ON A CERTAIN VOLTERRA-FREDHOLM TYPE INTEGRAL EQUATION

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Volterra-Fredholm Type Integral Equation

B.G. Pachpatte vol. 9, iss. 4, art. 116, 2008

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ON A CERTAIN VOLTERRA-FREDHOLM TYPE INTEGRAL EQUATION

B.G. PACHPATTE

57 Shri Niketan Colony Near Abhinay Talkies Aurangabad 431 001 (Maharashtra) India

EMail:bgpachpatte@gmail.com

Received: 10 February, 2008

Accepted: 18 October, 2008

Communicated by: C.P. Niculescu 2000 AMS Sub. Class.: 34K10, 35R10.

Key words: Volterra-Fredholm type, integral equation, Banach fixed point theorem, integral inequality, Bielecki type norm, existence and uniqueness, continuous dependence.

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.

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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 num- bers and the derivative of a function. We denote by I = [a, b], R+ = [0,∞) the given subsets of Rand assume that f ∈ C(I,Rⁿ), g, h ∈ C(I²×Rⁿ×Rⁿ,Rⁿ) and are continuously differentiable with respect to t on 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 = 0 using 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 assump- tions 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 continuously differentiable with respect tot and satisfies the equation (1.1). For every continuous function u(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)inRⁿtogether with the continuous first derivativeu⁰(t)in Rⁿfort∈I which fulfil the condition

(2.1) |u(t)|₁ =O(exp (λ(t−a))),

for t ∈ I, where λ is a positive constant. In the spaceS we 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 thatSwith 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 com- pleteness.

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Lemma 2.1. Letu(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 ∈I wherec≥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 functionsg, hin equation (1.1) and their derivatives with respect totsatisfy 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|]¯ ,

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(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 constantβsuch 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 .

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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 thatT xmapsSinto itself. Evidently,T x,(T x)⁰ are continuous onI andT 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

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+ 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. Let x(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

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+ 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 inS. The fixed point ofT 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.

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The following theorem holds concerning the uniqueness of solutions of equation (1.1) inRⁿwithout 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 functions p_i(t, s), q_i(t, s) (i= 1,2) in (2.4) – (2.7) are nondecreasing in t ∈ I for each s∈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))|

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+ 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 Lemma2.1to (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 .

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3. Bounds on Solutions

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 functionsf, g, h in equation (1.1) and their deriva- tives 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≥0is a constant and fori= 1,2,m_i(t, s), n_i(t, s)∈C(I²,R+)and they are nondecreasing int ∈Ifor 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 ∈I whered¯≥0is a constant such thatd <¯ 1and

m(t, s) =m1(t, s) +m2(t, s), n(t, s) =n1(t, s) +n2(t, s).

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Ifx(t), t∈I is 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 thatx(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 Lemma2.1to (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

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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 functionsg, hand their derivatives with respect totsatisfy Lipschitz type conditions.

Theorem 3.2. Assume that the hypotheses of Theorem2.3hold. Suppose that (3.11)

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

(3.12) |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 (3.13)

+ Z t

a

|g(t, s, f(s), f⁰(s))|ds

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+ 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 (3.13) we observe that (3.14) 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 Lemma2.1to (3.14) yields (3.12).

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4. Continuous Dependence

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, for t ∈ I, where y, F, G, H are in Rⁿ. We assume that F ∈ C(I,Rⁿ), G, H ∈ C(I² ×Rⁿ× Rⁿ,Rⁿ) and are continuously differentiable with respect to t on 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 Theorem2.3hold. Suppose that

|f(t)−F (t)|+|f⁰(t)−F⁰(t)|

(4.2)

+|g(t, t, y(t), y⁰(t))−G(t, t, y(t), y⁰(t))|

+ 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

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+ 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) and ε > 0is an arbitrary small constant. Then the solution x(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

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+ 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 Lemma2.1to (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,

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fort ∈ I, wherez, f, g, h are inRⁿand 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 de- pendency 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|,

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(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|,

where k_i(t, s), r_i(t, s) ∈ C(I²,R+) (i= 1,2) are nondecreasing in t ∈ I, for eachs∈I andδ_i(t, s), γ_i(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 ∈I whereλ, 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.

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Proof. Letu(t) = |z₁(t)−z₂(t)|+|z⁰₁(t)−z₂⁰ (t)|fort ∈ I.Using the facts that z₁(t)andz₂(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), z⁰₁(s), µ)−g(t, s, z₂(s), z₂⁰ (s), µ)|ds (4.20)

+ Z t

a

|g(t, s, z₂(s), z₂⁰ (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), z₁⁰ (s), µ)− ∂

∂th(t, s, z₂(s), z₂⁰ (s), µ)

ds +

Z b

a

∂

∂th(t, s, z₂(s), z₂⁰ (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

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+ 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 depen- dency 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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References

[1] A.M. BICA, V.A. C ˇAU ¸S AND S. MURE ¸SAN, Application of a trapezoid inequality to neutral Fredholm integro-differential equations in Banach spaces, J. Inequal. Pure and Appl. Math., 7(5) (2006), Art. 173. [ONLINE: http:

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[2] A. BIELECKI, Un remarque sur l’application de la méthode de Banach- Cacciopoli-Tikhonov dans la théorie des équations differentilles ordinaires, Bull. Acad. Polon. Sci. Sér. Sci. Math.Phys. Astr., 4 (1956), 261–264.

[3] T.A. BURTON, Volterra Integral and Differential Equations, Academic Press, New York, 1983.

[4] C. CORDUNEANU, Bielecki’s method in the theory of integral equations, Ann. Univ.Mariae Curie-Sklodowska, Section A, 38 (1984), 23–40.

[5] C. CORDUNEANU, Integral Equations and Applications, Cambridge Univer- sity Press, 1991.

[6] M.A. KRASNOSELSKII, Topological Methods in the Theory of Nonlinear In- tegral Equations, Pergamon Press, Oxford, 1964.

[7] M. KWAPISZ, Bieleck’s method, existence and uniqueness results for Volterra integral equations inL^pspaces, J. Math. Anal. Appl., 154 (1991), 403–416.

[8] R.K. MILLER, Nonlinear Volterra Integral Equations, W.A. Benjamin, Menlo Park CA, 1971.

[9] B.G. PACHPATTE, On a nonlinear Volterra Integral-Functional equation, Funkcialaj Ekvacioj, 26 (1983), 1–9.

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[10] B.G. PACHPATTE, Integral and Finite Difference Inequalities and Applica- tions, North-Holland Mathematics Studies, Vol. 205 Elsevier Science, Amster- dam, 2006.

[11] B.G. PACHPATTE, On higher order Volterra-Fredholm integrodifferential equation, Fasciculi Mathematici, 37 (2007), 35–48.

[12] W. WALTER, Differential and Integral Inequalities, Springer-Verlag, Berlin, New York, 1970.