Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces
Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces are considered. We obtain necessary and sufficient conditions for finding solutions of these problems and construct converging iterative procedures of determination of solutions of these...
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irk-123456789-1773322021-02-15T01:26:56Z Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces Zhuravlev, V.F. Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces are considered. We obtain necessary and sufficient conditions for finding solutions of these problems and construct converging iterative procedures of determination of solutions of these boundaryvalue problems. Розглянуто слабконелiнiйнi крайовi задачi для iнтегральних рiвнянь Фредгольма з виродженим ядром у банахових просторах. Одержано необхiднi та достатнi умови iснування розв’язкiв таких задач, а також побудовано збiжнi iтерацiйнi процедури для знаходження розв’язкiв зазначених задач. 2018 Article Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces / V.F. Zhuravlev // Нелінійні коливання. — 2018. — Т. 21, № 3. — С. 347-357 — Бібліогр.: 17 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177332 517.983 en Нелінійні коливання Інститут математики НАН України |
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Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces are considered. We obtain necessary and sufficient conditions for finding solutions of these problems and construct converging iterative procedures of determination of solutions of these boundaryvalue problems. |
| format |
Article |
| author |
Zhuravlev, V.F. |
| spellingShingle |
Zhuravlev, V.F. Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces Нелінійні коливання |
| author_facet |
Zhuravlev, V.F. |
| author_sort |
Zhuravlev, V.F. |
| title |
Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces |
| title_short |
Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces |
| title_full |
Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces |
| title_fullStr |
Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces |
| title_full_unstemmed |
Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces |
| title_sort |
weakly nonlinear boundary-value problems for fredholm integral equations with degenerate kernel in banach spaces |
| publisher |
Інститут математики НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/177332 |
| citation_txt |
Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces / V.F. Zhuravlev // Нелінійні коливання. — 2018. — Т. 21, № 3. — С. 347-357 — Бібліогр.: 17 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT zhuravlevvf weaklynonlinearboundaryvalueproblemsforfredholmintegralequationswithdegeneratekernelinbanachspaces |
| first_indexed |
2025-07-15T15:22:58Z |
| last_indexed |
2025-07-15T15:22:58Z |
| _version_ |
1837726935660101632 |
| fulltext |
UDC 517.983
WEAKLY NONLINEAR BOUNDARY-VALUE PROBLEMS
FOR FREDHOLM INTEGRAL EQUATIONS
WITH DEGENERATE KERNEL
IN BANACH SPACES
СЛАБКОНЕЛIНIЙНI КРАЙОВI ЗАДАЧI
ДЛЯ IНТЕГРАЛЬНИХ РIВНЯНЬ ФРЕДГОЛЬМА
З ВИРОДЖЕНИМ ЯДРОМ
У БАНАХОВИХ ПРОСТОРАХ
V. F. Zhuravlev
Zhytomyr National Agricultural-Ecological University
Staryi Blvd, 7, Zhytomyr, 10008, Ukraine
e-mail: vfz2008@ukr.net
Weakly nonlinear boundary-value problems for Fredholm integral equations with degenerate kernel in
Banach spaces are considered. We obtain necessary and sufficient conditions for finding solutions of these
problems and construct converging iterative procedures of determination of solutions of these boundary-
value problems.
Розглянуто слабконелiнiйнi крайовi задачi для iнтегральних рiвнянь Фредгольма з виродженим
ядром у банахових просторах. Одержано необхiднi та достатнi умови iснування розв’язкiв таких
задач, а також побудовано збiжнi iтерацiйнi процедури для знаходження розв’язкiв зазначених
задач.
This paper is a continuation of the research on the conditions of solvability and the construction
of solutions of weakly nonlinear integral Fredholm equations with a degenerate kernel in Banach
spaces that were started in [1].
Constructive methods for the analysis of weakly nonlinear boundary-value problems for the
systems of functional-differential and other equations traditionally occupy one of the important
places in the qualitative theory of differential equations and continue the development of methods
of perturbation theory, in particular, the methods of Lyapunov – Poincare small parameter [2, 3].
