Solvability of Systems of Volterra Integral Equations of ... - Springer Link

c Allerton Press, Inc., 2013. ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2013, Vol. 57, No. 1, pp. 54–63. c D.N. Sidorov, 2013, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2013, No. 1, pp. 62–72. Original Russian Text

Solvability of Systems of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernels D. N. Sidorov1* 1

Senior Research Fellow, Energy Systems Institute of the Siberian Branch of the Russian Academy of Sciences, ul. Lermontova 130, Irkutsk, 664033 Russia; Irkutsk State University, bulv. Gagarina 20, Irkutsk, 664003 Russia Received December 14, 2011

Abstract—We construct an asymptotic approximation for solutions of systems of Volterra integral equations of the first kind with piecewise continuous kernels. We use the asymptotics as an initial approximation in the proposed method of successive approximations to the desired solutions. We prove the existence of a continuous solution depending on free parameters and establish sufficient conditions for the existence of a unique continuous solution. We illustrate the proved existence theorems with examples. DOI: 10.3103/S1066369X13010064 Keywords and phrases: systems of Volterra integral equations of the first kind, asymptotics, continuous kernel, successive approximations.

1. INTRODUCTION Consider the following system of integral equations: t K(t, s)x(s)ds = f (t), 0 < t ≤ T.

(1)

0

Assume that the matrix kernel K(t, s) of dimension m × m has discontinuity points of the first kind on curves s = αi (t), i = 1, . . . , n − 1, lying in the compact 0 ≤ s ≤ t ≤ T . Therefore, ⎧ K1 (t, s), 0 ≤ s ≤ α1 (t); ⎪ ⎪ ⎪ ⎨ K2 (t, s), α1 (t) < s ≤ α2 (t); (2) K(t, s) = ⎪ ...... ⎪ ⎪ ⎩ Kn (t, s), αn−1 (t) < s ≤ t, f (t) = (f1 (t), . . . , fm (t)) , and x(t) = (x1 (t), . . . , xm (t)) . Matrices Ki (t, s) of dimension m × m are defined; they are continuous and have continuous derivatives with respect to t in the corresponding domains Di = {(s, t) | αi−1 (t) < s ≤ αi (t)}, i = 1, . . . , n, α0 = 0, αn (t) = t. Functions fi (t) and αi (t) have continuous derivatives, fi (0) = 0, αi (0) = 0, 0 < α1 (0) < α2 (0) < · · · < αn−1 (0) < 1, 0 < α1 (t) < α2 (t) < · · · < αn−1 (t) < t for t ∈ (0, T ]. It is needed to construct continuous solutions to Eq. (1) for t ∈ (0, T ], where 0 < T ≤ T , and lim x(t) may be infinite. We assume that every matrix t→+0

Ki (t, s), i = 1, . . . , n has a continuous differentiable with respect to t extension to the set 0 ≤ s ≤ t ≤ T . The homogeneous system may have nontrivial solutions. Integral systems, particularly, those of type (1)–(2) with discontinuous kernels occur in some applications (see [1–3]). Differentiating system eqrefeq1 with respect to t, in contrast to the classical case [2, 4], we obtain a new class of Volterra integral equations with functionally perturbed argument. Therefore, the standard methods are inapplicable to system eqrefeq1, and the study of systems with kernels of type (2) has a theoretical *

E-mail: [email protected].

54

SOLVABILITY OF SYSTEMS OF VOLTERRA INTEGRAL EQUATIONS

55

interest. Operator-integral equations, including those with functionally perturbed argument, are considered in [5, 6, 7]. An integral equation with kernel of type (2) is studied in [8]. In this paper we use methods proposed in [6–8]. The paper has the following structure. In Section 2 we describe a method for constructing logarithmic-power asymptotics x (t) =

