ON A CERTAIN VOLTERRA-FREDHOLM TYPE INTEGRAL EQUATION
B.G. PACHPATTE 57 SHRINIKETANCOLONY
NEARABHINAYTALKIES
AURANGABAD431 001 (MAHARASHTRA) INDIA
bgpachpatte@gmail.com
Received 10 February, 2008; accepted 18 October, 2008 Communicated by C.P. Niculescu
ABSTRACT. The aim of this paper is to study the existence, uniqueness and other properties of solutions of a certain Volterra-Fredholm type integral equation. The main tools employed in the analysis are based on applications of the Banach fixed point theorem and a certain integral inequality with explicit estimate.
Key words and phrases: Volterra-Fredholm type, integral equation, Banach fixed point theorem, integral inequality, Bielecki type norm, existence and uniqueness, continuous dependence.
2000 Mathematics Subject Classification. 34K10, 35R10.
1. INTRODUCTION
Consider the following Volterra-Fredholm type integral equation (1.1) x(t) =f(t) +
Z t
a
g(t, s, x(s), x0(s))ds+ Z b
a
h(t, s, x(s), x0(s))ds,
for−∞< a≤t≤b < ∞, wherex, f, g, hare inRn, then-dimensional Euclidean space with appropriate norm denoted by |·|. LetRand0 denote the set of real numbers and the derivative of a function. We denote byI = [a, b], R+ = [0,∞) the given subsets ofRand assume that f ∈ C(I,Rn), g, h ∈ C(I2×Rn×Rn,Rn)and are continuously differentiable with respect toton the respective domains of their definitions.
The literature provides a good deal of information related to the special versions of equation (1.1), see [3, 5, 6, 8, 12] and the references cited therein. Recently, in [1] the authors studied a Fredholm type equation similar to equation (1.1) forg = 0using Perov’s fixed point theorem, the method of successive approximations and the trapezoidal quadature rule. The purpose of this paper is to study the existence, uniqueness and other properties of solutions of equation (1.1) under various assumptions on the functions involved and their derivatives. The well known Banach fixed point theorem (see [5, p. 37]) coupled with a Bielecki type norm (see [2]) and an integral inequality with an explicit estimate given in [10, p. 44] are used to establish the results.
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2. EXISTENCE AND UNIQUENESS
By a solution of equation (1.1) we mean a continuous functionx(t)fort ∈ I which is con- tinuously differentiable with respect tot and satisfies the equation (1.1). For every continuous functionu(t)in Rn together with its continuous first derivative u0(t) fort ∈ I we denote by
|u(t)|1 =|u(t)|+|u0(t)|.LetS be a space of those continuous functionsu(t)inRn together with the continuous first derivativeu0(t)inRnfort ∈Iwhich fulfil the condition
(2.1) |u(t)|1 =O(exp (λ(t−a))),
fort∈I, whereλis a positive constant. In the spaceSwe define the norm (see [2, 4, 7, 9, 11])
(2.2) |u|S = sup
t∈I
{|u(t)|1exp (λ(t−a))}.
It is easy to see that S with its norm defined in (2.2) is a Banach space. We note that the condition (2.1) implies that there exists a nonnegative constantN such that
|u(t)|1 ≤Nexp (λ(t−a)). Using this fact in (2.2) we observe that
(2.3) |u|S ≤N.
We need the following special version of the integral inequality given in [10, Theorem 1.5.2, part(b2), p. 44]. We shall state it in the following lemma for completeness.
Lemma 2.1. Let u(t) ∈ C(I,R+), k(t, s), r(t, s) ∈ C(I2,R+)be nondecreasing int ∈ I for eachs ∈Iand
u(t)≤c+ Z t
a
k(t, s)u(s)ds+ Z b
a
r(t, s)u(s)ds, fort ∈Iwherec≥0is a constant. If
d(t) = Z b
a
r(t, s) exp Z s
a
k(s, σ)dσ
ds <1, fort ∈I, then
u(t)≤ c
1−d(t)exp Z t
a
k(t, s)ds
, fort ∈I.
The following theorem ensures the existence of a unique solution to equation (1.1).