These methods were successfully developed in [4, 5] and applied in the study of weakly
nonlinear boundary-value problems for systems of ordinary differential equations [6] and the
construction of bounded solutions of weakly nonlinear differential equations [7] in Banach
spaces.
In finite-dimensional Euclidean spaces, weakly nonlinear integral-differential equations and
Fredholm integral equations with a nondegenerate kernel, which are not always solvable, were
studied in [8, 9].
The specific nature of the study of boundary-value problems for systems of integral equations
in Banach spaces lies in the fact that their linear part is an operator that does not have an
inverse [10], which considerably complicates the study of boundary-value problems for such
equations. Therefore, the problem of studying the conditions of existence and constructing the
general solutions of weakly nonlinear boundary-value problems for not always solvable integral
Fredholm equations with a degenerate kernel in Banach spaces is topical.
© V. F. Zhuravlev, 2018
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3 347
348 V. F. ZHURAVLEV
Statement of the problem. We consider a weakly nonlinear boundary-value problem
(Lz)(t) := z(t)−M(t)
b∫
a
N(s)z(s)ds = f(t) + ε
b∫
a
K(t, s)Z(z(s, ε), s, ε)ds, (1)
`z(·) = α+ εJ(z(·, ε), ε), (2)
where the operator-valued functions M(t) and N(t) are defined on the finite interval I = [a, b],
act from the Banach space B into the same space, are strongly continuous with the norms
|||M ||| = sup
t∈I
‖M(t)‖B = M0 < ∞ и |||N ||| = sup
t∈I
‖N(t)‖B = N0 < ∞; the operator-valued
function K(t, s) is defined in the square I × I and acts from the Banach space B into the same
space with respect to each variable, is strongly continuous with respect to each variable with
the norm |||K||| = sup
t∈I
‖K(t, s)‖B = K0 < ∞; Z(z(t, ε), t, ε) is nonlinear z bounded operator
function, J(z(·, ε), ε) is nonlinear z vector-functional that in the neighborhood of the generating
solution ‖z − z0‖ ≤ q have a strongly continuous Frechet derivative with respect to z and are
continuous for the set of variables z, t, ε, q and ε0, that are rather small constants; Z(0, t, 0) = 0,
Z ′z(0, t, 0) = 0, J(0, 0) = 0, J ′z(0, 0) = 0; f(t) is a vector-valued function in the Banach space
C(I,B) that are continuous vector functions on the interval I; α is an element of the Banach
space B1 : α ∈ B1.
Along with the problem (1), (2) we consider the linear generating boundary-value problem
z0(t)−M(t)
b∫
a
N(s)z0(s)ds = f(t), (3)
`z0(·) = α, (4)
which is obtained from (1), (2) for ε = 0.
The problem is to find the necessary and sufficient conditions for the existence of the solutions
of the weakly nonlinear boundary-value problem (1), (2). We seek solutions in the class of vector-
valued functions z(t, ε), that are continuous with respect to the variable t and with respect to
the parameter ε, and turning at ε = 0 into the generating solution of the linear boundary-value
problem (3), (4).
Auxiliary information. Suppose that a bounded linear operator D = IB−
∫ b
a
N(s)M(s) ds,
D : B → B is generalized invertible. Then [11, 12] there is a bounded projector PN(D) : B →
→ N(D), that projects a Banach space B onto the null space N(D) of D operator, a bounded
projector PYD
: B→ YD, that projects a Banach space B on the subspace YD = B R(D) and
D− is a bounded generalized inverse operator to the operator D [4, 5, 13].
The class of bounded linear generalized invertible operators that act from the Banach space
B to the Banach space B will be denoted as GI(B,B). It is obvious that the operator belonging
to GI(B,B) is normally solvable [14].
