Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales
UDC 517.9 We consider a three-point boundary-value problem for p-Laplacian dynamic equation on time scales. We show the existence at least three positive solutions of the boundary-value problem by using the Avery and Peterson fixed point theorem. The conditions we used here differ from those in the...
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507072685146112 |
|---|---|
| author | Dogan, A. Dogan, A. |
| author_facet | Dogan, A. Dogan, A. |
| author_sort | Dogan, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:46Z |
| description | UDC 517.9
We consider a three-point boundary-value problem for p-Laplacian dynamic equation on time scales. We show the existence at least three positive solutions of the boundary-value problem by using the Avery and Peterson fixed point theorem. The conditions we used here differ from those in the majority of papers as we know. The interesting point is that the nonlinear term $ f$ involves the first derivative of the unknown function. As an application, an example is given to illustrate our results. |
| doi_str_mv | 10.37863/umzh.v72i6.646 |
| first_indexed | 2026-03-24T02:03:30Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i6.646
UDC 517.9
A. Dogan (Dep. Appl. Math., Abdullah Gul Univ., Kayseri, Turkey)
POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY-VALUE PROBLEM
FOR \bfitp -LAPLACIAN DYNAMIC EQUATION ON TIME SCALES
ДОДАТНI РОЗВ’ЯЗКИ ТРИТОЧКОВОЇ КРАЙОВОЇ ЗАДАЧI
ДЛЯ ДИНАМIЧНОГО РIВНЯННЯ IЗ \bfitp -ЛАПЛАСIАНОМ
НА ЧАСОВИХ ШКАЛАХ
We consider a three-point boundary-value problem for p-Laplacian dynamic equation on time scales. We show the existence
at least three positive solutions of the boundary-value problem by using the Avery and Peterson fixed point theorem. The
conditions we used here differ from those in the majority of papers as we know. The interesting point is that the nonlinear
term f involves the first derivative of the unknown function. As an application, an example is given to illustrate our results.
Розглядається триточкова крайова задача для динамiчного рiвняння iз p-лапласiаном на часових шкалах. За допо-
могою теореми Ейвери та Петерсона про нерухому точку доведено iснування принаймнi трьох додатних розв’язкiв
такої крайової задачi. Умови, якi використовуються тут, вiдрiзняються вiд умов, якi використано у бiльшостi вiдомих
нам робiт. Цiкавим моментом є те, що нелiнiйний член f мiстить першу похiдну невiдомої функцiї. Як застосування
наведено приклад для iлюстрацiї отриманих результатiв.
1. Introduction. This paper is concerned with the existence of positive solutions of the p-Laplacian
dynamic equation on time scales
\bigl(
\phi p(u
\Delta (t))
\bigr) \nabla
+ g(t)f
\bigl(
t, u(t), u\Delta (t)
\bigr)
= 0, t \in [0, T ]\BbbT , (1.1)
u(0) - B0
\bigl(
u\Delta (\nu )
\bigr)
= 0, u\Delta (T ) = 0, (1.2)
or
u\Delta (0) = 0, u(T ) +B1
\bigl(
u\Delta (\nu )
\bigr)
= 0, (1.3)
where \phi p(s) is p-Laplacian operator, i.e., \phi p(s) = | s| p - 2s for p > 1, with (\phi p)
- 1 = \phi q and
1/p+ 1/q = 1, \nu \in (0, \rho (T ))\BbbT . Some basic knowledge and definitions about time scales, which can
be found in [7, 8]. As we know, when the nonlinear term f is involved in the first-order derivative,
difficulties arise immediately. In this work, we use a fixed point theorem because of Avery and
Peterson to overcome the difficulties.
Throughout the paper, we will suppose that the following conditions are satisfied:
(H1) \BbbT is a time scales with 0, T \in \BbbT , \nu \in (0, \rho (T ))\BbbT ;
(H2) let \zeta \geq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
t \in \BbbT : t \geq T
2
\biggr\}
, and there exists \tau \in \BbbT such that \zeta < \tau < T holds;
(H3) f : [0, T ]\BbbT \times \BbbR + \times \BbbR \rightarrow \BbbR + is continuous, and does not vanish identically on any closed
subinterval of [0, T ]\BbbT ;
(H4) g : \BbbT \rightarrow \BbbR + is left dense continuous
\bigl(
i.e., g \in Cld(\BbbT ,\BbbR +)
\bigr)
, and does not vanish identically
on any closed subinterval of [0, T ]\BbbT ;
c\bigcirc A. DOGAN, 2020
790 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY-VALUE PROBLEM FOR p-LAPLACIAN . . . 791
(H5) B0(\upsilon ) and B1(\upsilon ) are both continuous odd functions defined on \BbbR and satisfy that there
exist A,B > 0 such that
B\upsilon \leq Bj(\upsilon ) \leq A\upsilon , \upsilon \geq 0 j = 0, 1.
In [3], Anderson established the existence of multiple positive solutions to the nonlinear second-
order three-point boundary-value problem (BVP) on time scale \BbbT given by
u\Delta \nabla (t) + f(t, u(t)) = 0, t \in (0, T ) \subset \BbbT ,
u(0) = 0, au(\eta ) = u(T ).
He employed the Leggett – Williams fixed point theorem in an appropriate cone to guarantee the
existence of at least three positive solutions to this nonlinear problem.
Anderson et al. [4] studied the time scale, delta-nabla dynamic equation
(g(u\Delta ))\nabla + c(t)f(u) = 0 for a < t < b
with boundary conditions
u(a) - B0
\bigl(
u\Delta (\nu )
\bigr)
= 0 and u\Delta (b) = 0.
