Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales
In the paper, the four functionals fixed-point theorem is used to study the existence of positive solutions for nonlinear thirdorder $m$-point impulsive boundary-value problems on time scales. As an application, we give an example demonstrating our results.
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| author | Fen, F. T. Karaca, I. Y. Фен, Ф. Т. Караца, І. Й. |
| author_facet | Fen, F. T. Karaca, I. Y. Фен, Ф. Т. Караца, І. Й. |
| author_sort | Fen, F. T. |
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| description | In the paper, the four functionals fixed-point theorem is used to study the existence of positive solutions for nonlinear thirdorder $m$-point impulsive boundary-value problems on time scales. As an application, we give an example demonstrating
our results. |
| first_indexed | 2026-03-24T02:13:51Z |
| format | Article |
| fulltext |
UDC 517.9
I. Y. Karaca (Ege Univ., Izmir, Turkey),
F. T. Fen (Gazi Univ., Ankara, Turkey)
EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR THIRD-ORDER
\bfitm -POINT IMPULSIVE BOUNDARY VALUE PROBLEMS ON TIME SCALES
IСНУВАННЯ ДОДАТНИХ РОЗВ’ЯЗКIВ НЕЛIНIЙНИХ \bfitm -ТОЧКОВИХ
IМПУЛЬСИВНИХ ГРАНИЧНИХ ЗАДАЧ ТРЕТЬОГO ПОРЯДКУ НА ЧАСОВИХ
МАСШТАБАХ
In the paper, the four functionals fixed-point theorem is used to study the existence of positive solutions for nonlinear third-
order m-point impulsive boundary-value problems on time scales. As an application, we give an example demonstrating
our results.
Теорема про нерухому точку для чотирьох функцiоналiв застосовано для дослiдження задачi iснування додатних
розв’язкiв нелiнiйних m-точкових iмпульсивних граничних задач третьогo порядку на часових масштабах. Як
застосування, наведено приклад, який iлюструє результати, що отриманi в роботi.
1. Introduction. Impulsive differential equations, which arise in physics, chemical technology,
population dynamics, biotechnology, economics and so on (see [3] and references therein), have
become more important in recent years in some mathematical models of real processes. There has
been a significant development in impulsive theory especially in the area of impulsive differential
equations with fixed moments; see the monographs of Bainov and Simeonov [2], Lakshmikantham
et al. [12], Samoilenko and Perestyuk [19] and the references therein.
The theory of time scales was introduced by Stefan Hilger [10] in his PhD thesis in 1988 in order
to unify continuous and discrete analysis. We refer to the books by Bohner and Peterson [5, 6] and
Lakshmikantham et al. [13].
Recently, the existence and multiplicity of positive solutions for linear and nonlinear second-order
impulsive differential equations have been studied extensively. To identify a few, we refer to the
reader to see [8, 9, 11, 16, 20]. However, there is not work on third-order with m-point impulsive
boundary-value problems except that in [17] by Liang and Zhang. On the other hand, there is not
much reported concerning the boundary-value problems for impulsive dynamic equations on time
scales, see [4, 7, 14, 15]. Especially the existence of positive solutions for third-order with m-point
impulsive boundary-value problems on time scales still remains unknown.
In [9], Guo studied the following two-point boundary-value problem:
- x\prime \prime = f(t, x, x\prime ), t \not = tk,
\Delta x| t=tk = Ik
\bigl(
x(tk)
\bigr)
,
\Delta x\prime | t=tk = \=Ik
\bigl(
x(tk)x
\prime (tk)
\bigr)
, k = 1, 2, . . . ,m,
ax(0) - bx\prime (0) = x0, cx(1) + dx\prime (1) = x\ast 0.
By using the Darbo fixed point theorem, Guo obtained the existence criteria of at least one solution.
In [11], Hu, Liu and Wu studied second-order two-point impulsive boundary-value problem
- u\prime \prime = h(t)f(t, u), t \in J \prime ,
c\bigcirc I. Y. KARACA, F. T. FEN, 2016
408 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR THIRD-ORDER m-POINT . . . 409
- \Delta u\prime | t=tk = Ik
\bigl(
u(tk)
\bigr)
,
\Delta u| t=tk = \=Ik
\bigl(
u(tk)
\bigr)
, k = 1, 2, . . . ,m,
\alpha u(0) - \beta u\prime (0) = 0
\gamma u(1) + \delta u\prime (1) = 0.
By using the fixed point theorem in cone, they obtained the existence criteria of one or two positive
solutions.
In [18], Ma considered the existence and multiplicity of positive solutions for the m-boundary-
value problems \bigl(
p(t)u\prime
\bigr) \prime - q(t)u+ f(t, u) = 0, 0 < t < 1,
au(0) - bp(0)u\prime (0) =
m - 2\sum
i=1
\alpha iu(\xi i),
cu(1) + dp(1)u\prime (1) =
m - 2\sum
i=1
\beta iu(\xi i).
The main tool is Guo – Krasnoselskii fixed point theorem.
