Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 145050, 12 pages
http://dx.doi.org/10.1155/2013/145050
Research Article
Existence of Three Positive Solutions to Some p-Laplacian
Boundary Value Problems
Moulay Rchid Sidi Ammi,1 and �el�m �. M. Torres2
1
Department of Mathematics, AMNEA Group, Faculty of Sciences and Technics, Moulay Ismail University,
P.O. Box 509, Errachidia, Morocco
2
Department of Mathematics, Center for Research and Development in Mathematics and Applications, University of Aveiro,
3810-193 Aveiro, Portugal
Correspondence should be addressed to Del�m F. M. Torres; del�m�ua.pt
Received 24 March 2012; Revised 5 August 2012; Accepted 19 October 2012
Academic Editor: Zengji Du
Copyright © 2013 M. R. Sidi Ammi and D. F. M. Torres. is is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We obtain, by using the Leggett-Williams �xed point theorem, su�cient conditions that ensure the existence of at least three positive
solutions to some p-Laplacian boundary value problems on time scales.
1. Introduction
e study of dynamic equations on time scales goes back
to the 1989 Ph.D. thesis of Hilger [1, 2] and is currently
an area of mathematics receiving considerable attention [3–
7]. Although the basic aim of the theory of time scales is
to unify the study of differential and difference equations
in one and the same subject, it also extends these classical
domains to hybrid and in-between cases. A great deal of
work has been done since the eighties of the XX century in
unifying the theories of differential and difference equations
by establishing more general results in the time scale setting
[8–12].
Boundary value 𝑝𝑝-Laplacian problems for differential
equations and �nite difference equations have been studied
extensively (see, e.g., [13] and references therein). Although
many existence results for dynamic equations on time scales
are available [14, 15], there are not many results concerning 𝑝𝑝-Laplacian problems on time scales [16–19]. In this
paper we prove new existence results for three classes of
𝑝𝑝-Laplacian boundary value problems on time scales. In
contrast with our previous works [17, 18], which make use
of the �rasnoselskii �xed point theorem and the �xed point
index theory, respectively, here we use the Leggett-Williams
�xed point theorem [20, 21] obtaining multiplicity of positive
solutions. e application of the Leggett-Williams �xed point
theorem for proving multiplicity of solutions for boundary
value problems on time scales was �rst introduced by Agarwal
and O’Regan [22] and is now recognized as an important tool
to prove existence of positive solutions for boundary value
problems on time scales [23–28].
e paper is organized as follows. In Section 2 we
present some necessary results from the theory of time
scales (Section 2.1) and the theory of cones in Banach spaces
(Section 2.2). We end Section 2.2 with the Leggett-Williams
�xed point theorem for a cone-preserving operator, which
is our main tool in proving existence of positive solutions
to the boundary value problems on time scales we consider
in Section 3. e contribution of the paper is Section 3,
which is divided into three parts. e purpose of the �rst part
(Section 3.1) is to prove existence of positive solutions to the
nonlocal 𝑝𝑝-Laplacian dynamic equation on time scales
−󶀢󶀢𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢Δ (𝑡𝑡)󶀲󶀲󶀲󶀲
∇
=
𝑇𝑇
󶀣󶀣∫0
𝜆𝜆𝜆𝜆 (𝑢𝑢 (𝑡𝑡))
𝑓𝑓 (𝑢𝑢 (𝜏𝜏)) ∇𝜏𝜏󶀳󶀳
satisfying the boundary conditions
2
,
𝑡𝑡 𝑡 (0, 𝑇𝑇)𝕋𝕋 , (1)
𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢Δ (0)󶀲󶀲 − 𝛽𝛽 󶀢󶀢𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢Δ 󶀡󶀡𝜂𝜂󶀱󶀱󶀱󶀱󶀱󶀱 = 0,
𝑢𝑢 (𝑇𝑇) − 𝛽𝛽𝛽𝛽 󶀡󶀡𝜂𝜂󶀱󶀱 = 0,
(2)
2
Discrete Dynamics in Nature and Society
where 𝜂𝜂 𝜂 𝜂𝜂𝜂 𝜂𝜂𝜂𝕋𝕋 , 𝜙𝜙𝑝𝑝 (⋅) is the 𝑝𝑝-Laplacian operator de�ned
𝑝𝑝𝑝𝑝
−1
by 𝜙𝜙𝑝𝑝 (𝑠𝑠𝑠 𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠, 𝑝𝑝 𝑝 𝑝, and (𝜙𝜙𝑝𝑝 ) = 𝜙𝜙𝑞𝑞 with 𝑞𝑞 the Holder
conjugate of 𝑝𝑝, that is, 1/𝑝𝑝 𝑝 𝑝𝑝𝑝𝑝𝑝 𝑝. e concrete value
of 𝑝𝑝 is connected with the application at hands. For 𝑝𝑝 𝑝 𝑝,
problem (1)-(2) describes the operation of a device �owed by
an electric current, for example, thermistors [29], which are
devices made from materials whose electrical conductivity
is highly dependent on the temperature. ermistors have
the advantage of being temperature measurement devices
of low cost, high resolution, and �exibility in size and
shape. Constant 𝜆𝜆 in (1) is a dimensionless parameter that
can be identi�ed with the square of the applied potential
difference at the ends of a conductor, 𝑓𝑓𝑓𝑓𝑓𝑓 is the temperaturedependent resistivity of the conductor, and 𝛽𝛽 in (2) is a
transfer coefficient supposed to verify 0 < 𝛽𝛽 𝛽𝛽. For
a more detailed discussion about the physical justi�cation
of (1)-(2) the reader is referred to [17]. eoretical analysis
(existence, uniqueness, regularity, and asymptotic results) for
thermistor problems with various types of boundary and
initial conditions has received signi�cant attention in the
last few years for the particular case 𝕋𝕋 𝕋 𝕋 [30–34]. e
second part of our results (Section 3.2) is concerned with the
following quasilinear elliptic problem:
∇
−󶀢󶀢𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢Δ (𝑡𝑡)󶀲󶀲󶀲󶀲 = 𝑓𝑓 (𝑢𝑢 (𝑡𝑡)) + ℎ (𝑡𝑡) ,
𝑡𝑡 𝑡 (0,𝑇𝑇)𝕋𝕋 ,
𝑢𝑢Δ (0) =0,𝑢𝑢 (𝑇𝑇) −𝑢𝑢 󶀡󶀡𝜂𝜂󶀱󶀱 =0,
(3)
where 𝜂𝜂 𝜂 𝜂𝜂𝜂 𝜂𝜂𝜂𝕋𝕋 . �esults on existence of in�nitely many
radial solutions to (3) are proved in the literature using (i)
variational methods, where solutions are obtained as critical
points of some energy functional on a Sobolev space, with 𝑓𝑓
satisfying appropriate conditions [35, 36]; (ii) methods based
on phase-plane analysis and the shooting method [37]; (iii)
the technique of time maps [38]. For 𝑝𝑝 𝑝 𝑝, ℎ ≡ 0, and 𝕋𝕋 𝕋 𝕋,
problem (3) becomes a well-known boundary value problem
of differential equations. Our results generalize earlier works
to the case of a generic time scale 𝕋𝕋, 𝑝𝑝 𝑝 𝑝, and ℎ not
identically zero. Finally, the third part of our contribution
(Section 3.3) is devoted to the existence of positive solutions
to the 𝑝𝑝-Laplacian dynamic equation
∇
󶀢󶀢𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢Δ (𝑡𝑡)󶀲󶀲󶀲󶀲 + 𝜆𝜆𝜆𝜆 (𝑡𝑡) 𝑓𝑓 (𝑢𝑢 (𝑡𝑡) ,𝑢𝑢 (𝜔𝜔 (𝑡𝑡))) =0,
𝑡𝑡 𝑡 (0,𝑇𝑇)𝕋𝕋 ,
𝑢𝑢 (𝑡𝑡) = 𝜓𝜓 (𝑡𝑡) ,
𝑡𝑡 𝑡 [−𝑟𝑟𝑟𝑟]𝕋𝕋 ,
𝑢𝑢 (0) − 𝐵𝐵0 󶀢󶀢𝑢𝑢Δ (0)󶀲󶀲 =0,𝑢𝑢Δ (𝑇𝑇) =0
(4)
on a time scale 𝕋𝕋 such that 0,𝑇𝑇𝑇𝑇𝑇𝜅𝜅𝜅𝜅 , −𝑟𝑟 𝑟𝑟𝑟 with −𝑟𝑟 𝑟 𝑟𝑟
𝑇𝑇, where 𝜆𝜆 𝜆𝜆. is problem is considered in [39] where the
author applies the �rasnoselskii �xed point theorem to obtain
one positive solution to (4). Here we use the same conditions
as in [39], but applying Leggett-Williams’ theorem we are able
to obtain more: we prove existence of at least three positive
solutions and we are able to localize them.
2. Preliminaries
Here we just recall the basic concepts and results needed in
the sequel. For an introduction to time scales the reader is
referred to [3, 8–10, 40, 41] and references therein; for a good
introduction to the theory of cones in Banach spaces we refer
the reader to the book [42].
2.1. Time Scales. A time scale 𝕋𝕋 is an arbitrary nonempty
closed subset of the real numbers ℝ. e operators 𝜎𝜎 and 𝜌𝜌
from 𝕋𝕋 to 𝕋𝕋 are de�ned in [1, 2] as
𝜎𝜎 (𝑡𝑡) = inf {𝜏𝜏 𝜏𝜏𝜏 𝜏 𝜏𝜏 𝜏𝜏𝜏} ∈ 𝕋𝕋𝕋
𝜌𝜌 (𝑡𝑡) = sup {𝜏𝜏 𝜏𝜏𝜏 𝜏 𝜏𝜏 𝜏𝜏𝜏} ∈ 𝕋𝕋
(5)
and are called the forward jump operator and the backward
jump operator, respectively. A point 𝑡𝑡 𝑡𝑡𝑡 is le-dense, lescattered, right-dense, and right-scattered if 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌,
and 𝜎𝜎𝜎𝜎𝜎𝜎𝜎𝜎𝜎𝜎 𝜎𝜎𝜎𝜎𝜎𝜎𝜎𝜎𝜎, respectively. If 𝕋𝕋 has a right-scattered
minimum 𝑚𝑚, de�ne 𝕋𝕋𝜅𝜅 = 𝕋𝕋 𝕋𝕋𝕋𝕋𝕋; otherwise set 𝕋𝕋𝜅𝜅 = 𝕋𝕋.
If 𝕋𝕋 has a le-scattered maximum 𝑀𝑀, de�ne 𝕋𝕋𝜅𝜅 = 𝕋𝕋 𝕋𝕋𝕋𝕋𝕋;
otherwise set 𝕋𝕋𝜅𝜅 = 𝕋𝕋. Following [43], we also introduce the
set 𝕋𝕋𝜅𝜅𝜅𝜅 = 𝕋𝕋𝜅𝜅 ∩ 𝕋𝕋𝜅𝜅 .
