Singular periodic impulse problems
We obtain an existence principle for the impulsive periodic boundary-value problem u’’ + cu’ = g(x) + e(t), u(ti+) = u(ti) + Ji(u, u’ ), u’(ti+) = u’(ti) + Mi(u, u’), i = 1, . . ., m, u(0) = u(T), u’(0) = u’(T), where g ∈ C(0,∞) can have a strong singularity at the origin. Furthermore, we assume tha...
Збережено в:
| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2008
|
| Назва видання: | Нелінійні коливання |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/178156 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Singular periodic impulse problems / Z. Halas, M. Tvrdy // Нелінійні коливання. — 2008. — Т. 11, № 1. — С. 32-44. — Бібліогр.: 24 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-178156 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1781562021-02-19T01:26:55Z Singular periodic impulse problems Halas, Z. Tvrdy, M. We obtain an existence principle for the impulsive periodic boundary-value problem u’’ + cu’ = g(x) + e(t), u(ti+) = u(ti) + Ji(u, u’ ), u’(ti+) = u’(ti) + Mi(u, u’), i = 1, . . ., m, u(0) = u(T), u’(0) = u’(T), where g ∈ C(0,∞) can have a strong singularity at the origin. Furthermore, we assume that 0 < t1 < . . . < tm < T, e ∈ L₁ [0, T], c ∈ R and Ji , Mi , i = 1, 2, . . . , m, are continuous mappings of G[0, T] × G[0, T] into R, where G[0, T] denotes the space of functions regulated on [0, T]. The presented principle is based on an averaging procedure similar to that introduced by Manasevich ´ and Mawhin for singular periodic problems with p-Laplacian. Отримано принцип iснування розв’язку перiодичної граничної задачi з iмпульсною дiєю, u’’ + c u’ = g(x) + e(t), u(ti+) = u(ti) + Ji(u, u’), u’(ti+) = u’(ti) + Mi(u, u’), i = 1, . . ., m, u(0) = u(T), u’(0) = u”(T), де g ∈ C(0,∞) може мати сильну особливiсть у нулi. Далi, припускається, що 0 < t1 < . . . < tm < T, e ∈ L₁ [0, T], c ∈ R i Ji , Mi , i = 1, 2, . . . , m, — неперервнi вiдображення з G[0, T] × G[0, T] в R, де G[0, T] — простiр функцiй, регульованих на [0, T]. Отримання принципу базується на процедурi усереднення, яка є аналогом процедури, запро- понованої Менасевiчем та Мавхiним, для сингулярних перiодичних задач iз p-лапласiаном. 2008 Article Singular periodic impulse problems / Z. Halas, M. Tvrdy // Нелінійні коливання. — 2008. — Т. 11, № 1. — С. 32-44. — Бібліогр.: 24 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178156 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We obtain an existence principle for the impulsive periodic boundary-value problem u’’ + cu’ = g(x) + e(t), u(ti+) = u(ti) + Ji(u, u’ ), u’(ti+) = u’(ti) + Mi(u, u’), i = 1, . . ., m, u(0) = u(T), u’(0) = u’(T), where g ∈ C(0,∞) can have a strong singularity at the origin. Furthermore, we assume that 0 < t1 < . . . < tm < T, e ∈ L₁ [0, T], c ∈ R and Ji , Mi , i = 1, 2, . . . , m, are continuous mappings of G[0, T] × G[0, T] into R, where G[0, T] denotes the space of functions regulated on [0, T]. The presented principle is based on an averaging procedure similar to that introduced by Manasevich ´ and Mawhin for singular periodic problems with p-Laplacian. |
| format |
Article |
| author |
Halas, Z. Tvrdy, M. |
| spellingShingle |
Halas, Z. Tvrdy, M. Singular periodic impulse problems Нелінійні коливання |
| author_facet |
Halas, Z. Tvrdy, M. |
| author_sort |
Halas, Z. |
| title |
Singular periodic impulse problems |
| title_short |
Singular periodic impulse problems |
| title_full |
Singular periodic impulse problems |
| title_fullStr |
Singular periodic impulse problems |
| title_full_unstemmed |
Singular periodic impulse problems |
| title_sort |
singular periodic impulse problems |
| publisher |
Інститут математики НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/178156 |
| citation_txt |
Singular periodic impulse problems / Z. Halas, M. Tvrdy // Нелінійні коливання. — 2008. — Т. 11, № 1. — С. 32-44. — Бібліогр.: 24 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT halasz singularperiodicimpulseproblems AT tvrdym singularperiodicimpulseproblems |
| first_indexed |
2025-07-15T16:31:23Z |
| last_indexed |
2025-07-15T16:31:23Z |
| _version_ |
1837731240286879744 |
| fulltext |
UDC 517 . 9
SINGULAR PERIODIC IMPULSE PROBLEMS
СИНГУЛЯРНI ПЕРIОДИЧНI IМПУЛЬСНI ЗАДАЧI
Z. Halas, M. Tvrdý
Mat. únstav AV ČR
Žitná 25, 11567, Praha 1, Česká Republika
e-mail: tvrdy@math.cas.cz
We obtain an existence principle for the impulsive periodic boundary-value problem u′′ + c u′ = g(x) + e(t),
u(ti+) = u(ti) +Ji(u, u′), u′(ti+)= u′(ti)+ Mi(u, u′), i = 1, . . .,m, u(0)= u(T ), u′(0)= u′(T ), where
g ∈ C(0,∞) can have a strong singularity at the origin. Furthermore, we assume that 0 <t1 < . . . < tm <T,
e ∈ L1[0, T ], c ∈ R and Ji, Mi, i = 1, 2, . . . ,m, are continuous mappings of G[0, T ] × G[0, T ] into R,
where G[0, T ] denotes the space of functions regulated on [0, T ].