In [17], Liang and Zhang studied the following third-order impulsive boundary-value problem\bigl(
\varphi ( - u\prime \prime (t))
\bigr) \prime
+ a(t)f
\bigl(
u(t)
\bigr)
= 0, t \not = tk, 0 < t < 1,
\Delta u| t=tk = Ik
\bigl(
u(tk)
\bigr)
, k = 1, 2, . . . , N,
u(0) =
m - 2\sum
i=1
\alpha iu(\xi i),
u\prime (1) = 0, u\prime \prime (0) = 0,
where \varphi : \BbbR \rightarrow \BbbR is the increasing homeomorphism and positive homomorphism with \varphi (0) = 0. By
using the five functionals fixed point theorem, they provided sufficient conditions for the existence
of three positive solutions.
In [15], Li and Li studied the following boundary-value problem for the nonlinear third-order
impulsive dynamic system on time scales
- u\bigtriangleup
3
(t) = f
\bigl(
t, u(t), u\bigtriangleup (t), u\bigtriangleup
2
(t)
\bigr)
, t \in [0, T ]\BbbT \setminus \Omega ,
\Delta u(tk) = Ik, \Delta u\bigtriangleup (tk) = Jk, \Delta u\bigtriangleup
2
(tk) = Lk, k = 1, 2, . . . ,m,
u(0) = \lambda u(\sigma (T )), u\bigtriangleup (0) = \lambda u\bigtriangleup (\sigma (T )), u\bigtriangleup
2
(0) = \lambda u\bigtriangleup
2
(\sigma (T )).
They obtained some sufficient conditions for the existence of solutions by using Schauder’s fixed
point theorem.
Motivated by the above results, in this study, we consider the following third-order impulsive
boundary-value problem (BVP) on time scales:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
410 I. Y. KARACA, F. T. FEN\Bigl(
\phi p
\Bigl(
u\bigtriangleup \bigtriangleup (t)
\Bigr) \Bigr) \bigtriangleup
+ q(t)f(t, u(t), u\bigtriangleup (t)) = 0, t \in J : = [0, 1]\BbbT , t \not = tk, k = 1, 2, . . . , n,
\Delta u (tk) = Ik
\bigl(
u(tk)
\bigr)
,
\Delta u\bigtriangleup (tk) = - Jk
\bigl(
u(tk), u
\bigtriangleup (tk)
\bigr)
,
(1.1)
au(0) - bu\bigtriangleup (0) =
m - 2\sum
i=1
\alpha iu(\xi i),
cu(1) + du\bigtriangleup (1) =
m - 2\sum
i=1
\beta iu(\xi i),
u\bigtriangleup \bigtriangleup (0) = 0,
where \BbbT is a time scale, 0, 1 \in \BbbT , [0, 1]\BbbT = [0, 1] \cap \BbbT , \phi p(s) is a p-Laplacian operator, i.e.,
\phi p(s) = | s| p - 2s for p > 1, (\phi p)
- 1(s) = \phi q(s), where
1
p
+
1
q
= 1, tk \in (0, 1)\BbbT , k = 1, 2, . . . , n with
0 < t1 < t2 < . . . < tn < 1, \Delta u(tk) and \Delta u\bigtriangleup (tk) denote the jump of u(t) and u\bigtriangleup (t) at t = tk, i.e.,
\Delta u(tk) = u(t+k ) - u(t - k ), \Delta u\bigtriangleup (tk) = u\bigtriangleup (t+k ) - u\bigtriangleup (t - k ),
where u(t+k ), u
\bigtriangleup (t+k ) and u(t - k ), u
\bigtriangleup (t - k ) represent the right-hand limit and left-hand limit of u(t)
and u\bigtriangleup (t) at t = tk, k = 1, 2, . . . , n, respectively.
Throughout this paper we assume that following conditions hold:
(C1) a, b, c, d \in [0,\infty ) with ac+ad+bc > 0; \alpha i, \beta i \in [0,\infty ), \xi i \in (0, 1)\BbbT for i \in \{ 1, 2, . . . ,m -
- 2\}
(C2) f \in \scrC ([0, 1]\BbbT \times \BbbR + \times \BbbR ,\BbbR +),
(C3) q \in \scrC ([0, 1]\BbbT ,\BbbR +),
(C4) Ik \in \scrC (\BbbR +,\BbbR +) is a bounded function, Jk \in \scrC (\BbbR + \times \BbbR ,\BbbR +) such that (c(1 - tk) +
+ d)Jk
\bigl(
u(tk), u
\bigtriangleup (tk)
\bigr)
> cIk
\bigl(
u(tk)
\bigr)
, k = 1, 2, . . . , n.
By using the four functional fixed point theorem [1], we get the existence of at least one positive
solution for the impulsive BVP (1.1). In fact, our result is also new when \BbbT = \BbbR (the differential
case) and \BbbT = \BbbZ (the discrete case). Therefore, the result can be considered as a contribution to this
field.
This paper is organized as follows. In Section 2, we provide some definitions and preliminary
lemmas which are key tools for our main result. We give and prove our main result in Section 3.
Finally, in Section 4, we give an example to demonstrate our result.
2. Preliminaries. In this section, we present auxiliary lemmas which will be used later.
Throughout the rest of this paper, we assume that the points of impulse tk are right dense for
each k = 1, 2, . . . , n. Let J = [0, 1]\BbbT , J \prime = J \setminus \{ t1, t2, . . . , tn\} .
