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volume 7, issue 5, article 173, 2006.

Received 27 January, 2006;

accepted 19 November, 2006.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

APPLICATION OF A TRAPEZOID INEQUALITY TO NEUTRAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS IN

BANACH SPACES

ALEXANDRU MIAHI BICA, VASILE AUREL C ˘AU ¸S AND SORIN MURE ¸SAN

Department of Mathematics University of Oradea, Str. Universitatii no.1 410087, Oradea, Romania EMail:abica@uoradea.ro EMail:vcaus@uoradea.ro

c

2000Victoria University ISSN (electronic): 1443-5756 028-06

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Application of a Trapezoid Inequality to Neutral Fredholm Integro-Differential Equations in

Banach Spaces

Alexandru Miahi Bica, Vasile Aurel C ˘au¸s and

Sorin Mure¸san

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Abstract

A new approach for neutral Fredholm integro-differential equations in Banach spaces, using the Perov’s fixed point theorem of existence, uniqueness and approximation is presented. The approximation of the solution and of its derivative is realized using the method of successive approximations and a trapezoidal quadrature rule in Banach spaces for Lipschitzian functions. The interest is focused on the error estimation.

2000 Mathematics Subject Classification:Primary 45J05, Secondary 45N05, 45B05, 65L05.

Key words: Nonlinear neutral Fredholm integro-differential equations in Banach spaces, Perov’s fixed point theorem, Method of successive approximations, Trapezoidal quadrature rule.

1. Introduction

Consider the neutral Fredholm integro-differential equation (1.1) x(t) =

Z b a

f(t, s, x(s), x⁰(s))ds+g(t), t∈[a, b]

where f : [a, b]×[a, b]×X×X →Xis continuous,Xis a Banach space and g ∈C¹([a, b], X).

To obtain the existence, uniqueness and global approximation of the solution of (1.1) we will use Perov’s fixed point theorem. To this purpose, we differentiate the equation (1.1) with respect to t and assume that f(·, s, u, v) ∈ C¹([a, b], X),∀s ∈ [a, b],∀u, v ∈ X, wherex⁰ = y.Hence (1.1) reduces to the following system of Fredholm integral equations:

(1.2)







x(t) = Rb

a f(t, s, x(s), y(s))ds+g(t) y(t) =Rb

a

∂f

∂t (t, s, x(s), y(s))ds+g⁰(t)

, t∈[a, b].

The Perov fixed point theorem will be applied to the system (1.2) obtaining also the approximation of the solution of (1.1) and its derivative.

The Perov fixed point theorem appeared for the first time in [16] and was later used for two point boundary value problems of second order differential equations in [5]. The Perov fixed point theorem was also used in [1], [10], [18] and [19]. Bica and Muresan have used the Perov fixed point theorem for delay neutral integro-differential equations in [6] and [7]. In this paper the authors have constructed a method of approximating the solution of (1.1) and

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its derivative using a sequence of successive approximations and a trapezoidal quadrature rule from [8]. Some of the existing numerical methods applied to Fredholm integro-differential equations can be found in the papers [2], [3], [4], [9], [11], [12], [13], [14], [15], [17]. The tools utilised in these papers are:

the tau method, direct methods, collocation methods, Runge-Kutta methods, wavelet methods and spline approximation.

In this paper, our interest will be focused on the error estimation of the method presented in the Section3.

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2. Existence, Uniqueness and Approximation

Consider the following conditions:

(i) (continuity): f ∈ C([a, b]×[a, b]×X×X, X), g ∈ C¹([a, b], X) and f(·, s, u, v)∈C¹([a, b], X),for anys∈[a, b], u, v ∈X

(ii) (Lipschitz conditions): there existα₁, α₂, β₁, β₂, γ₁, γ₂, δ₁, δ₂, η∈R^∗+such that for anyt, t⁰, s, s₁, s₂ ∈[a, b]andu, v, u₁, u₂, v₁, v₂ ∈X,we have:

(2.1) kf(t, s, u₁, v₁)−f(t, s, u₂, v₂)k_X

≤α₁ku₁−u₂k_X +β₁kv₁−v₂k_X,

(2.2)