The presented principle is based on an averaging procedure similar to that introduced by Manásevich
and Mawhin for singular periodic problems with p-Laplacian.
Отримано принцип iснування розв’язку перiодичної граничної задачi з iмпульсною дiєю, u′′ +
+ c u′ = g(x) + e(t), u(ti+) = u(ti) +Ji(u, u′), u′(ti+)= u′(ti) +Mi(u, u′), i = 1, . . .,m, u(0)= u(T ),
u′(0)= u′(T ), де g ∈ C(0,∞) може мати сильну особливiсть у нулi. Далi, припускається, що
0 <t1 < . . . < tm <T, e ∈ L1[0, T ], c ∈ R i Ji, Mi, i = 1, 2, . . . ,m, — неперервнi вiдображення з
G[0, T ]×G[0, T ] в R, де G[0, T ] — простiр функцiй, регульованих на [0, T ].
Отримання принципу базується на процедурi усереднення, яка є аналогом процедури, запро-
понованої Менасевiчем та Мавхiним, для сингулярних перiодичних задач iз p-лапласiаном.
1. Preliminaries. Starting with Hu and Lakshmikantham [1], periodic boundary-value prob-
lems for nonlinear second order impulsive differential equations of the form
u′′ = f(t, u, u′), (1.1)
u(ti+) = u(ti) + Ji(u, u′),
u′(ti+) = u′(ti) + Mi(u, u′), i = 1, 2, . . . ,m,
(1.2)
u(0) = u(T ), u′(0) = u′(T ) (1.3)
have been studied by many authors. Usually it is assumed that the function f : [0, T ]×R2 → R
fulfils the Carathéodory conditions,
0 <t1 <t2 < . . . < tm <T are fixed points of the interval [0, T ] (1.4)
and Ji, Mi : R2 → R, i = 1, 2, . . . ,m, are continuous functions. A rather representative
(however not complete) list of related papers is given in references. In particular, in [2 – 6] the
existence results in terms of lower/upper functions obtained by the monotone iterative method
can be found. All of these results impose monotonicity of the impulse functions and existence
of an associated pair of well-ordered lower/upper functions. The papers [7] and [8] are based
c© Z. Halas, M. Tvrdý, 2008
32 ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
SINGULAR PERIODIC IMPULSE PROBLEMS 33
on the method of bound sets, however the effective criteria contained therein correspond to
the situation when there is a well-ordered pair of constant lower and upper functions. The ex-
istence results which apply also to the case when there is a pair of lower and upper functions,
which need not be well-ordered, were provided only by Rachůnková and Tvrdý, see [9 –12].
Analogous results for impulsive problems with quasilinear differential operator were delivered
by Rachůnková and Tvrdý in [13 –15]. When no impulses are acting, periodic problems with
singularities have been treated by many authors. For rather representative overview and refer-
ences, see e.g. [16] or [17]. To our knowledge, up to now singular periodic impulsive problems
have not been treated. For singular Dirichlet impulsive problems we refer to the papers by
Rachůnková [18], Rachůnková and Tomeček [19] and Lee and Liu [20].
In this paper we establish an existence principle suitable for solving singular impulsive pe-
riodic problems.
Notations. Throughout the paper we keep the following notation and conventions: for
a real-valued function u defined a.e. on [0, T ], we put
‖u‖∞ = sup esst∈[0,T ] |u(t)| and ‖u‖1 =
T∫
0
|u(s)| ds.
For a given interval J ⊂ R, by C(J) we denote the set of real-valued functions which are
continuous on J. Furthermore, C1(J) is the set of functions having continuous first derivatives
on J and L1(J) is the set of functions which are Lebesgue integrable on J.
Any function x : [0, T ] → R which possesses finite limits
x(t+) = lim
τ→t+
x(τ) and x(s−) = lim
τ→s−
x(τ)
for all t ∈ [0, T ) and s ∈ (0, T ] is said to be regulated on [0, T ]. The linear space of functions
regulated on [0, T ] is denoted by G[0, T ]. It is well known that G[0, T ] is a Banach space with
respect to the norm x∈G[0, T ]→‖x‖∞ (cf. [21], Theorem I.3.6).
