ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)
EXISTENCE AND UNIQUENESS OF SOLUTIONS OF NONLINEAR MIXED INTEGRODIFFERENTIAL EQUATIONS
WITH NONLOCAL CONDITION IN BANACH SPACES
MACHINDRA B. DHAKNE, HARIBHAU L. TIDKE
Abstract. In this article, we study the existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition in Banach spaces. Furthermore, we study continuous dependence of mild solutions. Our analysis is based on semigroup theory and Banach fixed point theorem.
1. Introduction
LetX be a Banach space with normk · k. Let Br ={x∈X :kxk ≤r} ⊂X be a closed ball inX andE=C([t0, t0+β];Br) denote the complete metric space with metric
d(x, y) =kx−ykE= sup
t∈[t0,t0+β]
{kx(t)−y(t)k:x, y∈E}.
Motivated by the work in [3, 7], we consider the nonlinear mixed Volterra- Fredholm integrodifferential equation
x0(t) +Ax(t) =f(t, x(t), Z t
t0
k(t, s, x(s))ds, Z t0+β
t0
h(t, s, x(s))ds), t∈[t0, t0+β]
(1.1) x(t0) +g(t1, t2, . . . , tp, x(·) =x0, (1.2) where 0 ≤ t0 < t1 < t2 < · · · < tp ≤ t0+β, −A is the infinitesimal generator of a C0 semigroup T(t), t≥0, in a Banach space X and the nonlinear functions f : [t0, t0+β]×X ×X×X →X, g : [t0, t0+β]p×X →X, k, h: [t0, t0+β]× [t0, t0+β]×X →X andx0 is a given element ofX.
The notion of “nonlocal condition” has been introduced to extend the study of the classical initial value problems and it is more precise for describing nature phe- nomena than the classical condition since more information is taken into account,
2000Mathematics Subject Classification. 45N05, 47B38, 47H10.
Key words and phrases. Existence and uniqueness; mild and strong solutions;
mixed Volterra-Fredholm; integrodifferential equation; Banach fixed point theorem; nonlocal condition.
c
2011 Texas State University - San Marcos.
Submitted April 28, 2010. Published February 18, 2011.
1
thereby decreasing the negative effects incurred by a possibly erroneous single mea- surement taken at the initial value. The importance of nonlocal conditions in many applications is discussed in [1, 4, 5, 8, 9, 10]. For example, in [10], the author used
g(t1, t2, . . . , tp, x(·)) =
p
X
i=1
cix(ti), (1.3)
where ci,(i = 1,2, . . . , p) are given constants and t = 0 < t1 < · · · < tp ≤ b to describe, for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give better result than using the usual local Cauchy problem with x(0) = x0. In this case, (1.3) allows the additional measurements at ti, i = 1,2, . . . p. Subsequently, several authors are devoted to studying of nonlocal problems by using different techniques, see [2, 6, 11, 13, 14, 16, 17] and the references given therein.
The objective of the present paper is to study the existence, uniqueness and other properties of solutions of the problem (1.1)–(1.2). The main tool employed in our analysis is based on the Banach fixed point theorem and the theory of semigroups.
Our results extend and improve the correspondence results in [12]. We indicate that the method used in this paper is different from that in [12].
This article is organized as follows. In section 2, we present the preliminaries and the statement of our main results. Section 3 deals with proof of the theorems.
Finally in section 4 we give example to illustrate the application of our results.
2. Preliminaries and Main Results
Before proceeding to the statement of our main results, we shall setforth some preliminaries and hypotheses that will be used in our subsequent discussion.
Definition 2.1. A continuous solutionxof the integral equation x(t) =T(t−t0)x0−T(t−t0)g(t1, t2, . . . , tp, x(·))
+ Z t
t0
T(t−s)f(s, x(s), Z s
t0
k(s, τ, x(τ))dτ, Z t0+β
t0
h(s, τ, x(τ))dτ)ds, (2.1) witht∈[t0, t0+β], is said to be a mild solution of (1.1)–(1.2) on [t0, t0+β].