They established the existence result of at least one positive solution by a fixed point theorem of
cone expansion and compression of functional type.
In [9], Dogan investigated the following p-Laplacian dynamic equation on time scales:
(\phi p(u
\Delta (t)))\nabla + a(t)f
\bigl(
t, u(t), u\Delta (t)
\bigr)
= 0, t \in [0, T ]\BbbT ,
u(0) - B0(u
\Delta (0)) = 0, u\Delta (T ) = 0,
where \phi p(u) = | u| p - 2u for p > 1. We proved the existence of triple positive solutions for the
one-dimensional p-Laplacian BVP by using the Leggett – Williams fixed point theorem.
In [10], Dogan studied the existence of positive solutions of the p-Laplacian dynamic equation
on time scales \bigl(
\phi p(y
\Delta (t))
\bigr) \nabla
= - w(t)f
\bigl(
t, y(t), y\Delta (t)
\bigr)
, t \in [0, T ]\BbbT ,
y(0) - B0(y
\Delta (\nu )) = 0, y\Delta (T ) = 0,
or
y\Delta (0) = 0, y(T ) +B1(y
\Delta (\nu )) = 0.
We proved the existence at least three positive solutions of the BVP by using the Avery and Peterson
fixed point theorem.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
792 A. DOGAN
In [11], Guo considered the following one-dimensional p-Laplacian three-point BVP on time
scales \bigl(
\varphi p(u
\Delta (t))
\bigr) \nabla
+ h(t)f
\bigl(
t, u(t), u\Delta (t)
\bigr)
= 0, t \in (0, T )\BbbT ,
u(0) - \beta u\Delta (0) = \gamma u\Delta (\eta ), u\Delta (T ) = 0.
He established existence criteria for at least three positive solutions by using a fixed point theorem
for operators on a cone.
In [12], He investigated the existence of positive solutions of the p-Laplacian dynamic equation
on a time scale \bigl[
\phi p(u
\Delta (t))
\bigr] \nabla
+ a(t)f(u(t)) = 0, t \in [0, T ]\BbbT ,
satisfying the boundary conditions
u(0) - B0
\bigl(
u\Delta (\nu )
\bigr)
= 0, u\Delta (T ) = 0,
or
u\Delta (0) = 0, u(T ) +B1
\bigl(
u\Delta (\nu )
\bigr)
= 0,
where \phi p(s) is p-Laplacian operator, i.e., \phi p(s) = | s| p - 2s, p > 1, (\phi p)
- 1 = \phi q, 1/p + 1/q = 1,
\nu \in (0, \rho (T ))\BbbT . By using a new double fixed point theorem due to Avery et al. [5] in a cone, he
proved that there exists at least double positive solutions of BVP.
In [19], Sun and Li studied the one-dimensional p-Laplacian BVP on time scales\bigl(
\varphi p(u
\Delta (t))
\bigr) \Delta
+ h(t)f(u\sigma (t)) = 0, t \in [a, b],
u(a) - B0(u
\Delta (a)) = 0, u\Delta (\sigma (b)) = 0,
where \varphi p(u) is p-Laplacian operator, i.e., \varphi p(u) = | u| p - 2u, p > 1. They found some new results
for the existence of at least single, twin or triple positive solutions of the above problem by using
Krasnosel’skii’s fixed point theorem, new fixed point theorem because of Avery and Henderson and
Leggett – Williams fixed point theorem.
Sun et al. [20] considered the eigenvalue problem for the following one-dimensional p-Laplacian
three-point BVP on time scales\bigl(
\varphi p(u
\Delta (t))
\bigr) \nabla
+ \lambda h(t)f(u(t)) = 0, t \in (0, T )\BbbT ,
u(0) - \beta u\Delta (0) = \gamma u\Delta (\eta ), u\Delta (T ) = 0.
They established some sufficient conditions for the nonexistence and existence of at least one or two
positive solutions for the BVP.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY-VALUE PROBLEM FOR p-LAPLACIAN . . . 793
In [23], Wang considered the existence of three positive solutions to the following BVPs for
p-Laplacian dynamic equations on time scales:\bigl[
\phi p(u
\Delta (t))
\bigr] \nabla
+ a(t)f(u(t)) = 0, t \in [0, T ]\BbbT ,
u\Delta (0) = 0, u(T ) +B1(u
\Delta (\eta )) = 0,
or
u(0) - B0(u
\Delta (\eta )) = 0, u\Delta (T ) = 0.
He established the existence result for at least three positive solutions by using the Leggett – Williams
fixed point theorem.
In recent years, there has been much current attention focused on study of positive solutions
of BVPs on time scales. When the nonlinear term f does not depend on the first-order derivative,
nonlinear BVPs on time scales have been studied extensively in the literature (see [1, 3, 4, 12 – 23]).
However, there are few papers dealing with the existence of positive solutions for BVPs on time
scales when the nonlinear term f is involved in the first-order derivative explicitly (see [9, 11]).
Compared with [9] and [11], in this paper, we remark that our boundary conditions are entirely
different from those used in [9, 11]. Dogan [9] studied the existence of positive solutions of a
two-point BVP on time scales by using Leggett – Williams fixed point theorem. Here we study the
existence of positive solutions of a three-point BVP on time scales by using Avery and Peterson fixed
point theorem. Guo [11] studied the existence of positive solutions for p-Laplacian three-point BVPs
on time scales by using the Avery and Peterson fixed point theorem. His method was the same as
ours. But the assumptions we used in the paper are different from those in [11]. We have defined
that Banach space E and the cone P are different from [11]. Guo [11] took t \in [0, T ]. We have
taken t \in [0, \sigma (T )] instead of t \in [0, T ].