∂f

∂t (t, s, u₁, v₁)− ∂f

∂t (t, s, u₂, v₂) X

≤α₂ku₁−u₂k_X +β₂kv₁−v₂k_X,

(2.3) kf(t, s₁, u, v)−f(t, s₂, u, v)k_X ≤γ₁|s₁−s₂|,

(2.4)

∂f

∂t (t, s₁, u, v)−∂f

∂t (t, s₂, u, v) X

≤γ₂|s₁−s₂|,

(2.5) kf(t, s, u, v)−f(t⁰, s, u, v)k_X ≤δ1|t−t⁰|,

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(2.6)

∂f

∂t (t, s, u, v)− ∂f

∂t (t⁰, s, u, v) X

≤δ₂|t−t⁰|,

(2.7) kg⁰(t)−g⁰(t⁰)k_X ≤η· |t−t⁰|. We use Perov’s fixed point theorem (see [16], [5] and [10]):

Theorem 2.1. Let (X, d) be a complete generalized metric space such that d(x, y) ∈ Rⁿ forx, y ∈ X. Suppose that there exists a functionA : X → X such that:

d(A(x), A(y))≤Q·d(x, y)

for anyx, y ∈X,whereQ ∈ Mn(R⁺). If all eigenvalues ofQlie in the open unit disc fromR² thenQ^m → 0form → ∞and the operatorA has a unique fixed pointx^∗ ∈X. Moreover, the sequence of successive approximationsx_m = A(xm−1)converges to x^∗ inX for anyx0 ∈ X and the following estimation holds:

(2.8) d(x_m, x^∗)≤Q^m(I_n−Q)⁻¹·d(x₀, x₁), for eachm∈N^∗ whereI_nis the unity matrix inM_n(R).

We recall the notion of generalized metric, which is a functiond:Y ×Y → Rⁿon a nonempty setY with the properties:

a) d(x, y)≥0,for anyx, y ∈Y;

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b) d(x, y) = 0⇔x=y,forx, y ∈Y; c) d(y, x) =d(x, y),for anyx, y ∈Y;

d) d(x, z)≤d(x, y) +d(y, z),for anyx, y, z ∈Y.

Here, the order relation onRⁿis defined by

x≤y⇔x_i ≤y_i, in R, for eachi= 1, n, x_i, y_i ∈R

forx= (x₁, . . . , x_n), y = (y₁, . . . , y_n)∈Rⁿ.The pair(Y, d)denote a generalized metric space.

In the following, we will use the notation

kuk_C = max{ku(t)k_X :t∈[a, b]}, foru∈C([a, b], X).

We consider the product spaceY =C([a, b], X)×C([a, b], X)and define the generalized metricd:Y ×Y →R² by,

d((u₁, v₁),(u₂, v₂)) = (ku₁−u₂k_C,kv₁−v₂k_C), for(u1, v1),(u2, v2)∈Y.

It is easy to prove that (Y, d) is a complete generalized metric space. We define the operatorA:Y →Y, A= (A₁A₂)by,

A₁(x, y) (t) = Z b

a

f(t, s, x(s), y(s))ds+g(t)

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and

(2.9) A₂(x, y) (t) = Z b

a

∂f

∂t (t, s, x(s), y(s))ds+gt(t), t∈[a, b]. The following result concerning the existence and uniqueness of the solution for the equation (1.1) holds.

Theorem 2.2. In the conditions (i), (2.1), (2.2), if(α₁+β₂) (b−a)<2and (2.10) 1 + (b−a)⁴(α₁β₂−α₂β₁)²+ 2α₂β₁(b−a)² >(b−a)² α²₁+β₂² then the operatorAhas a unique fixed point(x^∗, y^∗)such that

x^∗ ∈C¹([a, b], X), y^∗ = (x^∗)⁰

and x^∗ is the unique solution of the equation (1.1). Moreover, the sequence of the successive approximations given by,

(2.11) (x₀(t), y₀(t)) = (g(t), g⁰(t)), t∈[a, b]

(2.12) x_m+1(t) = Z b

a

f(t, s, x_m(s), y_m(s))ds+g(t), t∈[a, b]

and

(2.13) y_m+1(t) = Z b

a

∂f

∂t (t, s, x_m(s), y_m(s))ds+g⁰(t), t ∈[a, b],

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converges inY to(x^∗, y^∗)and the following error estimation holds:

(2.14) d((x_m, y_m),(x^∗, y^∗))≤Q^m(I_n−Q)⁻¹·d((x₀, y₀),(x₁, y₁)), for anym ∈N^∗,

where

Q= (b−a)

α₁ β₁ α₂ β₂

.

