On mild solutions of a semilinear mixed Volterra-Fredholm functional integrodifferential evolution nonlocal problem
in Banach spaces
1Kamalendra Kumar,2Rakesh Kumar and3Manoj Karnatak
1Department of Mathematics, SRMS College of Engineering & Technology, Bareilly-243001, India
2,3Department of Mathematics,Hindu College, Moradabad 244001, India
e-mail: 1[email protected],2[email protected],3[email protected]
Abstract The existence, uniqueness and continuous dependence on initial data of mild solutions of nonlocal mixed Volterra-Fredholm functional integrodifferential equa- tions with delay in Banach spaces has been discussed and proved in the present paper.
The results are established by using the semigroup theory and modified version of Banach contraction theorem.
Keywords Mixed Volterra-Fredholm functional integrodifferential equation; fixed point; semigroup theory; nonlocal condition.
2010 Mathematics Subject Classification 47H10, 45J05, 47B38.
1 Introduction
Byszewski and Acka [1] established the existence, uniqueness and continuous dependence of a mild solution of a semilinear functional differential equation with nonlocal condition of the form
du (t)
dt + Au (t) = f (t, ut) , t ∈ [0, a] , u (s) +
g ut1, . . . ., utp
(s) = φ (s) , s ∈ [−r, 0] ,
where 0 < t1< . . . < tp≤ a (p ∈ N) , −A is the infinitesimal generator of a C0 semigroup of operators on a general Banach space, f, g and φ are given functions and ut(s) = u (t + s) for t ∈ [0, a] , s ∈ [−r, 0]. The problems of existence, uniqueness and other qualitative properties of solutions for semilinear differential equations in Banach spaces has been studied extensively in the literature for last many years, see [2-5], [6], [7-9], [10].
Theorem about the existence, uniqueness and stability of solutions of differential, inte- grodifferential equations and functional differential abstract evolution equations with non- local conditions were studied by Byszewski [1, 11], Balachandran and Chandrasekaran [12], Lin and Liu [13] and Balachandran and Park [14].
In the present paper, we consider semilinear mixed Volterra-Fredholm functional inte- grodifferential equations of the form
dx (t)
dt = Ax (t) + f t, xt, Z t
0
k (t, s, xs) ds, Z T
0
h (t, s, xs) ds
!
, t ∈ [0, T ] , (1) x (t) + g xt1, . . . ., xtp
(t) = φ (t) , t ∈ [−r, 0] , (2)
where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators T (t) , t ≥ 0 on X, f, k, h, g and φ are given functions satisfying some assumptions and xt(θ) = x (t + θ), for θ ∈ [−r, 0] and t ∈ [0, T]. The objective of this paper is to improve
the results of [2] and also generalize the results of Jain and Dhakne [15]. We are finding the results with less restriction by using modified version of Banach contraction principle.
The rest of the paper is organised as follows: In section 2 we give preliminaries and hypotheses. In section 3 we prove existence and uniqueness of solutions. In section 4 we deal with continuous dependence of initial data of mild solutions. Finally in section 5 we give application to illustrate the theory.
2 Preliminaries and hypotheses
Let X be Banach space with the norm k.k. Let C = C ([−r, 0] , X) , 0 < r < ∞ be the Banach space of all continuous functions ψ : [−r, 0] → X endowed with the supremum norm
kψkC= sup {kψ (t)k : −r < t < 0} .
Let B = C ([−r, T ] , X) , T > 0 be the Banach space of all continuous functions x : [−r, T ] → X with the supremum norm kxkB = sup {kx (t)k : −r ≤ t ≤ T } . For any x ∈ B and t ∈ [0, T], we denote xtthe element of C given by xt(θ) = x (t + θ), for θ ∈ [−r, 0] and φ is given element of C.
In this paper, we assume that there exists positive constant M ≥ 1 such that kT (t)k ≤ M , for every t ∈ [0, T].