Let 𝑓𝑓 𝑓 𝑓𝑓 𝑓 𝑓 and 𝑡𝑡 𝑡𝑡𝑡𝜅𝜅 (assume 𝑡𝑡 is not le-scattered
if 𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡), then the delta derivative of 𝑓𝑓 at the point 𝑡𝑡 is
de�ned to be the number 𝑓𝑓Δ (𝑡𝑡𝑡 (provided it exists) with the
property that for each 𝜖𝜖 𝜖𝜖 there is a neighborhood 𝑈𝑈 of 𝑡𝑡
such that
󶙢󶙢𝑓𝑓 (𝜎𝜎 (𝑡𝑡)) − 𝑓𝑓 (𝑠𝑠) − 𝑓𝑓Δ (𝑡𝑡)(𝜎𝜎 (𝑡𝑡) − 𝑠𝑠)󶙢󶙢 ≤ |𝜎𝜎 (𝑡𝑡) − 𝑠𝑠|
∀𝑠𝑠 𝑠𝑠𝑠𝑠
(6)
󶙢󶙢𝑓𝑓 󶀡󶀡𝜌𝜌 (𝑡𝑡)󶀱󶀱 − 𝑓𝑓 (𝑠𝑠) − 𝑓𝑓∇ (𝑡𝑡) 󶀡󶀡𝜌𝜌 (𝑡𝑡) − 𝑠𝑠󶀱󶀱󶀱󶀱 ≤ 󶙡󶙡𝜌𝜌 (𝑡𝑡) − 𝑠𝑠󶙡󶙡
∀𝑠𝑠 𝑠𝑠𝑠𝑠
(7)
Similarly, for 𝑡𝑡 𝑡𝑡𝑡𝜅𝜅 (assume 𝑡𝑡 is not right-scattered if 𝑡𝑡 𝑡
inf 𝕋𝕋), the nabla derivative of 𝑓𝑓 at the point 𝑡𝑡 is de�ned in [44]
to be the number 𝑓𝑓∇ (𝑡𝑡𝑡 (provided it exists) with the property
that for each 𝜖𝜖 𝜖𝜖 there is a neighborhood 𝑈𝑈 of 𝑡𝑡 such that
If 𝕋𝕋 𝕋 𝕋, then 𝑓𝑓Δ (𝑡𝑡𝑡𝑡𝑡𝑡∇ (𝑡𝑡𝑡𝑡𝑡𝑡′ (𝑡𝑡𝑡. If 𝕋𝕋 𝕋 𝕋, then 𝑓𝑓Δ (𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 is the forward difference operator while 𝑓𝑓∇ (𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 is the backward difference operator.
A function 𝑓𝑓 is le-dense continuous (i.e., 𝑙𝑙𝑙𝑙continuous), if 𝑓𝑓 is continuous at each le-dense point
in 𝕋𝕋 and its right-sided limit exists at each right-dense point
in 𝕋𝕋. If 𝑓𝑓 is 𝑙𝑙𝑙𝑙-continuous, then there exists 𝐹𝐹 such that
𝐹𝐹∇ (𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 for any 𝑡𝑡 𝑡𝑡𝑡𝜅𝜅 . We then introduce the nabla
integral by
𝑏𝑏
󵐐󵐐 𝑓𝑓 (𝑡𝑡) ∇𝑡𝑡 𝑡𝑡𝑡 (𝑏𝑏) − 𝐹𝐹 (𝑎𝑎) .
𝑎𝑎
(8)
We de�ne right-dense continuous (𝑟𝑟𝑟𝑟-continuous) functions
in a similar way. If 𝑓𝑓 is 𝑟𝑟𝑟𝑟-continuous, then there exists 𝐹𝐹
such that 𝐹𝐹Δ (𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 for any 𝑡𝑡 𝑡𝑡𝑡𝜅𝜅 , and we de�ne the delta
integral by
𝑏𝑏
󵐐󵐐 𝑓𝑓 (𝑡𝑡) Δ𝑡𝑡 𝑡𝑡𝑡 (𝑏𝑏) − 𝐹𝐹 (𝑎𝑎) .
𝑎𝑎
(9)
Discrete Dynamics in Nature and Society
3
2.2. Cones in Banach Spaces. In this paper 𝕋𝕋 is a time scale
with 0 ∈ 𝕋𝕋𝜅𝜅 and 𝑇𝑇 𝑇𝑇𝑇𝜅𝜅 . We use ℝ+ and ℝ+0 to denote,
respectively, the set of positive and nonnegative real numbers.
By [0, 𝑇𝑇𝑇𝕋𝕋 we denote the set [0, 𝑇𝑇𝑇 𝑇 𝑇𝑇. Similarly, (0, 𝑇𝑇𝑇𝕋𝕋 =
(0, 𝑇𝑇𝑇 𝑇 𝑇𝑇. Let 𝐸𝐸 𝐸 𝐸𝑙𝑙𝑙𝑙 ([0, 𝑇𝑇𝑇𝕋𝕋 , ℝ). It follows that 𝐸𝐸 is a
Banach space with the norm ‖𝑢𝑢𝑢𝑢 𝑢𝑢𝑢[0,𝑇𝑇𝑇𝕋𝕋 |𝑢𝑢𝑢𝑢𝑢𝑢𝑢.
�e�nition 1. Let 𝐸𝐸 be a real Banach space. A nonempty,
closed, and convex set 𝑃𝑃 𝑃 𝑃𝑃 is called a cone if it satis�es
the following two conditions:
(i) 𝑢𝑢 𝑢𝑢𝑢, 𝜆𝜆 𝜆 𝜆, implies 𝜆𝜆𝜆𝜆𝜆𝜆𝜆;
(ii) 𝑢𝑢 𝑢𝑢𝑢, −𝑢𝑢 𝑢𝑢𝑢, implies 𝑢𝑢 𝑢𝑢.
Every cone 𝑃𝑃 𝑃 𝑃𝑃 induces an ordering in 𝐸𝐸 given by
𝛼𝛼 󶀡󶀡𝑡𝑡𝑡𝑡 𝑡 (1 − 𝑡𝑡) 𝑦𝑦󶀱󶀱 ≥𝑡𝑡𝑡𝑡 (𝑥𝑥) + (1 − 𝑡𝑡) 𝛼𝛼 󶀡󶀡𝑦𝑦󶀱󶀱
(11)
�e�nition 2. Let 𝐸𝐸 be a real Banach space and 𝑃𝑃 𝑃 𝑃𝑃 a cone.
A function 𝛼𝛼 𝛼 𝛼𝛼 𝛼 𝛼+0 is called a nonnegative continuous
concave functional if 𝛼𝛼 is continuous and
for all 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 and 0 ≤𝑡𝑡𝑡𝑡.
Let 𝑎𝑎𝑎 𝑎𝑎, and 𝑟𝑟 be positive constants, 𝑃𝑃𝑟𝑟 = {𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑢
𝑟𝑟𝑟, 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃. e following
�xed point theorem provides the existence of at least three
positive solutions. e origin in 𝐸𝐸 is denoted by ∅. e proof
of the Leggett-Williams �xed point theorem can be found in
Guo and Lakshmikantham [42] or Leggett and Williams [45].
eorem 3 (Leggett-Williams’ eorem). Let 𝑃𝑃 be a cone in
a real Banach space 𝐸𝐸. Let 𝐺𝐺 𝐺 𝑃𝑃𝑐𝑐 → 𝑃𝑃𝑐𝑐 be a completely
continuous map and 𝛼𝛼 a nonnegative continuous concave
functional on 𝑃𝑃 such that 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 for all 𝑢𝑢 𝑢 𝑃𝑃𝑐𝑐 . Suppose
there exist positive constants 𝑎𝑎𝑎 𝑎𝑎, and 𝑑𝑑 with 0 < 𝑎𝑎 𝑎 𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎
such that
(i) {𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢 𝑢 𝑢 and 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 for all
𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢;
(ii) ‖𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 for all 𝑢𝑢 𝑢 𝑃𝑃𝑎𝑎 ;
(iii) 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 for all 𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 with ‖𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺.
en 𝐺𝐺 has at least three �xed points 𝑢𝑢1 , 𝑢𝑢2 , and 𝑢𝑢3 satisfying
3. Main Results
𝑡𝑡
(13)
𝑢𝑢 (𝑡𝑡) = − 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠 𝑠𝑠𝑠
0
where
𝑠𝑠
0
(10)
𝑎𝑎 𝑎𝑎𝑎 󶀡󶀡𝑢𝑢2 󶀱󶀱 ,
Lemma 4 (see Lemma 3.1 of [17]). Assume that hypothesis
(H1) on function 𝑓𝑓 is satis�ed. en 𝑢𝑢 is a solution to (1)-(2)
if and only if 𝑢𝑢 𝑢𝑢𝑢 is a solution to the integral equation
𝑔𝑔 (𝑠𝑠) = 󵐐󵐐 𝜆𝜆𝜆 (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟 𝑟 𝑟𝑟𝑟
𝑢𝑢 𝑢 𝑢𝑢 iff 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
󶙱󶙱𝑢𝑢1 󶙱󶙱 < 𝑎𝑎𝑎
(H1) 𝑓𝑓 𝑓𝑓𝑓𝑓+ is a continuous function.
󶙱󶙱𝑢𝑢3 󶙱󶙱 > 𝑎𝑎𝑎𝑎𝑎 󶀡󶀡𝑢𝑢3 󶀱󶀱 <𝑏𝑏𝑏
(12)
We prove existence of three positive solutions to different 𝑝𝑝Laplacian problems on time scales: in Section 3.1 we study
problem (1)-(2), in Section 3.2 problem (3), and �nally (4) in
Section 3.3.
3.1. Nonlocal ermistor Problem. By a solution 𝑢𝑢 𝑢𝑢𝑢𝑢𝑢
of (1)-(2) we mean a delta differentiable function such that 𝑢𝑢Δ
∇
and (|𝑢𝑢Δ |𝑝𝑝𝑝𝑝 𝑢𝑢Δ ) are both continuous on 𝕋𝕋𝜅𝜅𝜅𝜅 and 𝑢𝑢 satis�es
(1)-(2). We consider the following hypothesis:
𝐴𝐴𝐴𝐴𝐴𝑝𝑝 󶀢󶀢𝑢𝑢Δ (0)󶀲󶀲 = −
ℎ (𝑢𝑢 (𝑡𝑡)) =
𝐵𝐵𝐵𝐵𝐵 (0) =
𝑇𝑇
󶀣󶀣∫0
𝜆𝜆𝜆𝜆 𝜂𝜂
󵐐󵐐 ℎ (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟𝑟
1 − 𝛽𝛽 0
𝜆𝜆𝜆𝜆 (𝑢𝑢 (𝑡𝑡))
𝑓𝑓 (𝑢𝑢 (𝜏𝜏)) ∇𝜏𝜏󶀳󶀳
2
,
𝑇𝑇
𝜂𝜂
1
󶁆󶁆󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠𝑠𝑠𝑠𝑠 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠󶁖󶁖 .
1 − 𝛽𝛽 0
0
(14)
Lemma 5. Suppose (H1) holds. en a solution 𝑢𝑢 to (1)-(2)
satis�es 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 for 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝕋𝕋 .
𝜂𝜂
Proof. We have 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴0 ℎ(𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢. en,
𝑠𝑠
𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔0 ℎ(𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢. It follows that 𝜙𝜙𝑝𝑝 (𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔.