Set
PC(J) =
\bigl\{
u : [0, 1]\BbbT \rightarrow \BbbR ;u \in C(J \prime ), u(t+k ) andu(t - k ) \mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}, andu(t - k ) = u(tk), 1 \leq k \leq n
\bigr\}
,
PC1(J) =
\bigl\{
u \in PC(J) : u\Delta \in C(J \prime ), u\Delta (t+k ) andu\Delta (t - k ) \mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}, andu\Delta (t - k ) = u\Delta (tk), 1 \leq k \leq n
\bigr\}
.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR THIRD-ORDER m-POINT . . . 411
Obviously, PC(J) and PC1(J) are Banach spaces with the norms
\| u\| PC = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
| u(t)| , \| u\| PC1 = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
\| u\| PC , \| u\Delta \| PC
\bigr\}
,
respectively. A function u \in PC1(J)\cap C2(J \prime ) is called a solution to (1.1) if it satisfies all equations
of (1.1).
Define the cone \scrP \subset PC1(J) by
\scrP =
\Biggl\{
u \in PC1(J) : u(t) is nonnegative,
nondecreasing on [0, 1]\BbbT andu\bigtriangleup (t) is nonincreasing on [0, 1]\BbbT ,
au(0) - bu\bigtriangleup (0) =
m - 2\sum
i=1
\alpha iu(\xi i)
\Biggr\}
.
Denote by \theta and \varphi , the solutions of the corresponding homogeneous equation\Bigl(
\phi p
\Bigl(
u\bigtriangleup \bigtriangleup (t)
\Bigr) \Bigr) \bigtriangleup
= 0, t \in J : = [0, 1]\BbbT , t \not = tk, k = 1, 2, . . . , n, (2.1)
under the initial conditions
\theta (0) = b, \theta \Delta (0) = a,
\varphi (1) = d, \varphi \Delta (1) = - c.
(2.2)
Using the initial conditions (2.2), we can deduce from equation (2.1) for \theta and \varphi the following
equations:
\theta (t) = b+ at, \varphi (t) = d+ c(1 - t). (2.3)
Set
\Delta :=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
-
m - 2\sum
i=1
\alpha i(b+ a\xi i) \rho -
m - 2\sum
i=1
\alpha i(d+ c(1 - \xi i))
\rho -
m - 2\sum
i=1
\beta i(b+ a\xi i) -
m - 2\sum
i=1
\beta i(d+ c(1 - \xi i))
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
, (2.4)
and
\rho := ad+ ac+ bc. (2.5)
Lemma 2.1. Let (C1) – (C4) hold. Assume that
(C5) \Delta \not = 0.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
412 I. Y. KARACA, F. T. FEN
If u \in PC1(J) \cap C2(J \prime ) is a solution of the equation
u(t) =
1\int
0
G (t, s)\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
n\sum
k=1
Wk(t, tk) +A(f)(b+ at) +B(f)(d+ c(1 - t)), (2.6)
where
Wk(t, tk) =
1
\rho
\left\{ (b+ at)
\bigl(
- cIk(u(tk)) + (d+ c(1 - tk))Jk
\bigl(
u(tk), u
\bigtriangleup (tk)
\bigr) \bigr)
, t < tk,
(d+ c(1 - t))
\bigl(
aIk(u(tk)) + (b+ atk)Jk
\bigl(
u(tk), u
\bigtriangleup (tk)
\bigr) \bigr)
, tk \leq t,
(2.7)
G(t, s) =
1
\rho
\left\{ (b+ a\sigma (s))(d+ c(1 - t)), \sigma (s) \leq t,
(b+ at)(d+ c(1 - \sigma (s))), t \leq s,
(2.8)
A(f) :=
1
\Delta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
m - 2\sum
i=1
\alpha i\scrK i \rho -
m - 2\sum
i=1
\alpha i(d+ c(1 - \xi i))
m - 2\sum
i=1
\beta i\scrK i -
m - 2\sum
i=1
\beta i(d+ c(1 - \xi i))
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
, (2.9)
B(f) :=
1
\Delta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
-
m - 2\sum
i=1
\alpha i(b+ a\xi i)
m - 2\sum
i=1
\alpha i\scrK i
\rho -
m - 2\sum
i=1
\beta i(b+ a\xi i)
m - 2\sum
i=1
\beta i\scrK i
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
, (2.10)
and
\scrK i :=
1\int
0
G(\xi i, s)\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
n\sum
k=1
Wk (\xi i, tk) , (2.11)
then u is a solution of the impulsive BVP (1.1).
Proof. Let u satisfies the integral equation (2.6), then u is a solution of the impulsive BVP (1.1).