Let m ∈ N and let 0 = t0 < t1 < t2 < . . . < tm < tm+1 = T be a division of the interval
[0, T ]. We denote D = {t1, t2, . . . , tm} and define C1
D[0, T ] as the set of functions u : [0, T ] → R
such that
u(t) =
u[0](t) if t ∈ [0, t1],
u[1](t) if t ∈ (t1, t2],
. . . . . . . . . . . .
u[m](t) if t ∈ (tm, T ],
where u[i] ∈ C1[ti, ti+1] for i = 0, 1, . . . ,m. In particular, if u ∈ C1
D[0, T ], then u′ possesses
finite one-sided limits
u′(t−) := lim
τ→t−
u(τ) and u′(s+) := lim
τ→s+
u(τ)
for each t ∈ (0, T ] and s ∈ [0, T ). Moreover, u′(t−) = u′(t) for all t ∈ (0, T ] and u′(0+) =
= u′(0). For u ∈ C1
D[0, T ] we put
‖u‖D = ‖u‖∞ + ‖u′‖∞.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
34 Z. HALAS, M. TVRDÝ
Then C1
D[0, T ] becomes a Banach space when endowed with the norm ‖.‖D. Furthermore, by
AC1
D[0, T ] we denote the set of functions u ∈ C1
D[0, T ] having first derivatives absolutely con-
tinuous on each subinterval (ti, ti+1), i = 1, 2, . . . ,m + 1.
We say that f : [0, T ] × R2 7→ R satisfies the Carathéodory conditions on [0, T ] × R2 if
(i) for each x ∈ R and y ∈ R the function f(., x, y) is measurable on [0, T ]; (ii) for almost
every t ∈ [0, T ] the function f(t, ., .) is continuous on R2; (iii) for each compact set K ⊂ R2
there is a function mK(t) ∈ L[0, T ] such that |f(t, x, y)| ≤ mK(t) holds for a.e. t ∈ [0, T ] and
all (x, y) ∈ K. The set of functions satisfying the Carathéodory conditions on [0, T ] × R2 is
denoted by Car([0, T ]× R2).
Given a subset Ω of a Banach space X, its closure is denoted by Ω. Finally, we will write ē
instead of
1
T
∫ T
0
e(s) ds and ∆+u(t) instead of u(t+)− u(t).
If f ∈ Car([0, T ]×R2), problem (1.1) – (1.3) is said to be regular and a function u∈AC1
D[0, T ]
is its solutions if
u′′(t) = f(t, u(t), u′(t)) holds for a.e. t ∈ [0, T ]
and conditions (1.2) and (1.3) are satisfied. If f /∈ Car([0, T ]×R2), problem (1.1) – (1.3) is said
to be singular.
In this paper we will deal with rather simplified, however the most typical, case of the sin-
gular problem with
f(t, x, y) = c y + g(x) + e(t) for x ∈ (0,∞), y ∈ R and a.e. t ∈ [0, T ],
where
c ∈ R, g ∈ C(0,∞), e ∈ L1[0, T ]. (1.5)
Definition 1.1. A function u∈AC1
D[0, T ] is called a solution of the problem
u′′ + c u′ = g(u) + e(t), (1.2), (1.3) (1.6)
if u > 0 a.e. on [0, T ],
u′′(t) + c u′(t) = g(u(t)) + e(t) for a.e. t ∈ [0, T ],
and conditions (1.2) and (1.3) are satisfied.
2. Green’s functions and operator representations for impulsive two-point boundary-value
problems. For our purposes an appropriate choice of the operator representation of (1.1) –
(1.3) is important. To this aim, let us consider the following impulsive problem with nonlinear
two-point boundary conditions,
u′′ + a2(t) u′ + a1(t) u = f(t, u, u′) a.e. on [0, T ], (2.1)
∆+u(ti) = Ji(u, u′), ∆+u′(ti) = Mi(u, u′), i = 1, 2, . . . ,m, (2.2)
P
(
u(0)
u′(0)
)
+ Q
(
u(T )
u′(T )
)
= R(u, u′), (2.3)
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
SINGULAR PERIODIC IMPULSE PROBLEMS 35
and its linearized version
u′′ + a2(t) u′ + a1(t) u = h(t) a.e. on [0, T ], (2.4)
∆+u(ti) = di, ∆+u′(ti) = d ′i, i = 1, 2, . . . ,m, (2.5)
P
(
u(0)
u′(0)
)
+ Q
(
u(T )
u′(T )
)
= c, (2.6)
where
a1, h ∈ L[0, T ], a2 ∈ C[0, T ], f ∈ Car([0, T ]× R2),
Ji and Mi : G[0, T ]×G[0, T ]→R, i=1, 2, . . . ,m, are continuous mappings,
c ∈ R2, di, d ′i ∈ R, i = 1, 2, . . . ,m,
P, Q are real 2× 2-matrices, rank(P,Q) = 2,
R : G[0, T ]×G[0, T ]→R2 is a continuous mapping.
