approximate solution of nonlinear integral equations of the theory of developing systems

Differential Equations, Vol. 39, No. 9, 2003, pp. 1277–1288. Translated from Differentsial’nye Uravneniya, Vol. 39, No. 9, 2003, pp. 1214–1223.Original Russian Text Copyright c© 2003 by Boikov, Tynda.

NUMERICALMETHODS

Approximate Solution of Nonlinear IntegralEquations of the Theory of Developing Systems

I. V. Boikov and A. N. TyndaPenza State University, Penza, Russia

Received March 3, 2003

A large class of problems in economics, ecology, medicine, and other fields can be described bymodels of developing systems introduced by Glushkov and described in detail in [1]. Systems ofVolterra nonlinear equations are the main tool for describing such models.

In the present paper, we suggest a numerical method for solving systems of nonlinear Volterraintegral equations of special form describing two- and n-commodity economy models.

1. STATEMENT OF THE PROBLEM

Consider the two-commodity model described by the system of nonlinear integral equations

x(t)−t∫

y(t)

h(t, τ)g(τ)x(τ)dτ = 0,

t∫y(t)

k(t, τ)[1 − g(τ)]x(τ)dτ = f(t), 0 < t0 ≤ t ≤ T, (1.1)

with the unknown functions

x(t) ∈ C[0,∞], y(t) ∈ C1[t0,∞] (y(t) < t)

and with given functions

h(t, τ), k(t, τ) ∈ C[0,∞]×[t0,∞], f(t), g(t) ∈ C[t0,∞] (0 < g(t) < 1)

defined on the interval [t0, T ].Integral equations of the form (1.1) have numerous applications, and the necessity of their

numerical solution was emphasized in [2]. A rather complicated scheme, which in particular requiresconstructing the resolvent of a linear Volterra equation in closed form, was suggested in [1] for theapproximate solution of system (1.1).

To construct an approximate method for system (1.1) on the interval [t0, T ], we rewrite thesystem in the form

P1(x(t), y(t)) ≡ x(t)−t∫

y(t)

H(t, τ)x(τ)dτ = 0,

P2(x(t), y(t)) ≡ f(t)−t∫

y(t)

K(t, τ)x(τ)dτ = 0, 0 < t0 ≤ t ≤ T,

or in the operator form

P (X) = (P1(X), P2(X)) = 0, X = (x(t), y(t)). (1.2)

HereH(t, τ) = h(t, τ)g(τ), K(t, τ) = k(t, τ)[1 − g(τ)];

0012-2661/03/3909-1277$25.00 c© 2003 MAIK “Nauka/Interperiodica”

1278 BOIKOV, TYNDA

moreover, the functions H(t, τ) and K(t, τ) are assumed to be zero for τ 6∈ [t0, T ]. We also considern-commodity models described by nonlinear systems of n = r + p+ 1 equations of the form

xi(t) =r∑j=1

t∫y(t)

Hij(t, τ)xj(τ)dτ, i = 1, . . . , r,

fi(t) =r∑j=1

t∫y(t)

Kij(t, τ)xj(τ)dτ, i = 1, . . . , p,

c(t) =r∑i=1

xi(t) +p∑i=1

fi(t), r + p+ 1 = n,

(1.3)

where xi(t), i = 1, . . . , r, fi(t), i = 1, . . . , p, and y(t), 0 ≤ t0 ≤ t ≤ T , y(t) < t, y(t) ≥ y (t0) = 0,are unknown functions, and xi(t) = ϕi(t), t ∈ [0, t0], is a given history. Here xi(t), i = 1, . . . , r, isthe reconstruction rate of the ith new commodity of kind I used for performing internal functionsof the system and for its development; fi(t), i = 1, . . . , p, is the reconstruction rate of the ithnew commodity of kind II used for performing external functions of the system, and Hij(t, τ) andKsj(t, τ), i, j = 1, . . . , r, s = 1, . . . , p, are the productivities of the generation of the ith commodityof kind I and the sth commodity of kind II with the use of the corresponding jth commodity ofkind I (nonnegative functions). The function y(t) corresponds to the intensity of using commoditiesof kind I at time t.