Then we have
u(t) =
1\int
0
G(t, s)\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
n\sum
k=1
Wk(t, tk) +A(f)(b+ at) +B(f)(d+ c(1 - t)),
i.e.,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR THIRD-ORDER m-POINT . . . 413
u(t) =
t\int
0
1
\rho
(b+ a\sigma (s))(d+ c(1 - t))\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
1\int
t
1
\rho
(b+ at)(d+ c(1 - \sigma (s)))\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
\sum
0<tk<t
(d+ c(1 - t))
\Bigl(
aIk(u(tk)) + (b+ atk)Jk
\Bigl(
u(tk), u
\bigtriangleup (tk)
\Bigr) \Bigr)
+
+
\sum
t<tk<1
(b+ at)
\Bigl(
- cIk(u(tk)) + (d+ c(1 - tk))Jk
\Bigl(
u(tk), u
\bigtriangleup (tk)
\Bigr) \Bigr)
+
+A(f)(b+ at) +B(f)(d+ c(1 - t)),
u\bigtriangleup (t) = -
t\int
0
c
\rho
(b+ a\sigma (s))\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau , u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
1\int
t
a
\rho
(d+ c(1 - \sigma (s)))\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s -
-
\sum
0<tk<t
c
\Bigl(
aIk(u(tk)) + (b+ atk)Jk
\Bigl(
u(tk), u
\bigtriangleup (tk)
\Bigr) \Bigr)
+
+
\sum
t<tk<1
a
\Bigl(
- cIk(u(tk)) + (d+ c(1 - tk))Jk
\Bigl(
u(tk), u
\bigtriangleup (tk)
\Bigr) \Bigr)
+A(f)a - B(f)c,
u\bigtriangleup \bigtriangleup (t) =
1
\rho
( - c (b+ a\sigma (t)) - a(d+ c(1 - \sigma (t))))\phi q
\left( t\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) -
- \phi q
\left( t\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) ,
u\bigtriangleup \bigtriangleup (0) = 0.
So that
\Bigl(
\phi p
\Bigl(
u\bigtriangleup \bigtriangleup (t)
\Bigr) \Bigr) \bigtriangleup
=
\left( -
t\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup
= - q(t)f(t, u(t), u\bigtriangleup (t)),
\Bigl(
\phi p
\Bigl(
u\bigtriangleup \bigtriangleup (t)
\Bigr) \Bigr) \bigtriangleup
+ q(t)f(t, u(t), u\bigtriangleup (t)) = 0.
Since
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
414 I. Y. KARACA, F. T. FEN
u(0) =
1\int
0
b
\rho
(d+ c(1 - \sigma (s)))\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
n\sum
k=1
b
\Bigl(
- cIk(u(tk)) + (d+ c(1 - tk))Jk
\Bigl(
u(tk), u
\bigtriangleup (tk)
\Bigr) \Bigr)
+A(f)b+B(f)(d+ c),
u\bigtriangleup (0) =
1\int
0
a
\rho
(d+ c(1 - \sigma (s)))\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
n\sum
k=1
a
\Bigl(
- cIk(u(tk)) + (d+ c(1 - tk))Jk
\Bigl(
u(tk), u
\bigtriangleup (tk)
\Bigr) \Bigr)
+A(f)a - B(f)c,
we have that
au(0) - bu\bigtriangleup (0) =
= B(f)(ad+ ac+ bc) =
m - 2\sum
i=1
\alpha i
\left[ 1\int
0
G(\xi i, s)\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
n\sum
k=1
Wk (\xi i, tk) +A(f)(b+ a\xi i) +B(f)(d+ c(1 - \xi i))
\Biggr]
. (2.12)
Since
u(1) =
1\int
0
d
\rho
(b+ a\sigma (s))\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau , u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
n\sum
k=1
d
\Bigl(
aIk(u(tk)) + (b+ atk)Jk
\Bigl(
u(tk), u
\bigtriangleup (tk)
\Bigr) \Bigr)
+A(f)(b+ a) +B(f)d,
u\bigtriangleup (1) = -
1\int
0
c
\rho
(b+ a\sigma (s))\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau , u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
n\sum
k=1
- c
\Bigl(
aIk(u(tk)) + (b+ atk)Jk
\Bigl(
u(tk), u
\bigtriangleup (tk)
\Bigr) \Bigr)
+A(f)a - B(f)c,
we have that
cu(1) + du\bigtriangleup (1) = A(f)(ad+ ac+ bc) =
=
m - 2\sum
i=1
\beta i
\left[ 1\int
0
G (\xi i, s)\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
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EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR THIRD-ORDER m-POINT . . . 415
+
n\sum
k=1
Wk(\xi i, tk) +A(f)(b+ a\xi i) +B(f)(d+ c(1 - \xi i))
\right] . (2.13)
From (2.5), (2.12) and (2.13), we get that\Biggl[
-
m - 2\sum
i=1
\alpha i(b+ a\xi i)
\Biggr]
A(f) +
\Biggl[
\rho -
m - 2\sum
i=1
\alpha i(d+ c(1 - \xi i))
\Biggr]
B(f) =
m - 2\sum
i=1
\alpha i\scrK i,
\Biggl[
\rho -
m - 2\sum
i=1
\beta i(b+ a\xi i)
\Biggr]
A(f) +
\Biggl[
-
m - 2\sum
i=1
\beta i(d+ c(1 - \xi i))
\Biggr]
B(f) =
m - 2\sum
i=1
\beta i\scrK i,
which implies that A(f) and B(f) satisfy (2.9) and (2.10), respectively.