(2.7)
Solutions of problems (2.1) – (2.3) and (2.4) – (2.6) are defined in a natural way quite analo-
gously to the above mentioned definition of regular periodic problems. Problem (2.4) – (2.6) is
equivalent to the two-point problem for a special case of generalized linear differential systems
of the form
x(t)− x(0)−
t∫
0
A(s) x(s) ds = b(t)− b(0) on [0, T ], (2.8)
P x(0) + Qx(T ) = c, (2.9)
where
x(t) =
(
x1(t)
x2(t)
)
=
(
u(t)
u′(t)
)
, A(t) =
(
0 1
−a1(t) −a2(t)
)
, (2.10)
b(t) =
t∫
0
(
0
h(s)
)
ds +
m∑
i=1
(
di
d ′i
)
χ(ti, T ](t), t ∈ [0, T ],
and χ(ti, T ](t) = 1 if t∈ (ti, T ], χ(ti, T ](t) = 0 otherwise. Solutions of (2.8), (2.9) are 2-vector
functions of bounded variation on [0, T ] satisfying the two-point condition (2.9) and fulfilling
the integral equation (2.8) for all t ∈ [0, T ], cf. e.g. [22]. Assume that the homogeneous problem
u′′ + a2(t) u′ + a1(t) u = 0, P
(
u(0)
u′(0)
)
+ Q
(
u(T )
u′(T )
)
= 0 (2.11)
has only the trivial solution. Then, obviously, the problem
x′ −A(t) x = 0, P x(0) + Qx(T ) = 0 (2.12)
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
36 Z. HALAS, M. TVRDÝ
has also only the trivial solution. In view of [23] (Theorems 4.2 and 4.3) (see also [24], Theorem
4.1), problem (2.8), (2.9) has a unique solution x and it is given by
x(t) =
T∫
0
Γ(t, s) d[b(s)] + x0(t), t ∈ [0, T ], (2.13)
where x0 is the uniquely determined solution of
x′ −A(t) x = 0, P x(0) + Qx(T ) = c,
and
Γ(t, s) = (γi,j(t, s))i,j=1,2
is Green’s matrix for (2.12). Recall that, for each s ∈ (0, T ), the matrix function t → Γ(t, s) is
absolutely continuous on [0, T ] \ {s} and
∂
∂ t
Γ(t, s)−A(t) Γ(t, s) = 0 for a.e. t ∈ [0, T ],
P Γ(0, s) + QΓ(T, s) = 0,
Γ(t+, t)− Γ(t−, t) = I,
where I stands for the identity 2×2-matrix. In particular, the component γ1,2 of Γ is absolutely
continuous on [0, T ] for each s ∈ (0, T ) and
∂
∂ t
γ1,2(t, s) = γ 2,2(t, s) for a.e. t ∈ [0, T ].
Denote G(t, s) = γ1,2(t, s). Then G(t, s) is Green’s function of (2.11). Furthermore, we have
∂
∂s
Γ(t, s) = −Γ(t, s) A(s) for all t ∈ (0, T ) and a.e. s ∈ [0, T ].
In particular,
γ1,1(t, s) = − ∂
∂s
G(t, s) + a1(s) G(t, s) for all t ∈ [0, T ] and a.e. s ∈ [0, T ].
Inserting (2.10) into (2.13) we get that, for each h∈L[0, T ], c, di, d ′i ∈R, i = 1, 2, . . . ,m, the
unique solution u of problem (2.4) – (2.6) is given by
u(t) = u0(t) +
t∫
0
G(t, s) h(s) ds+
+
m∑
i=1
(
− ∂
∂s
G(t, ti) + a1(t) G(t, ti)
)
di +
m∑
i=1
G(t, ti) d ′i for t ∈ [0, T ],
where u0 is the uniquely determined solution of the problem
u′′ + a2(t) u′ + a1(t) u = 0, (2.6). (1)
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
SINGULAR PERIODIC IMPULSE PROBLEMS 37
Now, choose an arbitrary w ∈ C1
D[0, T ] and put
h(t) = f(t, w(t), w′(t)) for a.e. t ∈ [0, T ],
di = Ji(w,w′), d ′i = Mi(w,w′), i = 1, 2, . . . ,m,
c = R(w,w′).
Then h ∈ L[0, T ], c, di, d ′i ∈ R, i = 1, 2, . . . ,m, and there is a unique u ∈ AC1
D[0, T ] fulfilling
(2.4) – (2.6) and it is given by (2.12). Therefore, assuming, in addition, that the problem
u′′ + a2(t) u′ + a1(t) u = 0, (2.14)
P
(
u(0)
u′(0)
)
+ Q
(
u(T )
u′(T )
)
= R(u, u′) (2.15)
has a unique solution u0, we conclude that u∈C1
D[0, T ] is a solution to (2.1) – (2.3) if and only if
u(t) = u0(t) +
t∫
0
G(t, s) f(s, u(s), u′(s)) ds+
+
m∑
i=1
(
− ∂
∂s
G(t, ti) + a1(t) G(t, ti)
)
Ji(u, u′) +
m∑
i=1
G(t, ti) Mi(u, u′) for t ∈ [0, T ].