Definition 2.2. A function x is said to be a strong solution of (1.1)–(1.2) on [t0, t0+β] ifxis differentiable almost everywhere on [t0, t0+β],x0∈L1([t0, t0+β], X) and satisfying (1.1)–(1.2) a.e. on [t0, t0+β].
We list the following hypotheses for our convenience.
(H1) There exists a constantG >0 such that
kg(t1, t2, . . . , tp, x1(·))−g(t1, t2, . . . , tp, x2(·))k ≤Gkx1−x2kE
forx1, x2∈E.
(H2) −Ais the infinitesimal generator of aC0 semigroupT(t),t≥0 in X such that
kT(t)k ≤M, for someM ≥1.
(H3) There are constantsL1, K1, H1 andG1such that L1= max
t0≤t≤t0+βkf(t,0,0,0)k,
K1= max
t0≤s≤t≤t0+βkk(t, s,0)k,
H1= max
t0≤s,t≤t0+βkh(t, s,0)k, G1= max
x∈Ekg(t1, t2, . . . , tp, x(·))k.
(H4) The constants kx0k, M, G1, L, K, K1, H, H1, β and r satisfy the following two inequalities:
M[kx0k+G1+Lrβ+LKrβ2+LK1β2+LHrβ2+LH1β2+L1β]≤r, [M G+M Lβ+M LKβ2+M LHβ2]<1.
With these preparations we are now in a position to state our main results to be proved in the present paper.
Theorem 2.3. Assume that (i) hypotheses(H1)–(H4) hold,
(ii) f : [t0, t0+β]×X×X×X→X is continuous inton[t0, t0+β]and there exists a constantL >0such that
kf(t, x1, y1, z1)−f(t, x2, y2, z2)k ≤L(kx1−x2k+ky1−y2k+kz1−z2k), forxi, yi, zi∈Br,i= 1,2.
(iii) k, h: [t0, t0+β]×[t0, t0+β]×X →X are continuous ins, ton[t0, t0+β]
and there exist positive constants K, H such that kk(t, s, x1)−k(t, s, x2)k ≤K(kx1−x2k), kh(t, s, x1)−h(t, s, x2)k ≤H(kx1−x2k), forxi, yi∈Br,i= 1,2.
Then problem (1.1)–(1.2)has a unique mild solution on [t0, t0+β].
Theorem 2.4. Assume that (i) hypotheses(H1)–(H4) hold,
(ii) X is a reflexive Banach space with norm k · k andx0∈D(A),the domain of A,
(iii) g(t1, t2, . . . , tp, x(·))∈D(A),
(iv) There exists a constantL >0such that
kf(t1, x1, y1, z1)−f(t2, x2, y2, z2)k ≤L(|t1−t2|+kx1−x2k+ky1−y2k +kz1−z2k),
(v) There exist constantsK, H >0such that
kk(t1, s, x1)−k(t2, s, x2)k ≤K(|t1−t2|+kx1−x2k), kh(t1, s, x1)−h(t2, s, x2)k ≤H(|t1−t2|+kx1−x2k), Thenxis a unique strong solution of (1.1)–(1.2) on[t0, t0+β].
Theorem 2.5. Suppose that the functions f, g, k and h satisfy hypotheses (H1)- (H4) and assumptions (ii), (iii) of Theorem 2.3. Then, for each pair of elements x∗0, x∗∗0 ∈X, and for the corresponding mild solutions x1, x2 of problem (1.1)with x1(t0) +g(t1, t2, . . . , tp, x1(·)) = x∗0 and x2(t0) +g(t1, t2, . . . , tp, x2(·)) = x∗∗0 , the inequality
kx1−x2kE ≤ M
(1−M G)kx∗0−x∗∗0 kexp ( M Lβ
(1−M G)(1 +Kβ+Hβ))
is true, wheneverG <1/M.