Theorem 2.2. Assume that
(i) the functions g, h in equation (1.1) and their derivatives with respect to t satisfy the conditions
(2.4) |g(t, s, u, v)−g(t, s,u,¯ ¯v)| ≤p1(t, s) [|u−u|¯ +|v−v|]¯ ,
(2.5)
∂
∂tg(t, s, u, v)− ∂
∂tg(t, s,u,¯ ¯v)
≤p2(t, s) [|u−u|¯ +|v−v|]¯ ,
(2.6) |h(t, s, u, v)−h(t, s,u,¯ v)| ≤¯ q1(t, s) [|u−u|¯ +|v−v¯|],
(2.7)
∂
∂th(t, s, u, v)− ∂
∂th(t, s,u,¯ v)¯
≤q2(t, s) [|u−u|¯ +|v−¯v|],
wherepi(t, s), qi(t, s)∈C(I2,R+) (i= 1,2), (ii) forλas in (2.1)
(a) there exists a nonnegative constantαsuch thatα <1and (2.8) p1(t, t) exp (λ(t−a)) +
Z t
a
p(t, s) exp (λ(s−a))ds
+ Z b
a
q(t, s) exp (λ(s−a))ds≤αexp (λ(t−a)), fort∈I, wherep(t, s) = p1(t, s) +p2(t, s), q(t, s) = q1(t, s) +q2(t, s), (b) there exists a nonnegative constantβ such that
(2.9) |f(t)|+|f0(t)|+|g(t, t,0)|+ Z t
a
|g(t, s,0,0)|+
∂
∂tg(t, s,0,0)
ds +
Z b
a
|h(t, s,0,0)|+
∂
∂th(t, s,0,0)
ds ≤βexp (λ(t−a)), wheref, g, hare the functions given in equation (1.1).
Then the equation (1.1) has a unique solutionx(t)inS onI . Proof. Letx(t)∈Sand define the operator
(2.10) (T x) (t) =f(t) + Z t
a
g(t, s, x(s), x0(s))ds+ Z b
a
h(t, s, x(s), x0(s))ds.
Differentiating both sides of (2.10) with respect totwe have (2.11) (T x)0(t) =f0(t) +g(t, t, x(t), x0(t)) +
Z t
a
∂
∂tg(t, s, x(s), x0(s))ds +
Z b
a
∂
∂th(t, s, x(s), x0(s))ds.
Now we show that T x maps S into itself. Evidently, T x,(T x)0 are continuous on I and T x,(T x)0 ∈ Rn. We verify that (2.1) is fulfilled. From (2.10), (2.11), using the hypotheses and (2.3) we have
|(T x) (t)|1 (2.12)
=|(T x) (t)|+
(T x)0(t)
≤ |f(t)|+|f0(t)|+|g(t, t, x(t), x0(t))−g(t, t,0,0)|+|g(t, t,0,0)|
+ Z t
a
|g(t, s, x(s), x0(s))−g(t, s,0,0)|ds+ Z t
a
|g(t, s,0,0)|ds
+ Z t
a
∂
∂tg(t, s, x(s), x0(s))− ∂
∂tg(t, s,0,0)
ds+ Z t
a
∂
∂tg(t, s,0,0)
ds +
Z b
a
|h(t, s, x(s), x0(s))−h(t, s,0,0)|ds+ Z b
a
|h(t, s,0,0)|ds
+ Z b
a
∂
∂th(t, s, x(s), x0(s))− ∂
∂th(t, s,0,0)
ds+ Z b
a
∂
∂th(t, s,0,0)
ds
≤βexp (λ(t−a)) +p1(t, t)|x(t)|1+ Z t
a
p(t, s)|x(s)|1ds+ Z b
a
q(t, s)|x(s)|1ds
≤βexp (λ(t−a)) +|x|S
p1(t, t) exp (λ(t−a)) + Z t
a
p(t, s) exp (λ(s−a))ds
+ Z b
a
q(t, s) exp (λ(s−a))ds
≤βexp (λ(t−a)) +|x|Sαexp (λ(t−a))
≤[β+N α] exp (λ(t−a)).