It is shown in [15] that if the operator D ∈ GI(B,B), then, under the condition
M(t)PYD
b∫
a
N(s)f(s)ds = 0
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
WEAKLY NONLINEAR BOUNDARY-VALUE PROBLEMS FOR FREDHOLM INTEGRAL EQUATIONS . . . 349
and only under it, the operator gather (3) is solvable and has a family of solutions
z0(t) = M(t)PN(D)c+
(
L−f
)
(t), (5)
where c is an arbitrary element of the Banach space B,
(
L−f
)
(t) = f(t) +M(t)D−
b∫
a
N(s)f(s)ds
is a bounded generalized inverse operator to the integral operator L [10].
Substituting the solution (5) of the inhomogeneous operator gather (3) into the boundary
condition (4), we obtain the operator gather
Qc+ `f(·) + `M(·)D−
b∫
a
N(s)f(s)ds = α,
where Q = `M(·)PN(D) : B→ B1 is a bounded linear operator.
Let the operator Q ∈ GI(B,B1). Denote PN(Q) : B → N(Q) is the bounded projector of
the Banach space B to the null space N(Q) of the operator Q, PYQ
: B1 → YQ is the bounded
projector of the Banach space B1 onto the subspace YQ = B1 R(Q), Q− is a bounded
generalized inverse operator to the operator Q.
Theorem 1 [15]. Let the operators D ∈ GI(B,B) and Q ∈ GI(B,B1).
Then the corresponding (3), (4) homogeneous (f(t) = 0, α = 0) boundary-value problem
has the family of solutions
z(t) = M̃(t)c,
where M̃(t) = M(t)PN(D)PN(Q), c is an arbitrary element of the Banach space B.
The nonhomogeneous boundary-value problem (3), (4) is solvable for those and only those
f(t) ∈ C(I,B) and α ∈ B1, that satisfy the system of conditions
M(t)PYD
∫ b
a
N(s)f(s)ds = 0,
PYQ
[
α− `f(·)− `M(·)D−
∫ b
a
N(s)f(s)ds
]
= 0
(6)
and it has the family of solutions
z0(t) = M̃(t)c+ (Gf)(t) +M(t)PN(D)Q
−α, (7)
where
(Gf)(t) :=
[
f(t)−M(t)PN(D)Q
−`f(·)
]
+
+M(t)
[
IB − PN(D)Q
−`M(·)
]
D−
b∫
a
N(s)f(s)ds (8)
is a generalized Green operator of the corresponding (3), (4) semi-homogeneous (α = 0)
boundary-value problem.
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
350 V. F. ZHURAVLEV
It is necessary to note that the first of the conditions (6) will always be satisfied if the condition
PYD
∫ b
a
N(s)f(s)ds = 0 is satisfied.
To solve the problem, we need the information about the solvability conditions and the
representation of the solutions of the operator gathers with linear operator B0 that is an operator
matrix
B0 =
[
B1
B2
]
,
where B1 : B→ B and B2 : B→ B1 are linear bounded generalized invertible operators [11].
In this case [11, 12] there are bounded projectors PN(B1) : B → N(B1) and PN(B2) : B →
→ N(B2) to the null spaces of the operators B1 and B2, and also the bounded projectors
PYB1
: B→ YB1 and PYB2
: B1 → YB2 to the subspaces YB1 = B R(B1) and YB2 = B1 R(B2),
respectively, and also bounded generalized inverse operators B−1 and B−2 .
Then, using [16] for the system of operator equations
B0c =
[
B1
B2
]
c =
[
b1
b2
]
, b1 ∈ B, b2 ∈ B1 (9)
the following theorem is true.
Theorem 2 [17]. Let B1 ∈ GI(B,B) and B2 ∈ GI(B,B1). Then the system of operator
gathers (9) is solvable for those and only those col [b1, b2] that satisfy the condition
PYB0
[
b1
b2
]
= 0,
under which it has a family of solutions
c = PN(B0)ĉ+B−0
[
b1
b2
]
,
where PYB0
=
[
IB −B1PN(B2)B
−
1 −B1B
−
2
0 PYB2
]
is a bounded projector onto the subspace YB0 =
= IB×B1 R(B0), PN(B0) = PN(B2)PN(B1) is a bounded projector onto the null space N(B0)
of the operator B0, ĉ is an arbitrary element of the Banach space B,
B−0 =
[
PN(B2)B
−
1 B−2
]
is a bounded generalized inverse operator to the operator B0.