To construct an approximate method for solving system (1.3) on the interval [t0, T ], we rewritethe system in the form

Pi(X,F, y) ≡ xi(t)−t∫

y(t)

r∑j=1

Hij(t, τ)xj(τ)dτ = 0, i = 1, . . . , r,

Pr+i(X,F, y) ≡ fi(t)−t∫

y(t)

r∑j=1

Kij(t, τ)xj(τ)dτ = 0, i = 1, . . . , p,

Pn(X,F, y) ≡ c(t)−r∑i=1

xi(t)−p∑i=1

fi(t) = 0,

(1.4)

where X = (x1(t), x2(t), . . . , xr(t)) and F = (f1(t), f2(t), . . . , fp(t)).

2. DESCRIPTION OF THE METHOD

We solve Eq. (1.2) by the modified Newton–Kantorovich method. To this end, we write out theapproximate equation

P ′(X0) (X −X0) + P (X0) = 0, X0 = (x0(t), y0(t)) . (2.1)

One can readily show that the derivative P ′(X0) of the nonlinear operator P (X) at the point X0

is determined by the matrix (∂P1/∂x|(x0,y0) ∂P1/∂y|(x0,y0)

∂P2/∂x|(x0,y0) ∂P2/∂y|(x0,y0)

).

Consequently, we have

∂Pi/∂x(t)|(x0,y0) (∆x(t)) + ∂Pi/∂y(t)|(x0,y0) (∆y(t)) = −Pi (x0(t), y0(t)) , i = 1, 2, (2.2)

DIFFERENTIAL EQUATIONS Vol. 39 No. 9 2003

APPROXIMATE SOLUTION OF NONLINEAR INTEGRAL EQUATIONS 1279

where ∆x(t) = x1(t)− x0(t), ∆y(t) = y1(t)− y0(t), (x0(t), y0(t)) is the initial approximation, and

∂P1/∂x|(x0,y0) = lims→0

(P1 (x0 + sx, y0)− P1 (x0, y0))/s

= lims→0

1s

x0(t) + sx(t)−t∫

y0(t)

H(t, τ) [x0(τ) + sx(τ)] dτ − x0(t) +

t∫y0(t)

H(t, τ)x0(τ)dτ

= x(t)−

t∫y0(t)

H(t, τ)x(τ)dτ,

∂P1

∂y

∣∣∣∣(x0,y0)

= lims→0

P1 (x0, y0 + sy)− P1 (x0, y0)s

= lims→0

1s

y0(t)+sy(t)∫y0(t)

H(t, τ)x0(τ)dτ

= lims→0

(sy(t)H(t, y0(t) + θsy(t))x0 (y0(t) + θsy(t)))/s

= H(t, y0(t)) x0(y0(t)) y(t), 0 ≤ θ ≤ 1.

In a similar way, we obtain

∂P2

∂x

∣∣∣∣(x0,y0)

= −t∫

y0(t)

K(t, τ)x(τ)dτ,∂P2

∂y

∣∣∣∣(x0,y0)

= K (t, y0(t)) x0 (y0(t)) y(t).

Therefore, system (2.2) acquires the form

∆x(t)−t∫

y0(t)

H(t, τ)∆x(τ)dτ +H (t, y0(t)) x0 (y0(t)) ∆y(t) =

t∫y0(t)

H(t, τ)x0(τ)dτ − x0(t),

−t∫

y0(t)

K(t, τ)∆x(τ)dτ +K (t, y0(t)) x0 (y0(t)) ∆y(t) =

t∫y0(t)

K(t, τ)x0(τ)dτ − f(t).

(2.3)

System (2.3) is linear, and by solving it for ∆x(t) and ∆y(t), we obtain (x1(t), y1(t)).By continuing this process, we obtain a sequence of approximate solutions (xm(t), ym(t)) found

from the systems

∆xm(t)−t∫

y0(t)

H(t, τ)∆xm(τ)dτ +H (t, y0(t)) x0 (y0(t)) ∆ym(t)

=

t∫ym−1(t)

H(t, τ)xm−1(τ)dτ − xm−1(t),

−t∫

y0(t)

K(t, τ)∆xm(τ)dτ +K (t, y0(t)) x0 (y0(t)) ∆ym(t)

=

t∫ym−1(t)

K(t, τ)xm−1(τ)dτ − f(t),

(2.4)

where ∆xm(t) = xm(t)− xm−1(t) and ∆ym(t) = ym(t)− ym−1(t), m = 2, 3, . . .