From (2.12) it follows thatT x∈S.This proves thatT mapsSinto itself.
Now, we verify that the operator T is a contraction map. Letx(t), y(t) ∈ S.From (2.10), (2.11) and using the hypotheses we have
|(T x) (t)−(T y) (t)|1 (2.13)
=|(T x) (t)−(T y) (t)|+
(T x)0(t)−(T y)0(t)
≤ |g(t, t, x(t), x0(t))−g(t, t, y(t), y0(t))|
+ Z t
a
|g(t, s, x(s), x0(s))−g(t, s, y(s), y0(s))|ds
+ Z t
a
∂
∂tg(t, s, x(s), x0(s))− ∂
∂tg(t, s, y(s), y0(s))
ds +
Z b
a
|h(t, s, x(s), x0(s))−h(t, s, y(s), y0(s))|ds
+ Z b
a
∂
∂th(t, s, x(s), x0(s))− ∂
∂th(t, s, y(s), y0(s))
ds
≤p1(t, t)|x(t)−y(t)|1+ Z t
a
p(t, s)|x(s)−y(s)|1ds
+ Z b
a
q(t, s)|x(s)−y(s)|1ds
≤ |x−y|S
p1(t, t) exp (λ(t−a)) + Z t
a
p(t, s) exp (λ(s−a))ds
+ Z b
a
q(t, s) exp (λ(s−a))ds
≤ |x−y|Sαexp (λ(t−a)). From (2.13) we obtain
|T x−T y|S ≤α|x−y|S.
Sinceα <1, it follows from the Banach fixed point theorem (see [5, p. 37]) thatT has a unique fixed point in S. The fixed point of T is however a solution of equation (1.1). The proof is
complete.
Remark 1. We note that in 1956 Bielecki [2] first used the norm defined in (2.2) for proving global existence and uniqueness of solutions of ordinary differential equations. It is now used very frequently to obtain global existence and uniqueness results for wide classes of differential and integral equations. For developments related to the topic, see [4] and the references cited therein.
The following theorem holds concerning the uniqueness of solutions of equation (1.1) inRn without the existence part.
Theorem 2.3. Assume that the functionsg, hin equation (1.1) and their derivatives with respect totsatisfy the conditions (2.4) – (2.7). Further assume that the functionspi(t, s), qi(t, s) (i= 1,2) in (2.4) – (2.7) are nondecreasing int∈I for eachs∈I,
(2.14) p1(t, t)≤d,
fort ∈I,whered≥0is a constant such thatd <1,
(2.15) e(t) =
Z b
a
1
1−dq(t, s) exp Z s
a
1
1−dp(s, σ)dσ
ds <1, where
p(t, s) =p1(t, s) +p2(t, s), q(t, s) =q1(t, s) +q2(t, s). Then the equation (1.1) has at most one solution onI .
Proof. Letx(t)andy(t)be two solutions of equation (1.1) and w(t) =|x(t)−y(t)|+|x0(t)−y0(t)|. Then by hypotheses we have
w(t)≤ Z t
a
|g(t, s, x(s), x0(s))−g(t, s, y(s), y0(s))|ds (2.16)
+ Z b
a
|h(t, s, x(s), x0(s))−h(t, s, y(s), y0(s))|ds +|g(t, t, x(t), x0(t))−g(t, t, y(t), y0(t))|
+ Z t
a
∂
∂tg(t, s, x(s), x0(s))− ∂
∂tg(t, s, y(s), y0(s))
ds +
Z b
a
∂
∂th(t, s, x(s), x0(s))− ∂
∂th(t, s, y(s), y0(s))
ds
≤ Z t
a
p1(t, s) [|x(s)−y(s)|+|x0(s)−y0(s)|]ds
+ Z b
a
q1(t, s) [|x(s)−y(s)|+|x0(s)−y0(s)|]ds +p1(t, t) [|x(t)−y(t)|+|x0(t)−y0(t)|]
+ Z t
a
p2(t, s) [|x(s)−y(s)|+|x0(s)−y0(s)|]ds
+ Z b
a
q2(t, s) [|x(s)−y(s)|+|x0(s)−y0(s)|]ds.