3. Proofs of theorems Proof of Theorem 2.3. Define an operatorF :E→E by
(F z)(t) =T(t−t0)x0−T(t−t0)g(t1, t2, . . . , tp, z(·)) +
Z t
t0
T(t−s)f(s, z(s), Z s
t0
k(s, τ, z(τ))dτ, Z t0+β
t0
h(s, τ, z(τ))dτ)ds, fort∈[t0, t0+β]. Now, we show thatFmapsEinto itself. Forz∈E, t∈[t0, t0+β]
and using hypotheses (H2)-(H4) and assumptions (ii), (iii), we have k(F z)(t)k
≤ kT(t−t0)x0k+kT(t−t0)g(t1, t2, . . . , tp, z(·))k +k
Z t
t0
T(t−s)f(s, z(s), Z s
t0
k(s, τ, z(τ))dτ, Z t0+β
t0
h(s, τ, z(τ))dτ)dsk
≤Mkx0k+M G1+M Z t
t0
[kf(s, z(s), Z s
t0
k(s, τ, z(τ))dτ, Z t0+β
t0
h(s, τ, z(τ))dτ)−f(s,0,0,0)k+kf(s,0,0,0)k]ds
≤Mkx0k+M G1+M Z t
t0
[L(kz(s)−0k+k Z s
t0
k(s, τ, z(τ))dτ−0k +k
Z t0+β
t0
h(s, τ, z(τ))dτ−0k) +kf(s,0,0,0)k]ds
≤Mkx0k+M G1+M Z t
t0
[Lr+L Z s
t0
kk(s, τ, z(τ))−k(s, τ,0) +k(s, τ,0)kdτ +L
Z t0+β
t0
kh(s, τ, z(τ))−h(s, τ,0) +h(s, τ,0)kdτ+L1]ds
≤Mkx0k+M G1+M Z t
t0
[Lr+Lβ(Kr+K1) +Lβ(Hr+H1) +L1]ds
≤M[kx0k+G1+Lrβ+LKrβ2+LK1β2+LHrβ2+LH1β2+L1β]≤r.
Thus,F mapsE into itself.
Now, for everyz1, z2∈E,t∈[t0, t0+β] and using hypotheses (H1), (H2), (H4) and assumptions (ii), (iii), we obtain
k(F z1)(t)−(F z2)(t)k
≤ kT(t−t0)kkg(t1, t2, . . . , tp, z1(·))−g(t1, t2, . . . , tp, z2(·))k +
Z t
t0
kT(t−s)kk[f(s, z1(s), Z s
t0
k(s, τ, z1(τ))dτ, Z t0+β
t0
h(s, τ, z1(τ))dτ)
−f(s, z2(s), Z s
t0
k(s, τ, z2(τ))dτ, Z t0+β
t0
h(s, τ, z2(τ))dτ)]kds
≤M Gkz1−z2kE+ Z t
t0
M L[kz1(s)−z2(s)k
+ Z s
t0
kk(s, τ, z1(τ))−k(s, τ, z2(τ))kdτ +
Z t0+β
t0
kh(s, τ, z1(τ))−h(s, τ, z2(τ))kdτ]ds
≤M Gkz1−z2kE+M Lkz1−z2kE
Z t
t0
[1 +K Z s
t0
dτ +H Z t0+β
t0
dτ]ds
≤M Gkz1−z2kE+M Lkz1−z2kEβ[1 +Kβ+Hβ]
≤qkz1−z2kE,
whereq=M G+M Lβ+M LKβ2+M LHβ2and hence, we obtain kF z1−F z2kE≤qkz1−z2kE,
with 0< q <1. This shows that the the operatorFis a contraction on the complete metric spaceE. By the Banach fixed point theorem, the function F has a unique fixed point in the spaceEand this point is the mild solution of problem (1.1)–(1.2) on [t0, t0+β]. This completes the proof of the Theorem 2.3.
Proof of Theorem 2.4. All the assumptions of Theorem 2.3 are being satisfied, then problem (1.1)–(1.2) has a unique mild solution belonging toE. Now we will show thatxis unique strong solution of (1.1)–(1.2) on [t0, t0+β]. Take
L2= max
t0≤t≤t0+βkf(t, x(t),0,0)k,
K2= max
t0≤s≤t≤t0+βkk(t, s, x(s))k,
H2= max
t0≤s,t≤t0+βkh(t, s, x(s))k.