Definition 2.1 A function x ∈ Bsatisfying the equations:
x (t) = T (t)
φ (0) − g xt1, ..., xtp
(0)
+ Z t
0 T (t − s) f s, xs, Z s
0
k (s, τ, xτ) dτ, Z T
0
h (s, τ, xτ) dτ
!
ds, t ∈ [0, T]
x (t) + g xt1, ..., xtp
(t) = φ (t) , −r ≤ t ≤ 0,
is called a mild solution of the initial value problem (1) – (2).
Our results are based on the modified version of Banach contraction principle.
Lemma 2.1 [16, p.196] Let X be a Banach space. Let D be an operator which maps the elements of X into itself for which Dr is a contraction, wherer is a positive integer. Then D has a unique fixed point.
We make the following assumptions:
(A1) Let f : [0, T ] × C × X × X → X such that for every w ∈ B, x, y ∈ X and t ∈ [0, T ] , f (., wt, x, y) ∈ B and there exists a constant L > 0 such that
kf (t, ψ, x, y) − f (t, φ, u, v)k ≤ L(kψ − φkC+ kx − uk + ky − vk), φ, ψ ∈ C, x, y, u, v ∈ X.
(A2) Let k : [0, T ] ×[0, T]×C → X such that for every w ∈ B and t ∈ [0, T] , k (., ., wt) ∈ B and there exists a constant K > 0 such that
kk (t, s, ψ) − k (t, s, φ)k ≤ K (kψ − φkC) , φ, ψ ∈ C.
(A3) Let h : [0, T ] × [0, T] × C → X such that for every w ∈ B and t ∈ [0, T] , h (., ., wt) ∈ B and there exists a constant H > 0 such that
kh (t, s, ψ) − h (t, s, φ)k ≤ H (kψ − φkC) , φ, ψ ∈ C.
(A4) Let g : Cp→ C such that there exists a constant G ≥ 0 such that g xt1, ..., xtp
(t) − g yt1, ..., ytp
(t)
≤ G (kx − ykB) , t ∈ [−r, 0] .
3 Existence of mild solution
Theorem 3.1 Consider that the assumptions (A1) − (A4) are satisfied. Then the initial value problem (1) – (2) has a unique mild solution x on [−r, T] .
Proof: Let x (t) be a mild solution of the problem (1) – (2). Then it satisfies the equivalent integral equation
x (t) = T (t) φ (0) − T (t) g xt1, ..., xtp
(0)
+ Z t
0 T (t − s) f s, xs, Z s
0
k (s, τ, xτ) dτ, Z T
0
h (s, τ, xτ) dτ
!
ds, t ∈ [0, T ] , (3) x (t) + g xt1, ..., xtp
(t) = φ (t) , −r ≤ t ≤ 0. (4)
Now we rewrite solution of initial value problem (1) – (2) as follows: For φ ∈ C, define φ ∈ B byb
φ (t) =b
( φ (t) − g xt1, ..., xtp
(t)
, if − r ≤ t ≤ 0, T (t)
φ (0) − g xt1, ..., xtp
(0)
, if 0 ≤ t ≤ T.
If y ∈ B and x (t) = y (t) + bφ (t) , t ∈ [−r, T] , then it is easy to see that y satisfies
y (t) = 0; t ∈ [−r, 0] (5)
and y (t) = Z t
0T (t − s) f s, ys+ cφs, Z s
0
k
s, τ, yτ+ cφτ
dτ, Z T
0
h
s, τ, yτ+ cφτ
dτ
!
ds, t ∈[0, T] (6)
if and only if x (t) satisfies the equations (3) – (4) . We define the operator F : B → B, by
(F y) (t) = (0, −r ≤ t ≤ 0,
Rt
0T (t − s) f
s, ys+ cφs,Rs 0 k
s, τ, yτ+ cφτ
dτ,RT 0h
s, τ, yτ+ cφτ
dτ
ds, if 0 ≤ t ≤ T. (7) From the definition of an operator F defined by the equation (7), it is to be noted that the equation (5) – (6) can be written as
y = F y.