Compared with [10], in this paper, we are concerned with same problem. So the papers all look
the same and both papers seem to achieve similar results. But here we have replaced \nu \in (0, T )\BbbT
with \nu \in (0, \rho (T ))\BbbT . We have also replaced t \in [0, T ]\BbbT with t \in [0, \sigma (T )]\BbbT . Therefore, Lemmas 2.1
and 2.2 and their proofs are different from Lemmas 2.2 and 2.3 in [10]. We define that the cones P
and P1 are different from the published paper [10]. Moreover, example is slightly different from the
published paper [10].
Motivated by works mentioned above, in this paper, we shall show that the BVP (1.1) and (1.2)
has a least three positive solutions by using the the fixed point theorem due to Avery and Peterson.
The interesting point is that the nonlinear term f is involved with the first-order derivative explicitly.
Our results are new for the special cases of difference equations and differential equations as well as
in the general time scale setting.
This paper is organized as follows. In Section 2, we state some definitions, notations, lemmas
and prove several preliminary results. In Sections 3 and 4, by defining an appropriate Banach space
and cones, we impose the growth conditions on f which allow us to apply the fixed point theorem
in finding existence of three positive solutions of (1.1), (1.2) (respectively, (1.1), (1.3)). In Section 5,
we give an example to demonstrate our results.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
794 A. DOGAN
2. Preliminaries and lemmas. In this section, we list the following well-known definitions
which can be found in [7, 8].
Definition 2.1. A time scale \BbbT is a closed nonempty subset of \BbbR . For t < \mathrm{s}\mathrm{u}\mathrm{p}\BbbT and r > \mathrm{i}\mathrm{n}\mathrm{f} \BbbT ,
define the forward jump operator \sigma and the backward jump operator \rho as, respectively,
\sigma (t) = \mathrm{i}\mathrm{n}\mathrm{f}\{ \tau \in \BbbT : \tau > t\} \in \BbbT ,
\rho (r) = \mathrm{s}\mathrm{u}\mathrm{p}\{ \tau \in \BbbT : \tau < r\} \in \BbbT
for all t, r \in \BbbT . If \sigma (t) > t, t is said to be right scattered, and if \sigma (t) = t, t is said to be right
dense (rd). If \rho (t) < t, t is said to be left scattered, and if \rho (t) = t, t is said to be left dense (ld). A
function f is left dense continuous, ld-continuous, f is continuous at each left dense point in \BbbT and
its right-hand sided limits exist at each right dense points in \BbbT .
Definition 2.2. For x : \BbbT \rightarrow \BbbR and t \in \BbbT (assume t is not left scattered if t = \mathrm{s}\mathrm{u}\mathrm{p}\BbbT ), we define
the delta derivative of x(t), x\Delta (t), to be the number (when it exists) with the property that, for each
\epsilon > 0, there is a neighborhood U of t such that\bigm| \bigm| x(\sigma (t)) - x(s) - x\Delta (t)(\sigma (t) - s)
\bigm| \bigm| < \epsilon | \sigma (t) - s|
for all s \in U. For x : \BbbT \rightarrow \BbbR and t \in \BbbT (assume t is not right scattered if t = \mathrm{i}\mathrm{n}\mathrm{f} \BbbT ), we define the
nabla derivative of x(t), x\nabla (t), to be the number (when it exists) with the property that, for each
\epsilon > 0, there is a neighborhood V of t such that\bigm| \bigm| x(\rho (t)) - x(s) - x\nabla (t)(\rho (t) - s)
\bigm| \bigm| < \epsilon | \rho (t) - s|
for all s \in V.
If \BbbT = \BbbR , then x\Delta (t) = x\nabla (t) = x\prime (t). If \BbbT = \BbbZ , then x\Delta (t) = x(t+ 1) - x(t) is the forward
difference operator while x\nabla (t) = x(t) - x(t - 1) is the backward difference operator.
Definition 2.3. If F\Delta (t) = f(t), then we define the delta integral by
t\int
a
f(s)\Delta s = F (t) - F (a).
If \Phi \nabla (t) = f(t), then we define the nabla integral by
t\int
a
f(s)\nabla s = \Phi (t) - \Phi (a).
We provide some background materials from the theory of cones in Banach spaces.
Definition 2.4. Let E be a real Banach space. A nonempty, closed, convex set P \subset E is a cone
if it satisfies the following two conditions:
(i) x \in P, \lambda \geq 0 imply \lambda x \in P ;
(ii) x \in P, - x \in P imply x = 0.
Every cone P \subset E induces an ordering in E given by x \leq y if and only if y - x \in P.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY-VALUE PROBLEM FOR p-LAPLACIAN . . . 795
Let \gamma and \theta be nonnegative continuous convex functionals on P, \alpha be a nonnegative continuous
concave functional on P, and \psi be a nonnegative continuous functional on P. Then, for positive real
numbers r1, r2, r3, and r4 we define the following sets:
P (\gamma , r4) = \{ x \in P : \gamma (x) < r4\} ,
P (\gamma , \alpha , r2, r4) = \{ x \in P : r2 \leq \alpha (x), \gamma (x) \leq r4\} ,
P (\gamma , \theta , \alpha , r2, r3, r4) = \{ x \in P : r2 \leq \alpha (x), \theta (x) \leq r3, \gamma (x) \leq r4\} ,
R(\gamma , \psi , r1, r4) = \{ x \in P : r1 \leq \psi (x), \gamma (x) \leq r4\} .
