Regularity of solutions of Sobolev type semilinear integrodifferential equations in Banach spaces

Abstract

In this article, we prove the existence of mild and classical solutions of Sobolev type semilinear integrodifferential equations of the form $$ frac{d}{dt}[Ex(t)] = A[x(t)+int_0^tF(t-s)x(s)ds]+f(t,x(t)) $$ in Banach spaces. The results are obtained by using the Banach contraction mapping principle and resolvent operator. An application is provided to illustrate the theory.

Full text

Page 1

Electronic Journal of Differential Equations, Vol. 2003(2003), No. 114, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
REGULARITY OF SOLUTIONS OF SOBOLEV TYPE
SEMILINEAR INTEGRODIFFERENTIAL EQUATIONS IN
BANACH SPACES
KRISHNAN BALACHANDRAN & SUBBARAYAN KARUNANITHI
Abstract. In this article, we prove the existence of mild and classical solu-
tions of Sobolev type semilinear integrodifferential equations of the form
?t
in Banach spaces. The results are obtained by using the Banach contraction
mapping principle and resolvent operator. An application is provided to illus-
trate the theory.
d
dt[Ex(t)] = A[x(t) +
0
F(t − s)x(s)ds] + f(t,x(t))
1. Introduction
Corduneanu [6] and Gripenberg et. al [10] studied the problem of existence of
solutions for Volterra integral equations of various types. Grimmer [9] introduced
the resolvent operators for integral equations in Banach spaces. Liu [17] studied
the weak solutions of integrodifferential equations by using resolvent operators and
semigroup theory. Fitzgibbon [8] investigated the existence problem for semilinear
integrodifferential equations in Banach spaces. Using the method of semigroups
and Banach’s fixed point theorem Byszewski [5] proved the existence and unique-
ness of mild, strong and classical solutions of nonlocal Cauchy problem. Lin and
Liu [16] investigated the nonlocal Cauchy problem of semilinear integrodifferential
equations by using resolvent operators. Brill [4] discussed the existence problem
for semilinear Sobolev type equations in Banach spaces. Balachandran et. al [2] es-
tablished the existence of solutions for Sobolev type integrodifferential equations in
Banach spaces. Recently Balachandran et. al [1] investigated the same problem for
Sobolev type delay integrodifferential equations. Several authors have studied the
problem of existence of solutions of semilinear differential equations and Sobolev
type equations [3, 7, 11, 12, 13, 14, 15, 18, 19, 22].
The purpose of this article is to study the regularity of solutions of Sobolev type
semilinear integrodifferential equations in Banach spaces by using semigroup theory
and the Banach fixed point theorem.
2000 Mathematics Subject Classification. 34G20.
Key words and phrases. Classical solution, semilinear integrodifferential equation,
resolvent operator, fixed point theorem.
c ?2003 Texas State University-San Marcos.
Submitted July 17, 2003. Published November 20, 2003.
1

Page 2

2 K. BALACHANDRAN & S. KARUNANITHIEJDE–2003/114
2. Preliminaries
Consider the Sobolev type semilinear integrodifferential equation
d
dt[Ex(t)] = A[x(t) +
?t
0
F(t − s)x(s)ds] + f(t,x(t)),
t ∈ J = [0,b],x(0) = x0,
(2.1)
where A and E are closed linear operators with domain contained in a Banach
space X and ranges contained in a Banach space Y , f : J ×X → Y is a continuous
function. F(t) ∈ B(X), 0 ≤ t ≤ b. F(t) : W → W and for x(·) continuous in
Y , AF(·)x(·) ∈ L1([0,b],X). For x ∈ X, F?(t)x is continuous in t ∈ [0,b], where
B(X) is the space of all bounded linear operators on X, and W is the Banach space
formed from D(A), the domain of A, endowed with the graph norm.
The operators A : D(A) ⊂ X → Y and E : D(E) ⊂ X → Y satisfy the
hypothesis:
(A1) A and E are closed linear operators,
(A2) D(E) ⊂ D(A) and E is bijective,
(A3) E−1: Y → D(E) is continuous and E−1F = FE−1,
(A4) AE−1generates a strongly continuous semigroup of bonded linear operators
in X.
Definition 2.1. A family of bounded linear operator R(t) ∈ B(X) for t ∈ [0,b] is
called a resolvent operator for
?t
if
(i) R(0) = I, (the identity operator on X).
(ii) For all x ∈ X, R(t)x is continuous for t ∈ J.
(iii) R(t) ∈ B(W), t ∈ J. For y ∈ W, R(·)y ∈ C1([0,b],X) ∩ C([0,b],W) and
d
dtR(t)y = AE−1[R(t)y +
0
?t
Definition 2.2. A function x(t) ∈ C([0,b],X) is called a mild solution of the
Cauchy problem (2.1) if it satisfies the integral equation
dx
dt= A[x(t) +
0
F(t − s)x(s)ds]
?t
F(t − s)R(s)yds]
= R(t)AE−1y +
0
R(t − s)AE−1F(s)yds,t ∈ J.
x(t) = E−1R(t)Ex0+ E−1
?t
0
R(t − s)f(s,x(s))ds.
(2.2)
Definition 2.3. A classical solution of (2.1) is a function x(·) ∈ C([0,b],W) ∩
C1([0,b],X) which satisfies the integrodifferential equation (2.1) on [0,b].
Assume the following conditions:
(A5) The resolvent operator R(t) is compact in X and there exists a constant
M1> 0, such that ?R(t)? ≤ M1.
(A6) The nonlinear operator f : J × X → X is continuous in t on J and there
exists a constant L > 0 such that
?f(t,x1) − f(t,x2)? ≤ L?x1− x2?X,t ∈ J, x1,x2∈ X,