1280 BOIKOV, TYNDA

Therefore, to find each successive approximation, one should solve a system of two linear Volterraintegral equations. Moreover, the kernels of the equations on the left-hand side remain the sameat each step.

3. THE CONVERGENCE THEOREM

On the basis of general theorems on the convergence of the Newton–Kantorovich method, onecan state the following theorem on the convergence of the process described in Section 2 in thevector space

C =X = x(t), y(t) : x(t), y(t) ∈ C[t0,T ]

with the norm ‖X‖C = max

‖x‖C[t0,T ] , ‖y‖C[t0 ,T ]

.

Theorem 3.1. Let the operator P have a continuous second derivative in the ball Ω0

(‖X −X0‖ ≤ r), and let the following conditions be satisfied :(1) system (2.3) has a unique solution in the domain [t0, T ] ; i.e., there exists Γ0 = [P ′ (X0)]−1 ;(2) ‖∆X‖ = max

‖∆x‖C[t0,T ] , ‖∆y‖C[t0,T ]

≤ η;

(3) ‖Γ0P′′(X)‖ ≤ L in Ω0.

If h = Lη < 1/2 and(1−√

1− 2h)η/h ≤ r ≤

(1 +√

1− 2h)η/h, then Eq. (1.2) has a unique

solution X∗ in the sphere Ω0. The process (2.4) converges to this solution, and the convergence ratecan be estimated as ‖X∗ −Xm‖ ≤ (η/h)

(1−√

1− 2h)m+1

, m = 0, 1, . . .

To prove the theorem, it suffices to verify assumption (1) and indicate the form of the secondderivative, which is necessary for estimating the constant L occurring in assumption (3).

Suppose that the functions H (t, y0(t)) and K (t, y0(t)) have no zeros on t ∈ [t0, T ]. (If this isnot the case, then the proof is slightly modified.) The validity of the condition x0 (y0(t)) 6= 0,t ∈ [t0, T ], is ensured by the choice of the initial approximation (x0(t), y0(t)). Let us eliminate∆y(t) from (2.3):

∆y(t) =1

H (t, y0(t)) x0 (y0(t))

t∫y0(t)

H(t, τ) [∆x(τ) + x0(τ)] dτ − [∆x(t) + x0(t)]

,G(t)

t∫y0(t)

H(t, τ) [∆x(τ) + x0(τ)] dτ − [∆x(t) + x0(t)]

=

t∫y0(t)

K(t, τ)∆x(τ)dτ +

t∫y0(t)

K(t, τ)x0(τ)dτ − f(t),

(3.1)

where G(t) = K (t, y0(t))/H (t, y0(t)). Let us rewrite the second equation in (3.1) in the form

G(t)∆x(t) −t∫

y0(t)

[K(t, τ)−G(t)H(t, τ)]∆x(τ)dτ

=

t∫y0(t)

[G(t)H(t, τ) −K(t, τ)]x0(τ)dτ + f(t)−G(t)x0(t).

By setting

F0(t) =

t∫y0(t)

[H(t, τ)−K(t, τ)/G(t)]x0(τ)dτ + f(t)/G(t)− x0(t),



we obtain

∆x(t)−t∫

y0(t)

[K(t, τ)G(t)

−H(t, τ)]

∆x(τ)dτ = F0(t). (3.2)

Equation (3.2) is a linear Volterra integral equation of the second kind with continuous coefficients.Since t0 ≤ y0(t) ≤ t (owing to the choice of the initial approximation), it follows that Eq. (3.2) hasa unique solution, which can be obtained by the method of successive approximations (e.g., see [6]).Then the function ∆y(t) can be uniquely determined from (3.1).

Therefore, system (2.3) has a unique solution, i.e., assumption (1) of the theorem is valid.The second derivative P ′′(X0) (X) is given by the following relations:

∂2Pi/∂x2∣∣(x0,y0)

= 0, i = 1, 2,

since ∂Pi/∂x|(x0,y0) are independent of x0(t), i = 1, 2;

∂2P1/∂x ∂y∣∣(x0,y0)

= lims→0

[P ′1x (x0, y0 + sy) (x, y)− P ′1x (x0, y0) (x, y)]/s

= lims→0

1s

x(t)−t∫

y0(t)+sy(t)

H(t, τ)x(τ)dτ − x(t) +

t∫y0(t)

H(t, τ)x(τ)dτ

= lim

s→0

1s

y0(t)+sy(t)∫y0(t)

H(t, τ)x(τ)dτ

= lims→0

1s

(sy(t)H (t, y0(t) + θsy(t)) x (y0(t) + θsy(t)))

= H (t, y0(t)) x (y0(t)) y(t), 0 ≤ θ ≤ 1.