Let the Banach space E = C1
ld([0, \sigma (T )]\BbbT \rightarrow \BbbR ) with the norm
\| u\| = \mathrm{m}\mathrm{a}\mathrm{x}
\Biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\sigma (T )]\BbbT
| u(t)| , \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
\bigm| \bigm| u\Delta (t)\bigm| \bigm| \Biggr\} ,
and define the cone P \subset E by
P =
\bigl\{
u \in E : u(t) \geq 0, t \in [0, \sigma (T )]\BbbT ; u
\Delta \nabla (t) \leq 0, u\Delta (t) \geq 0, t \in [0, T ]\BbbT , u
\Delta (T ) = 0
\bigr\}
.
We note that u(t) is a solution to the BVP (1.1), (1.2) if and only if
u(t) =
t\int
0
\phi q
\left( T\int
s
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \Delta s+
+B0
\left( \phi q
\left( T\int
\nu
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \right) .
Define the operator F : P \rightarrow E by
(Fu)(t) =
t\int
0
\phi q
\left( T\int
s
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \Delta s+
+B0
\left( \phi q
\left( T\int
\nu
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \right) .
Lemma 2.1. If u \in P, then:
(i) u(t) \geq t
\sigma (T )
u(\sigma (T )) for t \in [0, \sigma (T )]\BbbT ;
(ii) tu(s) \geq su(t) for t, s \in [0, \sigma (T )]\BbbT with s \leq t.
Proof. (i) Since u\Delta \nabla (t) \leq 0, it follows that u\Delta (t) is nonincreasing. Thus, for 0 < t < \sigma (T ),
u(t) - u(0) =
t\int
0
u\Delta (s)\Delta s \geq tu\Delta (t)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
796 A. DOGAN
and
u(\sigma (T )) - u(t) =
\sigma (T )\int
t
u\Delta (s)\Delta s \leq (\sigma (T ) - t)u\Delta (t)
from which we have
u(t) \geq tu(\sigma (T )) + (\sigma (T ) - t)u(0)
\sigma (T )
\geq t
\sigma (T )
u(\sigma (T )).
(ii) If t = s, then the conclusion holds. If s < t, since u(t) is concave, nonnegative on
[0, \sigma (T )]\BbbT and u\Delta (\sigma (T )) = 0, hence, we get
u(t) - u(0)
t
\leq u(s) - u(0)
s
.
Thus,
tu(s) \geq su(t) + (t - s)u(0) \geq su(t).
Lemma 2.1 is proved.
Lemma 2.2. For any u \in P, there exists a real number M > 0 such that \mathrm{s}\mathrm{u}\mathrm{p}t\in [0,\sigma (T )]\BbbT u(t) \leq
\leq M \mathrm{s}\mathrm{u}\mathrm{p}t\in [0,T ]\BbbT u
\Delta (t), where M = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
1,
\sigma (T )
T
(B + T )
\biggr\}
.
Proof. Because u(t) = u(0) +
t\int
0
u\Delta (t)\Delta t and u\Delta (t) \geq u\Delta (T ) = 0, we obtain
u(T ) = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
u(t) \leq \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
\left\{ B0(u
\Delta (\nu )) +
t\int
0
u\Delta (t)\Delta t
\right\} \leq
\leq (B + T ) \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
u\Delta (t).
From Lemma 2.1, we have
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\sigma (T )]\BbbT
u(t) = u(\sigma (T )) \leq \sigma (T )
T
u(T ) \leq \sigma (T )
T
(B + T ) \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
u\Delta (t).
Lemma 2.2 is proved.
The next theorem from Theorem 1.3 in [17] is stated in context of \BbbT \subseteq \BbbR . The proof is, therefore,
omitted.
Theorem 2.1 (Arzela – Ascoli theorem on \BbbT ). Let D \subseteq C([a, b]\BbbT ;\BbbR ). Then D is relatively com-
pact if and only if it is bounded and equicontinuous.
Lemma 2.3. F : P \rightarrow P is completely continuous.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY-VALUE PROBLEM FOR p-LAPLACIAN . . . 797
Proof. Firstly, we verify that F : P \rightarrow P. From (H3), it is obvious that (Fu)(t) \geq 0 for
t \in [0, T ]\BbbT \subset t \in [0, \sigma (T )]\BbbT and (Fu)\Delta (T ) = 0. Moreover,
(Fu)\Delta (t) = \phi q
\left( T\int
t
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \geq 0
is continuous and nonincreasing in [0, T ]\BbbT ,\left( T\int
t
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \nabla = - g(t)f(t, u(t), u\Delta (t)) \leq 0, t \in [0, T ]\BbbT .
In addition, \phi q(u) is a monotone increasing continuously differentiable function for u > 0. If
T\int
t
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r > 0 for t \in [0, T ]\BbbT ,
we find (Fu)\Delta \nabla (t) \leq 0 for t \in [0, T ]\BbbT . If
T\int
t
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r = 0 for t \in [0, T ]\BbbT ,
then (Fu)\Delta \nabla (t) = 0 for t \in [0, T ]\BbbT .
Secondly, we prove that F maps a bounded set into itself. Suppose that c > 0 is a constant and
u \in Pc =
\biggl\{
u \in P : \| u\| = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\sigma (T )]\BbbT
| u(t)| , \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
\bigm| \bigm| u\Delta (t)\bigm| \bigm| \biggr\} \leq c
\biggr\}
.