N

xi (ln t)ti

i=0

of the desired continuous solutions to system (1). Note that logarithmic-power approximations are used to solve other classes of equations, for example, in [4, 9, 10] etc. In Section 3 we prove the existence theorem for parametric families of solutions to system (1). In Section 4 we establish some sufficient conditions for the unique existence of a continuous solution to this system. Note that generalized solutions of Volterra equations of the first kind are constructed in [11–13]. 2. CONSTRUCTION OF ASYMPTOTIC APPROXIMATION x (t) OF A SOLUTION TO A NONHOMOGENEOUS SYSTEM Let the following condition be fulfilled: N

(A) There exist matrices Pi =

Kiνμ tν sμ , i = 1, . . . , n, a vector function f N (t) =

ν+μ=0

and polynomials αN i (t) =

N

N

f ν tν ,

ν=1

αIν tν , i = 1, . . . , n − 1, where 0 < α11 < α21 < · · · < αn−1,1 < 1,

ν=1

such that Ki (t, s) − Pi (t, s) = O((t + s)N +1 ), i = 1, . . . , n, f (t) − f N (t) = O(tN +1 ), αi (t) − N +1 ), i = 1, . . . , n − 1, as t → +0 and s → +0. αN i (t) = O(t Further we treat the power expansions with respect to t and s in condition (A) as “Taylor polynomials” of the corresponding elements. Introduce the matrix B(j) = Kn (0, 0) +

n−1

(αi (0))1+j (Ki (0, 0) − Ki+1 (0, 0))

i=1

and consider the algebraic equation def

L(j) = det B(j) = 0. We call it the characteristic equation of the system of integral equations (1). Since f (0) = 0 and matrices Ki (t, s) and the vector f (t) have continuous derivatives with respect to t, differentiating both sides of system (1), we obtain the following equivalent system of integral-functional equations: def

F (x) = Kn (t, t)x(t) +

n−1

αi (t) Ki (t, αi (t))

i=1 n − Ki+1 (t, αi (t)) x(αi (t)) +

i=1

αi (t) αi−1 (t)

Ki (t, s)x(s)ds − f (t) = 0, (3) (1)

where α0 = 0 and αn (t) = t. We do not assume that the homogeneous system corresponding to (1) has only the trivial solution. Therefore, the homogeneous integro-functional system corresponding to (3) may have nontrivial solutions. Using the method proposed in [6, 7], we seek for an asymptotic approximation of a particular solution to the nonhomogeneous equation (3) in the form of the polynomial x (t) =

N

xj (ln t)tj .

j=0

RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 57 No. 1 2013

(4)

56

SIDOROV

Let us show that in the general irregular case the coefficients xj depend on ln t and free parameters; this agrees with the possibility of the existence of nontrivial solutions to the homogeneous system. When calculating the coefficients xj , the regular and irregular cases are possible. 2.1. The regular case. L(j) = 0, j ∈ {0, 1, . . . , N }. In this case the coefficients xj are constant vectors from Rm . Really, substituting (4) in system (3), using the method of indefinite coefficients, taking into account (A), we obtain the following recurrent sequence of linear systems of algebraic equations with respect to vectors xj : B(0)x0 = f (0), B(j)xj = Mj (x0 , . . . , xj−1 ), j = 1, . . . , N.

(5) (6)

The vector Mj is expressed in a certain way in terms of solutions x0 , . . . , xj−1 to the previous systems and coefficients of the “Taylor polynomials” from condition (A). Since in the regular case det B(j) = 0, the vectors x0 , . . . , xN are defined uniquely, and one can find the asymptotic (4). 2.2. The irregular case. The equation L(j) = 0 has integer roots. Definition 1. A number j ∗ is a regular point of the matrix B(j), if the matrix B(j ∗ ) is invertible. Definition A number j ∗ is a simple singular point of the matrix B(j), if det B(j ∗ ) = 0 and

(1) 2. r det (B (j ∗ )φi , ψk ) i,k=1 = 0, where {φi }r1 is a basis in N (B(j ∗ )), {ψi }r1 is a basis in N (B (j ∗ )),

B (j ∗ ) is the transposed matrix, and B (1) (j) is the derivative of the matrix B(j) with respect to j.