Now we show that Fn is a contraction on B for some positive integer n. Let y, w ∈ B and using assumptions (A1) − (A4), we get
k(F y) (t) − (F w) (t)k
≤ Z t
0
kT (t − s)k
f
s, ys+ cφs,Rs 0k
s, τ, yτ+ cφτ
dτ,RT 0 h
s, τ, yτ+ cφτ
dτ
−f
s, ws+ cφs,Rs 0k
s, τ, wτ+ cφτ
dτ,RT 0 h
s, τ, wτ+ cφτ
dτ ds
≤ Z t
0
M L ys+ cφs
−
ws+ cφs
C+ Z s
0
K( yτ+ cφτ
−
wτ+ cφτ
C
dτ
+ Z T
0
H
yτ+ cφτ
− (wτ+ cφτ)
C
dτ
# ds
≤ ML Z t
0 kys− wskCds + M L Z t
0
K Z s
0 kyτ− wτkCdτ ds + M L Z t
0
H Z T
0 kyτ− wτkCdτ ds
≤ ML Z t
0 ky − wkBds + M L Z t
0
K Z s
0 ky − wkBdτ ds + M L Z t
0
H Z T
0 ky − wkBdτ ds
≤ ML ky − wkBt + M LK ky − wkB
t2
2 + M LH ky − wkB
t2 2
≤ ML ky − wkBt + M LKT ky − wkB t
2+ M LHT ky − wkB t 2
≤ ML ky − wkBt + M LKT ky − wkBt + M LHT ky − wkBt.
≤ ML (1 + KT + HT ) ky − wkBt.
F2y
(t) − F2w (t)
= k(F (F y)) (t) − (F (F w)) (t)k = k(F (y1)) (t) − (F (w1)) (t)k
≤ Z t
0kT (t − s)k
f
s, y1s+ cφs,Rs 0k
s, τ, y1τ+ cφτ
dτ,RT 0 h
s, τ, y1τ+ cφτ
dτ
−f
s, w1s+ cφs,Rs 0 k
s, τ, w1τ+ cφτ
dτ,RT 0 h
s, τ, w1τ+ cφτ
dτ ds
≤ Z t
0
M L
y1s+ cφs
−
w1s+ cφs C+
Z s 0
K
y1τ + cφτ
−
w1τ+ cφτ C
dτ +
Z T 0
H
y1τ+ cφτ
−
w1τ+ cφτ C
dτ
# ds
≤ Z t
0 M L ky1s− w1skCds + M L Z t
0
K Z s
0 ky1τ− w1τkCdτ ds + M L
Z t 0
H Z T
0 ky1τ− w1τkCdτ ds
≤ ML Z t
0 ky1− w1kC([−r,s],X)ds + M L Z t
0
K Z s
0 ky1− w1kC([−r,τ ],X)dτ ds + M L
Z t 0
H Z T
0 ky1− w1kC([−r,τ ],X)dτ ds
≤ ML Z t
0
sup
τ ∈[−r,s]ky1(τ ) − w1(τ )k ds + MLK Z t
0
Z s 0
sup
η∈[−r,τ]ky1(η) − w1(η)k dτ ds + M LH
Z t 0
Z T 0
sup
η∈[−r,τ]ky1(η) − w1(η)k dτ ds
≤ ML Z t
0
sup
τ ∈[−r,s]kF y (τ ) − F w (τ )k ds + MLK Z t
0
Z s 0
sup
η∈[−r,τ ]kF y (η) − F w (η)k dτ ds
+ M LH Z t
0
Z T 0
sup
η∈[−r,τ]kF y (η) − F w (η)k dτ ds
≤ ML Z t
0
sup
τ ∈[−r,s](M L (1 + KT + HT ) ky − wkBτ ) ds + M LK
Z t 0
Zs
0
sup
η∈[−r,τ](M L (1 + KT + HT ) ky − wkBη) dτ ds + M LH
Z t 0
Z T 0
sup
η∈[−r,τ](M L (1 + KT + HT ) ky − wkBη) dτ ds
≤ M2L2(1 + KT + HT ) ky − wkB
"Z t 0
sup
τ ∈[−r,s]
τ
! ds +
Z t 0
K Z s
0
sup
η∈[−r,τ]
η
! dτ ds
+ Z t
0
H Z T
0
sup
η∈[−r,τ]
η
! dτ ds
#
≤ M2L2(1 + KT + HT ) ky − wkB
"Z t 0
sds + Z t
0
K Z s
0
τ dτ ds + Z t
0
H Z T
0
τ dτ ds
#
≤ M2L2(1 + KT + HT ) ky − wkB
t2 2 + Kt3
3!+ Ht3 3!