Notice that f(t, u, v) is continuous, so there exists a constant C > 0 such that f(t, u, v) \leq \phi p(C)
for (t, u, v) \in [0, T ]\BbbT \times [0, c]\times [0, c]. From here, t \in [0, T ]\BbbT ,\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \phi q
\left( T\int
t
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < +\infty (2.1)
and \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
0
\phi q
\left( T\int
s
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \Delta s+
+ B0
\left( \phi q
\left( T\int
\nu
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < +\infty . (2.2)
Consequently, F maps a bounded set into a bounded set.
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798 A. DOGAN
Thirdly, if t1, t2 \in [0, T ]\BbbT and t1 < t2, then we have
\bigm| \bigm| (Fu)(t1) - (Fu)(t2)
\bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t2\int
t1
\phi q
\left( T\int
s
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \Delta s
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t2\int
t1
\phi q
\left( T\int
0
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \Delta s
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq C| t1 - t2| \phi q
\left( T\int
0
g(r)\nabla r
\right) .
Therefore, by Theorem 2.1, we see that FPc is relatively compact.
We next claim that F : Pc \rightarrow P is continuous. Suppose that \{ un\} \infty n=1 \subset Pc and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| un -
- u0\| \rightarrow 0. This means that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
| un - u0| \rightarrow 0 and \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
| u\Delta n - u\Delta 0 | \rightarrow 0.
Since
\bigl\{
(Fun)(t)
\bigr\} \infty
n=1
is uniformly bounded and equicontinuous on [0, T ]\BbbT , there exists a uniformly
convergent subsequence in
\bigl\{
(Fun)(t)
\bigr\} \infty
n=1
. Let
\bigl\{
(Fun(m))(t)
\bigr\} \infty
m=1
be a subsequence which con-
verges to w(t) uniformly on [0, T ]\BbbT . Examine that
(Fun)(t) =
t\int
0
\phi q
\left( T\int
s
g(r)f(r, un(r), u
\Delta
n (r))\nabla r
\right) \Delta s+
+B0
\left( \phi q
\left( T\int
\nu
g(r)f(r, un(r), u
\Delta
n (r))\nabla r
\right) \right) .
From (2.1) and (2.2), inserting un(m) into the above and then letting m\rightarrow \infty , we find
w(t) =
t\int
0
\phi q
\left( T\int
s
g(r)f(r, u0(r), u
\Delta
0 (r))\nabla r
\right) \Delta s+
+B0
\left( \phi q
\left( T\int
\nu
g(r)f(r, u0(r), u
\Delta
0 (r))\nabla r
\right) \right) .
From the definition of F, we know that w(t) = Fu0(t) on [0, T ]\BbbT . This shows that each subsequence
of
\bigl\{
(Fun)(t)
\bigr\} \infty
n=1
uniformly converges to (Fu0)(t). So, the sequence
\bigl\{
(Fun)(t)
\bigr\} \infty
n=1
uniformly
converges to (Fu0)(t). This means that F is continuous at u0 \in Pc. Therefore, F is continuous on
Pc since u0 is arbitrary. Thus, F is completely continuous.
Lemma 2.3 is proved.
The following fixed point theorem due to Avery and Peterson is fundamental in the proofs our
main results.
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Theorem 2.2 [6]. Let P be a cone in a real Banach space E. Let \gamma and \theta be nonnegative
continuous convex functionals on P, \alpha be a nonnegative continuous concave functional on P, and \psi
be a nonnegative continuous functional on P satisfying \psi (\lambda u) \leq \lambda \psi (u) for 0 \leq \lambda \leq 1 such that
for some positive numbers h and r4,
\alpha (u) \leq \psi (u) and \| u\| \leq h\gamma (u)
for all u \in P (\gamma , r4). Suppose F : P (\gamma , r4) \rightarrow P (\gamma , r4) is completely continuous and there exist
positive real numbers r1, r2, and r3 with r1 < r2 such that:
(S1) \{ u \in P (\gamma , \theta , \alpha , r2, r3, r4) : \alpha (u) > r2\} \not = \varnothing and \alpha (Fu) > r2 for u \in P (\gamma , \theta , \alpha , r2, r3, r4);
(S2) \alpha (Fu) > r2 for u \in P (\gamma , \alpha , r2, r4) with \theta (Fu) > r3;
(S3) 0 /\in R(\gamma , \psi , r1, r4) and \psi (Fu) < r1 for all u \in R(\gamma , \psi , r1, r4) with \psi (u) = r1.
Then F has at least three fixed points u1, u2, u3 \in P (\gamma , r4) such that
\gamma (ui) \leq r4 for i = 1, 2, 3;
r2 < \alpha (u1);
r1 < \psi (u2) with \alpha (u2) < r2;
\psi (u3) < r1.
3. Solutions of (1.1) and (1.2) in a cone. Let \zeta \in \BbbT be such that 0 < \nu < \zeta < T, and define
the nonnegative continuous convex functionals \gamma and \theta , nonnegative continuous concave functional
\alpha , and nonnegative continuous functional \psi , respectively, on P by
\alpha (u) = \mathrm{i}\mathrm{n}\mathrm{f}
t\in [\zeta ,T ]\BbbT
u(t) = u(\zeta ),
\psi (u) = \mathrm{i}\mathrm{n}\mathrm{f}
t\in [\zeta ,T ]\BbbT
u(t) = u(\zeta ),
\gamma (u) = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
u\Delta (t) = u\Delta (0),
\theta (u) = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [\tau ,T ]\BbbT
u\Delta (t) = u\Delta (\tau ).