Page 3

EJDE–2003/114REGULARITY OF SOLUTIONS OF SOBOLEV TYPE EQUATIONS3
(A7) Let α = |E−1| and 0 < αLbM1< 1.
3. Main Results
Theorem 3.1. If the hypothesis (A1) to (A7) are satisfied, then problem (2.1) has
a mild solution on J.
Proof. Let Z = C(J,X). Then define an operator Φ : Z → Z by
(Φx)(t) = E−1R(t)Ex0+ E−1
?t
0
R(t − s)f(s,x(s))ds.
Now for every x1,x2∈ Z and t ∈ J, we have
?(Φx1)(t) − (Φx2)(t)? = ?E−1
?t
?t
?t
0
R(t − s)[f(s,x1(s)) − f(s,x2(s))]ds?
≤ |E−1|
0
?R(t − s)??f(s,x1(s)) − f(s,x2(s))?ds
≤ αM1
≤ αM1Lb?x1(t) − x2(t)?X.
0
L?x1(s) − x2(s)?Xds
Since αLbM1< 1, the operator Φ is a contraction on E. Applying Banach’s fixed
point theorem we get a unique fixed point for Φ and this point is the mild solution
of (2.1) on J.
?
Next we prove that mild solutions are classical solutions when f ∈ C1(J ×X,Y ).
Theorem 3.2. Let assumptions (A1)–(A7) be satisfied and let x(·) be the unique
mild solution of (2.1). Assume further that x0∈ D(A), f ∈ C1(J × X,Y ). Then
x(·) is a unique classical solution of equation (2.1).
Proof. Since (A1)–(A7) are satisfied, problem (2.1) possesses a unique mild solution
which is denoted by x(·). We will show that x(·) ∈ C1(J,X).
Next we shall show that the mild solution is a classical solution of (2.1) on J.
To this end, let
B(s) =
∂xf(s,x(s)),
and
k(t) = E−1R(t)f(0,x0) + AR(t)x0+
?t
Note that x0∈ W, from Definition 2.1 and our assumptions, k(·) ∈ E. Thus the
method used in Pazy [20] or in the proof of Theorem 3.2 can be applied here to
show that the integral equation
?t
has a unique solution w(·) ∈ E. Moreover, from the assumptions we have
f(s,x(s + h)) − f(s,x(s)) = B(s)[x(s + h) − x(s)] + w1(s,h),
f(s + h,x(s + h)) − f(s,x(s + h)) =
∂
s ∈ J,
?t
(3.1)
?
0
F(t − s)R(s)x0ds
?
+ E−1
0
R(t − s)∂
∂sf(s,x(s))ds.
(3.2)
w(t) = k(t) + E−1
0
R(t − s)B(s)w(s)ds, t ∈ J,
(3.3)
∂
∂sf(s,x(s + h))h + w2(s,h),