≤ M2L2(1 + KT + HT ) ky − wkB
t2
2 + KTt2
3!+ HTt2 3!
≤ M2L2(1 + KT + HT ) ky − wkB
t2
2!+ KTt2
2!+ HTt2 2!
≤ M2L2(1 + KT + HT )2ky − wkB
t2 2!. Continuing in this way, we get
k(Fny) (t) − (Fnw) (t)k ≤ [M L (1 + KT + HT ) t]n
n! ky − wkB.
For n large enough,
[M L (1 + KT + HT ) t]n
n! < 1.
Thus there exists a positive integer nsuch that Fn is a contraction in B. By virtue of Lemma 2.1, the operator F has a unique fixed point ey in B. Then ex = ey + bφ is solution of the initial value problem (1) – (2). This completes the proof of the theorem (3.1). 2
4 Continuous dependence of a mild solution
Theorem 4.1Suppose that the functionf, k, h and gsatisfies the assumption (A1) − (A4).
Then for each φ1, φ2∈ Cand for the corresponding mild solutions x1, x2 of the problems dx (t)
dt = Ax (t) + f t, xt, Z t
0
k (t, s, xs) ds, Z T
0
h (t, s, xs) ds
!
, t ∈ [0, T] , (8) x (t) + g xt1, . . . ., xtp
(t) = φi(t) , t ∈ [−r, 0] , (i = 1, 2) . (9)
The following inequality kx1− x2kB≤
kφ1− φ2kC+ G + LHT2
kx1− x2kB
M eM LT(1+KT )
is true.
Additionally, if M G + LHT2
eM LT(1+KT )< 1 then
kx1− x2kB ≤ M eM LT(1+KT )
1 − M (G + LHT2) eM LT(1+KT )kφ1− φ2kC. Proof was given in paper [17], so we omit details here.
5 Applications
To illustrate the application of our result in section 3, consider the following semilinear partial functional mixed integrodifferential equation of the form
∂w (u, t)
∂t =∂2w (u, t)
∂u2 + F t, w (u, t − r) ,
Z t 0
k1(t, w (u, s − r))ds, Z T
0
h1(t, w (u, s − r)) ds
!
, t ∈ [0, T] , (10)
w (0, t) = w (π, t) = 0, 0 ≤ t ≤ T, (11)
w (u, t) + Xp i=1
w (u, ti+ t) = φ (u, t) , 0 ≤ u ≤ π, −r ≤ t ≤ 0, (12)
where 0 < t1 ≤ . . . ≤ tp≤ T, the function F : [0, T] × R × R × R → R is continuous. We assume that the function F, k1and h1satisfying the following conditions:
For every t ∈ [0, T] and u, v, x1, x2, y1, y2∈ R, there exists a constant
|F (t, u, x1, x2) − F (t, v, y1, y2)| ≤ l (|u − v| + |x1− y1| + |x2− y2|) ;
|k1(t, s, u) − k1(t, s, v)| ≤ p (|u − v|) ;
|h1(t, s, u) − h1(t, s, v)| ≤ q (|u − v|) .
Let us take X = L2[0, π] . Define the operator A : X → X by A (z) = z00 with domain D (A) = {z ∈ X : z, z0are absolutely continuous, z00∈ X and z (0) = z (π) = 0} .
Then the operator A can be written as
Az = X∞ n=1
−n2(z, zn) zn, z ∈ D (A)
where zn(u) = √ 2/π
sin nu, n = 1, 2, ... is the orthogonal set of eigenvectors of A and A is the infinitesimal generator of an analytic semigroup T (t) , t ≥ 0 and is given by
T (t) z = X∞ n=1
exp −n2t
(z, zn) zn, z ∈ X.