Likewise,

∂2P2/∂x ∂y∣∣(x0,y0)

= −K (t, y0(t)) x (y0(t)) y(t);

∂2P1/∂y2∣∣(x0,y0)

= lims→0

[P ′1y (x0, y0 + sy) (x, y)− P ′1y (x0, y0) (x, y)

]/s

= lims→0

[H (t, y0(t) + sy(t)) x0 (y0(t) + sy(t))−H (t, y0(t)) x0 (y0(t))] y(t)/s

= lims→0

y(t)[H (t, y0(t) + sy(t)) x0 (y0(t) + sy(t))−H (t, y0(t)) x0 (y0(t) + sy(t))

+H (t, y0(t)) x0 (y0(t) + sy(t))−H (t, y0(t)) x0 (y0(t))]/s

= lims→0

y(t) [H ′τ (t, y0(t) + sθy(t)) x0 (y0(t) + sy(t)) +H (t, y0(t)) x′0 (y0(t) + sθy(t))]

= [H ′τ (t, y0(t)) x0 (y0(t)) +H (t, y0(t)) x′0 (y0(t))] y(t), 0 ≤ θ ≤ 1.

(3.3)

In a similar way,

∂2P2/∂y2∣∣(x0,y0)

= [K ′τ (t, y0(t)) x0 (y0(t)) +K (t, y0(t)) x′0 (y0(t))] y(t) (3.4)

[formulas (3.3) and (3.4) are valid since y(t) is bounded];

∂2P1/∂y ∂x∣∣(x0,y0)

= lims→0

[H (t, y0(t)) (x0 (y0(t) + sx (y0(t)))) y(t)−H (t, y0(t)) x0 (y0(t)) y(t)]/s

= H (t, y0(t)) y(t)x (y0(t)) ;

∂2P2/∂y ∂x∣∣(x0,y0)

= −K (t, y0(t)) y(t)x (y0(t)) .


1282 BOIKOV, TYNDA

Therefore, for the derivatives (3.3) and (3.4) to be bounded, it is necessary that the initialapproximation x0(t) be differentiable.

To estimate the convergence rate, we suggest an alternative theorem, which does not require theexistence of the second derivative [3].

Theorem 3.2. Let

‖∆X‖ = max‖∆x‖C[t0,T ] , ‖∆y‖C[t0,T ]

≤ B0, ‖P (X0)‖ = η0,

and let the condition ‖P ′(X1)− P ′(X2)‖ ≤ q/B0 be satisfied in the ball

Ω = X : ‖X −X0‖ ≤ B0η0/(1 − q) (q < 1).

Then Eq. (1.2) has a unique solution X∗ in Ω. The approximations (2.4) converge to this solution,and ‖X∗ −Xm‖ ≤ qmη0B0/(1− q).

4. DISCRETIZATION OF THE METHOD

At each step of the iterative process (2.4), one has to solve the linear Volterra integral equationof the second kind

∆xm(t)−t∫

y0(t)

K1(t, τ)∆xm(τ)dτ = Fm−1(t), (4.1)

where K1(t, τ) = K(t, τ)/G(t) −H(t, τ) and

Fm−1(t) =

t∫ym−1(t)

[H(t, τ)−K(t, τ)/G(t)]xm−1(τ)dτ + f(t)/G(t)− xm−1(t).

The approximate solution of this equation is sought in the form of a local spline x∗N(t) of order rby the method in [4]. Then the estimate for the error of the approximate method has the form‖x∗N(t)−∆x∗m(t)‖ ≤ A/N r+1, where ∆x∗m(t) is the exact solution of Eq. (4.1).

In particular, in practice, it is convenient to use the approximation of the exact solution bya piecewise linear function (r = 1). Consider this case in detail. To this end, on the closedinterval [t0, T ], we introduce the grid (w) of the points ti = t0 + (T − t0) i/N , i = 1, . . . , N .