Since 0 <𝛽𝛽𝛽𝛽, we also have
𝑢𝑢 (0) = 𝐵𝐵
=
≥
𝑇𝑇
𝜂𝜂
1
󶁆󶁆󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠𝑠𝑠𝑠𝑠 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠󶁖󶁖
1 − 𝛽𝛽 0
0
𝑇𝑇
𝜂𝜂
1
󶁆󶁆𝛽𝛽 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠𝑠𝑠 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠󶁖󶁖
1 − 𝛽𝛽
0
0
≥ 0,
𝑇𝑇
𝑢𝑢 (𝑇𝑇) = 𝑢𝑢 (0) − 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠
=
=
=
0
𝜂𝜂
−𝛽𝛽
󵐐󵐐 𝜙𝜙 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠
1 − 𝛽𝛽 0 𝑞𝑞
+
𝑇𝑇
𝑇𝑇
1
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠
1 − 𝛽𝛽 0
0
𝑇𝑇
𝛽𝛽
−𝛽𝛽 𝜂𝜂
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠
1 − 𝛽𝛽 0
1 − 𝛽𝛽 0
𝑇𝑇
𝜂𝜂
𝛽𝛽
󶁆󶁆󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠󶁖󶁖
1 − 𝛽𝛽 0
0
≥ 0.
(15)
4
Discrete Dynamics in Nature and Society
If 𝑡𝑡 𝑡 𝑡𝑡𝑡 𝑡𝑡𝑡𝕋𝕋 , then
Proof. (i) holds clearly from above. (ii) Suppose that 𝐷𝐷 𝐷𝐷𝐷 is
a bounded set and let 𝑢𝑢 𝑢𝑢𝑢. en,
𝑡𝑡
𝑡𝑡
𝑢𝑢 (𝑡𝑡) = 𝑢𝑢 (0) − 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠
𝑇𝑇
0
≥ − 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠 𝑠𝑠 (0) = 𝑢𝑢 (𝑇𝑇)
≥ 0.
(16)
0
monotonicity of 𝜙𝜙𝑝𝑝 that 𝑢𝑢Δ is a nonincreasing function on
(0,𝑇𝑇𝑇𝕋𝕋 . Hence, 𝑢𝑢 is concave. In order to apply eorem 3, let
us de�ne the cone 𝑃𝑃 𝑃 𝑃𝑃 by
𝑃𝑃 𝑃 󶁁󶁁𝑢𝑢 𝑢𝑢𝑢 𝑢 𝑢𝑢 is nonnegative,
decreasing on [0,𝑇𝑇]𝕋𝕋 and concave on 𝐸𝐸󶁑󶁑 .
(17)
We also de�ne the nonnegative continuous concave functional 𝛼𝛼 𝛼 𝛼𝛼 𝛼 𝛼+0 by
𝜉𝜉 𝜉 󶀤󶀤0,
𝑡𝑡𝑡󶁡󶁡𝜉𝜉𝜉𝜉𝜉𝜉𝜉𝜉󶁱󶁱𝕋𝕋
𝑇𝑇
󶀴󶀴 ,
2
∀𝑢𝑢 𝑢𝑢𝑢𝑢
(18)
𝐺𝐺𝐺𝐺 (𝑡𝑡) = − 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠 𝑠𝑠𝑠
0
where 𝑔𝑔 and 𝐵𝐵 are as in Lemma 4.
Lemma 6. Let 𝐺𝐺 �e �e�ne� �y (19). en,
(i) 𝐺𝐺𝐺𝐺𝐺𝐺 𝐺 𝐺𝐺;
(ii) 𝐺𝐺 𝐺𝐺𝐺𝐺𝐺𝐺 is completely continuous.
𝑠𝑠
|𝐴𝐴| = 󶙥󶙥
0
0
𝑠𝑠
𝑇𝑇
0
󶀣󶀣∫0
𝜆𝜆𝜆𝜆 (𝑢𝑢 (𝑟𝑟))
(19)
𝑓𝑓 (𝑢𝑢 (𝜏𝜏)) ∇𝜏𝜏󶀳󶀳
𝜆𝜆 𝜆𝜆𝜆𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)
󶀡󶀡𝑇𝑇𝑇𝑇𝑇𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)󶀱󶀱
𝜆𝜆𝜆𝜆 𝜂𝜂
󵐐󵐐 ℎ (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟󶙥󶙥
1 − 𝛽𝛽 0
2
2
󶙀󶙀
∇𝑟𝑟𝑟𝑟𝑟󶀺󶀺 Δ𝑠𝑠󶙐󶙐 +|𝐵𝐵|
󶙘󶙘
∇𝑟𝑟𝑟𝑟𝑟󶀷󶀷 Δ𝑠𝑠 𝑠 |𝐵𝐵| ,
𝜂𝜂
𝑓𝑓 (𝑢𝑢 (𝑟𝑟))
󶙀󶙀 𝜆𝜆𝜆𝜆
󶙀󶙀
= 󶙐󶙐
∇𝑟𝑟󶙐󶙐
󵐐󵐐
2
𝑇𝑇
1 − 𝛽𝛽 0
󶀣󶀣∫0 𝑓𝑓 (𝑢𝑢 (𝜏𝜏)) ∇𝜏𝜏󶀳󶀳
󶙘󶙘
󶙘󶙘
≤
sup𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)
𝜆𝜆𝜆𝜆
𝜂𝜂𝜂
1 − 𝛽𝛽 󶀡󶀡𝑇𝑇𝑇𝑇𝑇𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)󶀱󶀱2
(20)
In the same way, we have
|𝐵𝐵| ≤
≤
𝑇𝑇
1
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠
1 − 𝛽𝛽 0
𝑇𝑇
𝜆𝜆 𝜆𝜆𝜆𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)
𝛽𝛽
1
𝜂𝜂󶀵󶀵󶀵󶀵 Δ𝑠𝑠𝑠
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀧󶀧
󶀥󶀥𝑠𝑠 𝑠
2
1 − 𝛽𝛽 0
1
−
𝛽𝛽
󶀡󶀡𝑇𝑇𝑇𝑇𝑇𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)󶀱󶀱
(21)
It follows that
𝑇𝑇
|𝐺𝐺𝐺𝐺 (𝑡𝑡)| ≤ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀧󶀧
0
𝜆𝜆 𝜆𝜆𝜆𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)
󶀡󶀡𝑇𝑇𝑇𝑇𝑇𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)󶀱󶀱
As a consequence, we get
It is easy to see that problem (1)-(2) has a solution 𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢
if and only if 𝑢𝑢 is a ��ed point of the operator 𝐺𝐺 𝐺𝐺𝐺𝐺𝐺𝐺
de�ned by
𝑡𝑡
𝑇𝑇
≤ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀧󶀧󵐐󵐐
∫0 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝜆 𝜆. Since 𝐴𝐴 𝐴𝐴𝐴𝑝𝑝 (𝑢𝑢Δ (0))≤0, then 𝑢𝑢Δ ≤
0. is means that ||𝑢𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢𝑢, inf𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝕋𝕋 𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢.
Moreover, 𝜙𝜙𝑝𝑝 (𝑢𝑢Δ (𝑠𝑠𝑠𝑠 is nonincreasing, which implies with the
min 𝑢𝑢 (𝑡𝑡) ,
0
󶙘󶙘
On the other hand, we have 𝜙𝜙𝑝𝑝 (𝑢𝑢Δ (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑝𝑝 (𝑢𝑢Δ (0)) −
𝛼𝛼 (𝑢𝑢) =
𝑡𝑡
󶙀󶙀
≤ 󶙐󶙐− 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀪󶀪󵐐󵐐
0
Consequently, 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 for 𝑡𝑡 𝑡 𝑡𝑡𝑡 𝑡𝑡𝑡𝕋𝕋 .
𝑠𝑠
|𝐺𝐺𝐺𝐺 (𝑡𝑡)| = 󶙥󶙥− 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠 𝑠𝑠󶙥󶙥
‖𝐺𝐺𝐺𝐺‖ ≤
≤
2
󶀥󶀥𝑠𝑠 𝑠
𝛽𝛽𝛽𝛽
󶀵󶀵󶀵󶀵 Δ𝑠𝑠 𝑠 |𝐵𝐵| .
1 − 𝛽𝛽
(22)
𝜆𝜆 𝜆𝜆𝜆𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)
𝛽𝛽𝛽𝛽
2 − 𝛽𝛽 𝑇𝑇
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀧󶀧
󶀵󶀵󶀵󶀵 Δ𝑠𝑠
󶀥󶀥𝑠𝑠 𝑠
2
1 − 𝛽𝛽 0
1 − 𝛽𝛽
󶀡󶀡𝑇𝑇𝑇𝑇𝑇𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)󶀱󶀱
𝑇𝑇
𝜆𝜆 𝜆𝜆𝜆𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)
𝛽𝛽𝛽𝛽
2
𝜙𝜙𝑞𝑞 󶀧󶀧
󶀵󶀵 Δ𝑠𝑠𝑠
󶀷󶀷 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀥󶀥𝑠𝑠 𝑠
2
1 − 𝛽𝛽
1 − 𝛽𝛽
0
󶀡󶀡𝑇𝑇𝑇𝑇𝑇𝑢𝑢𝑢𝑢𝑢 𝑓𝑓 (𝑢𝑢)󶀱󶀱
(23)
en 𝐺𝐺𝐺𝐺𝐺𝐺 is bounded on the whole bounded set 𝐷𝐷.
Moreover, if 𝑡𝑡1 , 𝑡𝑡2 ∈ [0,𝑇𝑇𝑇𝕋𝕋 and 𝑢𝑢 𝑢𝑢𝑢, then we have for a
positive constant 𝑐𝑐
𝑡𝑡2
󶙡󶙡𝐺𝐺𝐺𝐺 󶀡󶀡𝑡𝑡2 󶀱󶀱 − 𝐺𝐺𝐺𝐺 󶀡󶀡𝑡𝑡1 󶀱󶀱󶀱󶀱 ≤ 󶙦󶙦󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠󶙦󶙦 ≤ 𝑐𝑐 󶙡󶙡𝑡𝑡2 − 𝑡𝑡1 󶙡󶙡 . (24)
𝑡𝑡1
We see that the right-hand side of the above inequality goes
uniformly to zero when |𝑡𝑡2 − 𝑡𝑡1 | →0. en by a standard
application of the Arzela-Ascoli theorem we have that 𝐺𝐺 𝐺
𝑃𝑃 𝑃 𝑃𝑃 is completely continuous.
Discrete Dynamics in Nature and Society
5
We can also easily obtain the following properties.
We have
Lemma 7. (i) 𝛼𝛼𝛼𝛼𝛼𝛼 𝛼 𝛼𝛼𝛼𝛼𝛼 𝛼 𝛼𝛼𝛼 𝛼 𝛼𝛼𝛼𝛼 for all 𝑢𝑢 𝑢 𝑢𝑢;
(ii) 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝛼 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝛼 𝛼𝛼𝛼;
(iii) ‖𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺.
ℎ (𝑢𝑢 (𝑡𝑡)) =
We now state the main result of Section 3.1.
eorem 8. Suppose that (𝐻𝐻𝐻𝐻 is veri�ed and there e�ist
positive constants 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, and 𝑑𝑑 such that 0 < 𝜁𝜁𝜁𝜁𝜁𝜁𝜁1 < 𝑏𝑏 𝑏
𝑑𝑑 𝑑𝑑𝑑2 𝑐𝑐𝑐𝑐𝑞𝑞 (1/T) < 𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑1 , with 𝜁𝜁 𝜁𝜁𝜁𝜁2 /(1− 𝛽𝛽𝛽𝛽𝛽𝛽𝑞𝑞 (1/𝑇𝑇𝑇.