Definition 3. We call j ∗ a (k + 1)-fold singular point of the matrix B(j), if det B(j ∗ ) = 0, the derivatives B (1) (j ∗ ), . . . , B (k) (j ∗ ) are null matrices,

r det (B (k+1) (j ∗ )φi , ψk ) i,k=1 = 0, k ≥ 1, {φi }r1 is a basis in N (B(j ∗ )), and {ψi }r1 is a basis in N (B (j ∗ )). Note that B (k) (j) =

n−1 i=1

(αi (0))1+j aki (Ki (0, 0) − Ki+1 (0, 0)), where ai = ln αi (0).

Remark 1. Let m = 1 (i.e., we consider one integral equation of type (1), rather than a system). Then B(j) = L(j). Therefore, for one equation Definition 2 means that j is a simple root of the characteristic equation L(j) = 0, and Definition 3 implies that j is a multiple root of order (k + 1) of the equation. Let us show that in the irregular case the coefficients xj are polynomials of the variable ln t and depend on arbitrary constants. The degrees of the polynomials and the numbers of the constants are connected with the multiplicities of singular points of the matrices B(j) and their ranks. Actually, since in the irregular case the coefficient x0 can depend on ln t, based on the method of indefinite coefficients we seek for x0 as a solution to the difference system Kn (0, 0)x0 (z) +

n−1

αi (0)(Ki (0, 0) − Ki+1 (0, 0))x0 (z + ai ) = f (0),

(7)

i=1

where ai = ln α (0) and z = ln t. Here the following three cases are possible. Case 1. L(0) = 0, i.e., det B(0) = 0. Then x0 is independent of z and it is uniquely defined from SLAE (5) with the invertible matrix B(0). Case 2. Let j = 0 be a simple singular point of the matrix B(j). We find the coefficient x0 (z) from the difference system (7) in the form of a linear vector function: x0 (z) = x01 z + x02 . RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 57 No. 1 2013

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57

Substituting (8) in (7), we obtain two SLAE with respect to vectors x01 and x02 : B(0)x01 = 0, B(0)x02 + B (1) (0)x01 = f (0). Here det B(0) = 0, and {φi }r1 is a basis in N (B(0)).

Therefore, x01 =

(9) r

ck φk . The vector

k=1

c = (c1 , . . . , cr ) is found uniquely from solvability conditions for system (9), i.e., from the SLAE r (B (1) (0)φk , ψi )ck = (f (0), ψi ), i = 1, . . . , r, k=1

with a non-degenerate matrix. Furthermore, the coefficient x02 is determined from system (9) accurate to span(φ1 , . . . , φr ). Thus, in Case 2 the coefficient x0 (z) depends on r arbitrary constants and is linear with respect to z. Case 3. Let j = 0 be a singular point of the matrix B(j) of multiplicity k + 1, where k ≥ 1. We seek for a solution x0 (z) to the difference system (7) in the form of the polynomial x0 (z) = x01 z k+1 + x02 z k + . . . + x0k+1 z + x0k+2 .

(10)

Substituting polynomial (10) in (7), taking into account the identity dk B(j) = (αi (0))1+j aki (Ki (0, 0) − Ki+1 (0, 0)), dj k n−1 i=1

where ai = ln αi (0), and equating the coefficients at z k+1 , z k , . . . , z, z 0 to zero, we obtain the following recurrent sequence of SLAE with respect to the coefficients x01 , x02 , . . . , x0k+2 : B(0)x01 = 0, ⎛ ⎞ k + 1⎠ x01 = 0, B(0)x02 + B (1) (0) ⎝ k ⎛ B(0)x0l+1 + B (l) (0) ⎝

⎞ k+1 k+1−l

⎛

⎠ x01 + B (l−1) (0) ⎝

⎞ k

⎠ x02

k+1−l ⎛ ⎞ k + 1 − l + 1⎠ + · · · + B (1) (0) ⎝ x0l = 0, l = 1, . . . , k, k+1−l

(11) B(0)x0k+2 + B (k+1) (0)x01 + B (k) x02 + · · · + B (1) (0)x0k+1 = f (0). i B(j) , i = 1, . . . , k, are Since in the considered case, according to Definition 3, the derivatives d dj i j=0 zero matrices, we have x0i =

r

cij φj , i = 1, . . . , k + 1.

j=1

Therefore, system (11) takes the form B(0)x0,k+2 + B (k+1) (0)x01 = f (0).