Lemma 2.1 is proved.
Lemma 2.2. Let (C1) – (C4) hold. Assume
(C6) \Delta < 0, \rho -
\sum m - 2
i=1
\beta i(b+ a\xi i) > 0, a -
\sum m - 2
i=1
\alpha i > 0.
Then for u \in PC1(J) \cap C2(J \prime ) with f, q \geq 0, the solution u of the problem (1.1) satisfies
u(t) \geq 0 for t \in [0, 1]\BbbT .
Proof. It is an immediate subsequence of the facts that G \geq 0 on [0, 1]\BbbT \times [0, 1]\BbbT and A(f) \geq 0,
B(f) \geq 0.
Lemma 2.3. Let (C1) – (C4) and (C6) hold. Assume
(C7) c -
\sum m - 2
i=1
\beta i < 0.
Then the solution u \in PC1(J) \cap C2(J \prime )of the problem (1.1) satisfies u\bigtriangleup (t) \geq 0 for t \in [0, 1]\BbbT .
Proof. Assume that the inequality u\bigtriangleup (t) < 0 holds. Since u\bigtriangleup (t) is nonincreasing on [0, 1]\BbbT ,
one can verify that
u\bigtriangleup (1) \leq u\bigtriangleup (t), t \in [0, 1]\BbbT .
From the boundary conditions of the problem (1.1), we have
- c
d
u(1) +
1
d
m - 2\sum
i=1
\beta iu(\xi i) \leq u\bigtriangleup (t) < 0.
The last inequality yields
- cu(1) +
m - 2\sum
i=1
\beta iu(\xi i) < 0.
Therefore, we obtain that
m - 2\sum
i=1
\beta iu(1) <
m - 2\sum
i=1
\beta iu(\xi i) < cu(1),
i.e.,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
416 I. Y. KARACA, F. T. FEN\Biggl(
c -
m - 2\sum
i=1
\beta i
\Biggr)
u(1) > 0.
According to Lemma (2.2), we have that u(1) \geq 0. So, c -
\sum m - 2
i=1
\beta i > 0. However, this contradicts
to condition (C7). Consequently, u\bigtriangleup (t) \geq 0 for t \in [0, 1]\BbbT .
Lemma 2.3 is proved.
Lemma 2.4. If (C1) – (C7) hold, then \mathrm{m}\mathrm{a}\mathrm{x}t\in [0,1]\BbbT u(t) \leq M \mathrm{m}\mathrm{a}\mathrm{x}t\in [0,1]\BbbT u
\Delta (t) for u \in \scrP , where
M = 1 +
b+
\sum m - 2
i=1
\alpha i\xi i
a -
\sum m - 2
i=1
\alpha i
. (2.14)
Proof. For u \in \scrP , since u\bigtriangleup (t) is nonincreasing on [0, 1]\BbbT one arrives at
u(\xi i) - u(0)
\xi i
\leq u\bigtriangleup (0),
i.e., u(\xi i) - u(0) \leq \xi iu
\bigtriangleup (0). Hence,
m - 2\sum
i=1
\alpha iu(\xi i) -
m - 2\sum
i=1
\alpha iu(0) \leq
m - 2\sum
i=1
\alpha i\xi iu
\bigtriangleup (0).
By au(0) - bu\bigtriangleup (0) =
\sum m - 2
i=1
\alpha iu(\xi i), we get
u(0) \leq
b+
\sum m - 2
i=1
\alpha i\xi i
a -
\sum m - 2
i=1
\alpha i
u\bigtriangleup (0).
Hence
u(t) =
t\int
0
u\bigtriangleup (s)\bigtriangleup s+ u(0) \leq tu\bigtriangleup (0) + u(0) \leq
\leq tu\bigtriangleup (0) +
b+
\sum m - 2
i=1
\alpha i\xi i
a -
\sum m - 2
i=1
\alpha i
u\bigtriangleup (0) \leq
\left( 1 +
b+
\sum m - 2
i=1
\alpha i\xi i
a -
\sum m - 2
i=1
\alpha i
\right) u\bigtriangleup (0) = Mu\bigtriangleup (0),
i.e.,
\| u\| PC = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u(t) \leq Mu\bigtriangleup (0) \leq M \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u\bigtriangleup (t).
Lemma 2.4 is proved.
From Lemma (2.4), we obtain
\| u\| PC1 = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
\| u\| PC , \| u\Delta \| PC
\bigr\}
= \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
| u(t)| , \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
| u\Delta (t)|
\biggr\}
\leq
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EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR THIRD-ORDER m-POINT . . . 417
\leq \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
M \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u\bigtriangleup (t), \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u\Delta (t)
\biggr\}
= M \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u\bigtriangleup (t).
Now define an operator T : \scrP - \rightarrow PC1(J) by
Tu(t) =
1\int
0
G(t, s)\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
n\sum
k=1
Wk (t, tk)+
+A(f)(b+ at) +B(f)(d+ c(1 - t)), (2.15)
where Wk, G, A(f), B(f) and \theta , \varphi are defined as in (2.7), (2.8), (2.9), (2.10) and (2.3) respectively.