Proof. From condition (i) we infer that A(Y)⊂ Y.For(u1, v1),(u2, v2)∈ Y, t ∈[a, b]we have

kA1(u1, v1) (t)−A1(u2, v2) (t)k_X

≤ Z b

a

[α₁ku₁(s)−u₂(s)k_X +β₁kv₁(s)−v₂(s)k_X]

≤(b−a)·[α₁ku₁−u₂k_C+β₁kv₁−v₂k_C], for anyt∈[a, b]

and

kA2(u1, v1) (t)−A2(u2, v2) (t)k_X

≤(b−a)·[α₂ku₁−u₂k_C +β₂kv₁−v₂k_C], for anyt∈[a, b]. We infer that

d(A(u₁, v₁), A(u₂, v₂))≤Q·d((u₁, v₁),(u₂, v₂)),

for any (u₁, v₁),(u₂, v₂)∈Y,

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where

Q=

(b−a)α₁ (b−a)β₁ (b−a)α₂ (b−a)β₂

! .

It is easy to see that the eigenvalues ofQ are real. The inequalities (2.10) and (α₁+β₂) (b−a) < 2 lead to µ₁, µ₂ ∈ (−1,1), where µ₁ and µ₂ are these eigenvalues. From the Perov fixed point theorem we infer that Q^m → 0 for m → ∞and the operatorAhas a unique fixed point(x^∗, y^∗)∈Y.Then,

x^∗(t) = Z b

a

f(t, s, x^∗(s), y^∗(s))ds+g(t), ∀t ∈[a, b]

and

y^∗(t) = Z b

a

∂f

∂t (t, s, x^∗(s), y^∗(s))ds+g⁰(t), ∀t∈[a, b].

Since f(·, s, u, v) ∈ C¹([a, b], X), for any s ∈ [a, b], u, v ∈ X and g ∈ C¹([a, b], X) we infer that x^∗ ∈ C¹([a, b], X). If we differentiate the first equality with respect totwe obtainy^∗ = (x^∗)⁰.Thenx^∗ is the unique solution of (1.1). From the relations (2.12) and (2.13) and from (2.9) we infer that the sequences given in (2.12), (2.13) fulfil the recurrence relation

(xm+1, ym+1) =A((xm, ym)), ∀m∈N.

Now, the inequality (2.14) follows from the estimation (2.8). SinceQ^m →0for m → ∞ inM₂(R)we infer that

m→∞lim d((x_m, y_m),(x^∗, y^∗)) = (0,0). This proves the theorem.

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3. The Main Result

To compute the terms of the sequence of successive approximations we use in the calculus of integrals in (2.12), (2.13) the trapezoidal quadrature rule for Lipschitzian functions from [8]:

(3.1) Z b

a

F (x)dx

= (b−a) 2n

"

F (a) + 2

n−1

X

i=1

F

a+i(b−a) n

+F (b)

#

+R_n(F)

with

(3.2) kR_n(F)k_X ≤ L(b−a)²

4n ,

whereLis the Lipschitz constant ofF : [a, b]→X.

In this respect consider the uniform partition of[a, b], (3.3) ∆ : a=t₀ < t₁ <· · ·< t_n−1 < t_n =b

witht_i = a+ ^i(b−a)_n , i = 0, n and compute x_m(t_i), y_m(t_i), i = 0, n, m∈ N^∗.