Main result. Using the generalized Green operator (8) of the linear semi-homogeneous
boundary-value problem, we seek the existence conditions for the solutions z = z(t, ε) of the
boundary-value problem (1), (2) that are defined in the class of vector functions: z(·, ε) ∈ C(I,B),
z(t, ·) ∈ C(0, ε0] and turn for ε = 0 to one of the generating solutions z0(t, c)
Performing in (1), (2) the change of the variable
z(t, ε) = z0(t, c) + x(t, ε)
for the deviation x(t, ε) from the generating solution, we obtain the boundary-value problem
x(t)−M(t)
b∫
a
N(s)x(s)ds = ε
b∫
a
K(t, s)Z(z0(s, c) + x(s, ε), s, ε) ds, (10)
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
WEAKLY NONLINEAR BOUNDARY-VALUE PROBLEMS FOR FREDHOLM INTEGRAL EQUATIONS . . . 351
`x(·) = εJ
(
z0(·, c) + x(·, ε), ε
)
. (11)
We find the necessary condition of the existence of solutions z(t, ε) of the boundary-value
problem (1), (2), which for ε = 0 become one of the generating solutions z0(t, c) ∈ C(I,B) of
the generating boundary-value problem (3), (4).
Suppose that the boundary-value problem (1), (2) has a solution z(t, ε), then by Theorem 1,
the system of solvability conditions must be valid
PYD
∫ b
a
N(s)
[
f(t) + ε
∫ b
a
K(s, τ)Z(z(τ, ε), τ, ε)dτ
]
ds = 0,
PYQ
[
α+ εJ(z(·, ε), ε)− `f(·)−
− `M(·)D−
∫ b
a
N(s)
[
f(s) + ε
∫ b
a
K(s, τ)Z(z(τ, ε), τ, ε)dτ
]
ds
]
= 0,
which, taking into account (6) and ε 6= 0, take the form
PYD
[∫ b
a
N(s)
∫ b
a
K(s, τ)Z(z(τ, ε), τ, ε)dτ
]
ds = 0,
PYQ
[
J(z(·, ε), ε)− `M(·)D−
∫ b
a
N(s)
∫ b
a
K(s, τ)Z(z(τ, ε), τ, ε)dτ ds
]
= 0.
(12)
Taking into account the continuity of the operator-valued functions Z(z, t, ε) and J(z(·, ε), ε)
with respect to the totality of the variables z, t, ε, passing to the limit at ε → 0 in the
system (12), we obtain the necessary condition of the existence of solutions of the boundary-
value problem (1), (2)
F (c) =
PYD
[∫ b
a
N(s)
∫ b
a
K(s, τ)Z(z0(τ, c), τ, 0)dτ
]
ds = 0,
PYQ
[
J(z0(·, c), 0)− `M(·)D−
∫ b
a
N(s)
∫ b
a
K(s, τ)Z(z0(τ, c), τ, 0)dτ ds
]
= 0.
Thus, the theorem is valid for the boundary-value problem (1), (2).
Theorem 3. Suppose that with respect to the above conditions, the boundary-value
problem (1), (2) has the solution z(t, ε), continuous on ε ∈ [0, ε0], which converts at ε = 0
to some generating solution z0(t, c) of the form (7) obtained at c = c0. Then the element c0 ∈ B1
satisfies the system of gathers
F (c0) =
PYD
∫ b
a
N(s)
∫ b
a
K(s, τ)Z((z0(τ, c0)), τ, ε)dτds = 0,
PYQ
[
J(z0(·, c0))− `M(·)D−
∫ b
a
N(s)
∫ b
a
K(s, τ)Z(z0(s, c0)), τ, ε)dτ ds
]
= 0.