By the collocation method, we require that the approximate solution satisfy Eq. (4.1) at thepoints ti, i = 0, . . . , N :

∆xm (t0) = Fm−1 (t0) ,

∆xm (ti)−ti∫

y0(ti)

K1 (ti, τ) ∆xm(τ)dτ = Fm−1 (ti) , i = 1, . . . , N. (4.2)

On the grid (w), we introduce the integer-valued function given by the formula vw(t) = l,l = 1, . . . , N , if tl−1 < t ≤ tl and set vi = vw (y0 (ti)). Then system (4.2) can be represented in theform

∆xm (ti)−i−1∑j=vi

tj+1∫tj

K1 (ti, τ) ∆xm(τ)dτ −tvi∫

y0(ti)

K1 (ti, τ) ∆xm(τ)dτ

= Fm−1 (ti) , i = 1, . . . , N.

(4.3)



By computing the integrals occurring in (4.3) by the trapezoidal formula, we obtain the system

∆xm (ti)− 0.5 (tvi − y0 (ti)) [K1 (ti, y0 (ti)) ∆xm (y0 (ti)) +K1 (ti, tvi) ∆xm (tvi)]

−i−2∑j=vi

0.5 (tj+1 − tj) [K1 (ti, tj) ∆xm (tj) +K1 (ti, tj+1) ∆xm (tj+1)]

− 0.5 (ti − ti−1) [K1 (ti, ti−1) ∆xm (ti−1) +K1 (ti, ti) ∆xm (ti)] = Fm−1 (ti) .

Therefore, if vi 6= i (vi < i), then

∆xm (ti) =Fm−1 (ti) +A+B + 0.5 (ti − ti−1)K1 (ti, ti−1) ∆xm (ti−1)

1− 0.5 (ti − ti−1)K1 (ti, ti), i = 1, . . . , N, (4.4)

whereA = 0.5 (tvi − y0 (ti))

[K1 (ti, y0 (ti))

(∆xm (tvi) [y0 (ti)− tvi−1]

−∆xm (tvi−1) [y0 (ti)− tvi ])/

(tvi − tvi−1) +K1 (ti, tvi) ∆xm (tvi)],

B =i−2∑j=vi

0.5 (tj+1 − tj) [K1 (ti, tj) ∆xm (tj) +K1 (ti, tj+1) ∆xm (tj+1)] .

But if vi = i, then

∆xm (ti) =Fm−1 (ti) + 0.5 [ti − y0 (ti)]

2K1 (ti, y0 (ti)) ∆xm (ti−1)/(ti − ti−1)

1− 0.5 (ti − y0 (ti))K1 (ti, ti)− 0.5 ((ti − y0 (ti))/(ti − ti−1))K1 (ti, y0 (ti)),

i = 1, . . . , N.(4.5)

Remark. The integrals occurring in Fm−1 (ti), i = 0, . . . , N , are also replaced by quadraturesums over the same nodes tk, k = 0, . . . , N .

Having determined ∆xm (ti), i = 0, . . . , N , from the relations (4.4) and (4.5), we obtain xm (ti) =∆xm (ti) + xm−1 (ti). Further, from (3.1), we have

∆ym(t) =1

H (t, y0(t)) x0 (y0(t))

t∫y0(t)

H(t, τ)∆xm(τ)dτ +

t∫ym−1(t)

H(t, τ)xm−1(τ)dτ − xm(t)

, (4.6)

which readily gives the values ∆ym (ti), i = 0, . . . , N .One can readily see that the error in xm (ti) and ym (ti), i = 0, . . . , N , is O (1/N 2).

A Model Problem

As a model problem, we consider the system

x(t)−t∫

y(t)

tτx(τ)dτ = 0,

t∫y(t)

τx(τ)dτ = 6, t ∈ [10, 15],

whose exact solution is given by the functions x∗(t) = 6t, y∗(t) = 3√t3 − 3.