(12)

58

SIDOROV

def Since det (B (k+1) (0)φi , ψk ) i,k=1,r = 0, the vector c1 = (c11 , . . . , c1r ) is determined uniquely from the solvability conditions for system (12). Thus, x0k+2 =

r

ck+2j φj + x k+2 ,

j=1 def

x k+2 is a particular solution to SLAE (12). The vector ck+2 = (ck+2,1 , . . . , ck+2,r ) , as well as the vectors c2 , . . . , ck+1 , remains arbitrary. Therefore, in Case 3 the coefficient x0 (z) is a polynomial of degree (k + 1) with respect to z and depends on r × (k + 1) arbitrary constants. Applying the method of indefinite coefficients, taking into account the identity k k(k − 1) . . . (k − (s − 1)) k−s j k j+1 (−1)s ln t, t ln t dt = t (j + 1)s+1 s=0

we can construct a system of difference equations in order to find the coefficient x1 (z), z = ln t. Actually, F (x)

def x=x0 +x1 t

= Kn (0, 0)x1 (z) +

n−1

(αi (0))2 (Ki (0, 0)

i=1

− Ki+1 (0, 0))x1 (z + ai ) + P1 (x0 (z)) t + r(t), r(t) = o(t). (13) Here P1 (x0 (z)) is a definite polynomial of z, whose degree equals the multiplicity of the singular point j = 0 of the matrix B(j). From (13), taking into account the estimate r(t) = o(t) as t → 0, it follows that the coefficient x1 (z) satisfies the following system of difference equations: Kn (0, 0)x1 (z) +

n−1

(α (0))2 Ki (0, 0) − Ki+1 (0, 0) x1 (z + ai ) + P1 (x0 (z)) = 0.

(14)

i=1

If j = 1 is a regular point of the matrix B(j), then system (14) has a solution x1 (z) in the form of a polynomial of the same degree as the multiplicity of the singular point j = 0 of the matrix B(0). If j = 1 is a singular point of the matrix B(j), then the solution x1 (z) is constructed as a polynomial of degree k0 + k1 , where k0 and k1 are the multiplicities of the singular points j = 0 and j = 1 of the matrix B(j). The coefficient x1 (z) depends on r0 k0 + r1 k1 arbitrary constants, where r0 = dim N (B(0)) and r1 = dim N (B(1)). Introduce the following condition: (B) The the matrix B(j) has only regular points or singular points of multiplicities kj in the array (0, 1, . . . , N ). Then we can similarly calculate the rest coefficients x2 (z), . . . , xN (z) of the asymptotic approximation x (t) of a solution to Eq. (1) from the sequence of difference equations Kn (0, 0)xj (z) +

n−1

(α (0))1+j Ki (0, 0) − Ki+1 (0, 0) xj (z + ai )

i=1

+ Pj (x0 (z), . . . , xj−1 (z))) = 0, j = 2, N . Hence we get the following assertion. Lemma 1. Let conditions (A) and (B) be fulfilled. Then there exists a vector function x (t) = N xi (ln t)ti such that |F ( x(t))|Rm = o(tN ) as t → +0. At the same time, the coefficients xi (ln t) i=0 kj of the are polynomials with respect to ln t of increasing degrees not exceeding the sum j



multiplicities of the singular points j of the matrix B(j) from the array (0, 1, . . . , i). i coefficients xi (ln t) depend on dim N (B(j))kj arbitrary constants.