Using (2.14) in (2.16) we observe that
(2.17) w(t)≤ 1
1−d Z t
a
p(t, s)w(s)ds+ 1 1−d
Z b
a
q(t, s)w(s)ds.
Now a suitable application of Lemma 2.1 to (2.17) yields
|x(t)−y(t)|+|x0(t)−y0(t)| ≤0,
and hencex(t) =y(t),which proves the uniqueness of solutions of equation (1.1) onI .
3. BOUNDS ONSOLUTIONS
In this section we obtain estimates on the solutions of equation (1.1) under some suitable conditions on the functions involved and their derivatives.
The following theorem concerning an estimate on the solution of equation (1.1) holds.
Theorem 3.1. Assume that the functions f, g, h in equation (1.1) and their derivatives with respect totsatisfy the conditions
|f(t)|+|f0(t)| ≤c,¯ (3.1)
|g(t, s, u, v)| ≤m1(t, s) [|u|+|v|], (3.2)
∂
∂tg(t, s, u, v)
≤m2(t, s) [|u|+|v|], (3.3)
|h(t, s, u, v)| ≤n1(t, s) [|u|+|v|], (3.4)
∂
∂th(t, s, u, v)
≤n2(t, s) [|u|+|v|], (3.5)
where ¯c ≥ 0 is a constant and for i = 1,2, mi(t, s), ni(t, s) ∈ C(I2,R+) and they are nondecreasing int∈I for eachs∈I.Further assume that
(3.6) m1(t, t)≤d,¯
(3.7) e¯(t) = Z b
a
1
1−d¯n(t, s) exp Z s
a
1
1−d¯m(s, σ)dσ
ds <1, fort ∈Iwhered¯≥0is a constant such thatd <¯ 1and
m(t, s) =m1(t, s) +m2(t, s), n(t, s) =n1(t, s) +n2(t, s). Ifx(t), t ∈Iis any solution of equation (1.1), then
(3.8) |x(t)|+|x0(t)| ≤ ¯c
1−d¯
1 1−e¯(t)
exp
Z t
a
m(t, s)ds
,
fort ∈I.
Proof. Letu(t) =|x(t)|+|x0(t)|fort ∈ I.Using the fact that x(t)is a solution of equation (1.1) and the hypotheses we have
u(t)≤ |f(t)|+|f0(t)|+ Z t
a
|g(t, s, x(s), x0(s))|ds (3.9)
+ Z b
a
|h(t, s, x(s), x0(s))|ds+|g(t, t, x(t), x0(t))|
+ Z t
a
∂
∂tg(t, s, x(s), x0(s))
ds+ Z b
a
∂
∂th(t, s, x(s), x0(s))
ds
≤¯c+ Z t
a
m1(t, s)u(s)ds+ Z b
a
n1(t, s)u(s)ds
+m1(t, t)u(t) + Z t
a
m2(t, s)u(s)ds+ Z b
a
n2(t, s)u(s)ds.
Using (3.6) in (3.9) we observe that (3.10) u(t)≤ ¯c
1−d¯+ 1 1−d¯
Z t
a
m(t, s)u(s)ds+ 1 1−d¯
Z b
a
n(t, s)u(s)ds.
Now an application of Lemma 2.1 to (3.10) yields (3.8).
Remark 2. We note that the estimate obtained in (3.8) yields not only the bound on the solution of equation (1.1) but also the bound on its derivative. If the estimate on the right hand side in (3.8) is bounded, then the solution of equation (1.1) and its derivative is bounded onI .
Now we shall obtain an estimate on the solution of equation (1.1) assuming that the functions g, hand their derivatives with respect totsatisfy Lipschitz type conditions.
Theorem 3.2. Assume that the hypotheses of Theorem 2.3 hold. Suppose that Z t
a
|g(t, s, f(s)), f0(s)|ds+ Z b
a
|h(t, s, f(s), f0(s))|ds
+|g(t, t, f(t), f0(t))|+ Z t
a
∂
∂tg(t, s, f(s), f0(s))
ds +
Z b
a
∂
∂th(t, s, f(s), f0(s))
ds≤D, fort ∈I,whereD≥0is a constant. Ifx(t), t∈I is any solution of equation (1.1), then (3.11) |x(t)−f(t)|+|x0(t)−f0(t)| ≤
D 1−d
1 1−e(t)
exp
Z t
a
p(t, s)ds
,
fort ∈I.