From (2.12) and (2.13) we have (3.4) x_m+1(t_i) =

Z b a

f(t_i, s, x_m(s), y_m(s))ds+g(t_i)

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and

(3.5) y_m+1(t_i) = Z b

a

∂f

∂t (t_i, s, x_m(s), y_m(s))ds+g⁰(t_i),

for anyi= 0, n, m ∈N. We define the functions

F_m,i, G_m,i : [a, b]→X, m∈N, i= 0, n by

(3.6)







F_m,i(s) = f(t_i, s, x_m(s), y_m(s)) G_m,i(s) = ^∂f_∂t (t_i, s, x_m(s), y_m(s))

, for anys∈[a, b].

Definition 3.1. A setZ ⊂C([a, b], X)is equally Lipschitz if there existsL≥0 such that for anyh∈Z,

kh(t)−h(t⁰)k_X ≤L· |t−t⁰|, for eacht, t⁰ ∈[a, b]. Theorem 3.1. The subsets

{{F_m,i}_m∈

N, i= 0, n} ⊂C([a, b], X) and

{{G_m,i}_m∈

N, i= 0, n} ⊂C([a, b], X)

defined in (3.6), are equally Lipschitz, if the conditions (i) and (2.1) – (2.7) are true.

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Proof. Let

µ=kg⁰k_C = max{kg⁰(t)k_X :t∈[a, b]}. Form∈N^∗andi= 0, nwe have

kF_m,i(s₁)−F_m,i(s₂)k_X

≤γ₁|s₁−s₂|+α₁kx_m(s₁)−x_m(s₂)k_X +β₁ky_m(s₁)−y_m(s₂)k_X

≤γ1|s1−s2|+α1[δ1(b−a) +µ]· |s1−s2|

+β₁[δ₂(b−a)· |s₁−s₂|+kg⁰(s₁)−g⁰(s₂)k_X]

≤[γ₁+α₁µ+β₁η+ (b−a) (α₁δ₁+β₁δ₂)]· |s₁−s₂|, for anys₁, s₂ ∈[a, b]and

kGm,i(s1)−Gm,i(s2)k_X

≤γ₂|s₁−s₂|+α₂kx_m(s₁)−x_m(s₂)k_X +β₂ky_m(s₁)−y_m(s₂)k_X

≤[γ₂+α₂µ+β₂η+ (b−a) (α₂δ₁+β₂δ₂)]· |s₁−s₂|, for anys₁, s₂ ∈[a, b].Moreover, for anyi= 0, nwe have,

kF_0,i(s₁)−F_0,i(s₂)k_X ≤(γ₁+α₁µ+β₁η)·|s₁ −s₂|, for eachs₁, s₂ ∈[a, b]

and

kG_0,i(s₁)−G_0,i(s₂)k_X ≤(γ₂+α₂µ+β₂η)·|s₁ −s₂|, for eachs₁, s₂ ∈[a, b]. Let

L1 =γ1+α1µ+β1η+ (b−a) (α1δ1+β1δ2),

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L₂ =γ₂+α₂µ+β₂η+ (b−a) (α₂δ₁+β₂δ₂). From the above, we infer that for anyi= 0, n, we have,

kF_m,i(s₁)−F_m,i(s₂)k_X ≤L₁· |s₁−s₂|, for eachs₁, s₂ ∈[a, b]

and

kG_m,i(s₁)−G_m,i(s₂)k_X ≤L₂·|s₁−s₂|, for eachs₁, s₂ ∈[a, b] andm∈N. This concludes the proof of the theorem.

Applying in (3.4), (3.5) the quadrature rule (3.1) – (3.2) we obtain the numerical method:

(3.7) x₀(t_i) = g(t_i), y₀(t_i) = g⁰(t_i), fori= 0, n

(3.8) x_m(t_i) =g(t_i) + (b−a) 2n ·

"

f(t_i, a, xm−1(a), ym−1(a))

+ 2

n−1

X

j=1

f(t_i;t_j, x_m−1(t_j), y_m−1(t_j)) +f(t_i, b, x_m−1(b), y_m−1(b))

#

+R_m,i, fori= 0, nandm∈N^∗ and

(3.9) ym(ti) = g⁰(ti) + (b−a) 2n ·

"

∂f

∂t (ti, a, xm−1(a), ym−1(a))

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+ 2

n−1

X

j=1

∂f

∂t (t_i;t_j, xm−1(t_j), ym−1(t_j)) + ∂f

∂t (t_i, b, xm−1(b), ym−1(b))

#

+R_m,i, fori= 0, nandm ∈N^∗. with the remainder estimations

(3.10) kRm,ik_X ≤ L1(b−a)²

4n , for anym ∈N^∗andi= 0, n,

(3.11) kω_m,ik_X ≤ L₂(b−a)²

4n , or anym∈N^∗ andi= 0, n.