Let us define operators F1 and F2 : C1
D[0, T ] → C1
D[0, T ] by
(F1 u)(t) =
T∫
0
G(t, s) f(s, u(s), u′(s)) ds, t ∈ [0, T ],
and
(F2 u)(t) = u0(t) +
m∑
i=1
(
− ∂
∂s
G(t, ti) + a1(t) G(t, ti)
)
Ji(u, u′)+
+
m∑
i=1
G(t, ti) Mi(u, u′), t ∈ [0, T ],
respectively. The former one, F1, is a composition of the Green type operator
h ∈ L1[0, T ] →
T∫
0
G(t, s) h(s) ds ∈ C1[0, T ],
which is known to map equiintegrable subsets1 of L1[0, T ] onto relatively compact subsets of
C1[0, T ] ⊂ C1
D[0, T ], and the superposition operator generated by f ∈ Car([0, T ]× R2), which
1I.e., sets of functions having a common integrable majorant.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
38 Z. HALAS, M. TVRDÝ
similarly to the classical setting maps bounded subsets of C1
D[0, T ] to equiintegrable subsets
of L1[0, T ]. Therefore, it is easy to see that F1 is completely continuous. Furthermore, since
Ji, Mi, i = 1, 2, . . . ,m, are continuous mappings, the operator F2 is continuous as well. Having
in mind that F2 maps bounded sets onto bounded sets and its values are contained in a (2m+1)-
dimensional subspace2 of C1
D[0, T ], we conclude that the operators F2 and F = F1 + F2 are
completely continuous as well.
So, we have the following assertion.
Proposition 2.1. Assume (1.4) and (2.7). Furthermore, let problem (2.11) have Green’s
function G(t, s) and let u0 ∈ AC1
D[0, T ] be a uniquely defined solution of problem (2.14), (2.15).
Then u ∈ AC1
D is a solution to (2.1) – (2.3) if and only if u = F u, where F : C1
D[0, T ] →
→ C1
D[0, T ] is the completely continuous operator given by
(Fu)(t) =u0(t) +
T∫
0
G(t, s)
(
f(t, u(s), u′(s))−a1(s) u(s)−a2(s) u′(s)
)
ds+
+
m∑
i=1
(
− ∂
∂s
G(t, ti)+a1(t)G(t, ti)
)
Ji(u, u′) +
m∑
i=1
G(t, ti) Mi(u, u′), t ∈ [0, T ].
In particular, if a1(t) = a2(t) = 0 on [0, T ],
P =
(
1 0
0 0
)
and Q =
(
0 0
1 0
)
,
then problem (2.11) reduces to the simple Dirichlet problem
u′′ = 0, u(0) = u(T ) = 0
and its Green’s function is well-known,
G(t, s) =
s (t− T )
T
if 0 ≤ s < t ≤ T,
t (s− T )
T
if 0 ≤ t ≤ s ≤ T
(2.16)
and
∂
∂s
G(t, s) =
T − t
T
if 0 ≤ s < t ≤ T,
− t
T
if 0 ≤ t ≤ s ≤ T.
Furthermore, let us notice that the periodic boundary conditions (1.3) can be reformulated as
u(0) = u(T ) = u(0) + u′(0)− u′(T ),
i.e., in the form (2.15), where
R(u, v) = u(0) + v(0)− v(T ) for u, v ∈ G[0, T ].
It is easy to see that, in such a case, for any c ∈ R the only solution to (2.14), (2.15) is u0(t) ≡ c.
Therefore, we have the following corollary of Proposition 2.1.
2I.e., spanned over the set
�
u0, G(., ti),
�
− ∂
∂s
G(., ti) + a1 G(., ti)
�
, i = 1, 2, . . . , m
�
.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
SINGULAR PERIODIC IMPULSE PROBLEMS 39
Corollary 2.1. Assume (1.4) and (2.7) and let the function G(t, s) be given by (2.16). Then
u ∈ AC1
D is a solution to (1.1) – (1.3) if and only if u = F u, where F : C1
D[0, T ] → C1
D[0, T ] is
the completely continuous operator given by
(Fu)(t) =u(0)+ u′(0)−u′(T ) +
T∫
0
G(t, s) f(t, u(s), u′(s)) ds−
−
m∑
i=1
∂
∂s
G(t, ti) Ji(u, u′) +
m∑
i=1
G(t, ti) Mi(u, u′), t ∈ [0, T ].
Remark 2.1. Similarly, u ∈ AC1
D is a solution to the impulsive Dirichlet problem (1.1), (1.2),
u(0) = u(T ) = c if and only if u = Fdir u, where
(Fdiru)(t) = c+
T∫
0
G(t, s) f(t, u(s), u′(s)) ds−
−
m∑
i=1
∂
∂s
G(t, ti) Ji(u, u′) +
m∑
i=1
G(t, ti) Mi(u, u′), t ∈ [0, T ].