Proof. Letu(t) = |x(t)−f(t)|+|x0(t)−f0(t)|fort∈I.Using the fact thatx(t)is a solution of equation (1.1) and the hypotheses we have
u(t)≤ Z t
a
|g(t, s, x(s), x0(s))−g(t, s, f(s), f0(s))|ds (3.12)
+ Z t
a
|g(t, s, f(s), f0(s))|ds
+ Z b
a
|h(t, s, x(s), x0(s))−h(t, s, f(s), f0(s))|ds
+ Z b
a
|h(t, s, f(s), f0(s))|ds
+|g(t, t, x(t), x0(t))−g(t, t, f(t), f0(t))|+|g(t, t, f(t), f0(t))|
+ Z t
a
∂
∂tg(t, s, x(s), x0(s))− ∂
∂tg(t, s, f(s), f0(s))
ds +
Z t
a
∂
∂tg(t, s, f(s), f0(s))
ds +
Z b
a
∂
∂th(t, s, x(s), x0(s))− ∂
∂th(t, s, f(s), f0(s))
ds +
Z b
a
∂
∂th(t, s, f(s), f0(s))
ds
≤D+ Z t
a
p1(t, s)u(s)ds+ Z b
a
q1(t, s)u(s)ds
+p1(t, t)u(t) + Z t
a
p2(t, s)u(s)ds+ Z b
a
q2(t, s)u(s)ds.
Using (2.14) in (3.12) we observe that (3.13) u(t)≤ D
1−d + 1 1−d
Z t
a
p(t, s)u(s)ds+ 1 1−d
Z b
a
q(t, s)u(s)ds.
Now an application of Lemma 2.1 to (3.13) yields (3.11).
4. CONTINUOUSDEPENDENCE
In this section we shall deal with continuous dependence of solutions of equation (1.1) on the functions involved therein and also the continuous dependence of solutions of equations of the form (1.1) on parameters.
Consider the equation (1.1) and the following Volterra-Fredholm type integral equation (4.1) y(t) =F (t) +
Z t
a
G(t, s, y(s), y0(s))ds+ Z b
a
H(t, s, y(s), y0(s))ds,
fort ∈ I, wherey, F, G, H are inRn.We assume thatF ∈ C(I,Rn), G, H ∈ C(I2×Rn× Rn,Rn)and are continuously differentiable with respect toton the respective domains of their definitions.
The following theorem deals with the continuous dependence of solutions of equation (1.1) on the functions involved therein.
Theorem 4.1. Assume that the hypotheses of Theorem 2.3 hold. Suppose that
|f(t)−F (t)|+|f0(t)−F0(t)|+|g(t, t, y(t), y0(t))−G(t, t, y(t), y0(t))|
(4.2)
+ Z t
a
|g(t, s, y(s), y0(s))−G(t, s, y(s), y0(s))|ds
+ Z t
a
∂
∂tg(t, s, y(s), y0(s))− ∂
∂tG(t, s, y(s), y0(s))
ds +
Z b
a
|h(t, s, y(s), y0(s))−H(t, s, y(s), y0(s))|ds
+ Z b
a
∂
∂th(t, s, y(s), y0(s))− ∂
∂tH(t, s, y(s), y0(s))
ds
≤ε,
where f, g, h and F, G, H are the functions involved in equations (1.1) and (4.1), y(t) is a solution of equation (4.1) andε >0is an arbitrary small constant. Then the solutionx(t), t∈ I of equation (1.1) depends continuously on the functions involved on the right hand side of equation (1.1).