We have obtained the following results for N = 50 : εx = 7.11 × 10−12 and εy = 6.31 × 10−3

for m = 1; εx = 8.14 × 10−12 and εy = 6.26 × 10−6 for m = 2; εx = 6.12 × 10−15 andεy = 6.19 × 10−9 for m = 3; εx = 3.29 × 10−15 and εy = 6.2 × 10−12 for m = 4; εx = 3.01 × 10−15

and εy = 7.8 × 10−14 for m = 5. Here N is the number of nodes, m is the number of iterations,and εx = maxi |xm (ti)− x∗ (ti)| and εy = maxi |ym (ti)− y∗ (ti)| characterize the accuracy at thenodes.


1284 BOIKOV, TYNDA

5. n-COMMODITY MODELS

Here we generalized the results obtained for two-commodity models to the general case ofn-commodity models. Just as above, to solve system (14), we use the modified Newton–Kantorovichmethod. To this end, we find the derivatives of the nonlinear operators Pi(X,F, y), i = 1, . . . , n, atthe point

(X0, F0, y0) = (x10(t), x20(t), . . . , xr0(t); f10(t), f20(t), . . . , fp0(t); y0(t)) :∂Pi(X,F, y)/∂xj(t)|(X0,F0,y0) = lim

s→0[Pi (x10, . . . , xj0 + sxj, . . . , xr0; f10, . . . , fp0; y0)

− Pi (x10, . . . , xr0; f10, . . . , fp0; y0)]/s

= lims→0

1s

xi0(t) + δijsxj(t)−t∫

y0(t)

Hij(t, τ)sxj(τ)dτ − xi0(t)

= δijxj(t)−

t∫y0(t)

Hij(t, τ)xj(τ)dτ, i, j = 1, . . . , r,

where δi,j is the Kronecker delta;

∂Pr+i(X,F, y)∂xj(t)

∣∣∣∣(X0,F0,y0)

= −t∫

y0(t)

Kij(t, τ)xj(τ)dτ, i = 1, . . . , p, j = 1, . . . , r,

∂Pn(X,F, y)/∂xj(t)|(X0,F0,y0) = −xj(t), j = 1, . . . , r,

∂Pi(X,F, y)/∂fj(t)|(X0,F0,y0) = 0, i = 1, . . . , r, j = 1, . . . , p,

∂Pi(X,F, y)∂y(t)

∣∣∣∣(X0,F0,y0)

= lims→0

1s

y0(t)+sy(t)∫y0(t)

r∑j=1

Hij(t, τ)xj0(τ)dτ

=

[r∑j=1

Hij (t, y0(t)) xj0 (y0(t))

]y(t), i = 1, . . . , r,

∂Pr+i(X,F, y)/∂fj(t)|(X0,F0,y0)= δijfj(t), i, j = 1, . . . , p,

∂Pn(X,F, y)/∂fj(t)|(X0,F0,y0) = −fj(t), j = 1, . . . , p,

∂Pr+i(X,F, y)∂y(t)

∣∣∣∣(X0,F0,y0)

=

[r∑j=1

Kij (t, y0(t)) xj0 (y0(t))

]y(t), i = 1, . . . , p,

∂Pn(X,F, y)/∂y(t)|(X0,F0,y0) = 0.

Thus we obtain the system of linear Volterra integral equations

∆xi(t)−r∑j=1

t∫y0(t)

Hij(t, τ)∆xj(τ)dτ + ∆y(t)r∑j=1

Hij (t, y0(t)) xj0 (y0(t))

=

t∫y0(t)

r∑j=1

Hij(t, τ)xj0(τ)dτ − xi0(t), i = 1, . . . , r,



−r∑j=1

t∫y0(t)

Kij(t, τ)∆xj(τ)dτ + ∆fi(t) + ∆y(t)r∑j=1

Kij (t, y0(t)) xj0 (y0(t))

=

t∫y0(t)

r∑j=1

Kij(t, τ)xj0(τ)dτ − fi0(t), i = 1, . . . , p,

−r∑j=1

∆xj(t)−p∑j=1

∆fj(t) =r∑j=1

xj0(t) +p∑j=1

fj0(t)− c(t).

(5.1)

Here

∆xj(t) = xj1(t)− xj0(t), j = 1, . . . , r,∆fj(t) = fj1(t)− fj0(t), j = 1, . . . , p, ∆y(t) = y1(t)− y0(t),

and (x10(t), x20(t), . . . , xr0(t); f10(t), f20(t), . . . , fp0(t); y0(t)) is the initial approximation. By solvingthis system for ∆xj(t), j = 1, . . . , r, ∆fj(t), j = 1, . . . , p, and ∆y(t), we obtain

(x11(t), x21(t), . . . , xr1(t); f11(t), f21(t), . . . , fp1(t); y1(t)) .