Page 4

4 K. BALACHANDRAN & S. KARUNANITHIEJDE–2003/114
where h−1?wi(s,h)? → 0, as h → 0, uniformly on s ∈ J for i = 1,2. Define
wh(t) =x(t + h) − x(t)
h
− w(t).
(3.4)
Then from (3.1)–(3.4) and the fact that x(·) is a mild solution, we obtain
wh(t)
= h−1E−1[R(t + h)Ex0− R(t)Ex0] − A
+ h−1E−1??t+h
− E−1?
− E−1
0
?
R(t)x0+
?t
R(t − s)f(s,x(s))ds
?
0
F(t − s)R(s)x0(s)ds
?
0
R(t + h − s)f(s,x(s))ds −
?t
R(t − s)∂
?t
0
?
R(t)f(0,x0) +
?t
0
R(t − s)∂
∂sf(s,x(s))ds
∂xf(s,x(s))w(s)ds
= h−1E−1[R(t + h)Ex0− R(t)Ex0] − A
+ h−1?
+ h−1E−1
0
?t
+ E−1
0
?t
= h−1E−1[R(t + h)Ex0− R(t)Ex0] − A
?h
+ h−1E−1
0
?t
+ E−1
0
?
R(t)x0+
?t
0
F(t − s)R(s)x0(s)ds
?
?
E−1
?h
?t
R(t − s)∂
R(t − s)∂
R(t − s)∂
0
R(t + h − s)f(s,x(s))ds − E−1R(t)f(0,x0)
R(t − s)[w1(s,h) + w2(s,h)]ds
+ E−1
0
∂sf(s,x(s + h))ds − E−1
∂xf(s,x(s))?x(s + h) − x(s)
∂xf(s,x(s))w(s)ds
?t
0
R(t − s)∂
?ds
∂sf(s,x(s))ds
?t
h
− E−1
0
?
R(t)x0+
?t
0
F(t − s)R(s)x0(s)ds
?
+ h−1E−1
0
R(t + h − s)f(s,x(s))ds − E−1R(t)f(0,x0))
?t
R(t − s)∂
R(t − s)∂
R(t − s)[w1(s,h) + w2(s,h)]ds
+ E−1
0
∂s[f(s,x(s + h)) − f(s,x(s))]ds
∂xf(s,x(s))?x(s + h) − x(s)
?t
h
− w(s)?ds
and
?wh(t)?
≤ |E−1|??h−1[R(t + h)Ex0− R(t)Ex0] − A?R(t)x0+
?t
0
F(t − s)R(s)x0(s)ds???

Page 5

EJDE–2003/114REGULARITY OF SOLUTIONS OF SOBOLEV TYPE EQUATIONS5
+ |E−1|??h−1
+ |E−1|??h−1
+ |E−1|??
+ |E−1|??
≤ |E−1|??h−1[R(t + h)Ex0− R(t)Ex0] − A[R(t)x0+
+ |E−1|??h−1
+ |E−1|
0
?t
where
N = αmax
?h
?t
R(t − s)∂
R(t − s)∂
0
R(t + h − s)f(s,x(s))ds − E−1R(t)f(0,x0))??
R(t − s)[w1(s,h) + w2(s,h)]ds??
∂s[f(s,x(s + h)) − f(s,x(s))]ds??
0
?t
?t
0
0
∂xf(s,x(s))?x(s + h) − x(s)
h
− w(s)?ds??
F(t − s)R(s)x0(s)ds]??
?t
0
?h
?R(t − s)??w1(s,h) + w2(s,h)?ds
?R(t − s)?∂
0
R(t + h − s)f(s,x(s))ds − E−1R(t)f(0,x0)??
?t
+ |E−1|
0
∂s?f(s,x(s + h)) − f(s,x(s))?ds + N
?t
0
?wh(s)?ds,
t>0?R(t − s)∂
∂xf(s,x(s))?B(X).
From the definition of resolvent operator and our assumptions, it is clear that the
norm of each one of the first four terms on the right hand side of the above equation
tends to zero as h → 0. Therefore, we have
?wh(t)?X≤ ?(h) + N
?t
0
?wh(s)?Xds,
(3.5)
and ?(h) → 0 as h → 0. From (3.5) it follows by Gronwall’s inequality that
?wh(t)?X≤ ?(h)eTN,
and, therefore, ?wh(t)?X→ 0 as h → 0, t ∈ J. This implies that x(t) is differen-
tiable on J and that w(t) is the derivative of x(t). Since w ∈ E, x is continuously
differentiable on J.
Finally, to show that x is the classical solution of problem (2.1). Observe that,
from the continuous differentiability of x and f ∈ C1(J × X,X), t → f(t,x(t)) is
continuously differentiable on J. As shown in [16], the linear Cauchy problem
v?(t) = AE−1?
v(0) = x0,
v(t) +
?t
o
F(t − s)v(s)ds
?
+ f(t,x(t)),
0 ≤ t ≤ b,
has a unique classical solution v(·) given by
v(t) = R(t) +
?t
0
R(t − s)f(s,x(s))ds.
(3.6)
The right hand side of (3.6) is x(t) since x(·) is the mild solution. So we have
v(t) = x(t), t ∈ J, and hence, x(·) is the classical solutions of (2.1). Hence the
theorem is proved.
?