3. Existence principle.
Theorem 3.1. Let assumptions (1.4) and (1.5) hold. Furthermore, assume that there exist
r ∈ (0,∞), R ∈ (r,∞) and R ′ ∈ (0,∞) such that
(i) r < v < R on [0, T ] and ||v′||∞ < R ′ for each λ ∈ (0, 1] and for each positive solution v
of the problem
v′′(t) = λ
(
−c v′(t) + g(v(t)) + e(t)
)
for a.e. t ∈ [0, T ], (3.1)
∆+v(ti) = λ Ji(v, v′), i = 1, 2, . . . ,m, (3.2)
∆+v′(ti) = λ Mi(v, v′), i = 1, 2, . . . ,m, (3.3)
v(0) = v(T ), v′(0) = v′(T ); (3.4)
(ii) (g(x) + ē = 0) =⇒ r < x < R;
(iii) (g(r) + ē) (g(R) + ē) < 0.
Then problem (1.6) has a solution u such that
r <u < R on [0, T ] and ‖u′‖∞<R ′.
Proof. Step 1. For λ ∈ [0, 1] and v ∈ C1
D[0, T ] denote
Ξλ(v) =
T∫
0
g(v(s)) ds +T ē+
m∑
i=1
Mi(v, v′) +λ c
m∑
i=1
Ji(v, v′). (3.5)
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
40 Z. HALAS, M. TVRDÝ
Notice that
Ξλ(v) = 0 holds for all solutions v ∈ C1
D[0, T ] of (3.1) – (3.4). (3.6)
Indeed, let v ∈ C1
D[0, T ] be a solution to (3.1) – (3.4). Then
T∫
0
v′′(s) ds =
m∑
i=0
ti+1∫
ti
v′′(s) ds =
m∑
i=0
[
v′(ti+1)− v′(ti+)
]
=
= v′(T )− v′(0)−
m∑
i=1
∆+v′(ti) = −λ
m∑
i=1
Mi(v, v′)
and
T∫
0
c v′(s) ds = c
m∑
i=0
ti+1∫
ti
v′(s) ds = c
m∑
i=0
[
v(ti+1)− v(ti+)
]
=
= c
[
v(T )− v(0)−
m∑
i=1
∆+v(ti)
]
= −λ c
m∑
i=1
Ji(v, v′).
Thus, integrating (3.1) over [0, T ] gives (3.6).
Step 2. Consider system (3.7), (3.2), (3.4), where (3.7) is the functional-differential equa-
tion:
v′′ = λ
[
−c v′ + g(v) + e(t)
]
+ (1−λ)
1
T
Ξλ(v). (3.7)
Due to (3.6), we can see that for each λ∈ [0, 1] the problems (3.1) – (3.4) and (3.7), (3.2) – (3.4)
are equivalent. Moreover, for λ =1, problem (3.7), (3.2), (3.4) reduces to the given problem
(1.6) (with u replaced by v).
Now, notice that in view of (2.16) we have
T∫
0
G(t, s) ds =
1
2
t (t− T ) for t ∈ [0, T ]
and define, for λ ∈ [0, 1], u ∈ C1
D[0, T ], u > 0 on [0, T ], and t ∈ [0, T ],
Fλ(u)(t) = u(0) + u′(0)− u′(T ) + λ
T∫
0
G(t, s)
[
− cu′(s) + g(u(s))+ e(s)
]
ds+
+ (1−λ)
t (t− T )
2 T
Ξλ(u)−λ
m∑
i=1
∂
∂s
G(t, ti) Ji(u, u′)+
+ λ
m∑
i=1
G(t, ti) Mi(u, u′). (3.8)
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
SINGULAR PERIODIC IMPULSE PROBLEMS 41
In particular, if λ = 0, then
F0(u)(t) = u(0) + u′(0)− u′(T ) +
t (t− T )
2 T
Ξ0(u) for t ∈ [0, T ].
Let us put
Ω = {u ∈ C1
D[0, T ] : r < u < R and |u′| < R ′ on [0, T ]}.
Arguing similarly to the regular case (see Corollary 2.1), we can conclude that for each λ ∈
∈ [0, 1] the operator Fλ : Ω ⊂ C1
D[0, T ] → C1
D is completely continuous and a function v ∈ Ω
is a solution of (3.7), (3.2) – (3.4) if and only if it is a fixed point of Fλ. In particular,
u∈Ω is a solution to (1.6) if and only if F1(u) =u. (3.9)
Step 3. We will show that
Fλ(u) 6= u for all u ∈ ∂ Ω and λ ∈ [0, 1]. (3.10)
Indeed, for λ ∈ (0, 1] relation (3.10) follows immediately from assumption (i), while for λ = 0
it is a corollary of assumption (ii) and of the following claim.
Claim. u ∈ Ω is a fixed point of F0 if and only if there is x ∈ R such that u(t) ≡ x on [0, T ],
x ∈ (r, R), and
g(x) + ē = 0. (3.11)
Proof of Claim. Let u ∈ Ω be a fixed point of F0(v), i.e.,
u(t) = u(0) + u′(0)− u′(T ) +
t (t− T )
2 T
Ξ0(u) for all t ∈ [0, T ]. (3.12)
Inserting t = 0 into (3.12), we get u(0) = u(0)+u′(0)−u′(T ), which implies that u′(0) = u′(T ).
