that A generates a strongly continuous semigroup that is analytic, and resolvent operator R ( t ) can be extracted this analytic semigroup and given by
PPT Slide
Lager Image
Since the analytic semigroup R ( t ) is compact, there exists a constant M 1 > 0 such that ∥ R ( t )∥ ≤ M 1 . Especially, the operator (− A ) ½ is given by
PPT Slide
Lager Image
with the domain
PPT Slide
Lager Image
Let
PPT Slide
Lager Image
and define
PPT Slide
Lager Image
Hence for ( t, ϕ ) ∈ [0, b ] × Bh , where ϕ ( θ )( x ) = ϕ ( θ, x ), ( θ, x ) ∈ (−∞, 0] × [0, π ]. Set
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Then, the system (7)-(9) is the abstract formulation of the system (1)-(2). Further, we can impose some suitable conditions on the above defined functions to verify the assumptions on Theorem 3.3. We can conclude that system (7)-(9) has at least one mild solution on J .
BIO
S. Chandrasekaran
Department of Mathematics, SNS College of Technology, Coimbatore - 641 035, Tamil Nadu, India.
e-mail: chandrusavc@gmail.com
S. Karunanithi
Department of Mathematics, Kongunadu Arts and Science College(Autonomous), Coimbatore - 641 029, Tamil Nadu, India.
e-mail: sknithi1957@yahoo.co.in
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