One further imposes 𝑓𝑓 to satisfy the following hypotheses:
(H2) min0≤𝑢𝑢𝑢𝑢𝑢1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑓 𝑓𝑓2 /(𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑝𝑝 (𝑎𝑎𝑎𝑎 uniformly for all
𝑡𝑡 𝑡 𝑡𝑡𝑡 𝑡𝑡𝑡𝕋𝕋 ;
(H3) min0≤𝑢𝑢𝑢𝑢𝑢1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑓 𝑓𝑓2 /(𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑝𝑝 (𝑐𝑐𝑐𝑐 uniformly for all
𝑡𝑡 𝑡 𝑡𝑡𝑡 𝑡𝑡𝑡𝕋𝕋 ;
(H4) min𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓𝑓𝑓 𝑓 𝑓𝑓𝑝𝑝 (𝑏𝑏𝑏𝑏1 ) uniformly for all 𝑡𝑡 𝑡 𝑡𝑡𝑡 𝑡𝑡𝑡𝕋𝕋 ,
where
𝐵𝐵1 =
󶀡󶀡1− 𝛽𝛽󶀱󶀱
𝜆𝜆
󶙢󶙢𝜙𝜙𝑝𝑝 (𝑇𝑇𝑇𝑇𝑇)󶙢󶙢 𝜙𝜙𝑝𝑝 󶀧󶀧
󶀷󶀷 . (25)
2
𝛽𝛽𝛽𝛽
󶀡󶀡𝑇𝑇 𝑇𝑇𝑇𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓 (𝑢𝑢)󶀱󶀱
en the boundary value problem (1)-(2) has at least three
positive solutions 𝑢𝑢1 , 𝑢𝑢2 , and 𝑢𝑢3 , verifying
󶙱󶙱𝑢𝑢1 󶙱󶙱 < 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 󶀡󶀡𝑢𝑢2 󶀱󶀱 ,
󶙱󶙱𝑢𝑢3 󶙱󶙱 > 𝑎𝑎𝑎𝑎𝑎 󶀡󶀡𝑢𝑢3 󶀱󶀱 < 𝑏𝑏𝑏
(26)
Proof. e proof passes by several lemmas. We have already
seen in Lemma 6 that the operator 𝐺𝐺 is completely continuous. We now show the following Lemma.
Lemma 9. e following relations hold:
𝐺𝐺𝑃𝑃𝑐𝑐1 ⊂ 𝑃𝑃𝑐𝑐1 ,𝐺𝐺𝑃𝑃𝑎𝑎1 ⊂ 𝑃𝑃𝑎𝑎1 .
(27)
Proof. Obviously, 𝐺𝐺𝑃𝑃𝑎𝑎1 ⊂ 𝑃𝑃. Moreover, for all 𝑢𝑢 𝑢 𝑃𝑃𝑎𝑎1 , we
have 0≤𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢1 . On the other hand we have
𝑡𝑡
𝐺𝐺𝐺𝐺 (𝑡𝑡) =− 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠
0
𝑇𝑇
1
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠
1− 𝛽𝛽 0
𝜂𝜂
𝛽𝛽
−
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠𝑠
1− 𝛽𝛽 0
and for all 𝑢𝑢 𝑢 𝑢𝑢𝑃𝑃𝑎𝑎1 we have 0≤𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢1 . en,
𝑇𝑇
1
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠𝑠
|𝐺𝐺𝐺𝐺| ≤
1− 𝛽𝛽 0
(28)
𝑇𝑇
󶀣󶀣∫0
𝑠𝑠
𝜆𝜆𝜆𝜆 (𝑢𝑢 (𝑡𝑡))
𝑓𝑓 (𝑢𝑢 (𝜏𝜏)) ∇𝜏𝜏󶀳󶀳
𝑔𝑔 (𝑠𝑠) = 󵐐󵐐 𝜆𝜆𝜆 (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟 𝑟
0
2
𝜆𝜆𝜆𝜆 𝜂𝜂
󵐐󵐐 ℎ (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟
1− 𝛽𝛽 0
𝜆𝜆𝜆𝜆 𝑇𝑇
≤ 󶀦󶀦𝜆𝜆𝜆
󵐐󵐐 ℎ (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟󶀶󶀶
1− 𝛽𝛽 0
≤
(30)
𝑇𝑇
𝜆𝜆
󵐐󵐐 ℎ (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟𝑟
1− 𝛽𝛽 0
Using (H2) it follows that
1
𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 ≤ 𝑎𝑎𝑎𝑎𝑎𝑎𝑞𝑞 󶀤󶀤 󶀴󶀴 .
𝑇𝑇
en we get
(31)
(32)
|𝐺𝐺𝐺𝐺| ≤ 𝑎𝑎1 ,𝐺𝐺𝑃𝑃𝑎𝑎1 ⊂ 𝑃𝑃𝑎𝑎1 .
Similarly, using (H3) we get 𝐺𝐺𝑃𝑃𝑐𝑐1 ⊂ 𝑃𝑃𝑐𝑐1 .
Lemma 10. e set
is nonempty, and
(33)
{𝑢𝑢 𝑢 𝑢𝑢 (𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼) ∣ 𝛼𝛼 (𝑢𝑢) > 𝑏𝑏}
if 𝑢𝑢 𝑢 𝑢𝑢 (𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼) .
𝛼𝛼 (𝐺𝐺𝐺𝐺) > 𝑏𝑏𝑏
(34)
Proof. Let 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢. en, 𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢, and
𝛼𝛼𝛼𝛼𝛼𝛼 𝛼 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼. e �rst part of the lemma is proved.
For 𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 we have 𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏. If 𝑡𝑡 𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝕋𝕋 , then
𝛼𝛼 (𝐺𝐺𝐺𝐺) = (𝐺𝐺𝐺𝐺)(𝑇𝑇𝑇𝑇𝑇)
=− 󵐐󵐐
≥
𝑇𝑇𝑇𝑇𝑇
0
𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠 𝑠𝑠
(35)
𝑇𝑇
𝛽𝛽
󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠𝑠
1− 𝛽𝛽 𝑇𝑇𝑇𝑇𝑇
Since 𝐴𝐴 𝐴𝐴, we have by using (H4)
𝑠𝑠
𝑔𝑔 (𝑠𝑠) =𝜆𝜆 󵐐󵐐 ℎ (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟 𝑟𝑟𝑟
0
𝑠𝑠
≥𝜆𝜆𝜆𝜆 ℎ (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟
0
≥𝜆𝜆𝜆𝜆
(29)
,
≥𝜆𝜆
𝑠𝑠
0
𝑓𝑓 (𝑢𝑢)
2
󶀡󶀡𝑇𝑇 𝑇𝑇𝑇𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓 (𝑢𝑢)󶀱󶀱
󶀡󶀡𝑏𝑏𝑏𝑏1 󶀱󶀱
𝑝𝑝𝑝𝑝
󶀡󶀡𝑇𝑇 𝑇𝑇𝑇𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓 (𝑢𝑢)󶀱󶀱
2
𝑠𝑠𝑠
∇𝑢𝑢
(36)
6
Discrete Dynamics in Nature and Society
Using the fact that 𝜙𝜙𝑞𝑞 is nondecreasing we get
𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 ≥ 𝜙𝜙𝑞𝑞 󶀧󶀧𝜆𝜆
𝑝𝑝𝑝𝑝
󶀡󶀡𝑏𝑏𝑏𝑏1 󶀱󶀱
󶀡󶀡𝑇𝑇 𝑇𝑇𝑇𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓 (𝑢𝑢)󶀱󶀱
≥ 𝑏𝑏𝑏𝑏1 𝜙𝜙𝑞𝑞 󶀧󶀧
𝜆𝜆
𝑠𝑠󶀷󶀷
2
󶀡󶀡𝑇𝑇 𝑇𝑇𝑇𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓 (𝑢𝑢)󶀱󶀱
Using the expression of 𝐵𝐵1
𝛼𝛼 (𝐺𝐺𝐺𝐺) ≥
2
satisfying the boundary conditions
(37)
󶀷󶀷 𝜙𝜙𝑞𝑞 (𝑠𝑠) .
𝑇𝑇
𝛽𝛽
𝜆𝜆
𝑏𝑏𝑏𝑏 𝜙𝜙 󶀧󶀧
󶀷󶀷 󵐐󵐐 𝜙𝜙𝑞𝑞 (𝑠𝑠) Δ𝑠𝑠
1 − 𝛽𝛽 1 𝑞𝑞 󶀡󶀡𝑇𝑇 𝑇𝑇𝑇𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓 (𝑢𝑢)󶀱󶀱2
𝑇𝑇𝑇𝑇𝑇
𝛽𝛽
𝜆𝜆
𝜙𝜙𝑞𝑞 󶀧󶀧
≥ 𝑏𝑏𝑏𝑏1
󶀷󶀷 𝜙𝜙𝑞𝑞 (𝑇𝑇 𝑇 𝑇𝑇) 𝜉𝜉
2
1 − 𝛽𝛽
󶀡󶀡𝑇𝑇 𝑇𝑇𝑇𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓 (𝑢𝑢)󶀱󶀱
≥ 𝑏𝑏𝑏
(38)
Lemma 11. For all 𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢𝑢𝑢 𝑢𝑢1 ) with ‖𝐺𝐺𝐺𝐺𝐺 𝐺 𝐺𝐺 one has
(39)
𝛼𝛼 (𝐺𝐺𝐺𝐺) > 𝑏𝑏𝑏
Proof. If 𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢𝑢𝑢 𝑢𝑢1 ) and ‖𝐺𝐺𝐺𝐺𝐺 𝐺 𝐺𝐺, then 0 ≤𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑢𝑢1 .