By continuing this process, we obtain a sequence of approximate solutions

(x1m(t), x2m(t), . . . , xrm(t); f1m(t), f2m(t), . . . , fpm(t); ym(t)) ,

determined from the systems

∆xmi (t)−r∑j=1

t∫y0(t)

Hij(t, τ)∆xmj (τ)dτ + ∆ym(t)r∑j=1

Hij (t, y0(t)) xj0 (y0(t))

=

t∫ym−1(t)

r∑j=1

Hij(t, τ)xj,m−1(τ)dτ − xi,m−1(t), i = 1, . . . , r,

−r∑j=1

t∫y0(t)

Kij(t, τ)∆xmj (τ)dτ + ∆fmi (t) + ∆ym(t)r∑j=1

Kij (t, y0(t)) xj0 (y0(t))

=

t∫ym−1(t)

r∑j=1

Kij(t, τ)xj,m−1(τ)dτ − fi,m−1(t), i = 1, . . . , p,

−r∑j=1

∆xmj (t)−p∑j=1

∆fmj (t) =r∑j=1

xj,m−1(t) +p∑j=1

fj,m−1(t)− c(t),

(5.2)

where

∆xmj (t) = xj,m(t)− xj,m−1(t), j = 1, . . . , r,∆fmj (t) = fj,m(t)− fj,m−1(t), j = 1, . . . , p, ∆ym(t) = ym(t)− ym−1(t).


1286 BOIKOV, TYNDA

Let us show that system (5.1) is uniquely solvable. We set

L1i (t) =

r∑j=1

Hij (t, y0(t)) xj0 (y0(t)) ,

L2i (t) =

r∑j=1

Kij (t, y0(t)) xj0 (y0(t)) ,

F 1i (t) =

t∫y0(t)

r∑j=1

Hij(t, τ)xj0(τ)dτ − xi0(t), i = 1, . . . , r,

F 2i (t) =

t∫y0(t)

r∑j=1

Kij(t, τ)xj0(τ)dτ − fi0(t), i = 1, . . . , p,

F 3(t) = c(t)−r∑j=1

xj0(t)−p∑j=1

fj0(t).

Then system (5.1) can be rewritten in the form

∆xi(t)−r∑j=1

t∫y0(t)

Hij(t, τ)∆xj(τ)dτ + ∆y(t)L1i (t) = F 1

i (t), i = 1, . . . , r,

∆fi(t)−r∑j=1

t∫y0(t)

Kij(t, τ)∆xj(τ)dτ + ∆y(t)L2i (t) = F 2

i (t), i = 1, . . . , p,

r∑j=1

∆xj(t) +p∑j=1

∆fj(t) = F 3(t).

(5.3)

We add the first r equations of system (5.3):

r∑i=1

∆xi(t)−r∑i=1

r∑j=1

t∫y0(t)

Hij(t, τ)∆xj(τ)dτ + ∆y(t)r∑i=1

L1i (t) =

r∑i=1

F 1i (t).

Next, we sum the equations with numbers r + 1, r + 2, . . . , r + p :

p∑i=1

∆fi(t)−p∑i=1

r∑j=1

t∫y0(t)

Kij(t, τ)∆xj(τ)dτ + ∆y(t)p∑i=1

L2i (t) =

p∑i=1

F 2i (t).

By expressing∑r

i=1 ∆xi(t) and∑p

i=1 ∆fi(t) and by substituting them into the last equation, we ob-tain

r∑i=1

r∑j=1

t∫y0(t)

Hij(t, τ)∆xj(τ)dτ −∆y(t)r∑i=1

L1i (t) +

r∑i=1

F 1i (t)

+p∑i=1

r∑j=1

t∫y0(t)

Kij(t, τ)∆xj(τ)dτ −∆y(t)p∑i=1

L2i (t) +

p∑i=1

F 2i (t) = F 3(t),



whence it follows that

∆y(t)

(r∑i=1

L1i (t) +

p∑i=1

L2i (t)

)=

r∑j=1

t∫y0(t)

[r∑i=1

Hij(t, τ)∆xj(τ) +p∑i=1

Kij(t, τ)∆xj(τ)

]dτ

+r∑i=1

F 1i (t) +

p∑i=1

F 2i (t)− F 3(t).