Similarly, inserting t = T we get u(T ) = u(0). Furthermore,
u′(t) =
2 t− T
2T
Ξ0(u) for t ∈ [0, T ].
Since u′(0) = u′(T ), it follows that Ξ0(u) = 0. This means that u is constant on [0, T ]. Denote
x = u(0). Then 0 = Ξ0(u) = T (g(x) + ē), i.e., (3.11) is true. On the other hand, it is easy to
see that if x ∈ R is such that (3.11) holds and u(t) ≡ x on [0, T ], then u ∈ Ω is a fixed point of
F0. This completes the proof of the claim.
Step 4. By Step 3 and by the invariance under the homotopy property of the topological
degree, we have
deg(I − F1,Ω) = deg(I − F0,Ω). (3.13)
Step 5. Let us denote
X = {u ∈ C1
D[0, T ] : u(t) ≡ u(0) on [0, T ]} and Ω0 = Ω ∩ X.
Notice that Ω0 = {u ∈ X : r <u(0)<R} and Ω0 = {u ∈ X : r≤u(0)<R} . By Claim in
Step 3, all fixed points of F0 belong to Ω0. Hence, by the excision property of the topologi-
cal degree we have
deg(I − F0,Ω) = deg(I − F0,Ω0). (3.14)
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
42 Z. HALAS, M. TVRDÝ
Step 6. Define
F̃µ(u)(t) = u(0) +
[
1− µ +
µ
2
t (t− T )
] (
g(u(0) + ē
)
(3.15)
for t ∈ [0, T ], u ∈ Ω0 and µ ∈ [0, 1].
We have
F̃0(u) = u(0) + g(u(0)) + ē and F̃1(u) = F0(u) for each u ∈ X.
Similarly to Fλ, the operators F̃µ, µ ∈ [0, 1], are also completely continuous and, by the Claim
in Step 3, we have
F̃1(u) 6= u for all u ∈ ∂ Ω0.
Let i and i−1 be respectively the natural isometrical isomorphism R → X and its inverse, i.e.,
i(x)(t) ≡ u for x∈R and i−1(u) = u(0) for u∈X,
and assume that µ ∈ [0, 1), x ∈ (0,∞), u = i(x) and F̃µ(u) = u. Then[
1− µ +
µ
2
t (T − t)
] (
g(x) + e
)
= 0 for all t ∈ [0, T ].
If t = 0, this relation reduces to g(x) + e = 0, which is, due to assumption (ii) possible only if
x ∈ (r, R). To summarize, we have
F̃µ(u) 6= u for all u ∈ ∂ Ω0 and all µ ∈ [0, 1].
Hence, using the homotopy invariance property of the topological degree and taking into ac-
count that dim X = 1, we conclude that
deg(I − F̃1,Ω0) = deg(I − F̃1,Ω0) = dB(I − F̃0,Ω0), (3.16)
where dB(I − F̃0,Ω0) stands for the Brouwer degree of I − F̃0 with respect to the set Ω0 (and
the point 0).
Step 7. Define Φ : x ∈ (0,∞) → g(x) + ē ∈ R. Then
(I − F̃0)(i(x)) = i(Φ(x)) for each x ∈ (0,∞).
In other words, Φ = i−1 ◦ (I − F̃0) ◦ i on (0,∞). Consequently,
dB(I − F̃0,Ω0) = dB(Φ, (r, R)). (3.17)
Now, put
Ψ(x) = Φ(r)
R− x
R− r
+ Φ(R)
x− r
R− r
.
We can see that Ψ has a unique zero x0 ∈ (r, R) and
Ψ′(x0) =
Φ(R)− Φ(r)
R− r
.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
SINGULAR PERIODIC IMPULSE PROBLEMS 43
Hence, by the definition of the Brouwer degree in R, we have
dB(Ψ, (r, R)) = signΨ′(x0) = sign (Φ(R)− Φ(r)) .
By the homotopy property and thanks to our assumption (iii), we conclude that
dB(Φ, (r, R)) = dB(Ψ, (r, R)) = sign (Φ(R)− Φ(r)) 6= 0. (3.18)
Step 8. To summarize, by (3.13) – (3.18) we have
deg(I − F1,Ω) 6= 0,
which, in view of the existence property of the topological degree, shows that F1 has a fixed
point u∈Ω. By Step 1 this means that problem (1.6) has a solution.
The theorem is proved.
1. Hu Shouchuan, Laksmikantham V. Periodic boundary value problems for second order impulsive differential
systems // Nonlinear Anal. — 1989. — 13. — P. 75 – 85.
2. Bainov D., Simeonov P. Impulsive differential equations: periodic solutions and applications. — Harlow:
Longman Sci. Tech., 1993.