59

The

j=0

3. THEOREM OF EXISTENCE OF CONTINUOUS PARAMETRIC FAMILIES OF SOLUTIONS Since 0 ≤ αi (0) < 1, αi (0) = 0, i = 1, . . . , n − 1, for any 0 < ε < 1 there exists T ∈ (0, T ] such that αi (t) max |αi (t)| ≤ ε and sup t ≤ ε.

i=1,...,n−1, t∈[0,T ]

i=1,...,n−1, t∈(0,T ]

Introduce the following condition: (C) Let det Kn (t, t) = 0, t ∈ [0, T ], and let N ∗ be chosen so large that ∗

max εN |Kn−1 (t, t)|L(Rm →Rm )

n−1

t∈[0,T ]

(1)

|αi (t)| |Ki (t, αi (t)) − Ki+1 (t, αi (t))|L(Rm →Rm ) ≤ q < 1;

i=1

here | · |L(Rm →Rm ) is the norm of an m × m matrix. Lemma 2. Let condition (C) be fulfilled. Assume that there exists an element x (t) from the class C(0,T ] of vector functions which are continuous for t ∈ (0, T ] and have a limit, possibly infinite, as t → +0, such that |F ( x(t))|Rm = o(tN ), N ≥ N ∗ . Then Eq. (3) has a solution ∗

x(t) = x (t) + tN u(t)

(15)

in C(0,T ] , where u(t) ∈ C[0,T ] is defined uniquely by successive approximations. Proof. Substituting (15) in Eq. (3), we obtain the following integro-functional system with respect to u(t): Kn (t, t)u(t) +

n−1

αi (t)

i=1

αi (t) t

N ∗

+

Ki (t, αi (t)) − Ki+1 (t, αi (t)) u(αi (t)) n i=1

Introduce the linear operators def

Lu =

def

Ku =

Kn−1 (t, t) n i=1

n−1

αi (t)

i=1 αi (t) αi−1 (t)

αi (t)

αi−1 (t)

αi (t) t

N ∗

N ∗ s ∗ u(s)ds + F ( x(t))/tN = 0. (16) t

(1) Ki (t, s)

Ki (t, αi (t)) − Ki+1 (t, αi (t)) u(αi (t)),

∗

Kn−1 (t, t)Ki (t, s)(s/t)N u(s)ds. (1)

Then we can write system (16) in the compact form u + (L + K)u = γ(t), N∗

where γ(t) = Kn−1 (t, t)F (xN (t))/t is a continuous vector function. Introduce the Banach space X of continuous vector functions u(t) with the norm ul = max e−lt |u(t)|Rm , l > 0. 0≤t≤T


60

SIDOROV

Then by virtue of the inequality sup t∈[0,T ]

αi (t) t

≤ ε < 1 and condition (C) for any l ≥ 0 the norm of the linear

functional operator L satisfies the estimate LL(X→X) ≤ q < 1. In addition, for the integral operator K with sufficiently large l we have KL(X→X) ≤ q1 < 1 − q. Consequently, with sufficiently large l > 0, L + KL(X→X) < 1, i.e., the linear operator L + K is contractive in the space X. Therefore, the sequence {un }, where un = −(L + K)un−1 + γ(t), u0 = γ(t), converges. Theorem 1 (the main theorem). Let conditions (A), (B), and (C) be fulfilled, f (0) = 0; let the matrix B(j) in the array (0, 1, . . . , N ) have exactly ν singular points j1 , . . . , jν of multiplicities ki , i = 1, . . . , ν, and let the rest numbers of the array be regular; let rankB(ji ) = ri , i = 1, . . . , ν. Then Eq. (1) with 0 < t ≤ T ≤ T has a solution in the form ∗

x(t) = x (t) + tN u(t), which depends on

ν

(m − ri )ki arbitrary constants.

i=1

Proof. Based on Lemma 1, taking into account the conditions of the theorem, it is possible to construct an asymptotic approximation x (t) of the desired solution in the form of a logarithmic-power polynomial N xi (ln t)ti . In addition, by construction, the coefficient at xi (ln t) depends on the indicated number i=0