Lemma 2.5. Let (C1) – (C7) hold. Then T : \scrP \rightarrow \scrP is completely continuous.
Proof. By Arzela – Ascoli theorem, we can easily prove that operator T is completely continuous.
3. Main Results. We are now ready to apply the four functionals fixed point theorem [1] to the
operator T in order to get sufficient conditions for the existence of at least one positive solution to
the problem (1.1).
Let \alpha and \Psi be nonnegative continuous concave functionals on \scrP , and let \beta and \Phi be nonnegative
continuous convex functionals on \scrP , then for positive numbers r, j, l and R, we define the sets:
Q(\alpha , \beta , r, R) = \{ u \in \scrP : r \leq \alpha (u), \beta (u) \leq R\} ,
U(\Psi , j) = \{ u \in Q(\alpha , \beta , r, R) : j \leq \Psi (u)\} , (3.1)
V (\Phi , l) = \{ u \in Q(\alpha , \beta , r, R) : \Phi (u) \leq l\} .
Lemma 3.1 [1]. If \scrP is a cone in a real Banach space \BbbB , \alpha and \Psi are nonnegative continuous
concave functionals on \scrP , \beta and \Phi are nonnegative continuous convex functionals on \scrP and there
exist positive numbers r, j, l and R, such that
T : Q(\alpha , \beta , r, R) \rightarrow \scrP
is a completely continuous operator, and Q(\alpha , \beta , r, R) is a bounded set. If
(i) \{ u \in U(\Psi , j) : \beta (u) < R\} \cap \{ u \in V (\Phi , l) : r < \alpha (u)\} \not = \varnothing ;
(ii) \alpha (Tu) \geq r, for all u \in Q(\alpha , \beta , r, R), with \alpha (u) = r and l < \Phi (Tu);
(iii) \alpha (Tu) \geq r, for all u \in V (\Phi , l), with \alpha (u) = r;
(iv) \beta (Tu) \leq R, for all u \in Q(\alpha , \beta , r, R), with \beta (u) = R and \Psi (Tu) < j;
(v) \beta (Tu) \leq R, for all u \in U(\Psi , j), with \beta (u) = R.
Then T has a fixed point u in Q(\alpha , \beta , r, R).
Suppose \omega , z \in \BbbT with 0 < \omega < z < 1. For the convenience, we take the notations
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
418 I. Y. KARACA, F. T. FEN
A :=
1
\Delta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
m - 2\sum
i=1
\alpha i
\left( 1\int
0
G (\xi i, s)\phi q\times
\times
\left( s\int
0
q(\tau )\bigtriangleup \tau
\right) \bigtriangleup s+
n
\rho
(c+ d)(2a+ b)
\right) \rho -
m - 2\sum
i=1
\alpha i(d+ c(1 - \xi i))
m - 2\sum
i=1
\beta i
\left( 1\int
0
G (\xi i, s)\phi q\times
\times
\left( s\int
0
q(\tau )\bigtriangleup \tau
\right) \bigtriangleup s+
n
\rho
(c+ d)(2a+ b)
\right) -
m - 2\sum
i=1
\beta i(d+ c(1 - \xi i))
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\Omega =
z\int
w
G(\omega , s)\phi q
\left( z\int
w
q(\tau )\bigtriangleup \tau
\right) \bigtriangleup s,
\Lambda =
1\int
0
1
\rho
(c(1 - \sigma (s)) + d)\phi q
\left( 1\int
0
q(\tau )\bigtriangleup \tau
\right) \bigtriangleup s+ na(c+ d) +Aa,
L =
a -
\sum m - 2
i=1
\alpha i
b+
\sum m - 2
i=1
\alpha i\xi i
,
and define the maps
\alpha (u) = \mathrm{m}\mathrm{i}\mathrm{n}
t\in [\omega ,z]\BbbT
u(t), \Phi (u) = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u(t), \beta (u) = \Psi (u) = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u\Delta (t). (3.2)
and let Q(\alpha , \beta , r, R), U(\Psi , j) and V (\Phi , l) be defined by (3.1).
Theorem 3.1. Assume (C1) – (C7) hold. If there exist constants r, j, l, R with \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ r
\omega
,R
\Bigr\}
\leq l,
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
L+ 1
L
j,
L+ 1
L\omega + 1
r
\biggr\}
< R and suppose that f satisfies the following conditions:
(C8) f(t, u, u\bigtriangleup ) \geq \phi p
\Bigl( r
\Omega
\Bigr)
for (t, u, u\bigtriangleup ) \in [\omega , z]\BbbT \times [r, l]\times [0, R];
(C9) f(t, u, u\bigtriangleup ) \leq \phi p
\biggl(
R
\Lambda
\biggr)
, Ik
\bigl(
u(tk)
\bigr)
\leq R
\Lambda
, Jk
\bigl(
u(tk), u
\bigtriangleup (tk)
\bigr)
\leq R
\Lambda
for (t, u, u\bigtriangleup ) \in [0, 1]\BbbT \times
\times [0,MR]\times [0, R].
Then the BVP (1.1) has at least one positive solution u \in \scrP such that
\mathrm{m}\mathrm{i}\mathrm{n}
t\in [\omega ,z]\BbbT
u(t) \geq r, \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u(t) \leq R.