Further, let the function G(t) =∑r

i=1 L1i (t) +

∑p

i=1 L2i (t) have no zeros on the interval [t0, T ];

then, by substituting the resulting expression for G(t)∆y(t) into each of the first r equations insystem (5.3), we obtain the system

G(t)∆xk(t) +r∑j=1

t∫y0(t)

Ψkj(t, τ)∆xj(τ)dτ = Φk(t), k = 1, . . . , r, (5.4)

where

Ψkj(t, τ) = L1k(t)

[r∑i=1

Hij(t, τ) +p∑i=1

Kij(t, τ)

]−G(t)Hkj(t, τ),

Φk(t) = F 1k (t)G(t) + L1

k(t)

[F 3(t)−

r∑i=1

F 1i (t)−

p∑i=1

F 2i (t)

].

System (5.4) is a standard system of linear Volterra integral equations of the second kind and hasa unique solution ∆x∗k(t), k = 1, . . . , r, t ∈ [t0, T ], under the preceding assumption. Then thefunctions ∆f ∗i (t), i = 1, . . . , r, and ∆y∗(t) are uniquely determined from the relations

∆y∗(t) =1

G(t)

[r∑j=1

t∫y0(t)

(r∑i=1

Hij(t, τ) +p∑i=1

Kij(t, τ)

)∆x∗j(τ)dτ

+r∑i=1

F 1i (t) +

p∑i=1

F 2i (t)− F 3(t)

],

∆f ∗i (t) =r∑j=1

t∫y0(t)

Kij(t, τ)∆x∗j (τ)dτ −∆y∗(t)L2i (t) + F 2

i (t), i = 1, . . . , p.

Therefore, the original system (5.1) has a unique solution on the interval [t0, T ], and the followingassertion is valid.

Theorem 5.1. Let the following conditions be satisfied :(1) ‖P (X0, F0, y0)‖ = η0;

(2) max‖∆x∗i ‖C[t0,T ]

,∥∥∆f ∗j

∥∥C[t0,T ]

(i = 1, . . . , r, j = 1, . . . , p), ‖∆y‖C[t0,T ]

≤ B0;

(3) the condition ‖P ′(X1, F1, y1)− P ′(X2, F2, y2)‖ ≤ q/B0 is satisfied in the ball

Ω = (X,F, y) : ‖(X,F, y) − (X0, F0, y0)‖ ≤ B0η0/(1− q) (q < 1).

Then system (1.4) has a unique solution (X∗, F ∗, y∗) in Ω, the approximations (5.2) converge tothis solution, and ‖(X∗, F ∗, y∗)− (Xm, Fm, ym)‖ ≤ qmη0B0/(1 − q).


1288 BOIKOV, TYNDA

Systems of the form (5.4) can be numerically solved with the use of a scheme similar to thatconstructed in Section 4 for the two-commodity model.

ACKNOWLEDGMENTS

The work was financially supported by the Russian Humanitarian Scientific Foundation (grantno. 01-02-00147a).

REFERENCES

1. Glushkov, V.M., Ivanov, V.V., and Yanenko, V.M., Modelirovanie razvivayushchikhsya sistem (Modelingof Developing Systems), Moscow, 1983.

2. Baker, C.T.H., J. of Comput. and Appl. Math., 2000, vol. 125, pp. 217–249.3. Boikov, I.V., in Sbornik aspirantskikh rabot. Tochnye nauki (Collection of Papers by Postgraduate

Students. Exact Sciences), Kazan, 1970, pp. 82–94.4. Boikov, I.V. and Tynda, A.N., Kubaturnye formuly i ikh prilozheniya : Materialy VI Mezhdunar. Semi-

nara-Soveshchaniya (Cubature Formulas and Their Applications: Proc. VI Int. Workshop), Ufa, 2001,pp. 27–34.

5. Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz (Functional Analysis), Moscow, 1979.6. Smirnov, V.I., Kurs vysshei matematiki (Course of Higher Mathematics), Moscow, 1957, vol. 4.


approximate solution of nonlinear integral equations of the theory of developing systems

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