For 0< θ < t−t0andt∈[t0, t0+β], we have x(t+θ)−x(t)
= [T(t+θ−t0)−T(t−t0)]x0
−[T(t+θ−t0)−T(t−t0)]g(t1, t2, . . . , tp, x(·)) +
Z t0+θ
t0
T(t+θ−s)f(s, x(s), Z s
t0
k(s, τ, x(τ))dτ, Z t0+β
t0
h(s, τ, x(τ))dτ)ds +
Z t+θ
t0+θ
T(t+θ−s)f(s, x(s), Z s
t0
k(s, τ, x(τ))dτ, Z t0+β
t0
h(s, τ, x(τ))dτ)ds
− Z t
t0
T(t−s)f(s, x(s), Z s
t0
k(s, τ, x(τ))dτ, Z t0+β
t0
h(s, τ, x(τ))dτ)ds
=T(t−t0)[T(θ)−I]x0−T(t−t0)[T(θ)−I]g(t1, t2, . . . , tp, x(·)) +
Z t0+θ
t0
T(t+θ−s)[f(s, x(s), Z s
t0
k(s, τ, x(τ))dτ, Z t0+β
t0
h(s, τ, x(τ))dτ)
−f(s, x(s),0,0) +f(s, x(s),0,0)]ds +
Z t
t0
T(t−s)[f(s+θ, x(s+θ), Z s+θ
t0
k(s+θ, τ, x(τ))dτ, Z t0+β
t0
h(s+θ, τ, x(τ))dτ)−f(s, x(s),
Z s
t0
k(s, τ, x(τ))dτ, Z t0+β
t0
h(s, τ, x(τ))dτ)]ds.
Using the assumptions and the factk[T(θ)−I]xk=θkAxk+o(θ), we obtain kx(t+θ)−x(t)k
≤M[θ1+θkAx0k] +M[θ2+θkAg(t1, t2, . . . , tp, x(·))k]
+ Z t0+θ
t0
M[kf(s, x(s), Z s
t0
k(s, τ, x(τ))dτ, Z t0+β
t0
h(s, τ, x(τ))dτ)−f(s, x(s),0,0)k+kf(s, x(s),0,0)k]ds +
Z t
t0
M[kf(s+θ, x(s+θ), Z s+θ
t0
k(s+θ, τ, x(τ))dτ, Z t0+β
t0
h(s+θ, τ, x(τ))dτ)
−f(s, x(s), Z s
t0
k(s, τ, x(τ))dτ, Z t0+β
t0
h(s, τ, x(τ))dτ)k]ds
≤M[θ1+θkAx0k] +M[θ2+θkAg(t1, t2, . . . , tp, x(·))k]
+ Z t0+θ
t0
M L[
Z s
t0
K2dτ+ Z t0+β
t0
H2dτ]ds +M
Z t0+θ
t0
L2ds+ Z t
t0
M L[θ+kx(s+θ)−x(s)k +
Z s
t0
K(s+θ−s|+kx(τ)−x(τ)k)dτ +
Z s+θ
s
K2dτ+ Z t0+β
t0
H(|s+θ−s|+kx(τ)−x(τ)k)dτ]ds
≤M[θ1+θkAx0k] +M[θ2+θkAg(t1, t2, . . . , tp, x(·))k] +M LK2θβ+M LH2θβ +M L2θ+M Lθβ+M L
Z t
t0
kx(s+θ)−x(s)kds +M LKθβ2+M LK2θβ+M LHθβ2
≤P θ+M L Z t
t0
kx(s+θ)−x(s)kds, where1, 2>0 and
P =M[1+kAx0k+2+kAg(t1, t2, . . . , tp, x(·))k+LK2β+LH2β +L2+Lβ+LKβ2+LHβ2+LK2β,
which is independent ofθ andt∈[t0, t0+β].
Thanks to Gronwall’s inequality, we obtain
kx(t+θ)−x(t)k ≤P θeM Lβ, fort∈[t0, t0+β].