Now, the analytic semigroup T (t)being compact, there exists constant M such that
|T (t)| ≤ M, for each t ∈ [0, T] . Define the function f : [0, T ] × C × X × X → X , as follows
f (t, ψ, x, y) (u) = F (t, ψ (−r) , x (u) , y (u)) , k (t, φ) (u) = k1(t, φ (−r) u) ,
h (t, φ) (u) = h1(t, φ (−r) u) .
For t ∈ [0, T] , ψ, φ ∈ C, x ∈ X and 0 ≤ u ≤ π. With these choices of the functions the equations (10)-(12) can be formulated as an abstract mixed integrodifferential equation in Banach space X :
dx (t)
dt = Ax (t) + f t, xt, Z t
0
k (t, s, xs) ds, Z T
0
h (t, s, xs) ds
!
, t ∈ [0, T] , x (t) + g xt1, . . . ., xtp
(t) = φ (t) , t ∈ [−r, 0] .
Since, all the hypotheses of theorem (3.1) are satisfied, the theorem (3.1), can be applied to guarantee the existence of mild solution w (u, t) = x (t) u, t ∈ [0, T] , u ∈ [0, π] , of semilinear partial integrodifferential equation (10) – (12).
6 Conclusion
In this paper, the existence, uniqueness and continuous dependence of initial data on a mild solution of semilinear mixed Volterra-Fredholm functional integrodifferential equations with nonlocal conditions in general Banach space are discussed. We apply the concepts of semigroup theory and modified version of Banach contraction theorem. We also give example to illustrate the theory.
References
[1] Byszewski, L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 1992. 162: 494-505.
[2] Byszewski, L. and Akca, H. On mild solution of a semilinear functional differential evolution nonlocal problem. J. Appl. Math. Stochastic Anal. 1997. 10(3): 265–271.
[3] Byszewski, L. and Akca, H. Existence of solutions of a semilinear functional evolution nonlocal problem. Nonlinear Anal. 1998. 34: 65–72.
[4] Byszewski, L. and Lakshamikantham, V. Theorems about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal.
1990. 40: 11–19.
[5] Corduneanu, C. Integral Equations and Stability of Feedback System. New York: Aca- demic Press. 1973.
[6] Hale, J. K. Theory of Functional Differential Equations. New York: Springer. 1977.
[7] Ntouyas, S. K. Initial and boundary value problems for functional differential equations via the topological transversality method: a survey. Bull. Greek Math. Soc. 1998. 40 3–41.
[8] Ntouyas, S. K. Nonlocal initial and boundary value problems: A survey. Handbook of Differential Equations: Ordinary Differential Equations, Vol. 2.2006. 461–557.
[9] Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equa- tions. New York: Springer Verlag. 1983.
[10] Quang, Y. Delay Differential Equations with Applications in Population Dynamics New York: Academic Press. 1993.
[11] Byszewski, L. Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problem. Dynamic Systems and Applications. 1996. 5: 595-605.
[12] Balachandran, K. and Chandrasekaran, M. Existence of solution of a delay differential equation with nonlocal condition. Indian J. Pure Appl. Math. 1996. 27: 443-449.
[13] Lin, Y and Liu, J. H. Semilinear integrodifferential equations with nonlocal Cauchy problem. Nonlinear Anal. 1996. 26: 1023-1033.
[14] Balachandran, K and Park, J. Y. Existence of a mild solution of a functional inte- grodifferential equation with nonlocal condition. Bull. Korean Math. Soc. 2001. 38(1):
175-182.
[15] Jain, R. S. and Dhakne, M. B. On some qualitative properties of mild solutions of nonlocal semilinear functional differential equations. Demonstratio Mathematica 2014.
XLVII (4).
[16] Siddiqi, A. H. Functional Analysis with Applications. New Delhi: Tata McGraw-Hill Publishing ltd. 1986.
[17] Kumar, K and Kumar, R. Nonlocal Cauchy problem for Sobolev type mixed Volterra- Fredholm functional integrodifferential equation. Journal of Physical Science. 2013. 17:
69-76.