Proof. The impulsive BVP (1.1) has a solution u = u(t) if and only if u solves the operator
equation u = Tu. Thus we set out to verify that the operator T satisfies four functionals fixed point
theorem which will prove the existence of a fixed point of T.
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EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR THIRD-ORDER m-POINT . . . 419
We first show that Q(\alpha , \beta , r, R) is bounded and T : Q(\alpha , \beta , r, R) \rightarrow \scrP is completely continuous.
For all u \in Q(\alpha , \beta , r, R) with Lemma 2.4, we have
\| u\| PC1 \leq M \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u\Delta (t) = M\beta (u) \leq MR,
which means that Q(\alpha , \beta , r, R) is a bounded set. According to Lemma (2.5), it is clear that T :
Q(\alpha , \beta , r, R) \rightarrow \scrP is completely continuous.
Let
u0 =
R
L+ 1
(Lt+ 1).
Clearly, u0 \in \scrP . By direct calculation,
\alpha (u0) = u0(\omega ) =
R
L+ 1
(L\omega + 1) > r,
\beta (u0) =
R
L+ 1
L < R,
\Psi (u0) = \beta (u0) =
R
L+ 1
L \geq j,
\Phi (u0) = u0(1) =
R
L+ 1
(L+ 1) = R \leq l.
So, u0 \in
\bigl\{
u \in U(\Psi , j) : \beta (u) < R\} \cap \{ u \in V (\Phi , l) : r < \alpha (u)
\bigr\}
, which means that (i) in
Lemma (3.1) is satisfied.
For all u \in Q(\alpha , \beta , r, R), with \alpha (u) = r and l < \Phi (Tu), since u\bigtriangleup is nonincreasing on [0, 1]\BbbT
we have
\alpha (Tu) = Tu(\omega ) \geq \omega Tu(1) = \omega \Phi (Tu) > \omega l \geq r.
So, \alpha (Tu) > r. Hence (ii) in Lemma 3.1 is fulfilled.
For all u \in V (\Phi , l), with \alpha (u) = r,
\alpha (Tu) = \mathrm{m}\mathrm{i}\mathrm{n}
t\in [\omega ,z]\BbbT
Tu(t) = (Tu)(\omega ) =
=
1\int
0
G (\omega , s)\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
n\sum
k=1
Wk (\omega , tk) +A(f)\theta (\omega ) +B(f)\varphi (\omega ) \geq
\geq
1\int
0
G (\omega , s)\phi q
\left( s\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s \geq
\geq
z\int
\omega
G (\omega , s)\phi q
\left( s\int
\omega
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s \geq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
420 I. Y. KARACA, F. T. FEN
\geq r
\Omega
z\int
\omega
G (\omega , s)\phi q
\left( s\int
\omega
q(\tau )\bigtriangleup \tau
\right) \bigtriangleup s = r,
and for all u \in U(\Psi , j), with \beta (u) = R,
\beta (Tu) = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
(Tu)\Delta (t) = (Tu)\Delta (0) \leq
\leq
1\int
0
1
\rho
a (c(1 - \sigma (s)) + d)\phi q
\left( 1\int
0
q(\tau )f
\Bigl(
\tau , u(\tau ), u\bigtriangleup (\tau )
\Bigr)
\bigtriangleup \tau
\right) \bigtriangleup s+
+
n\sum
k=1
a
\Bigl(
- cIk(u(tk)) + (c(1 - tk) + d) Jk(u(tk), u
\bigtriangleup (tk))
\Bigr)
+A(f)a \leq
\leq R
\Lambda
1\int
0
1
\rho
a (c(1 - \sigma (s)) + d)\phi q
\left( 1\int
0
q(\tau )\bigtriangleup \tau
\right) \bigtriangleup s+ na(c+ d)
R
\Lambda
+ a
R
\Lambda
A =
=
R
\Lambda
\left( 1\int
0
1
\rho
a (c(1 - \sigma (s)) + d)\phi q
\left( 1\int
0
q(\tau )\bigtriangleup \tau
\right) \bigtriangleup s+ na(c+ d) + aA
\right) = R.
Thus (iii) and (v) in Lemma 3.1 hold. We finally prove that (iv) in Lemma 3.1 holds.
For all u \in Q(\alpha , \beta , r, R), with \beta (u) = R and \Psi (Tu) < j, we have
\beta (Tu) = \Psi (Tu) < j <
L
L+ 1
R < R.
Thus, all conditions of Lemma 3.1 are satisfied. T has a fixed point u in Q(\alpha , \beta , r, R). Therefore,
the BVP (1.1) has at least one positive solution u \in \scrP such that
\mathrm{m}\mathrm{i}\mathrm{n}
t\in [\omega ,z]\BbbT
u(t) \geq r, \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]\BbbT
u(t) \leq R.
Theorem 3.1 is proved.