Therefore, xis Lipschitz continuous on [t0, t0+β]. The Lipschitz continuity ofx on [t0, t0+β] combined with (iv) and (v) of Theorem 2.4 implies
t→f(t, x(t), Z t
t0
k(t, s, x(s))ds, Z t0+β
t0
h(t, s, x(s))ds)
is Lipschitz continuous on [t0, t0+β]. By [15, Corollary 4.2.11], we observe that the equation
y0(t) +Ay(t) =f(t, x(t), Z t
t0
k(t, s, x(s))ds, Z t0+β
t0
h(t, s, x(s))ds), t∈[t0, t0+β] y(t0) =x0−g(t1, t2, . . . , tp, x(·))
has a unique strong solutiony(t) on [t0, t0+β] satisfying the equation y(t) =T(t−t0)x0−T(t−t0)g(t1, t2, . . . , tp, x(·))
+ Z t
t0
T(t−s)f(s, x(s), Z s
t0
k(s, τ, x(τ))dτ, Z t0+β
t0
h(s, τ, x(τ))dτ)ds
=x(t), t∈[t0, t0+β].
Consequently, x(t) is the strong solution of initial value problem (1.1)–(1.2) on [t0, t0+β]. This completes the proof of Theorem 2.4.
Proof of Theorem 2.5. Suppose that x1(t) and x2(t) satisfy (1.1) on [t0, t0+β]
with x1(t0) +g(t1, t2, . . . , tp, x1(·)) = x∗0 and x2(t0) +g(t1, t2, . . . , tp, x2(·)) =x∗∗0 , respectively and x1, x2 ∈E. Using the equation (2.1), hypotheses (H1)–(H4) and assumptions (ii), (iii), we obtain
kx1(t)−x2(t)k
≤Mkx∗0−x∗∗0 k+M Gkx1−x2kE+ Z t
t0
M Lh
kx1(s)−x2(s)k +
Z s
t0
Kkx1(τ)−x2(τ)kdτ+ Z t0+β
t0
Hkx1(τ)−x2(τ)kdτi ds
≤Mkx∗0−x∗∗0 k+M Gkx1−x2kE
+ Z t
t0
M Lh
kx1(s)−x2(s)k+ Z s
t0
K sup
τ∈[t0,s]
kx1(τ)−x2(τ)kdτ
+ Z t0+β
t0
H sup
τ∈[t0,t0+β]
kx1(τ)−x2(τ)kdτi ds
≤Mkx∗0−x∗∗0 k+M Gkx1−x2kE+ Z t
t0
M L
1 +βK+βH
kx1−x2kEds.
Therefore, we obtain kx1−x2kE ≤ M
(1−M G)kx∗0−x∗∗0 k+ Z t
t0
M Lβ
(1−M G)(1 +Kβ+Hβ)kx1−x2kEds.
Using Gronwall’s inequality, we obtain kx1−x2kE ≤ M
(1−M G)kx∗0−x∗∗0 kexp ( M Lβ
(1−M G)(1 +Kβ+Hβ)), provided thatG < M1. From this inequality, it follows that the continuous depen- dence of solutions depends upon the initial data. This completes the proof of the
Theorem 2.5.
4. Application
To illustrate the applications of some of our main results, we consider the non- linear mixed Volterra- Fredholm partial integrodifferential equation
wt(u, t)−wuu(u, t) =P(t, w(u, t), Z t
0
k1(t, s, w(u, s))ds, Z β
0
h1(t, s, w(u, s))ds), 0< u <1, 0≤t≤β
(4.1) with initial and boundary conditions
w(0, t) =w(1, t) = 0, 0≤t≤β, (4.2) w(u,0) +
p
X
i=1
w(u, ti) =w0(u), 0< t1< t2<· · ·< tp≤β. (4.3) where P : [0, β]×R×R×R→R, k1, h1 : [0, β]×[0, β]×R→R are continuous functions. We assume that the functions P, k1 and h1 in (4.1)–(4.3) satisfy the following conditions:
(1) There exists a constantG∗>0 such that
|
p
X
i=1
w(u, ti)−
p
X
i=1
w(v, ti)| ≤G∗ sup
t∈[0,β]
|u(t)−v(t)|
foru, v∈E1=C([0, β];Br∗∗), whereBr∗∗ ={x∈R:|x| ≤r∗}.