These lead to the following algorithm:

x₀(t_i) =g(t_i), y₀(t_i) =g⁰(t_i), x₁(t_i)

(3.12)

=g(t_i) + (b−a) 2n ·

"

f(t_i, a, g(a), g⁰(a))

+ 2

n−1

X

j=1

f(t_i;t_j, g(t_j), g⁰(t_j)) +f(t_i, b, g(b), g⁰(b))

# +R_1,i

=x₁(t_i) +R_1,i,

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y₁(t_i) = g⁰(t_i) + (b−a) 2n ·

∂f

∂t (t_i, a, g(a), g⁰(a)) (3.13)

+ 2

n−1

X

j=1

∂f

∂t (t_i;t_j, g(t_j), g⁰(t_j)) +∂f

∂t (t_i, b, g(b), g⁰(b))

+ω_1,i

=y₁(t_i) +ω_1,i,

x2(ti) =g(ti) + (b−a) 2n ·

f(ti, t0, x1(t0) +R1,0, y1(t0) +ω1,0) (3.14)

+ 2

n−1

X

j=1

f(ti;tj, x1(tj) +R1,j, y1(tj) +ω1,j) +f(t_i, t_n, x₁(t_n) +R_1,n, y₁(t_n) +ω_1,n)

+R_2,i

=g(t_i) + (b−a) 2n ·

f(t_i, t₀, x₁(t₀), y₁(t₀)) + 2

n−1

X

j=1

f(t_i;t_j, x₁(t_j), y₁(t_j))

+f(t_i, t_n, x₁(t_n), y₁(t_n))

+R_2,i

=x₂(t_i) +R_2,i,

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y₂(t_i) =g⁰(t_i) + (b−a) 2n ·

"

∂f

∂t (t_i, t₀, x₁(t₀) +R_1,0, y₁(t₀) +ω_1,0) (3.15)

+ 2

n−1

X

j=1

∂f

∂t (ti;tj, x1(tj) +R1,j, y1(tj) +ω1,j) + ∂f

∂t(t_i, t_n, x₁(t_n) +R_1,n, y₁(t_n) +ω_1,n)

# +ω_2,i

=g⁰(ti) + (b−a) 2n ·

"

∂f

∂t (ti, t0, x1(t0), y1(t0)) + 2

n−1

X

j=1

∂f

∂t (t_i;t_j, x₁(t_j), y₁(t_j)) + ∂f

∂t (t_i, t_n, x₁(t_n), y₁(t_n))

# +ω_2,i

=y2(ti) +ω2,i, wheni= 0, n.

By induction, form≥3,we obtain fori= 0, nthat (3.16) x_m(t_i)

=g(t_i) + (b−a) 2n ·

"

f t_i, t₀, xm−1(t₀) +Rm−1,0, ym−1(t₀) +ωm−1,0

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+ 2

n−1

X

j=1

f t_i;t_j, xm−1(t_j) +Rm−1,j, ym−1(t_j) +ωm−1,j

+f t_i, t_n, xm−1(t_n) +Rm−1,n, ym−1(t_n) +ωm−1,n

#

+R_m,i

=g(t_i) + (b−a) 2n ·

"

f(t_i, t₀, xm−1(t₀), ym−1(t₀))

+ 2

n−1

X

j=1

f(t_i;t_j, xm−1(t_j), ym−1(t_j))

+f(t_i, t_n, x_m−1(t_n), y_m−1(t_n))

#

+R_m,i

=xm(ti) +Rm,i

and

y_m(t_i) (3.17)