Proof. Letz(t) =|x(t)−y(t)|+|x0(t)−y0(t)|fort ∈ I.Using the facts thatx(t)andy(t) are the solutions of equations (1.1) and (4.1) and the hypotheses we have
z(t)≤ |f(t)−F(t)|+|f0(t)−F0(t)|
(4.3)
+|g(t, t, x(t), x0(t))−g(t, t, y(t), y0(t))|
+|g(t, t, y(t), y0(t))−G(t, t, y(t), y0(t))|
+ Z t
a
|g(t, s, x(s), x0(s))−g(t, s, y(s), y0(s))|ds
+ Z t
a
|g(t, s, y(s), y0(s))−G(t, s, y(s), y0(s))|ds
+ Z b
a
|h(t, s, x(s), x0(s))−h(t, s, y(s), y0(s))|ds
+ Z b
a
|h(t, s, y(s), y0(s))−H(t, s, y(s), y0(s))|ds
+ Z t
a
∂
∂tg(t, s, x(s), x0(s))− ∂
∂tg(t, s, y(s), y0(s))
ds +
Z t
a
∂
∂tg(t, s, y(s), y0(s))− ∂
∂tG(t, s, y(s), y0(s))
ds +
Z b
a
∂
∂th(t, s, x(s), x0(s))− ∂
∂th(t, s, y(s), y0(s))
ds +
Z b
a
∂
∂th(t, s, y(s), y0(s))− ∂
∂tH(t, s, y(s), y0(s))
ds
≤ε+p1(t, t)z(t) + Z t
a
p(t, s)z(s)ds+ Z b
a
q(t, s)z(s)ds.
Using (2.14) in (4.3) we observe that (4.4) z(t)≤ ε
1−d + 1 1−d
Z t
a
p(t, s)z(s)ds+ 1 1−d
Z b
a
q(t, s)z(s)ds.
Now an application of Lemma 2.1 to (4.4) yields (4.5) |x(t)−y(t)|+|x0(t)−y0(t)| ≤
ε 1−d
1 1−e(t)
exp
Z t
a
p(t, s)ds
, fort∈ I.From (4.5) it follows that the solutions of equation (1.1) depend continuously on the
functions involved on the right hand side of equation (1.1).
Next, we consider the following Volterra-Fredholm type integral equations (4.6) z(t) =f(t) +
Z t
a
g(t, s, z(s), z0(s), µ)ds+ Z b
a
h(t, s, z(s), z0(s), µ)ds, and
(4.7) z(t) =f(t) + Z t
a
g(t, s, z(s), z0(s), µ0)ds+ Z b
a
h(t, s, z(s), z0(s), µ0)ds, for t ∈ I, where z, f, g, h are in Rn and µ, µ0 are real parameters. We assume that f ∈ C(I,Rn); g, h ∈ C(I2 ×Rn×Rn×R,Rn)and are continuously differentiable with respect toton the respective domains of their definitions.
Finally, we present the following theorem which deals with the continuous dependency of solutions of equations (4.6) and (4.7) on parameters.
Theorem 4.2. Assume that the functionsg, hin equations (4.6) and (4.7) and their derivatives with respect totsatisfy the conditions
(4.8) |g(t, s, u, v, µ)−g(t, s,u,¯ v, µ)| ≤¯ k1(t, s) [|u−u|¯ +|v −v¯|],
(4.9) |g(t, s, u, v, µ)−g(t, s, u, v, µ0)| ≤δ1(t, s)|µ−µ0|,
(4.10) |h(t, s, u, v, µ)−h(t, s,u,¯ v, µ)| ≤¯ r1(t, s) [|u−u|¯ +|v−v|]¯ ,
(4.11) |h(t, s, u, v, µ)−h(t, s, u, v, µ0)| ≤γ1(t, s)|µ−µ0|,
(4.12)
∂
∂tg(t, s, u, v, µ)− ∂
∂tg(t, s,u,¯ ¯v, µ)
≤k2(t, s) [|u−u|¯ +|v−¯v|],
(4.13)
∂
∂tg(t, s, u, v, µ)− ∂
∂tg(t, s, u, v, µ0)
≤δ2(t, s)|µ−µ0|,