(13)
By analogy with weakly nonlinear problems for ordinary differential gathers [2, 4, 5], the
system of gathers (13) will be called a system of equations for generating constants.
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
352 V. F. ZHURAVLEV
Therefore, if the system of equations (13) has a solution c = c0 ∈ B, then the element c0
determines the generating solution z0(t, c0) to which the solution z(t, ε) of the original nonlinear
boundary-value problem (1), (2) can correspond. If the system of equations (13) does not have
solutions, then the boundary-value problem (1), (2) does not have the desired solution. Thus,
the necessary condition of the existence of a solution of the boundary-value problem (1), (2) is
satisfied by choosing the constant c in the generating solution (7), as the real root of the system
of equations (13).
To prove the sufficiency, using the conditions on the nonlinear operator-valued functions
Z(z, t, ε) and J(z, ε), we single out the linear parts with respect to x and terms of order zero
with respect to ε. As a result, we obtain expansions
Z(z0(t, c0) + x(t, ε), t, ε) = Z0(t, c0) + T (t)x(t, ε) +R(x(t, ε), t, ε),
J(z0(·, c0) + x(·, ε), ε) = J0(·, c0) + `1x(·, ε) +R1(x(·, ε), ε),
where
Z0(t, c0) = Z(z0(t, c0), t, 0) ∈ C(I,B),
J0(·, c0) = J0(z0(·, c0), 0) ∈ B1;
T (t) = T (t, c0) =
∂Z(z, t, 0)
∂z
∣∣∣∣
z=z(t,c0)
∈ C(I,B),
`1 =
∂J(z, 0)
∂z
∣∣∣∣
z=z(·,c0)
, `1 : C(I,B)→ B1;
R(x(t, ε), t, ε) is nonlinear vector-valued function, R1(x(·, ε), ε) is nonlinear vector-valued functi-
onal.
Considering the nonlinearities in the boundary-value problem (10), (11) as inhomogeneities
and applying theorem 1 to it, we obtain the following expression for the representation of its
solution x(t, ε) :
x(t, ε) = M̃(t)c+ x̄(t, ε).
In this case, the unknown vector c = c(ε) ∈ B1 is determined from the solvability conditions of
the type (12)
PYD
∫ b
a
N(s)
∫ b
a
K(s, τ)
{
Z0(τ, c0) + T (τ)x(τ, ε) +
+R(x(τ, ε), τ, ε)
}
dτ ds = 0,
PYQ
[
J0(·, c0) + `1x(·, ε) +R1(x(·, ε), ε)−
−`M(·)D−
∫ b
a
N(s)
∫ b
a
K(s, τ)
{
Z0(τ, c0) +
+ T (τ)x(τ, ε) +R(x(τ, ε), τ, ε)
}
dτds
]
= 0.
(14)
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
WEAKLY NONLINEAR BOUNDARY-VALUE PROBLEMS FOR FREDHOLM INTEGRAL EQUATIONS . . . 353
The unknown vector function x̄(t, ε) is defined by the formula
x̄(t, ε) = ε
G b∫
a
K(·, s) {Z0(s, c0) + T (s)x(s, ε) +R(x(s, ε), s, ε)} ds
(t)+
+M(t)Q−
[
J0(·, c0) + `1x(·, ε) +R1(x(·, ε), ε)
]
,
where the operator G acts on the vector function
ϕ(t, ε) =
b∫
a
K(t, s)
{
Z0(s, c0) + T (s)x(s, ε) +R(x(s, ε), s, ε)
}
ds
by the rule (8).