In view of Lemma 2.2, we find
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\sigma (T )]\BbbT
u(t) \leq M \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
u\Delta (t) =M\gamma (u) for all u \in P.
We also see that \theta (\lambda u) = \lambda \theta (u) for \lambda \in [0, 1].
For notational convenience, we denote \lambda 0, \mu and \delta by
\lambda 0 = \phi q
\left( T\int
0
g(r)\nabla r
\right) ,
\mu = (\zeta +B)\phi q
\left( T\int
\zeta
g(r)\nabla r
\right) ,
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800 A. DOGAN
\delta = (\zeta +A)\phi q
\left( T\int
0
g(r)\nabla r
\right) .
Theorem 3.1. Assume that there exist constants r1, r2, r4 such that 0 < r1 <
\zeta
T
r2 <
\zeta \mu
T\lambda 0
r4,
\lambda 0\zeta > \mu , and suppose that f satisfies the following conditions:
(A1) f(t, u, v) \leq \phi p
\biggl(
r4
\lambda 0
\biggr)
for (t, u, v) \in [0, T ]\BbbT \times [0,Mr4]\times [ - r4, r4];
(A2) f(t, u, v) > \phi p
\biggl(
r2
\mu
\biggr)
for (t, u, v) \in [\zeta , T ]\BbbT \times [r2,Mr4]\times [ - r4, r4];
(A3) f(t, u, v) < \phi p
\Bigl( r1
\delta
\Bigr)
for (t, u, v) \in [0, T ]\BbbT \times
\biggl[
0,
T
\zeta
r1
\biggr]
\times [ - r4, r4].
Then the BVP (1.1), (1.2) has at least three positive solutions u1, u2, and u3 such that
\| ui\| \leq r4 for i = 1, 2, 3, r2 < u1(\zeta ), r1 < u2(\zeta ) and u2(\zeta ) < r2 with u3(\zeta ) < r1.
(3.1)
Proof. The BVP (1.1), (1.2) has a solution u = u(t) if and only if u solves the operator
equation u = Fu. Thus we set out to verify that the operator F satisfies Avery and Peterson’s fixed
point theorem which will prove the existence of three fixed points of F which satisfy the conclusion
of the theorem.
Firstly, we will show that
F : P (\gamma , r4) \rightarrow P (\gamma , r4). (3.2)
For any u \in P (\gamma , r4), we have \gamma (u) = \mathrm{s}\mathrm{u}\mathrm{p}t\in [0,T ]\BbbT u
\Delta (t) \leq r4. From Lemma 2.2, we get
\mathrm{s}\mathrm{u}\mathrm{p}t\in [0,\sigma (T )]\BbbT u(t) \leq Mr4. From (A1), we obtain f(t, u, v) \leq \phi p
\biggl(
r4
\lambda 0
\biggr)
, and so
\gamma (Fu) = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
(Fu)\Delta (t) = \phi q
\left( T\int
0
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \leq
\leq \phi q
\left( T\int
0
g(r)\nabla r
\right) r4
\lambda 0
= r4.
Thus (3.2) holds.
Secondly, we prove that condition (S1) in Theorem 2.2 holds. Let u =
\lambda 0r2
\mu
t - \lambda 0r2
\mu
\zeta + 2r2.
Then \alpha (u) = 2r2 > r2, \theta (u) =
\lambda 0r2
\mu
and \gamma (u) =
\lambda 0r2
\mu
< r4. So,
\biggl\{
u \in P
\biggl(
\gamma , \theta , \alpha , r2,
\lambda 0r2
\mu
, r4
\biggr)
:
\alpha (u) > r2
\biggr\}
\not = \varnothing . On the other hand, for any
\biggl\{
u \in P
\biggl(
\gamma , \theta , \alpha , r2,
\lambda 0r2
\mu
, r4
\biggr)
: \alpha (u) > r2
\biggr\}
, it
follows from Lemma 2.2 that r2 \leq u(t) \leq Mr4, - r4 \leq u\Delta (t) \leq r4, and for all t \in [\zeta , T ]\BbbT . From
(A2), we get
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POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY-VALUE PROBLEM FOR p-LAPLACIAN . . . 801
\alpha (Fu) = Fu(\zeta ) =
\zeta \int
0
\phi q
\left( T\int
s
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \Delta s+
+B0
\left( \phi q
\left( T\int
\nu
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \right) >
> (\zeta +B)\phi q
\left( T\int
\zeta
g(r)\phi p
\biggl(
r2
\mu
\biggr)
\nabla r
\right) =
= (\zeta +B)\phi q
\left( T\int
\zeta
g(r)\nabla r
\right) r2
\mu
= r2.
Therefore, we have \alpha (u) > r2 for all u \in P
\biggl(
\gamma , \theta , \alpha , r2,
\lambda 0r2
\mu
, r4
\biggr)
. Consequently, condition (S1) in
Theorem 2.2 is satisfied.
Thirdly, we verify that condition (S2) of Theorem 2.2 holds. For any u \in P (\gamma , \alpha , r2, r4) with
\theta (Fu) >
\lambda 0r2
\mu
that is
\theta (Fu) = (Fu)\Delta (\tau ) = \phi q
\left( T\int
\tau
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) >
\lambda 0r2
\mu
,
we obtain
\alpha (Fu) = \mathrm{i}\mathrm{n}\mathrm{f}
t\in [\zeta ,T ]\BbbT
(Fu)(t) \geq
\zeta \int
0
\phi q
\left( T\int
s
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \Delta s+
+A
\left( \phi q
\left( T\int
\nu
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \right) >
> \zeta \phi q
\left( T\int
\tau
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) > \zeta
\lambda 0r2
\mu
> r2.
