SOP TRANSACTIONS ON APPLIED MATHEMATICS
ISSN(Print): 2373-8472 ISSN(Online): 2373-8480
DOI: 10.15764/AM.2015.01002
Volume 2, Number 1, January 2015
SOP TRANSACTIONS ON APPLIED MATHEMATICS
Existence and uniqueness of solution of
integrodifferential equation in cone metric space
H. L. Tidke1 , R. T. More2 *
1
Department of Mathematics, North Maharashtra University, Jalgaon-425 001, India.
2
Department of Mathematics, Arts, Commerce and Science College, Bodwad, Jalgaon-425 310, India.
*Corresponding author: [email protected].
Abstract:
In this paper, we study the existence and uniqueness of solution of Volterra integrodifferential equation with
nonlocal condition in cone metric space. The result is obtained by using the some extensions of Banach’s
contraction principle in complete cone metric space. Finally an application of the established result is
demonstrated.
Keywords:
Integrodifferential Equation; Cone Metric Space; Fixed Point Theorem; Nonlocal Condition
1. INTRODUCTION
The purpose of this paper is study the existence and uniqueness of solution of Volterra integrodifferential equation
with nonlocal condition in cone metric space of the form:
Zt
x0 (t) = A(t)x(t) + f t, x(t) + k s, x(s) ds,
t ∈ J = [0, b]
(1.1)
0
x(0) + g(x) = x0 ,
(1.2)
where A(t) is a bounded linear operator on a Banach space X with domain D(A(t)), the unknown x(·) takes values in
the Banach space X; f : J × X × X → X, k : J × X → X,g : C(J, X) → X are appropriate continuous functions and x0
is given element of X.
Many authors have been studied the problems of existence, uniqueness, continuation and other properties of
solutions of these type or special forms of the equations (1.1)–(1.2) are studied by different techniques, for example,
see [3–5, 8, 10, 14] and the references given therein.
The objective of the present paper is to study the existence and uniqueness of solution of the evolution equation
(1.1)–(1.2) under the conditions in respect of the cone metric space and fixed point theory. Hence we extend and
improve some results reported in [2, 8, 10, 11, 14, 15].
The paper is organized as follows: Section 2, we discuss the preliminaries. Section 3, we dealt with study of
existence and uniqueness of solution of integrodifferential equation with nonlocal condition in cone metric space.
Finally in Section 4, we give example to illustrate the application of our result.
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2. PRELIMINARIES
Let us recall the concepts of the cone metric space and we refer the reader to [1, 6, 7, 9, 12, 13] for the more details.
Let E be a real Banach space and P is a subset of E. Then P is called a cone if and only if,
1. P is closed, nonempty and P 6= {0};
2. a, b ∈ R, a, b ≥ 0, x, y ∈ P ⇒ ax + by ∈ P;
3. x ∈ P and −x ∈ P ⇒ x = 0.
For a given cone P ⊂ E, we define a partial ordering relation ≤ with respect to P by x ≤ y if and only if y − x ∈ P.
We shall write x < y to indicate that x ≤ y but x 6= y, while x << y will stand for y − x ∈ intP, where intP denotes the
interior of P.
The cone P is called normal if there is a number K > 0 such that 0 ≤ x ≤ y implies kxk ≤ Kkyk, for every x, y ∈ E.
The least positive number satisfying above is called the normal constant of P.
In the following we always suppose E is a real Banach space , P is a cone in E with intP 6= φ , and ≤ is partial
ordering with respect to P.
Definition 2.1.
Let X be a nonempty set. Suppose that the mapping d : X × X → E satisfies:
(d1 ) 0 ≤ d(x, y) for all x, y ∈ X and d(x, y) = 0 if and only if x = y;
(d2 ) d(x, y) = d(y, x), for all x, y ∈ X;
(d3 ) d(x, y) ≤ d(x, z) + d(z, y), for all x, y, z ∈ X.
Then d is called a cone metric on X and (X, d) is called a cone metric space. The concept of cone metric space is more general
than that of metric space.
The following example is a cone metric space, see [11].
Example 2.2.
Let E = R2 , P = {(x, y) ∈ E : x, y ≥ 0}, X = R, and d : X × X → E such that d(x, y) = (|x − y|, α|x − y|), where α ≥ 0 is a
constant. Then (X, d) is a cone metric space.
Definition 2.3.
Let X be a an ordered space. A function Φ : X → X is said to a comparison function if for every x, y ∈ X, x ≤ y, implies that
Φ(x) ≤ Φ(y), Φ(x) ≤ x and limn→∞ kΦn (x)k = 0, for every x ∈ X.