∗

of arbitrary constants. By Lemma 2, applying the substitution x(t) = x (t) + tN u(t), it is possible to construct a continuous function u(t) by the method of successive approximations. Remark 2. Under conditions of the main theorem we have the following asymptotic estimate for the ∗ asymptotic approximation: x (t): x(t) − x (t)Rn = O(tN ), t → +0. Corollary. Let αi (t) = αi t, i = 1, . . . , n − 1, 0 < α1 < α2 < · · · < αn−1 < 1; assume that elements of the matrices Ki (t, s), i = 1, . . . , n − 1 and the vector function f (t) can be extended analytically to the domain |s| < T , |t| < T , f (0) = 0. Let the matrix Kn (t, t)−1 be analytic for |t| < T . Let det B(j) = 0, ∞ xi ti for 0 ≤ t < T . j ∈ N ∪ {0}. Then Eq. (1) has a unique solution x(t) = i=0

In some cases under conditions of the theorem one can construct a parametric family of solutions in the closed form. Example. The system

t/2

Kx(s)ds +

0

t

(K − 2E)x(s)ds = bt, 0 < t < ∞, where K is a symmetric

t/2

constant m × m matrix, b ∈ Rm , x(t) = (x1 (t), . . . , xm (t)) , 1 is an eigenvalue of the matrix K of rank r, and {φ1 , . . . , φr } is the corresponding orthonormal system of eigenvectors, has the following parametric family of solutions: x(t) = − ln t

r (d, φi ) i=1

ln 2

φi + c1 φ1 + · · · + cn φr + a.

a satisfies the SLAE Here c1 , . . . , cr are arbitrary constants, and the vector r (b, φi )φi . (K − E) a=b− i=1



61

4. SUFFICIENT CONDITIONS FOR EXISTENCE OF A UNIQUE CONTINUOUS SOLUTION For simplicity of computation, in this Section we put αi (t) = αi t, i = 1, . . . , n − 1, 0 < α1 < α2 < · · · < αn−1 < 1. Introduce the function def

D(t) =

n−1

αi |Kn−1 (t, t)|L(Rm →Rm ) |(Ki (t, αi t) − Ki+1 (t, αi t))|L(Rm →Rm ) .

i=1

Let the following conditions be fulfilled: (S) D(0) < 1;

sup 0≤s≤t≤T

|Kn−1 (t, t)K(t, s)|L(Rm →Rm ) ≤ c < ∞.

Here and further the matrix K(t, s) is defined by (2). Theorem 2 (sufficient conditions for the unique existence of a solution). Assume that conditions (S) are fulfilled, all matrices Ki (t, s) in representation (2) are continuous and have continuous derivatives with respect to t, the vector f (t) also has a continuous derivative, and f (0) = 0. Then Eq. (1) has a unique solution in the class of continuous functions C[0,T ] . Moreover, the solution could be found by the step method combined with the method of successive approximations. Proof. We rewrite Eq. (3), which is equivalent to (1), in the form (17)

x(t) + Ax + Kx = f (t), def

where Ax = Kn−1 (t, t) def

and Kx =

n α i t i=1 αi−1 t

n−1

αi (Ki (t, αi t) − Ki+1 (t, αi t))x(αi t)

i=1

Kn−1 (t, t)Kt (t, s)x(s)ds, f (t) = Kn−1 (t, t)f (1) (t). (1)

We fix q < 1 and choose h1 > 0 such that max |D(t)|L(Rm →Rm ) = q < 1. By virtue of condition (S) such h1 > 0 exists. Put 0 < h < into segments

0≤t≤h1 1−q min{h1 , c }, where the constant c is defined in (S). Let us divide [0, T ]

(18)

[0, h], [h, h + εh], [h + εh, h + 2εh], . . . ,

1 . We denote by x0 (t) the restriction of the unknown where ε is chosen in (0, 1] so that αn−1 ≤ 1+ε solution x(t) to the segment [0, h] and we do by xn (t) its restriction to