3. Cabada A., Nieto J. J., Franco D., Trofimchuk S. I. A generalization of the monotone method for second order
periodic boundary value problem with impulses at fixed points // Dynam. Contin. Discrete Impuls. Syst. —
2000. — 7. — P. 145 – 158.
4. Erbe L. H., Liu Xinzhi. Existence results for boundary value problems of second order impulsive differential
equations // J. Math. Anal. and Appl. — 1990. — 149. — P. 56 – 69.
5. Liz E., Nieto J. J. Periodic solutions of discontinuous impulsive differential systems // Ibid. — 1991. — 161.
— P. 388 – 394.
6. Liz E., Nieto J. J. The monotone iterative technique for periodic boundary value problems of second order
impulsive differential equations // Comment. math. Univ. carol. — 1993. — 34. — P. 405 – 411.
7. Dong Yujun. Periodic solutions for second order impulsive differential systems // Nonlinear Anal. — 1996.
— 27. — P. 811 – 820.
8. Zhang Zhitao. Existence of solutions for second order impulsive differential equations // Appl. Math. Ser.
B (Engl. Ed.) — 1997. — 12. — P. 307 – 320.
9. Rachůnková I., Tvrdý M. Impulsive periodic boundary value problem and topological degree // Funct. Dif-
ferent. Equat. — 2002. — 9, № 3 – 4. — P. 471 – 498.
10. Rachůnková I., Tvrdý M. Non-ordered lower and upper functions in second order impulsive periodic prob-
lems // Dynam Contin. Discrete Impuls. Syst. Ser. A. Math. Anal. — 2005. — 12. — P. 397 – 415.
11. Rachůnková I., Tvrdý M. Existence results for impulsive second order periodic problems // Nonlinear Anal:
Theory, Methods and Appl. — 2004. — 59. — P. 133 – 146.
12. Rachůnková I., Tvrdý M. Method of lower and upper functions in impulsive periodic boundary value prob-
lems // EQUADIFF 2003. Proc. Int. Conf. Different. Equat. (Hasselt, Belgium, July 22-26, 2003) / Eds
F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, S. Verduyn. — Hackensack, NJ: World Sci., 2005.
— P. 252 – 257.
13. Rachůnková I., Tvrdý M. Second order periodic problem with φ-Laplacian and impulses. Pt I. — Math. Inst.
Acad. Sci. Czech Republic. Preprint 155/2004 [available as \http: //www.math.cas.cz/ e tvrdy/lapl1.pdf or
\http: //www.math.cas.cz/ e tvrdy/lapl1.ps].
14. Rachůnková I., Tvrdý M. Second order periodic problem with φ-Laplacian and impulses. Pt II. — Math. Inst.
Acad. Sci. Czech Republic, Preprint. 156/2004 [available as \http: //www.math.cas.cz/ e tvrdy/lapl2.pdf or
\http: //www.math.cas.cz/ e tvrdy/lapl2.ps].
15. Rachůnková I., Tvrdý M. Second order periodic problem with φ-Laplacian and impulses // Nonlinear Anal.
— 2005. — 63. — P. 257 – 266.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
44 Z. HALAS, M. TVRDÝ
16. Rachůnková I., Staněk S., Tvrdý M. Singularities and Laplacians in boundary value problems for nonlinear
ordinary differential equations // Handbook Different. Equat. Ordinary Different. Equat. / Eds A. Caňada,
P. Drábek, A. Fonda. — Elsevier, 2006. — Vol. 3. — P. 607 – 723.
17. Rachůnková I., Staněk S., Tvrdý M. Solvability of nonlinear singular problems for ordinary differential equa-
tions // Hindawi Cotemp. Math. and Appl. (to appear).
18. Rachůnková I. Singular Dirichlet second order boundary value problems with impulses // J. Different. Equat.
— 2003. — 193. — P. 435 – 459.
19. Rachůnková I., Tomeček J. Singular Dirichlet problem for ordinary differential equations with impulses //
Nonlinear Anal.: Theory, Methods and Appl. — 2006. — 65. — P. 210 – 229.
20. Lee Yong-Hoon, Liu Xinzhi. Study of singular boundary value problem for second order impulsive differen-
tial equations // J. Math. Anal. and Appl. — 2007. — 331. — P. 159 – 176.
21. Ch. S. Hönig. Volterra Stieltjes-integral equations // Math. Stud. — Amsterdam; New York: North Holland
and Amer. Elsevier, 1975. — 16.
22. Schwabik Š., Tvrdý M., Vejvoda O. Differential and integral equations: boundary value problems and adjoint.
— Praha, Dordrecht: Academia and D. Reidel, 1979.
23. Tvrdý M.. Fredholm-Stieltjes integral equations with linear constraints: duality theory and Green’s function
// Čas. pěstov. mat. — 1979. — 104. — P. 357 – 369.
24. Schwabik Š., Tvrdý M. Boundary value problems for generalized linear differential equations // Czech. Math.
J. — 1979. — 29, № 104. — P. 451 – 477.
Received 19.10.07
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
|