Example 2.4.
Let E = R2 , P = {(x, y) ∈ E : x, y ≥ 0}. It is easy to check that Φ : E → E, with Φ(x, y) = (ax, ay), for some a ∈ (0, 1) is a
comparison function. Also if Φ1 , Φ2 are two comparison functions over R, then Φ(x, y) = (Φ1 (x), Φ2 (y)) is also a comparison
function over E.
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Existence and uniqueness of solution of integrodifferential equation in cone metric space
3. EXISTENCE AND UNIQUENESS OF SOLUTION
Let X is a Banach space with norm k · k. Let B = C(J, X) be the Banach space of all continuous functions from J
into X endowed with supremum norm
kxk∞ = sup{kx(t)k : t ∈ J}.
Let P = {(x, y) : x, y ≥ 0} ⊂ E = R2 be a cone and define d( f , g) = (k f − gk∞ , αk f − gk∞ ), for every f , g ∈ B. Then
it is easily seen that (B, d) is a cone metric space.
Definition 3.1.
The function x ∈ B satisfies the integral equation
x(t) = x0 − g(x) +
Z t
h
Zs
i
A(s) f s, x(s) +
k τ, x(τ) dτ ds,
0
0
t ∈J
(3.1)
is called the solution of the evolution equation (1.1)–(1.2).
We need the following lemma for further discussion:
Lemma 3.2.
[11] Let (X, d) be a complete cone metric space, where P is a normal cone with normal constant K. Let f : X → X be a function
such that there exists a comparison function Φ : P → P such that
d( f (x), f (y)) ≤ Φ(d(x, y)),
for every x, y ∈ X. Then f has a unique fixed point.
We list the following hypotheses for our convenience:
(H1 ) A(t) is a bounded linear operator on X for each t ∈ J, the function t → A(t) is continuous in the uniform
operator topology and hence there exists a constant K such that
K = sup kA(t)k.
t∈J
(H2 ) Let Φ : R2 → R2 be a comparison function.
(i) There exists continuous function p : J → R+ such that
k f (t, x) − f (t, y)k, αk f (t, x) − f (t, y)k ≤ p(t)Φ d(x, y) ,
for every t ∈ J and x, y ∈ X.
(ii) There exists continuous function q : J → R+ such that
kk(t, x) − k(t, y)k, αkk(t, x) − k(t, y)k ≤ q(t)Φ d(x, y) ,
for every t ∈ J and x, y ∈ X.
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(iii) There exists a positive constant G such that
kg(x) − g(y)k, αkg(x) − g(y)k ≤ GΦ d(x, y) ,
for every x, y ∈ X.
Z th
Z s
n
i o
(H3 ) sup G + K
p(s) + q(τ)dτ ds = 1.
0
t∈J
0
Theorem 3.3.
Assume that hypotheses (H1 ) − (H3 ) hold. Then the evolution equation (1.1)–(1.2) has a unique solution x on J.
Proof: The operator F : B → B is defined by
Fx(t) = x0 − g(x) +
Z t
h
i
Zs
A(s) f s, x(s) + k τ, x(τ) dτ ds,
0
0
t ∈ J.
(3.2)
By using the hypotheses (H1 ) − (H3 ), we have
kFx(t) − Fy(t)k, αkFx(t) − Fy(t)k
Z t
Z s
h
i
≤ kg(x) − g(y)k + kA(s)k k f (s, x(s)) − f (s, y(s))k + kk(τ, x(τ)) − k(τ, y(τ))kdτ ds,
0
0
Z t
Z s
h
i αkg(x) − g(y)k + α kA(s)k k f (s, x(s)) − f (s, y(s))k + kk(τ, x(τ)) − k(τ, y(τ))kdτ ds
0
0
≤ kg(x) − g(y)k, αkg(x) − g(y)k
Z t + K k f (s, x(s)) − f (s, y(s))k, αk f (s, x(s)) − f (s, y(s))k ds
0
Z t Z s
+ K
kk(τ, x(τ)) − k(τ, y(τ))k, αkk(τ, x(τ)) − k(τ, y(τ))k dτds
0
0
Zt
≤ GΦ kx − yk, αkx − yk + K p(s)Φ kx(s) − y(s)k, αkx(s) − y(s)k ds
0
Z t Z s
+ K
q(τ)Φ kx(τ) − y(τ)k, αkx(τ) − y(τ)k dτds
0
0
Z s
Z t h
i
≤ GΦ kx − yk∞ , αkx − yk∞ + Φ kx − yk∞ , αkx − yk∞
K p(s) + q(τ)dτ ds
0
0
Z s
Z t h
i
≤ GΦ d(x, y) + Φ d(x, y)
K p(s) + q(τ)dτ ds
0
0
Z t h
Z s
i o
n
≤ Φ d(x, y) G + K p(s) + q(τ)dτ ds
0
0
≤ Φ d(x, y) ,
(3.3)
for every x, y ∈ B. This implies that d Fx, Fy ≤ Φ d(x, y) , for every x, y ∈ B. Now an application of Lemma 3.2,
the operator has a unique point in B. This means that the equation (1.1)–(1.2) has unique solution. This completes the
proof of the Theorem 3.3.