Using hypothesis (H3) and the fact that 0 < 𝛽𝛽 𝛽𝛽, it follows
that
𝛼𝛼 (𝐺𝐺𝐺𝐺) = 𝐺𝐺𝐺𝐺 (𝑇𝑇 𝑇 𝑇𝑇)
= − 󵐐󵐐
𝑇𝑇𝑇𝑇𝑇
0
𝑇𝑇
𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠 𝑠𝑠
≥ − 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀡󶀡𝑔𝑔 (𝑠𝑠)󶀱󶀱 Δ𝑠𝑠 𝑠 𝑠𝑠𝑠𝑠 (0)
0
1
≥ ‖𝐺𝐺𝐺𝐺‖ − 𝑇𝑇2 𝑐𝑐𝑐𝑐𝑞𝑞 󶀤󶀤 󶀴󶀴
𝑇𝑇
(40)
1
≥ 𝑑𝑑𝑑𝑑𝑑2 𝑐𝑐𝑐𝑐𝑞𝑞 󶀤󶀤 󶀴󶀴
𝑇𝑇
> 𝑏𝑏𝑏
Gathering Lemmas 4 to 11 and applying eorem 3, there
exist at least three positive solutions 𝑢𝑢1 , 𝑢𝑢2 , and 𝑢𝑢3 to (1)-(2)
verifying
󶙱󶙱𝑢𝑢1 󶙱󶙱 < 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 󶀡󶀡𝑢𝑢2 󶀱󶀱 ,
󶙱󶙱𝑢𝑢3 󶙱󶙱 > 𝑎𝑎𝑎𝑎𝑎 󶀡󶀡𝑢𝑢3 󶀱󶀱 < 𝑏𝑏𝑏
(41)
Example 12. Let 𝕋𝕋 𝕋 𝕋𝕋𝕋𝕋𝕋𝕋𝕋𝕋ℕ0 } ∪ {1}, where ℕ0 denotes
the set of all nonnegative integers. Consider the 𝑝𝑝-Laplacian
dynamic equation
Δ
∇
−󶀢󶀢𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢 (𝑡𝑡)󶀲󶀲󶀲󶀲 =
𝑇𝑇
𝜆𝜆𝜆𝜆 (𝑢𝑢 (𝑡𝑡))
󶀣󶀣∫0 𝑓𝑓 (𝑢𝑢 (𝜏𝜏)) ∇𝜏𝜏󶀳󶀳
2
,𝑡𝑡𝑡 (0, 𝑇𝑇)𝕋𝕋 , (42)
1
𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢Δ (0)󶀲󶀲 − 𝛽𝛽 󶀤󶀤𝜙𝜙𝑝𝑝 󶀤󶀤𝑢𝑢Δ 󶀤󶀤 󶀴󶀴󶀴󶀴󶀴󶀴 = 0,
4
(43)
1
𝑢𝑢 (1) − 𝛽𝛽𝛽𝛽 󶀤󶀤 󶀴󶀴 = 0,
4
where 𝑝𝑝 𝑝𝑝𝑝𝑝, 𝑞𝑞 𝑞𝑞, 𝜂𝜂 𝜂𝜂𝜂𝜂, 𝛽𝛽 𝛽𝛽𝛽𝛽, 𝜆𝜆 𝜆𝜆, 𝑇𝑇 𝑇𝑇, and
2√2,
0 ≤𝑢𝑢𝑢𝑢𝑢
󶀂󶀂
󶀒󶀒
󶀒󶀒
󶀒󶀒
3
󶀒󶀒
󶀒󶀒
󶀒󶀒 4 (𝑢𝑢𝑢𝑢) + 2√2, 1 ≤𝑢𝑢𝑢 ,
󶀒󶀒
󶀒󶀒
2
󶀒󶀒
󶀒󶀒
𝑓𝑓 (𝑢𝑢) = 󶀊󶀊
3
󶀒󶀒
≤𝑢𝑢𝑢𝑢𝑢𝑢
2 + 2√2,
󶀒󶀒
󶀒󶀒
󶀒󶀒
2
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒 2𝑢𝑢𝑢𝑢√2 − 18, 10 ≤𝑢𝑢𝑢𝑢𝑢𝑢
󶀚󶀚
(44)
Choose 𝑎𝑎1 = 1 = 2𝑎𝑎, 𝑏𝑏 𝑏𝑏𝑏𝑏, 𝑐𝑐1 = 16 = 2𝑐𝑐, and 𝑑𝑑𝑑𝑑𝑑. It is
easy to see that 𝜁𝜁 𝜁𝜁, 𝐵𝐵1 = 1/(2(2 + √2)) and
min 󶁁󶁁𝑓𝑓 (𝑢𝑢) ∶ 𝑢𝑢 𝑢 󶁡󶁡0, 𝑎𝑎1 󶁱󶁱󶁱󶁱 = 2√2
≥
𝜆𝜆2
2
=
= 2√2,
𝑇𝑇 󶀡󶀡1 − 𝛽𝛽󶀱󶀱 𝜙𝜙𝑝𝑝 (𝑎𝑎) √𝑎𝑎
≥
𝜆𝜆2
2
1
=
=
,
√
√
𝑇𝑇 󶀡󶀡1 − 𝛽𝛽󶀱󶀱 𝜙𝜙𝑝𝑝 (𝑐𝑐)
𝑐𝑐
2
min 󶁁󶁁𝑓𝑓 (𝑢𝑢) ∶ 𝑢𝑢 𝑢 󶁡󶁡0,𝑐𝑐1 󶁱󶁱󶁱󶁱 = 2√2
min 󶁁󶁁𝑓𝑓 (𝑢𝑢) ∶ 𝑢𝑢 𝑢 [𝑏𝑏𝑏 𝑏𝑏]󶁑󶁑 = 2 + 2√2
≥ 𝜙𝜙𝑝𝑝 󶀡󶀡𝑏𝑏𝑏𝑏1 󶀱󶀱 = 󵀄󵀄𝑏𝑏𝑏𝑏1 = 󵀌󵀌
3
4 󶀢󶀢2 + 2√2󶀲󶀲
(45)
.
en, hypotheses (H1)�(H4) are satis�ed. erefore, by
eorem 8, problem (42)-(43) has at least three positive
solutions.
3.2. Quasilinear Elliptic Problem. We are interested in this
section in the study of the following quasilinear elliptic
problem:
∇
−󶀢󶀢𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢Δ (𝑡𝑡)󶀲󶀲󶀲󶀲 = 𝑓𝑓 (𝑢𝑢 (𝑡𝑡)) + ℎ (𝑡𝑡) ,𝑡𝑡𝑡 (0, 𝑇𝑇)𝕋𝕋 ,
Δ
(46)
𝑢𝑢 (0) = 0,𝑢𝑢 (𝑇𝑇) −𝑢𝑢 󶀡󶀡𝜂𝜂󶀱󶀱 = 0,
where 𝜂𝜂 𝜂𝜂𝜂𝜂𝜂𝜂𝜂𝕋𝕋 . We assume the following hypotheses:
(A1) function 𝑓𝑓 𝑓 𝑓 𝑓 𝑓+0 is continuous;
(A2) function ℎ ∶ (0, 𝑇𝑇𝑇𝕋𝕋 →ℝ+0 is le dense continuous,
that is,
ℎ ∈ ℂ𝑙𝑙𝑙𝑙 󶀡󶀡(0, 𝑇𝑇)𝕋𝕋 ,ℝ+0 󶀱󶀱 ,
ℎ ∈ 𝐿𝐿∞ .
(47)
Similarly as in Section 3.1, we prove existence of solutions
by constructing an operator whose �xed points are solutions
Discrete Dynamics in Nature and Society
7
to (46). e main ingredient is, again, the Leggett-Williams
�xed point theorem (eorem 3). We can easily see that (46)
is equivalent to the integral equation
𝑇𝑇
𝑢𝑢 (𝑡𝑡) = 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶
𝜂𝜂
𝑡𝑡
𝑇𝑇
(48)
+ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠𝑠
0
𝑠𝑠
On the other hand, we have −(𝜙𝜙𝑝𝑝 (𝑢𝑢Δ ))∇ = 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓. Since
𝑓𝑓, ℎ ≥ 0, we have (𝜙𝜙𝑝𝑝 (𝑢𝑢Δ ))∇ ≤ 0 and 𝜙𝜙𝑝𝑝 (𝑢𝑢Δ (𝑡𝑡2 )) ≤ 𝜙𝜙𝑝𝑝 (𝑢𝑢Δ (𝑡𝑡1 ))
for any 𝑡𝑡1 , 𝑡𝑡2 ∈ [0, 𝑇𝑇𝑇𝕋𝕋 with 𝑡𝑡1 ≤ 𝑡𝑡2 . It follows that 𝑢𝑢Δ (𝑡𝑡2 ) ≤
𝑢𝑢Δ (𝑡𝑡1 ) for 𝑡𝑡1 ≤ 𝑡𝑡2 . Hence, 𝑢𝑢Δ (𝑡𝑡𝑡 is a decreasing function on
[0, 𝑇𝑇𝑇𝕋𝕋 . en, 𝑢𝑢 is concave. In order to apply eorem 3 we
de�ne the cone
𝑃𝑃 𝑃 󶁁󶁁𝑢𝑢 𝑢 𝑢𝑢 𝑢 𝑢𝑢 is nonnegative,
increasing on [0, 𝑇𝑇]𝕋𝕋 , and concave on 𝐸𝐸󶁑󶁑 .
(49)
For 𝜉𝜉 𝜉𝜉𝜉𝜉𝜉𝜉𝜉𝜉𝜉 we also de�ne the nonnegative continuous
concave functional 𝛼𝛼 𝛼 𝛼𝛼 𝛼 𝛼+0 by
𝛼𝛼 (𝑢𝑢) =
min 𝑢𝑢 (𝑡𝑡) ,
𝑡𝑡𝑡󶁡󶁡𝜉𝜉𝜉𝜉𝜉𝜉𝜉𝜉󶁱󶁱𝕋𝕋
and the operator 𝐹𝐹 𝐹𝐹𝐹𝐹𝐹𝐹 by
𝑢𝑢 𝑢𝑢𝑢
𝑇𝑇
𝐹𝐹𝐹𝐹 (𝑡𝑡) = 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶
𝑡𝑡
𝜂𝜂
𝑇𝑇
(50)
(51)
𝑠𝑠
It is easy to see that (46) has a solution 𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 if and only
if 𝑢𝑢 is a �xed point of the operator 𝐹𝐹. For convenience, we
introduce the following notation:
𝐴𝐴 𝐴
𝑎𝑎
𝑎𝑎𝑎‖ℎ‖1/(𝑝𝑝𝑝𝑝𝑝
∞
𝛼𝛼𝛼𝛼
𝛾𝛾 𝛾 (1 + 𝑇𝑇) 𝜙𝜙𝑞𝑞 (𝑇𝑇) ,
, where 𝛼𝛼 𝛼𝛼𝛼𝑞𝑞 󶀢󶀢2𝑝𝑝𝑝𝑝 󶀲󶀲 𝜙𝜙𝑞𝑞 (𝑇𝑇) (𝑇𝑇 𝑇𝑇) ,
𝐵𝐵 𝐵𝐵𝐵𝑝𝑝 󶀡󶀡𝑇𝑇 𝑇𝑇𝑇󶀱󶀱 .
(52)
eorem 13. Suppose that hypotheses (A1) and (A2) are
satis�ed; there e�ist positive �onstants 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, and 𝑑𝑑 with
0 < 𝛾𝛾𝛾𝛾
= 𝑎𝑎1 < 𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑝𝑝𝑝𝑝 (𝑇𝑇 𝑇𝑇𝑇) 𝜙𝜙𝑞𝑞 (𝑇𝑇 𝑇𝑇𝑇) 󶀢󶀢󶀢ℎ‖1/(𝑝𝑝𝑝𝑝𝑝 + 𝑏𝑏𝑏𝑏󶀲󶀲
< 𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑1
and, in addition to (A1) and (A2), that 𝑓𝑓 satis�es
(A3) max0≤𝑢𝑢𝑢𝑢𝑢 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑝𝑝 (𝑎𝑎𝑎𝑎𝑎;
(A5) min𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑝𝑝 (𝑏𝑏𝑏𝑏𝑏.
en problem (46) has at least three positive solutions 𝑢𝑢1 , 𝑢𝑢2 ,
and 𝑢𝑢3 , verifying
󶙱󶙱𝑢𝑢1 󶙱󶙱 < 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 󶀡󶀡𝑢𝑢2 󶀱󶀱 ,
(53)
󶙱󶙱𝑢𝑢3 󶙱󶙱 > 𝑎𝑎𝑎𝑎𝑎 󶀡󶀡𝑢𝑢3 󶀱󶀱 < 𝑏𝑏𝑏
(54)
Proof. As done for eorem 8, the proof is divided in several
steps. We �rst show that 𝐹𝐹 𝐹𝐹𝐹𝐹𝐹𝐹 is completely
continuous. Indeed, 𝐹𝐹 is obviously continuous. Let
𝑈𝑈𝛿𝛿 = {𝑢𝑢 𝑢𝑢𝑢 𝑢 ‖𝑢𝑢‖ ≤ 𝛿𝛿} .