Substituting the expression x(t, ε) in (14) for M̃(t)c+x̄(t, ε) and isolating the terms containing
the constant c, taking (13) into account, we obtain the operator equation
B0c = −
PYD
∫ b
a
N(s)
∫ b
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds,
PYQ
[
R1(x(·, ε), x̄(·, ε), ε)−
− `M(·)D−
∫ b
a
N(s)
∫ b
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds
]
,
where
B0 =
[
B1
B2
]
,
B1 = PYD
b∫
a
N(s)
b∫
a
K(s, τ)T (τ)M̃(τ)dτds,
B2 = PYQ
`1M̃(·)− `M(·)D−
b∫
a
N(s)
b∫
a
K(s, τ)T (τ)M̃(τ)dτds
,
R(x(s, ε), x̄(s, ε), s, ε) := T (s)x̄(s, ε) +R(x(s, ε), s, ε),
R1(x(·, ε), x̄(·, ε), ε) := `1x̄(·, ε) +R1(x(·, ε), ε).
The operator B0 acts from the Banach space B to the direct product of the Banach spaces B
and B1, B0 : B→ B×B1.
Using the fact that the vector constant c0 ∈ B1 satisfies the system of equations for the
generating constants (13), to find a continuous in ε solution of x(·, ε) ∈ C(I,B), x(t, 0) = 0 of
the weakly nonlinear boundary-value problem (1), (2), go to the equivalent operator system
x(t, ε) = M̃(t)c(ε) + x̄(t, ε),
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
354 V. F. ZHURAVLEV
[
B1
B2
]
c = −
PYD
∫ b
a
N(s)
∫ b
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτds,
PYQ
[
R1(x(·, ε), x̄(·, ε), ε)−
− `M(·)D−
∫ b
a
N(s)
∫ b
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds
]
, (15)
x̄(t, ε) = ε
G b∫
a
K(·, s) [Z0(s, c0) + T (s)x(s, ε) +R(x(s, ε), s, ε)] ds
(t)+
+M(t)Q−
[
J0(·, c0) + `1x(·, ε) +R1(x(·, ε), ε)
]
,
Let B1 ∈ GI(B,B) and B2 ∈ GI(B,B1). Then, byTheorem2, because of normal solvability,
the second equation of the operator system (15) is solvable if and only if its right-hand side satisfies
the condition
[
P̃YB1
B12
0 PYB2
]
PYD
∫ b
a
N(s)
∫ b
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds,
PYQ
[
R1(x(·, ε), x̄(·, ε), ε)−
−`M(·)D−
∫ b
a
N(s)
∫ b
a
K(s, τ)×
×R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds
]
= 02×1, (16)
where 02×1 is a dimensional zero matrix, P̃YB1
= IB −B1PN(B2)B
−
1 , B12 = −B1B
−
2 .
At [
P̃YB1
B12
0 PYB2
][
PYD
PYQ
]
= 02×1 (17)
the condition (16) will always be satisfied and, by Theorem 2, the second equation of the operator
system (15) will have a family of solutions
c(ε) = PN(B0)ĉ−
[
PN(B2)B
−
1 B−2
]
×
×
PYD
∫ b
a
N(s)
∫ b
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds,
PYQ
[
R1(x(·, ε), x̄(·, ε), ε)−
−`M(·)D−
∫ b
a
N(s)
∫ b
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds
]
,
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
WEAKLY NONLINEAR BOUNDARY-VALUE PROBLEMS FOR FREDHOLM INTEGRAL EQUATIONS . . . 355
where PN(B0) = PN(B2)PN(B1) is the projector onto the zero-space N(B0) of the operator B0, ĉ
is an arbitrary element of the Banach space B,
[
PN(B2)B
−
1 B−2
]
is a generalized inverse operator
to the operator B0 =
[
B1
B2
]
.
Setting ĉ ≡ 0, when conditions (17) are satisfied, one of the solutions of the second equation
of the operator system (15) will have the form
c(ε) = B̃−1
b∫
a
N(s)
b∫
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds+
+ B̃−2
[
R1(x(·, ε), x̄(·, ε), ε)−
− `M(·)D−
b∫
a
N(s)
b∫
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds
]
,
where B̃−1 = −PN(B2)B
−
1 PYD
, B̃−2 = −B−2 PYQ
.