(4.14)
∂
∂th(t, s, u, v, µ)− ∂
∂th(t, s,u,¯ v, µ)¯
≤r2(t, s) [|u−u|¯ +|v−v|]¯ ,
(4.15)
∂
∂th(t, s, u, v, µ)− ∂
∂th(t, s, u, v, µ0)
≤γ2(t, s)|µ−µ0|,
whereki(t, s), ri(t, s)∈C(I2,R+) (i= 1,2)are nondecreasing int ∈I,for eachs∈Iand δi(t, s), γi(t, s)∈C(I2,R+) (i= 1,2).Further, assume that
(4.16) k1(t, t)≤λ,
(4.17) e0(t) = Z b
a
1
1−λr¯(t, s) exp Z s
a
1 1−λ
¯k(s, σ)dσ
ds <1,
(4.18) δ1(t, t) + Z t
a
[δ1(t, s) +δ2(t, s)]ds+ Z b
a
[γ1(t, s) +γ2(t, s)]ds≤M, fort ∈Iwhereλ, M are nonnegative constants such thatλ <1and
¯k(t, s) =k1(t, s) +k2(t, s), r¯(t, s) = r1(t, s) +r2(t, s). Letz1(t)andz2(t)be the solutions of equations (4.6) and (4.7) respectively. Then (4.19) |z1(t)−z2(t)|+|z10 (t)−z20 (t)|
≤
|µ−µ0|M 1−λ
1 1−e0(t)
exp
Z t
a
k¯(t, s)ds
, fort ∈I.
Proof. Letu(t) = |z1(t)−z2(t)|+|z01(t)−z20 (t)|for t ∈ I. Using the facts thatz1(t)and z2(t)are the solutions of the equations (4.6) and (4.7) and the hypotheses we have
u(t)≤ Z t
a
|g(t, s, z1(s), z10 (s), µ)−g(t, s, z2(s), z02(s), µ)|ds (4.20)
+ Z t
a
|g(t, s, z2(s), z02(s), µ)−g(t, s, z2(s), z20 (s), µ0)|ds
+ Z b
a
|h(t, s, z1(s), z10 (s), µ)−h(t, s, z2(s), z20 (s), µ)|ds
+ Z b
a
|h(t, s, z2(s), z20 (s), µ)−h(t, s, z2(s), z20 (s), µ0)|ds +|g(t, t, z1(t), z10 (t), µ)−g(t, t, z2(t), z20 (t), µ)|
+|g(t, t, z2(t), z20 (t), µ)−g(t, t, z2(t), z20 (t), µ0)|
+ Z t
a
∂
∂tg(t, s, z1(s), z10 (s), µ)− ∂
∂tg(t, s, z2(s), z20 (s), µ)
ds +
Z t
a
∂
∂tg(t, s, z2(s), z20 (s), µ)− ∂
∂tg(t, s, z2(s), z20 (s), µ0)
ds +
Z b
a
∂
∂th(t, s, z1(s), z01(s), µ)− ∂
∂th(t, s, z2(s), z20 (s), µ)
ds +
Z b
a
∂
∂th(t, s, z2(s), z02(s), µ)− ∂
∂th(t, s, z2(s), z20 (s), µ0)
ds
≤ Z t
a
k1(t, s)u(s)ds+ Z t
a
δ1(t, s)|µ−µ0|ds
+ Z b
a
r1(t, s)u(s)ds+ Z b
a
γ1(t, s)|µ−µ0|ds +k1(t, t)u(t) +δ1(t, t)|µ−µ0|
+ Z t
a
k2(t, s)u(s)ds+ Z t
a
δ2(t, s)|µ−µ0|ds
+ Z b
a
r2(t, s)u(s)ds+ Z b
a
γ2(t, s)|µ−µ0|ds.
Using (4.16), (4.18) in (4.20) we observe that (4.21) u(t)≤ |µ−µ0|M
1−λ + 1
1−λ Z t
a
¯k(t, s)u(s)ds+ 1 1−λ
Z b
a
¯
r(t, s)u(s)ds.
Now an application of Lemma 2.1 to (4.21) yields (4.19), which shows the dependency of
solutions to equations (4.6) and (4.7) on parameters.
Remark 3. We note that our approach to the study of the more general equation (1.1) is different from those used in [1] and we believe that the results obtained here are of independent interest.
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