(55)
󶙡󶙡𝐹𝐹𝐹𝐹 󶀡󶀡𝑡𝑡2 󶀱󶀱 − 𝐹𝐹𝐹𝐹 󶀡󶀡𝑡𝑡1 󶀱󶀱󶀱󶀱 ≤ 𝑐𝑐 󶙡󶙡𝑡𝑡2 − 𝑡𝑡1 󶙡󶙡 ,
(56)
It is easy to see that for 𝑢𝑢 𝑢𝑢𝑢𝛿𝛿 there exists a constant 𝑐𝑐 𝑐𝑐
such that |𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹. On the other hand, let 𝑡𝑡1 , 𝑡𝑡2 ∈ (0, 𝑇𝑇𝑇𝕋𝕋 ,
𝑢𝑢 𝑢𝑢𝑢𝛿𝛿 . en there exists a positive constant 𝑐𝑐 such that
which converges uniformly to zero when |𝑡𝑡2 −𝑡𝑡1 | tends to zero.
Using the Arzela-Ascoli theorem we conclude that 𝐹𝐹 𝐹𝐹𝐹𝐹
𝑃𝑃 is completely continuous.
We now show that
𝐹𝐹𝑃𝑃𝑐𝑐1 ⊂ 𝑃𝑃𝑐𝑐1 ,
+ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠𝑠
0
(A4) max0≤𝑢𝑢𝑢𝑢𝑢 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑝𝑝 (𝑐𝑐𝑐𝑐𝑐;
𝐹𝐹𝑃𝑃𝑎𝑎1 ⊂ 𝑃𝑃𝑎𝑎1 .
(57)
For all 𝑢𝑢 𝑢 𝑃𝑃𝑎𝑎1 we have 0 ≤ 𝑢𝑢 𝑢𝑢𝑢1 and
𝑇𝑇
‖𝐹𝐹 (𝑢𝑢)‖ ≤ 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀢󶀢󶀢𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 + ‖ℎ‖∞ 󶀲󶀲 ∇𝑟𝑟󶀶󶀶
𝑇𝑇
𝜂𝜂
𝑇𝑇
+ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀢󶀢(𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 + ‖ℎ‖∞ 󶀲󶀲 ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠
0
𝑠𝑠
≤ 𝜙𝜙𝑞𝑞 󶀢󶀢󶀢󶀢󶀢𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 + ‖ℎ‖∞ 󶀲󶀲 󶀲󶀲𝑇𝑇 𝑇𝑇𝑇󶀱󶀱󶀱󶀱
𝑇𝑇
+ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀢󶀢󶀢󶀢(𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 + ‖ℎ‖∞ 󶀲󶀲 (𝑇𝑇 𝑇𝑇𝑇)󶀲󶀲 Δ𝑠𝑠
0
≤ 𝜙𝜙𝑞𝑞 󶀢󶀢(𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 + ‖ℎ‖∞ 󶀲󶀲 𝜙𝜙𝑞𝑞 (𝑇𝑇)
𝑇𝑇
+ 𝜙𝜙𝑞𝑞 󶀢󶀢(𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 + ‖ℎ‖∞ 󶀲󶀲 󵐐󵐐 𝜙𝜙𝑞𝑞 (𝑇𝑇 𝑇𝑇𝑇) Δ𝑠𝑠
0
≤ 𝜙𝜙𝑞𝑞 󶀤󶀤(𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 + 󶀢󶀢‖ℎ‖1/(𝑝𝑝𝑝𝑝𝑝
󶀲󶀲
∞
𝑝𝑝 −1
𝑝𝑝 −1
+ 𝜙𝜙𝑞𝑞 󶀤󶀤(𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 + 󶀢󶀢‖ℎ‖1/(𝑝𝑝𝑝𝑝𝑝
󶀲󶀲
∞
≤ 𝜙𝜙𝑞𝑞 󶀤󶀤󶀤𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 + 󶀢󶀢󶀢ℎ‖1/(𝑝𝑝𝑝𝑝𝑝
󶀲󶀲
∞
󶀴󶀴 𝜙𝜙𝑞𝑞 (𝑇𝑇)
𝑝𝑝 −1
󶀴󶀴 𝜙𝜙𝑞𝑞 (𝑇𝑇) 𝑇𝑇
󶀴󶀴 𝜙𝜙𝑞𝑞 (𝑇𝑇) (𝑇𝑇 𝑇𝑇) .
(58)
Using the elementary inequality
𝑝𝑝
𝑥𝑥𝑝𝑝 + 𝑦𝑦𝑝𝑝 ≤ 2𝑝𝑝𝑝𝑝 󶀡󶀡𝑥𝑥 𝑥𝑥𝑥󶀱󶀱
(59)
8
Discrete Dynamics in Nature and Society
𝑇𝑇
and the form of 𝐴𝐴, it follows that
𝑝𝑝𝑝𝑝
‖𝐹𝐹 (𝑢𝑢)‖ ≤ 𝜙𝜙𝑞𝑞 (𝑇𝑇 𝑇 𝑇) 󶀢󶀢2
󶀲󶀲 󶀲󶀲𝑎𝑎𝑎𝑎𝑎
‖ℎ‖1/(𝑝𝑝𝑝𝑝𝑝
󶀲󶀲
∞
≤ 𝜙𝜙𝑞𝑞 (𝑇𝑇 𝑇 𝑇) 󶀢󶀢2𝑝𝑝𝑝𝑝 󶀲󶀲 𝑎𝑎 𝑎 𝑎𝑎𝑎𝑎 𝑎 𝑎𝑎1 .
{𝑢𝑢 𝑢 𝑢𝑢 (𝛼𝛼𝛼 𝛼𝛼𝛼 𝛼𝛼) ∣ 𝛼𝛼 (𝑢𝑢) > 𝑏𝑏} ≠ ∅,
if 𝑢𝑢 𝑢 𝑢𝑢 (𝛼𝛼𝛼 𝛼𝛼𝛼 𝛼𝛼) .
(61)
𝑇𝑇
𝜉𝜉
𝑇𝑇
+ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠
0
𝑇𝑇
𝑠𝑠
(62)
≥ 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟󶀶󶀶
𝜂𝜂
≥ 𝑏𝑏𝑏𝑏𝑏𝑏𝑞𝑞 (𝑇𝑇 𝑇𝑇𝑇)
≥ 𝑏𝑏𝑏
𝑇𝑇
= 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶
𝜉𝜉
𝜂𝜂
𝑇𝑇
+ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠
0
𝑇𝑇
𝑠𝑠
= 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶
𝜂𝜂
𝑠𝑠
𝑇𝑇
𝑇𝑇
𝑇𝑇
𝜉𝜉
𝑠𝑠
𝜉𝜉
𝜉𝜉
𝑇𝑇
𝑇𝑇
≥ ‖𝐹𝐹𝐹𝐹‖ − 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠
𝑇𝑇
≥ ‖𝐹𝐹𝐹𝐹‖ − 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀢󶀢(𝑇𝑇 𝑇𝑇𝑇) 󶀢󶀢󶀢ℎ‖ + 𝜙𝜙𝑝𝑝 (𝑏𝑏𝑏𝑏)󶀲󶀲󶀲󶀲 Δ𝑠𝑠
𝜉𝜉
≥ ‖𝐹𝐹𝐹𝐹‖ − (𝑇𝑇 𝑇𝑇𝑇) 𝜙𝜙𝑞𝑞 (𝑇𝑇 𝑇𝑇𝑇) 𝜙𝜙𝑞𝑞 󶀢󶀢‖ℎ‖ + 𝜙𝜙𝑝𝑝 (𝑏𝑏𝑏𝑏)󶀲󶀲 .
(63)
Using again the elementary inequality 𝑥𝑥𝑝𝑝 + 𝑦𝑦𝑝𝑝 ≤ 2𝑝𝑝𝑝𝑝 (𝑥𝑥 𝑥𝑥𝑥𝑥𝑝𝑝
we get that
𝛼𝛼 (𝐹𝐹𝐹𝐹) ≥ ‖𝐹𝐹𝐹𝐹‖ − (𝑇𝑇 𝑇𝑇𝑇) 𝜙𝜙𝑞𝑞 (𝑇𝑇 𝑇𝑇𝑇) 󶀢󶀢󶀢ℎ‖1/(𝑝𝑝𝑝𝑝𝑝 +𝑏𝑏𝑏𝑏󶀲󶀲
≥ 𝑑𝑑𝑑𝑑𝑝𝑝𝑝𝑝 (𝑇𝑇 𝑇𝑇𝑇) 𝜙𝜙𝑞𝑞 (𝑇𝑇 𝑇𝑇𝑇) 󶀢󶀢󶀢ℎ‖1/(𝑝𝑝𝑝𝑝𝑝 +𝑏𝑏𝑏𝑏󶀲󶀲 (64)
By eorem 3 there exist at least three positive solutions 𝑢𝑢1 ,
𝑢𝑢2 , and 𝑢𝑢3 to (46) satisfying ‖𝑢𝑢1 ‖ < 𝑎𝑎, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏2 ), ‖𝑢𝑢3 ‖ > 𝑎𝑎,
and 𝛼𝛼𝛼𝛼𝛼3 ) < 𝑏𝑏.
Example 14. Let 𝕋𝕋 𝕋𝕋𝕋𝕋𝕋𝕋𝕋𝕋𝕋ℕ0 } ∪ {1}, where ℕ0 denotes
the set of all nonnegative integers. Consider the 𝑝𝑝-Laplacian
dynamic equation
∇
󶁢󶁢𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢Δ (𝑡𝑡)󶀲󶀲󶀲󶀲 + 𝑓𝑓 (𝑢𝑢 (𝑡𝑡)) = 0,
satisfying the boundary conditions
Finally we prove that 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼 for all 𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢1 ) and
‖𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹:
𝛼𝛼 (𝐹𝐹𝐹𝐹) = 𝐹𝐹𝐹𝐹 (𝜉𝜉)
𝜉𝜉
𝑇𝑇
≥ 𝑏𝑏𝑏
≥ 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) ∇𝑟𝑟󶀶󶀶
𝜂𝜂
𝑠𝑠
≥ ‖𝐹𝐹𝐹𝐹‖ − 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠
e �rst point is obvious. Let us prove the second part of (61).
For 𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 we have 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏, if 𝑡𝑡 𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝕋𝕋 . en,
using (A2) we have
𝛼𝛼 (𝐹𝐹𝐹𝐹) = 𝐹𝐹𝐹𝐹 (𝜉𝜉)
0
− 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠
(60)
en 𝐹𝐹𝑃𝑃𝑎𝑎1 ⊂ 𝑃𝑃𝑎𝑎1 . In a similar way we prove that 𝐹𝐹𝑃𝑃𝑐𝑐1 ⊂ 𝑃𝑃𝑐𝑐1 .