Hence, condition (S2) in Theorem 2.2 is satisfied.
Finally, we prove that (S3) in Theorem 2.2 is satisfied. Since \psi (0) = 0 < r1, so,
0 /\in R(\gamma , \psi , r1, r4). Suppose that u \in R(\gamma , \psi , r1, r4) with \psi (u) = \mathrm{i}\mathrm{n}\mathrm{f}t\in [\zeta ,T ]\BbbT u(t) = u(\zeta ) = r1,
Lemma 2.1 implies that
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
u(t) = u(T ) \leq T
\zeta
u(\zeta ) =
T
\zeta
r1,
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802 A. DOGAN
this yields 0 \leq u(t) \leq T
\zeta
r1 for all t \in [0, T ]\BbbT . Moreover, \gamma (u) = \mathrm{s}\mathrm{u}\mathrm{p}t\in [0,T ]\BbbT u
\Delta (t) \leq r4. Hence, by
the condition (A3) of this theorem, we have
\alpha (Fu) = \mathrm{i}\mathrm{n}\mathrm{f}
t\in [\zeta ,T ]\BbbT
(Fu)(t) \leq
\zeta \int
0
\phi q
\left( T\int
0
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \Delta s+
+A
\left( \phi q
\left( T\int
\nu
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \right) <
< (\zeta +A)\phi q
\left( T\int
0
g(r)\phi p
\Bigl( r1
\delta
\Bigr)
\nabla r
\right) =
= (\zeta +A)\phi q
\left( T\int
0
g(r)\nabla r
\right) r1
\delta
= r1.
Thus, condition (S3) in Theorem 2.2 holds. As a result, all the conditions of Theorem 2.2 are
satisfied.
Theorem 3.1 is proved.
4. Solutions of (1.1) and (1.3) in a cone. Let the Banach space E = C1
ld
\bigl(
[0, \sigma (T )]\BbbT \rightarrow \BbbR
\bigr)
with the norm
\| u\| = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\sigma (T )]\BbbT
| u(t)| , \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
\bigm| \bigm| u\Delta (t)\bigm| \bigm| \biggr\} ,
and define the cone P1 \subset E by
P1 =
\bigl\{
u \in E : u(t) \geq 0, t \in [0, \sigma (T )]\BbbT ; u\Delta \nabla (t) \leq 0, u\Delta (t) \leq 0, t \in [0, T ]\BbbT , u
\Delta (0) = 0
\bigr\}
.
Fix \tau \in \BbbT such that 0 < \tau < \nu , and define the nonnegative continuous convex functionals \gamma and \theta ,
nonnegative continuous concave functional \alpha , and nonnegative continuous functional \psi , respectively,
on P1 by
\alpha (u) = \mathrm{i}\mathrm{n}\mathrm{f}
t\in [\tau ,T ]\BbbT
u(t) = u(T ),
\psi (u) = \mathrm{i}\mathrm{n}\mathrm{f}
t\in [\tau ,T ]\BbbT
u(t) = u(T ),
\gamma (u) = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]\BbbT
| u\Delta (t)| = u\Delta (T ),
\theta (u) = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [\tau ,T ]\BbbT
| u\Delta (t)| = u\Delta (T ).
Set
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POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY-VALUE PROBLEM FOR p-LAPLACIAN . . . 803
\lambda 1 = \phi q
\left( T\int
0
g(r)\nabla r
\right) ,
\mu 1 = B\phi q
\left( \tau \int
0
g(r)\nabla r
\right) ,
\delta 1 = A\phi q
\left( T\int
0
g(r)\nabla r
\right) .
We note that u(t) is a solution of (1.1), (1.3) if and only if
u(t) =
T\int
t
\phi q
\left( s\int
0
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \Delta s+
+B1
\left( \phi q
\left( \nu \int
0
g(r)f
\bigl(
r, u(r), u\Delta (r)
\bigr)
\nabla r
\right) \right) .
Theorem 4.1. Assume that conditions (H1) – (H5) are satisfied. Let 0 < r1 <
\zeta
T
r2 <
\zeta \mu 1
T\lambda 1
r4,
\lambda 1\zeta > \mu 1, and suppose that f satisfies the following conditions:
(C1) f(t, u, v) \leq \phi p
\biggl(
r4
\lambda 1
\biggr)
for (t, u, v) \in [0, T ]\BbbT \times [0,Mr4]\times [ - r4, r4];
(C2) f(t, u, v) > \phi p
\biggl(
r2
\mu 1
\biggr)
for (t, u, v) \in [\zeta , T ]\BbbT \times [r2,Mr4]\times [ - r4, r4];
(C3) f(t, u, v) < \phi p
\biggl(
r1
\delta 1
\biggr)
for (t, u, v) \in [0, T ]\BbbT \times
\biggl[
0,
T
\zeta
r1
\biggr]
\times [ - r4, r4].
Then the BVP (1.1) and (1.3) has at least three positive solutions u1, u2, and u3 such that
\| ui\| \leq r4 for i = 1, 2, 3, r2 < u1(\zeta ), r1 < u2(\zeta ) and u2(\zeta ) < r2 with u3(\zeta ) < r1.