=g⁰(t_i) + (b−a) 2n

∂f

∂t t_i, t₀, xm−1(t₀) +Rm−1,0, ym−1(t₀) +ωm−1,0

+ 2

n−1

X

j=1

∂f

∂t t_i;t_j, xm−1(t_j) +Rm−1,j, ym−1(t_j) +ωm−1,j

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+∂f

∂t t_i, t_n, xm−1(t_n) +Rm−1,n, ym−1(t_n) +ωm−1,n

+ω_m,i

=g⁰(t_i) + (b−a) 2n ·

"

∂f

∂t (t_i, t₀, x_m−1(t₀), y_m−1(t₀)) + 2

n−1

X

j=1

∂f

∂t (ti;tj, xm−1(tj), ym−1(tj)) +∂f

∂t (t_i, t_n, xm−1(t_n), ym−1(t_n))

#

+ω_m,i

=y_m(t_i) +ω_m,i.

At the remainder estimation we have for anyi= 0, n kR_1,ik_X ≤ L₁(b−a)²

4n , kω_1,ik_X ≤ L₂(b−a)²

4n .

Using in (3.14) the Lipschitz property (2.1) we obtain:

R_2,i

≤[1 + (b−a) (α₁+β₁)]·L1(b−a)²

4n , for anyi= 0, n.

Using in (3.15) the Lipschitz property (2.2) we obtain:

kω_2,ik ≤[1 + (b−a) (α₂+β₂)]· L₂(b−a)²

4n , for eachi= 0, n.

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By induction, we obtain form≥2andi= 0, n, R_m,i

≤

1 + (b−a) (α₁ +β₁) +· · ·+ (b−a)^m−1(α₁+β₁)^m−1 (3.18)

· L1(b−a)² 4n

= 1−(b−a)^m(α₁+β₁)^m

1−(b−a) (α₁+β₁) · L₁(b−a)² 4n and

kωm,ik ≤

1 + (b−a) (α2+β2) +· · ·+ (b−a)^m−1(α2+β2)^m−1 (3.19)

· L2(b−a)² 4n

= 1−(b−a)^m(α₂+β₂)^m

1−(b−a) (α₂+β₂) · L₂(b−a)²

4n .

Finally, we can state the following result:

Theorem 3.2. With the conditions (i), (2.1) – (2.7), (2.10), and if(b−a) (α₁+β₁)

<1and(b−a) (α₂+β₂)<1, then the solution of the system (1.2) is approxi- mated on the knots of the uniform partition∆given in (3.3), by the sequence

{(x_m(t_i), y_m(t_i))}_m∈

N, i= 0, n

obtained in (3.12) – (3.17) and the following error estimation holds:

(3.20)

kx^∗(t_i)−x_m(t_i)k_X ky^∗(ti)−ym(ti)k_X

!

≤Q^m(I₂ −Q)⁻¹d(x₀, x₁)

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+





L1(b−a)² 4n[1−(b−a)(α₁+β1)]

L2(b−a)² 4n[1−(b−a)(α₂+β2)]



, for eachm∈N^∗andi= 0, n.

Proof. Follows from (2.14), (3.18) and (3.19) since

kx^∗(t_i)−x_m(t_i)k_X ≤ kx^∗(t_i)−x_m(t_i)k_X +kx_m(t_i)−x_m(t_i)k_X for eachm∈N^∗ andi= 0, nand,

ky^∗(t_i)−y_m(t_i)k_X ≤ ky^∗(t_i)−y_m(t_i)k_X +ky_m(t_i)−y_m(t_i)k_X for eachm∈N^∗ and∀i= 0, n.

Remark 1. Whenf(t, s, u, v) =H(t, s)·f(s, u, v)withH¹ ∈C [a, b]², X we obtain the existence, uniqueness and approximation of the solution for Hammerstein-Fredholm integro-differential equations in Banach spaces. More- over, in the particular case H(t, s) = G(t, s), the Green function, we obtain a new approach for two point boundary value problems associated to second order differential equations in Banach spaces.

Remark 2. ForX =Rⁿwe obtain a new method in analysing systems of Fred- holm integro-differential equations and for X = R we obtain similar results for the approximation of the solution of a scalar Fredholm integro-differential equation.

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References

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