In = [(1 + (n − 1)ε)h, (1 + nε)h], n = 1, 2, . . . Due to the choice of ε, for t ∈ In the “perturbed” argument αi t ∈

n−1

Ik . This inclusion gives us

k=1

a possibility to apply the step method (which is well known in the theory of functional-differential equations) for constructing the solution x(t) [14]. For calculating the element x0 (t) ∈ C[0,h] we construct a sequence {xn0 (t)} as follows: − Kxn−1 + f (t), xn0 (t) = −Axn−1 0 0 x00 (t) = f (t), t ∈ [0, h]. By the choice of h we have the estimate A + KL(C[0,h] →C[0,h] ) < 1. Therefore, there exists a unique solution x0 (t) to Eq. (17) for t ∈ [0, h]. The sequence xn0 (t) converges uniformly to it. We continue the process of constructing the desired solution for t ≥ h, i.e., in In , n = 1, 2, . . . For definiteness, let further ε = 1 in (18).


62

SIDOROV

Then, after the calculation of the element x0 (t) ∈ C[0,h] we will seek for x1 (t) in the space C[h,2h] of continuous vector functions. We find x1 (t) from the Volterra integral equation of the second kind h t −1 Kn (t, t)Kt (t, s)x(s)ds = f (t) − Ax0 − Kn−1 (t, t)Kt (t, s)x0 (s)ds x(t) + h

0

by successive approximations. Here x0 (h) = x1 (h). Introduce the following continuous function: x0 (t), x1 (t) = x1 (t),

0 ≤ t ≤ h; h ≤ t ≤ 2h,

which is the restriction of the desired continuous solution x(t) to the segment [0, 2h]. Then the element x2 (t) ∈ C[2h,3h] can be calculated by successive approximations from the Volterra integral equation of the second kind 2h t −1 Kn (t, t)Kt (t, s)x(s)ds = f (t) − Ax1 − Kn−1 (t, t)Kt (t, s)x1 (s)ds. x(t) + 2h

0

Continuing this process, we will construct the unknown solution x(t) ∈ C[0,T ] to Eq. (1) after N steps (N ≥ Th ). Example. Assume that K1 (t − s) = K2 (t − s) + E are (m × m)-matrices, where E is the identity matrix, and |K2−1 (0)|L(Rm →Rm ) < 2. Assume that the matrix K2 (t) and the vector function f (t) have continuous derivatives with respect to t, and f (0) = 0. Then the integral equation t t/2 K1 (t − s)x(s)ds + K2 (t − s)x(s)ds = f (t), 0 < t ≤ T, 0

t/2

satisfies conditions of Theorem 2 and has a unique continuous solution. 5. CONCLUSION In the case of one equation (m = 1) the proposed method for solving difference systems coincides with the known method proposed by A. O. Gelfond ([15], P. 338) for constructing partial solutions to nonhomogeneous difference equations with a polynomial right-hand side. If we weaken the conditions of Definition 3 by assuming that the matrix B(j ∗ ) has B (k+1) (j ∗ )-adjoint elements, then Theorem 1 (t) of the desired can be enhanced. In addition, the coefficients xj (z) of the asymptotic approximation x solution must be constructed in the form of polynomials of degree p + k. Here p is the maximum of lengths of the corresponding generalized in the sense of V. A. Trenogin [16] Jordan chains, and k is the order of the polynomial in the right-hand side of the corresponding difference equation. Some results from functional analysis [17] allow us to generalize the main theorem of this paper for the case when K(t, s) in (1) is a linear piecewise continuous map between Banach spaces. This generalization is of interest for solving some classes of degenerate integral-differential systems [18–20]. If f (0) = 0, then Eq. (1) has no continuous solutions but may have generalized ones. The methods proposed in this paper and those in [11–13, 21] allow us to construct generalized solutions to Eq. (1) in the class of the Sobolev–Schwartz distributions. The development of stable numerical methods for solving the Volterra integral equations with piecewise continuous kernels is possible with the use of results of papers [22, 2], as well as the paper [23]. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project No. 11-08-00109), the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of the Innovative Russia,” contract No. 14.B.37.21.0365, and DAAD (project No. A1200665).



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Translated by S. R. Nasyrov