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Existence and uniqueness of solution of integrodifferential equation in cone metric space
4. APPLICATION
In this section, we give an example to illustrate the usefulness of our result discussed in previous section. Let us
consider the following evolution equation:
dx 240 −t
te−t x(t)
=
e x(t) +
+
dt
16
(9 + et )(1 + x(t))
x
x(0) +
= x0 ,
8+x
Z t
sx(s)ds,
t ∈ J = [0, 1],
x ∈ X,
(4.1)
0
(4.2)
Therefore, we have
A(t) =
240 −t
e ,
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f (t, x(t)) =
t ∈J
te−t x(t)
,
(9 + et )(1 + x(t))
(t, x) ∈ J × X
k(t, x(t)) = tx(t),
(t, x) ∈ J × X
x
, x ∈ X.
g(x) =
8+x
Now for x, y ∈ C(J, X) and t ∈ J, we have
k f (t, x) − f (t, y)k, αk f (t, x) − f (t, y)k
te−t x(t)
y(t)
x(t)
y(t) =
k
−
k,
αk
−
k
9 + et 1 + x(t) 1 + y(t)
1 + x(t) 1 + y(t)
te−t x(t) − y(t)
x(t) − y(t)
=
k
k,
αk
k
9 + et (1 + x(t))(1 + y(t))
(1 + x(t))(1 + y(t))
te−t kx(t)
−
y(t)k,
αkx(t)
−
y(t)k
≤
9 + et
te−t ≤
kx − yk∞ , αkx − yk∞
t
9+e
te−t
d(x, y)
≤
9 + et
t
≤ Φ d(x, y) ,
10
(4.3)
t
where p(t) = , which is continuous function of J into R+ and a comparison function Φ : R2 → R2 such that
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Φ d(x, y) = d(x, y).
Similarly, we can have
tx(t) ty(t)
tx(t) ty(t) kk(t, x) − k(t, y)k, αkk(t, x) − k(t, y)k = k
−
k, αk
−
k
20
20
20
20
t
kx(t) − y(t)k, αkx(t) − y(t)k
≤
20
t
≤
kx − yk∞ , αkx − yk∞
20
t
≤ d(x, y)
20
t ≤ Φ d(x, y) ,
20
(4.4)
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SOP TRANSACTIONS ON APPLIED MATHEMATICS
t
where q(t) = , which is continuous function of J into R+ and the comparison function Φ defined as above.
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Also,
kx − yk
kx − yk
,α
(8 + kxk)(8 + kyk) (8 + kxk)(8 + kyk)
8
kx − yk, αkx − yk
≤
64
1
≤
kx − yk∞ , αkx − yk∞
8
1 ≤ Φ d(x, y) ,
8
kg(x) − g(y)k, αkg(x) − g(y)k ≤ 8
(4.5)
1
240
where G = , and the comparison function Φ defined as above. Hence the condition (H1 ) holds with K =
.
8
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Moreover,
Z t h
Z s
n
i o
n 1 240 Z t h s Z s τ
i o
sup G + K p(s) + q(τ)dτ ds = sup
+
+
dτ ds
16 0 10
0
0
0 20
t∈J
t∈J 8
n 1 240 Z t h s
s2 i o
= sup
+
+
ds
16 0 10 40
t∈J 8
n 1 240 h s2
s3 io
= sup
+
+
16 20 120
t∈J 8
n 1 240 h 1
1 io
=
+
+
8
16 20 120
n1
240 h
1 io
=
+
1+
8 20 × 16
6
n1
240 h 7 io
=
+
8 20 × 16 6
h1
240 × 7 i
=
+
8 20 × 16 × 6
h1 7i
=
+
= 1.
8 8
(4.6)
Since all the conditions of Theorem 3.3 are satisfied, the problem (4.1)–(4.2) has a unique solution x on J.
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Existence and uniqueness of solution of integrodifferential equation in cone metric space
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