Thus, if conditions (17) are satisfied, the operator system (15) will have the form
x(t, ε) = M̃(t)c(ε) + x̄(t, ε),
c(ε) = B̃−1
b∫
a
N(s)
b∫
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds+
+ B̃−2
[
R1(x(·, ε), x̄(·, ε), ε)−
− `M(·)D−
b∫
a
N(s)
b∫
a
K(s, τ)R(x(τ, ε), x̄(τ, ε), τ, ε)dτ ds
]
,
(18)
x̄(t, ε) = ε
G b∫
a
K(·, s) {Z0(s, c0) + T (s)x(s, ε) +R(x(s, ε), s, ε)} ds
(t)+
+M(t)Q−
[
J0(·, c0) + `1x(·, ε) +R1(x(·, ε), ε)
]
,
By analogy with [1, 4, 5, 9], it can be shown that the operator system (18) belongs to the class
of the systems for which the convergent method of simple iterations is applicable.
Theorem 4. Suppose that the generating boundary-value problem (3), (4) under the condi-
tions (6) has a family of generating solutions (7). Then for each element c0 ∈ B1, which satisfies
the system of equations for the generating constants (13), under the conditions
PN(B0) 6= 0,
[
P̃YB1
B12
0 PYB2
][
PYD
PYQ
]
= 02×1
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
356 V. F. ZHURAVLEV
the boundary-value problem (1), (2) has at least one solution z(t, ε) = z0(t, c0) + x(t, ε) conti-
nuous with respect to ε, which turns into a generating solution z0(t, c0) at ε = 0. This solution
is found by converging to [0, ε∗] ⊂ [0, ε0] of the iterative process
zk+1(t, ε) = z0(t, c0) + xk+1(t, ε),
xk+1(t, ε) = M̃(t)ck(ε) + x̄k+1(t, ε), k = 0, 1, 2, . . . ,
ck(ε) = B̃−1
b∫
a
N(s)
b∫
a
K(s, τ)R(xk(τ, ε), x̄k(τ, ε), τ, ε)dτ ds+
+ B̃−2
[
R1(xk(·, ε), x̄k(·, ε), ε)−
− `M(·)D−
b∫
a
N(s)
b∫
a
K(s, τ)R(xk(τ, ε), x̄k(τ, ε), τ, ε)dτ ds
]
,
(19)
x̄k+1(t, ε) = ε
G
b∫
a
K(·, s)
{
Z0(s, c0)+
+ T (s)
[
M̃(s)ck(ε) + x̄k(s, ε)
]
+R(xk(s, ε), s, ε)ds)
(t)+
+M(t)Q−
{
J0(·, c0) + `1
[
M̃(·)ck(ε) + x̄k(·, ε)
]
+R1(xk(·, ε), ε)
}.
Remark 1. If PN(B0) 6= 0 and
[
P̃YB1
B12
0 PYB2
]
= 02×1, then the operator B0 is d-normal.
In this case, the condition (17) will always be satisfied and the second equation of the operator
system (15) will be always solvable, and the generalized inverse operator B−0 will be a right
inverse operator (B0)
−1
r [13]. Then the boundary-value problem (1), (2) will have at least one
solution that is found by means of the convergent iterative process (19), in which B−0 = (B0)
−1
r .
Remark 2. If PN(B0) = 0 and PYB0
=
[
P̃YB1
B12
0 PYB2
]
6= 02×1, then the operator B0 is n-
normal. In this case, the generalized inverse operator B−0 is the left inverse operator (B0)
−1
l and
under the condition (17) the second equation of the operator system (15) is definitely solvable [13].
Then for each c0 of the system for the generating constants (13) the boundary-value problem (1),
(2) has the only solution that is found by means of a convergent iterative process (19), in which
B−0 = (B0)
−1
l .
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
WEAKLY NONLINEAR BOUNDARY-VALUE PROBLEMS FOR FREDHOLM INTEGRAL EQUATIONS . . . 357
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Received 21.05.18
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
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