4. An example.
Example 4.1. In BVP (1.1), suppose that \BbbT = [0, 1], p = 2, m = 3, n = 1, q(t) = 1,
a = b = c = d = 1, \xi 1 =
1
4
, \alpha 1 =
1
10
, \beta 1 = 2, t1 =
1
2
, i.e.,
\Bigl(
u\bigtriangleup \bigtriangleup (t)
\Bigr) \bigtriangleup
+ f(t, u(t), u\bigtriangleup (t)) = 0, t \in [0, 1], t \not = 1
2
,
\Delta u
\biggl(
1
2
\biggr)
= I1
\biggl(
u
\biggl(
1
2
\biggr) \biggr)
,
(4.1)
\Delta u\bigtriangleup
\biggl(
1
2
\biggr)
= - J1
\biggl(
u
\biggl(
1
2
\biggr)
, u\bigtriangleup
\biggl(
1
2
\biggr) \biggr)
,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR THIRD-ORDER m-POINT . . . 421
u(0) - u\bigtriangleup (0) =
1
10
u
\biggl(
1
4
\biggr)
,
u(1) + u\bigtriangleup (1) = 2u
\biggl(
1
4
\biggr)
,
u\bigtriangleup \bigtriangleup (0) = 0,
where
f(t, u, u\bigtriangleup ) =
\left\{
0,09, u \in
\biggl[
0,
1
100
\biggr]
,
11
1799
u+
809
8995
, u \geq 1
100
,
I1(u) =
1
45
u, u \geq 0,
J1(u, u
\bigtriangleup ) =
3
180
u, (u, u\bigtriangleup ) \in [0,\infty )\times [0,\infty ).
Set \omega =
1
5
, z =
1
3
, by simple calculation we get
A = 14,64957265, \Omega = 0,03375593836, \Lambda = 17,14957265, L =
36
41
, M =
77
36
,
and
G(t, s) =
1
3
\left\{ (1 + \sigma (s))(2 - t), \sigma (s) \leq t,
(1 + t)(2 - \sigma (s)), t \leq s.
Choose r =
1
100
, l = 10, j = 2 and R = 8, it is easy to check that \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
1
20
, 8
\biggr\}
\leq 10,
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
154
36
,
77
4820
\biggr\}
< 8,
f(t, u(t), u\bigtriangleup (t)) = 0,09 \geq \phi p
\Bigl( r
\Omega
\Bigr)
= 0,08776060928
for (t, u(t), u\bigtriangleup (t)) \in
\biggl[
1
5
,
1
3
\biggr] \biggl[
1
100
, 10
\biggr]
\times [0, 8] ;
f(t, u(t), u\bigtriangleup (t)) \leq 0,2 \leq \phi p
\biggl(
R
\Lambda
\biggr)
= 0,2176072544,
I1
\biggl(
u
\biggl(
1
2
\biggr) \biggr)
= 0,3802469136 \leq 0,4532687651 =
R
\Lambda
,
J1
\biggl(
u
\biggl(
1
2
\biggr)
, u\bigtriangleup
\biggl(
1
2
\biggr) \biggr)
=
= 0,2851851852 \leq 0,4532687651 =
R
\Lambda
for(t, u(t), u\bigtriangleup (t)) \in [0, 1]\times
\biggl[
0,
154
9
\biggr]
\times [2, 8].
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
422 I. Y. KARACA, F. T. FEN
So, all conditions of Theorem 3.1 hold. Thus by Theorem 3.1, the BVP (4.1) has at least one positive
solution u such that
\mathrm{m}\mathrm{i}\mathrm{n}
t\in [ 15 ,
1
3 ]
u(t) \geq 1
100
, \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
u(t) \leq 8.
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Received 24.10.13
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
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| id | umjimathkievua-article-1848 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:13:51Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d7/628b06a00401ee6387b3600933344cd7.pdf |
| spelling | umjimathkievua-article-18482019-12-05T09:29:34Z Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales Існування додатних розв’язкiв нелiнiйних $m$ -точкових iмпульсивних граничних задач третьогo порядку на часових масштабах Fen, F. T. Karaca, I. Y. Фен, Ф. Т. Караца, І. Й. In the paper, the four functionals fixed-point theorem is used to study the existence of positive solutions for nonlinear thirdorder $m$-point impulsive boundary-value problems on time scales. As an application, we give an example demonstrating our results. Теорема про нерухому точку для чотирьох функцiоналiв застосовано для дослiдження задачi iснування додатних розв’язкiв нелiнiйних $m$-точкових iмпульсивних граничних задач третьогo порядку на часових масштабах. Як застосування, наведено приклад, який iлюструє результати, що отриманi в роботi. Institute of Mathematics, NAS of Ukraine 2016-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1848 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 3 (2016); 408-422 Український математичний журнал; Том 68 № 3 (2016); 408-422 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1848/830 Copyright (c) 2016 Fen F. T.; Karaca I. Y. |
| spellingShingle | Fen, F. T. Karaca, I. Y. Фен, Ф. Т. Караца, І. Й. Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales |
| title | Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales |
| title_alt | Існування додатних розв’язкiв нелiнiйних $m$ -точкових iмпульсивних граничних задач третьогo порядку на часових масштабах |
| title_full | Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales |
| title_fullStr | Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales |
| title_full_unstemmed | Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales |
| title_short | Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales |
| title_sort | existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1848 |
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