(2) There are constantsL∗1, K1∗, H1∗ andG∗1such that L∗1= max
0≤t≤β|P(t,0,0,0)|, K1∗= max
t0≤s≤t≤t0+β|k1(t, s,0)|, H1∗= max
t0≤s,t≤t0+β|h1(t, s,0)|, G∗1= max
x∈E1
|
p
X
i=1
w(u, ti)|, 0< u <1.
(3) P : [0, β]×R×R×R→Ris continuous int on [0, β] and there exists a constantL∗>0 such that
|P(t, x1, y1, z1)−P(t, x2, y2, z2)| ≤L∗(|x1−x2|+|y1−y2|+|z1−z2|), forxi, yi, zi∈Br∗∗,i= 1,2.
(4) k, h: [0, β]×[0, β]×R→Rare continuous ins, ton [0, β] and there exist respectively constantsK∗, H∗>0 such that
|k1(t, s, x1)−k1(t, s, x2)| ≤K∗(|x1−x2|),
|h1(t, s, x1)−h1(t, s, x2)| ≤H∗(|x1−x2|), forxi, yi∈Br∗∗,i= 1,2.
(5) −Ais the infinitesimal generator of aC0 semigroupT(t),t≥0 in X such that
kT(t)k ≤M∗, for someM∗≥1.
(6) The constants |w0(u)|, M∗, G∗1, L∗, K∗, K1∗, H∗, H1∗, β andrsatisfy the fol- lowing two inequalities:
M∗[|w0(u)|+G∗1+L∗rβ+L∗K∗rβ2+L∗K1∗β2 +L∗H∗rβ2+L∗H1∗β2+L∗1β]≤r∗,
and
[M∗G∗+M∗L∗β+M∗L∗K∗β2+M∗L∗H∗β2]<1.
First, we reduce the equations (4.1)–(4.3) into (1.1)–(1.2) by making suitable choices of A, f, g, k and h. Let X = L2[0,1]. Define the operator A : X → X by Az = −z00 with domain D(A) = {z ∈ X : z, z0 are absolutely continuous, z00∈X andz(0) =z(1) = 0}. Define the functions f : [0, β]×X×X×X →X, k: [0, β]×[0, β]×X →X,h: [0, β]×[0, β]×X →X andg: [0, β]p×X →X as follows
f(t, x, y, z)(u) =P(t, x(u), y(u), z(u)), k(t, s, x)(u) =k1(t, s, x(u)), h(t, s, x)(u) =h1(t, s, x(u)), g(t1, t2, . . . , tp, x(·)u=
p
X
i=1
w(u, ti)
fort ∈[0, β], x, y, z ∈X and 0< u <1. Then the above problem (4.1)–(4.3) can be formulated abstractly as nonlinear mixed Volterra-Fredholm integrodifferential equation in Banach spaceX:
x0(t) +Ax(t) =f(t, x(t), Z t
t0
k(t, s, x(s))ds, Z t0+β
t0
h(t, s, x(s))ds), t∈[t0, t0+β]
(4.4) x(t0) +g(t1, t2, . . . , tp, x(·) =x0. (4.5) Since all the hypotheses of the Theorem 2.3 are satisfied, the Theorem 2.3 can be applied to guarantee the mild solution of the nonlinear mixed Volterra-Fredholm partial integrodifferential equations (4.1)–(4.3).
Acknowledgements. We are grateful to Professor Julio G. Dix and to the anony- mous referee for their helpful comments that improved this article.
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Machindra B. Dhakne
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004, India
E-mail address:[email protected]
Haribhau L. Tidke
Department of Mathematics, School of Mathematical Sciences, North Maharashtra University, Jalgaon-425 001, India
E-mail address:[email protected]
![Introduction In a very recent paper [2], the existence and uniqueness of almost automorphic mild solutions with values in Banach spaces, for the differential equation (1.1) x0(t) =Ax(t) +f(t, x(t), x[α(x(t), t](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)