(4.1)
5. Example. Let
\BbbT =
\biggl\{
2 -
\biggl(
1
3
\biggr) \BbbN 0
\biggr\}
\cup
\biggl\{
0,
1
8
,
1
4
,
1
6
,
1
2
, 1,
5
4
,
3
2
,
7
4
, 2
\biggr\}
,
where \BbbN 0 denotes the set of all nonnegative integers. Take T = 2, p = 7, \nu =
1
2
, \zeta = 1, \tau =
3
2
,
A = B =
1
100000
and choose
g(t) = t+ \rho (t)
and
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804 A. DOGAN
f(t, u, v) =
\left\{
t+ 1 + | v| for (t, u, v) \in [0, 2]\BbbT \times [0, 4]\times [ - 6, 6],
t+ | v| + p(u) for (t, u, v) \in [0, 2]\BbbT \times [4, 4.1]\times [ - 6, 6],
t+ 1584 + | v| for (t, u, v) \in [0, 2]\BbbT \times [4.1, 20]\times [ - 6, 6].
Here p(u) satisfies p(4) = 1, p(4.1) = 1584, p(u) : \BbbR \rightarrow \BbbR + is continuous and p\Delta (u) > 0. Choose
r1 = 2, r2 = 4.1, r4 = 6. Then
\lambda 0 =
\left( 2\int
0
g(r)\nabla r
\right)
1
6
\approx 1.260,
\mu =
\biggl(
1 +
1
100000
\biggr) \left( 2\int
1
g(r)\nabla r
\right)
1
6
\approx 1.201,
\delta =
\biggl(
1 +
1
100000
\biggr) \left( 2\int
0
g(r)\nabla r
\right)
1
6
\approx 1.25993.
It is easy to see that 0 < r1 <
\zeta
T
r2 <
\zeta \mu
T\lambda 0
r4, \lambda 0\zeta > \mu , and f(t, u, v) satisfies that
f(t, u, v) < \phi p
\biggl(
r4
\lambda 0
\biggr)
=
\biggl(
6
1.260
\biggr) 6
\approx 11659.6 for 0 \leq t \leq 2, 0 \leq u \leq 10.1, | v| \leq 6,
f(t, u, v) > \phi p
\biggl(
r2
\mu
\biggr)
=
\biggl(
4.1
1.201
\biggr) 6
\approx 1583.26 for 1 \leq t \leq 2, 4.1 \leq u \leq 10.1, | v| \leq 6,
f(t, u, v) < \phi p
\Bigl( r1
\delta
\Bigr)
=
\biggl(
2
1.25993
\biggr) 6
\approx 15.9993 for 0 \leq t \leq 2, 0 \leq u \leq 4, | v| \leq 6.
Then all conditions of Theorem 3.1 hold. Thus by Theorem 3.1, the BVP (1.1), (1.2) has at least
three positive solutions u1, u2, u3 such that
\| ui\| \leq 6 for i = 1, 2, 3, 4.1 < u1(1), 2 < u2(1) and u2(1) < 4.1 with u3(1) < 2.
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POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY-VALUE PROBLEM FOR p-LAPLACIAN . . . 805
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Received 10.04.17,
after revision — 18.06.19
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
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| id | umjimathkievua-article-646 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:30Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c6/4dbc80d4337307c788796c9d7bee57c6.pdf |
| spelling | umjimathkievua-article-6462022-03-26T11:01:46Z Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales Dogan, A. Dogan, A. Time scales Boundary value problem p-Laplacian Positive solutions Fixed point theorem UDC 517.9 We consider a three-point boundary-value problem for p-Laplacian dynamic equation on time scales. We show the existence at least three positive solutions of the boundary-value problem by using the Avery and Peterson fixed point theorem. The conditions we used here differ from those in the majority of papers as we know. The interesting point is that the nonlinear term $ f$ involves the first derivative of the unknown function. As an application, an example is given to illustrate our results. УДК 517.9 Розглядається триточкова крайова задача для динамiчного рiвняння iз $p$-лапласiаном на часових шкалах. За допомогою теореми Ейвери та Петерсона про нерухому точку доведено iснування принаймнi трьох додатних розв’язкiв такої крайової задачi. Умови, якi використовуються тут, вiдрiзняються вiд умов, якi використано у бiльшостi вiдомих нам робiт. Цiкавим моментом є те, що нелiнiйний член $f$ мiстить першу похiдну невiдомої функцiї. Як застосування наведено приклад для iлюстрацiї отриманих результатiв. &nbsp; Institute of Mathematics, NAS of Ukraine 2020-06-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/646 10.37863/umzh.v72i6.646 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 6 (2020); 790-805 Український математичний журнал; Том 72 № 6 (2020); 790-805 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/646/8715 |
| spellingShingle | Dogan, A. Dogan, A. Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales |
| title | Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales |
| title_alt | Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales |
| title_full | Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales |
| title_fullStr | Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales |
| title_full_unstemmed | Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales |
| title_short | Positive solutions of a three-point boundary-value problem for $\mathcal {p}$-Laplacian dynamic equation on time scales |
| title_sort | positive solutions of a three-point boundary-value problem for $\mathcal {p}$-laplacian dynamic equation on time scales |
| topic_facet | Time scales Boundary value problem p-Laplacian Positive solutions Fixed point theorem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/646 |
| work_keys_str_mv | AT dogana positivesolutionsofathreepointboundaryvalueproblemformathcalplaplaciandynamicequationontimescales AT dogana positivesolutionsofathreepointboundaryvalueproblemformathcalplaplaciandynamicequationontimescales |