Our following step is to show that
𝛼𝛼 (𝐹𝐹𝐹𝐹) > 𝑏𝑏𝑏
𝑇𝑇
+ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 󶀡󶀡𝑓𝑓 (𝑢𝑢 (𝑟𝑟)) + ℎ (𝑟𝑟)󶀱󶀱 ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠
1
𝑢𝑢 (1) − 𝑢𝑢 󶀤󶀤 󶀴󶀴 = 0,
2
𝑡𝑡 𝑡 [0,1]𝕋𝕋 ,
𝑢𝑢▵ (0) = 0,
(65)
(66)
where 𝑝𝑝 𝑝𝑝𝑝𝑝, 𝑞𝑞 𝑞𝑞, 𝑎𝑎𝑎𝑎𝑎𝑎 𝑎 𝑎, ℎ ≡ 0, 𝑇𝑇 𝑇 𝑇, and
√2
1
󶀂󶀂
,
0 ≤ 𝑢𝑢 𝑢 ,
󶀒󶀒
󶀒󶀒
󶀒󶀒
2
2
󶀒󶀒
󶀒󶀒
󶀒󶀒
√2 1
3
1
󶀒󶀒
󶀒󶀒
,
≤ 𝑢𝑢 𝑢 ,
󶀒󶀒 4 󶀤󶀤𝑢𝑢 𝑢 󶀴󶀴 +
2
2
2
2
𝑓𝑓 (𝑢𝑢) = 󶀊󶀊
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
√2
󶀒󶀒
3
󶀒󶀒
󶀒󶀒
,
≤ 𝑢𝑢 𝑢𝑢𝑢𝑢
󶀒󶀒 4 +
2
2
󶀚󶀚
(67)
Discrete Dynamics in Nature and Society
9
Choose 𝑎𝑎 𝑎 𝑎𝑎𝑎, 𝑏𝑏 𝑏 𝑏𝑏𝑏, 𝑐𝑐 𝑐𝑐𝑐, and 𝑑𝑑 𝑑𝑑. It is easy to see
that 𝛾𝛾 𝛾𝛾, 𝐴𝐴 𝐴𝐴, 𝐵𝐵 𝐵 √2/2, 𝛼𝛼 𝛼𝛼, and
√2
√2
1
≤ (𝑎𝑎𝑎𝑎)𝑝𝑝𝑝𝑝 = √𝑎𝑎 𝑎
,
max 󶁄󶁄𝑓𝑓 (𝑢𝑢) ∶ 𝑢𝑢 𝑢 󶁤󶁤0, 󶁴󶁴󶁴󶁴 =
2
2
2
max 󶁁󶁁𝑓𝑓 (𝑢𝑢) ∶ 𝑢𝑢 𝑢 [0, 25]󶁑󶁑 = 4 +
≤ (𝑐𝑐𝑐𝑐)
𝑝𝑝𝑝𝑝
𝑝𝑝𝑝𝑝
= 𝑐𝑐
= √𝑐𝑐 𝑐 𝑐𝑐
√2
2
≃ 4, 707
√2
3
min 󶁄󶁄𝑓𝑓 (𝑢𝑢) ∶ 𝑢𝑢 𝑢 󶁤󶁤 , 3󶁴󶁴󶁴󶁴 = 4 +
≃ 4, 707
2
2
≥ (𝑏𝑏𝑏𝑏)𝑝𝑝𝑝𝑝 = √𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
(68)
3.3. A p-Laplacian Functional Dynamic Equation on Time
Scales with Delay. Let 𝕋𝕋 be a time scale with 0, 𝑇𝑇 𝑇𝑇𝑇𝜅𝜅𝜅𝜅 ,
−𝑟𝑟 𝑟𝑟𝑟 with −𝑟𝑟 𝑟𝑟 𝑟 𝑟𝑟. We are concerned in this section
with the existence of positive solutions to the 𝑝𝑝-Laplacian
dynamic equation
󶀢󶀢𝜙𝜙𝑝𝑝 󶀢󶀢𝑢𝑢Δ (𝑡𝑡)󶀲󶀲󶀲󶀲 +𝜆𝜆𝜆𝜆 (𝑡𝑡) 𝑓𝑓 (𝑢𝑢 (𝑡𝑡) , 𝑢𝑢 (𝜔𝜔 (𝑡𝑡))) = 0,
𝑢𝑢 (𝑡𝑡) = 𝜓𝜓 (𝑡𝑡) ,
Δ
𝑡𝑡 𝑡 [−𝑟𝑟𝑟𝑟]𝕋𝕋 ,
𝑢𝑢 (0) − 𝐵𝐵0 󶀢󶀢𝑢𝑢 (0)󶀲󶀲 = 0,
𝑡𝑡 𝑡 (0, 𝑇𝑇)𝕋𝕋 ,
Δ
𝑢𝑢 (𝑇𝑇) = 0,
𝑇𝑇
󶀂󶀂
𝐵𝐵0 󶀦󶀦𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝜆𝜆𝜆𝜆 (𝑟𝑟)
󶀒󶀒
󶀒󶀒
󶀒󶀒
0
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
×𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀵󶀵 󶀵󶀵
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
𝑡𝑡
𝑇𝑇
= 󶀊󶀊
󶀒󶀒 +󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝜆𝜆𝜆𝜆 (𝑟𝑟)
󶀒󶀒
󶀒󶀒
0
𝑠𝑠
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
×𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀵󶀵 Δ𝑠𝑠 if 𝑡𝑡 𝑡 [0, 𝑇𝑇]𝕋𝕋 ,
󶀒󶀒
󶀒󶀒
󶀒󶀒
󶀒󶀒
if 𝑡𝑡 𝑡 [−𝑟𝑟𝑟𝑟]𝕋𝕋 .
󶀚󶀚𝜓𝜓 (𝑡𝑡)
(70)
Let
𝐾𝐾 𝐾 󶁁󶁁𝑢𝑢 𝑢 𝑢𝑢 𝑢 𝑢𝑢 is nonnegative and concave on 𝐸𝐸󶁑󶁑 .
(71)
Clearly 𝐾𝐾 is a cone in the Banach space 𝑋𝑋. For each 𝑢𝑢 𝑢 𝑢𝑢,
we extend 𝑢𝑢 to [−𝑟𝑟𝑟𝑟𝑟𝕋𝕋 with 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 for 𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝕋𝕋 . We
also de�ne the nonnegative continuous concave functional
𝛼𝛼 𝛼 𝛼𝛼 𝛼 𝛼+0 by
𝛼𝛼 (𝑢𝑢) =
min 𝑢𝑢 (𝑡𝑡) ,
𝑡𝑡𝑡󶁡󶁡𝜉𝜉𝜉𝜉𝜉𝜉𝜉𝜉󶁱󶁱𝕋𝕋
𝜉𝜉 𝜉 󶀤󶀤0,
𝑇𝑇
󶀴󶀴 ,
2
∀𝑢𝑢 𝑢 𝑢𝑢𝑢
0
𝑡𝑡
𝑇𝑇
+ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝜆𝜆𝜆𝜆 (𝑟𝑟) 𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠𝑠
0
𝑠𝑠
(73)
𝑢𝑢 ,
𝑡𝑡 𝑡 [0, 𝑇𝑇]𝕋𝕋 ,
𝑢𝑢 (𝑡𝑡) = 󶁆󶁆 1
𝜓𝜓 (𝑡𝑡) , 𝑡𝑡 𝑡 [−𝑟𝑟𝑟𝑟]𝕋𝕋 .
(72)
(74)
It follows that (74) is a positive solution to (69) satisfying
‖𝑄𝑄𝑄𝑄‖ ≤ 󶀡󶀡𝑇𝑇 𝑇𝑇𝑇󶀱󶀱 𝜆𝜆𝑞𝑞𝑞𝑞
𝑇𝑇
× 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝑎𝑎(𝑟𝑟) 𝑓𝑓(𝑢𝑢(𝑟𝑟) , 𝑢𝑢(𝜔𝜔(𝑟𝑟))) ∇𝑟𝑟󶀶󶀶
0
for 𝑡𝑡 𝑡 [0, 𝑇𝑇]𝕋𝕋 .
(75)
Proof.
‖𝑄𝑄𝑄𝑄‖ = (𝑄𝑄𝑄𝑄) (𝑇𝑇)
𝑇𝑇
= 𝐵𝐵0 󶀦󶀦𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝜆𝜆𝜆𝜆 (𝑟𝑟) 𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀶󶀶󶀶󶀶
𝑇𝑇
0
𝑇𝑇
+ 󵐐󵐐 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝜆𝜆𝜆𝜆 (𝑟𝑟) 𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀶󶀶 Δ𝑠𝑠
0
(69)
where 𝜆𝜆 𝜆 𝜆. We de�ne 𝑋𝑋 𝑋 𝑋𝑙𝑙𝑙𝑙 ([0, 𝑇𝑇𝑇𝕋𝕋 , ℝ), which is a
Banach space with the maximum norm ‖𝑢𝑢𝑢𝑢𝑢𝑢𝑢[0,𝑇𝑇𝑇𝕋𝕋 |𝑢𝑢𝑢𝑢𝑢𝑢𝑢.
We note that 𝑢𝑢 is a solution to (69) if and only if
𝑢𝑢 (𝑡𝑡)
𝑇𝑇
𝑄𝑄𝑄𝑄 (𝑡𝑡) = 𝐵𝐵0 󶀦󶀦𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝜆𝜆𝜆𝜆 (𝑟𝑟) 𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀶󶀶󶀶󶀶
Lemma 15. Let 𝑢𝑢1 �e a �xed point of 𝑄𝑄 in the cone 𝐾𝐾. De�ne
erefore, by eorem 13, problem (65)-(66) has at least
three positive solutions.
∇
For 𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝕋𝕋 , de�ne 𝑄𝑄 𝑄𝑄𝑄𝑄𝑄𝑄 as
𝑠𝑠
𝑇𝑇
≤ 󶀡󶀡𝑇𝑇 𝑇𝑇𝑇󶀱󶀱 𝜆𝜆𝑞𝑞𝑞𝑞 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝑎𝑎 (𝑟𝑟) 𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀶󶀶 .
0
(76)
From (73) and (75) it follows that
(i) 𝑄𝑄𝑄𝑄𝑄𝑄 𝑄 𝑄𝑄;
(ii) 𝑄𝑄 𝑄𝑄𝑄𝑄𝑄𝑄 is completely continuous;
(iii) 𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢, 𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝕋𝕋 .
Depending on the signature of the delay 𝜔𝜔, we set the
following two subsets of [0, 𝑇𝑇𝑇𝕋𝕋 :
𝑌𝑌1 ∶= 󶁁󶁁𝑡𝑡 𝑡 [0, 𝑇𝑇]𝕋𝕋 ∣ 𝜔𝜔 (𝑡𝑡) < 0󶁑󶁑 ;
(77)
𝑌𝑌2 ∶= 󶁁󶁁𝑡𝑡 𝑡 [0, 𝑇𝑇]𝕋𝕋 ∣ 𝜔𝜔 (𝑡𝑡) ≥ 0󶁑󶁑 .
In the remainder of this section, we suppose that 𝑌𝑌1 is
nonempty and ∫𝑌𝑌 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎. For convenience we also
1
denote
𝑙𝑙 𝑙𝑙
𝑇𝑇
𝜙𝜙𝑝𝑝 󶀣󶀣∫0 𝑎𝑎 (𝑟𝑟) ∇𝑟𝑟󶀳󶀳
𝜆𝜆𝑞𝑞𝑞𝑞 󶀡󶀡𝑇𝑇 𝑇𝑇𝑇󶀱󶀱
,
𝑚𝑚 𝑚𝑚
𝑇𝑇
𝜙𝜙𝑝𝑝 󶀣󶀣∫0 𝑎𝑎 (𝑟𝑟) ∇𝑟𝑟󶀳󶀳
𝛿𝛿𝛿𝛿𝑞𝑞𝑞𝑞
.
(78)
eorem 16. Suppose that there exist positive constants 𝑎𝑎, 𝑏𝑏,
𝑐𝑐, and 𝑑𝑑 such that 0 < 𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎. Assume
that the following hypotheses (C1)–(C8) hold:
10
Discrete Dynamics in Nature and Society
(C1) 𝑓𝑓 𝑓 𝑓+0 × ℝ+0 → ℝ+0 is continuous;
(C2) function 𝑎𝑎 𝑎 𝑎𝑎𝑎 𝑎𝑎𝑎𝕋𝕋 → ℝ+0 is le-dense continuous;
(C3) 𝜓𝜓 𝜓 𝜓𝜓𝜓𝜓𝜓𝜓𝜓𝕋𝕋 → ℝ+0 is continuous;
(C4) 𝜔𝜔 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝕋𝕋 → [−𝑟𝑟𝑟𝑟𝑟𝑟𝕋𝕋 is continuous, 𝜔𝜔𝜔𝜔𝜔𝜔 𝜔 𝜔𝜔 for all
𝑡𝑡;
(C5) 𝐵𝐵0 ∶ℝ → ℝ is continuous and there are 0 < 𝛿𝛿 𝛿 𝛿𝛿
such that
𝛿𝛿𝛿𝛿 𝛿𝛿𝛿0 (𝑠𝑠) ≤𝛾𝛾𝛾𝛾
(C6) lim𝑥𝑥 𝑥𝑥+ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑝𝑝𝑝𝑝 <
[−𝑟𝑟𝑟𝑟𝑟𝕋𝕋 ;
for 𝑠𝑠 𝑠 𝑠+0 ;
(79)
𝑙𝑙𝑝𝑝𝑝𝑝 , uniformly in 𝑠𝑠 𝑠
𝑝𝑝𝑝𝑝
𝑝𝑝𝑝𝑝
(C7) lim𝑥𝑥1 → 0+ ;𝑥𝑥2 → 0+ 𝑓𝑓𝑓𝑓𝑓1 , 𝑥𝑥2 )/ max{𝑥𝑥1 , 𝑥𝑥2 } < 𝑙𝑙𝑝𝑝𝑝𝑝 ;
(C8) lim𝑥𝑥 𝑥 𝑥 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑝𝑝𝑝𝑝 > 𝑚𝑚𝑝𝑝𝑝𝑝 , uniformly in 𝑠𝑠 𝑠
[−𝑟𝑟𝑟𝑟𝑟𝕋𝕋 .
en, for each 0 < 𝜆𝜆 𝜆𝜆 the boundary value problem (69)
has at least three positive solutions 𝑢𝑢1 , 𝑢𝑢2 , and 𝑢𝑢3 verifying
󶙱󶙱𝑢𝑢1 󶙱󶙱 < 𝑎𝑎𝑎
𝑎𝑎 𝑎 𝑎𝑎 󶀡󶀡𝑢𝑢2 󶀱󶀱 ,
󶙱󶙱𝑢𝑢3 󶙱󶙱 > 𝑎𝑎𝑎
Proof. e proof passes by three lemmas.
𝑎𝑎 󶀡󶀡𝑢𝑢3 󶀱󶀱 < 𝑏𝑏𝑏
(80)
Lemma 17. e following relations hold:
𝑄𝑄𝑃𝑃𝑎𝑎 ⊂ 𝑃𝑃𝑎𝑎 ,
𝑄𝑄𝑃𝑃𝑐𝑐 ⊂ 𝑃𝑃𝑐𝑐 .
(81)
Proof. Using condition (C6) for 𝜀𝜀1 > 0 such that 0 < 𝑥𝑥 𝑥𝑥𝑥1 ,
we have
𝑓𝑓 󶀡󶀡𝑥𝑥𝑥𝑥𝑥 (𝑠𝑠)󶀱󶀱 < (𝑙𝑙𝑙𝑙)𝑝𝑝𝑝𝑝
for each 𝑠𝑠 𝑠 [−𝑟𝑟𝑟𝑟]𝕋𝕋 .
(82)
Applying condition (C7) we get
𝑝𝑝𝑝𝑝
𝑝𝑝𝑝𝑝
𝑓𝑓 󶀡󶀡𝑥𝑥1 , 𝑥𝑥2 󶀱󶀱 < max 󶁂󶁂𝑥𝑥1 , 𝑥𝑥2 󶁒󶁒 𝑙𝑙𝑝𝑝𝑝𝑝
(83)
for 𝜀𝜀2 > 0 such that 0 < 𝑥𝑥1 ≤ 𝜀𝜀2 , 0 < 𝑥𝑥2 ≤ 𝜀𝜀2 . Put 𝜀𝜀 𝜀
min{𝜀𝜀1 , 𝜀𝜀2 }. en for ‖𝑢𝑢𝑢𝑢𝑢𝑢 and from (75) we have
|𝑄𝑄𝑄𝑄| ≤ ‖𝑄𝑄𝑄𝑄‖
𝑇𝑇
≤ 󶀡󶀡𝑇𝑇𝑇𝑇𝑇󶀱󶀱 𝜆𝜆𝑞𝑞𝑞𝑞 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝑎𝑎 (𝑟𝑟) 𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀶󶀶
0
= 󶀡󶀡𝑇𝑇𝑇𝑇𝑇󶀱󶀱 𝜆𝜆𝑞𝑞𝑞𝑞 󶁦󶁦𝜙𝜙𝑞𝑞 󶀥󶀥󵐐󵐐 𝑎𝑎 (𝑟𝑟) 𝑓𝑓 󶀡󶀡𝑢𝑢 (𝑟𝑟) , 𝜓𝜓 (𝜔𝜔 (𝑟𝑟))󶀱󶀱 ∇𝑟𝑟
𝑌𝑌1
+ 󵐐󵐐 𝑎𝑎 (𝑟𝑟) 𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀵󶀵󶁶󶁶
𝑌𝑌2
𝑇𝑇
≤ 𝑙𝑙 󶀡󶀡𝑇𝑇𝑇𝑇𝑇󶀱󶀱 𝜆𝜆𝑞𝑞𝑞𝑞 ‖𝑢𝑢‖ 𝜙𝜙𝑞𝑞 󶀦󶀦󵐐󵐐 𝑎𝑎 (𝑟𝑟) ∇𝑟𝑟󶀶󶀶
𝑇𝑇
0
≤ 𝑙𝑙 󶀡󶀡𝑇𝑇𝑇𝑇𝑇󶀱󶀱 𝜆𝜆𝑞𝑞𝑞𝑞 𝑎𝑎𝑎𝑎𝑞𝑞 󶀦󶀦󵐐󵐐 𝑎𝑎 (𝑟𝑟) ∇𝑟𝑟󶀶󶀶
= 𝑎𝑎𝑎
0
en 𝑄𝑄𝑃𝑃𝑎𝑎 ⊂ 𝑃𝑃𝑎𝑎 . Similarly one can show that 𝑄𝑄𝑃𝑃𝑐𝑐 ⊂ 𝑃𝑃𝑐𝑐 .
(84)
Lemma 18. e set
is nonempty, and
{𝑢𝑢 𝑢𝑢𝑢 (𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼) ∣ 𝛼𝛼 (𝑢𝑢) > 𝑏𝑏}
if 𝑢𝑢 𝑢𝑢𝑢 (𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼) .
𝛼𝛼 (𝑄𝑄𝑄𝑄) > 𝑏𝑏𝑏
Proof. Applying hypothesis (𝐶𝐶𝐶𝐶 we have
𝑓𝑓 󶀡󶀡𝑢𝑢𝑢𝑢𝑢 (𝑠𝑠)󶀱󶀱 > 𝜙𝜙𝑝𝑝 (𝑚𝑚𝑚𝑚)
for each 𝑠𝑠 𝑠 [−𝑟𝑟𝑟𝑟]𝕋𝕋 .
(85)
(86)
(87)
We also have 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢. Let 𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢. en,
𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏. Hence,
𝛼𝛼 (𝑄𝑄𝑄𝑄) = (𝑄𝑄𝑄𝑄)(𝑇𝑇𝑇 𝑇𝑇)
≥
𝛿𝛿
‖𝑄𝑄𝑄𝑄‖
𝑇𝑇𝑇𝑇𝑇
𝑇𝑇
≥ 𝛿𝛿𝛿𝛿𝑞𝑞 󶀦󶀦󵐐󵐐 𝜆𝜆𝜆𝜆 (𝑟𝑟) 𝑓𝑓 (𝑢𝑢 (𝑟𝑟) , 𝑢𝑢 (𝜔𝜔 (𝑟𝑟))) ∇𝑟𝑟󶀶󶀶
0
≥ 𝛿𝛿𝛿𝛿𝑞𝑞𝑞𝑞 𝜙𝜙𝑞𝑞 󶀥󶀥󵐐󵐐 𝑎𝑎 (𝑟𝑟) 𝑓𝑓 󶀡󶀡𝑢𝑢 (𝑟𝑟) , 𝜓𝜓 (𝜔𝜔 (𝑟𝑟))󶀱󶀱 ∇𝑟𝑟
𝑌𝑌1
+ 󵐐󵐐 𝑎𝑎 (𝑟𝑟) 𝑓𝑓 󶀡󶀡𝑢𝑢 (𝑟𝑟) , 𝜓𝜓 (𝜔𝜔 (𝑟𝑟))󶀱󶀱 ∇𝑟𝑟󶀵󶀵
𝑌𝑌2
≥ 𝛿𝛿𝛿𝛿𝑞𝑞𝑞𝑞 𝜙𝜙𝑞𝑞 󶀥󶀥󵐐󵐐 𝑎𝑎 (𝑟𝑟) 𝑓𝑓 󶀡󶀡𝑢𝑢 (𝑟𝑟) , 𝜓𝜓 (𝜔𝜔 (𝑟𝑟))󶀱󶀱 ∇𝑟𝑟󶀵󶀵
𝑌𝑌1
≥ 𝑚𝑚𝑚𝑚𝑚𝑚𝑞𝑞𝑞𝑞 min {𝑢𝑢 (𝑡𝑡)} 𝜙𝜙𝑞𝑞 󶀥󶀥󵐐󵐐 𝑎𝑎 (𝑟𝑟) ∇𝑟𝑟󶀵󶀵
𝑡𝑡𝑡𝑡𝑡1
𝑌𝑌1
≥𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑞𝑞𝑞𝑞 𝜙𝜙𝑞𝑞 󶀥󶀥󵐐󵐐 𝑎𝑎 (𝑟𝑟) ∇𝑟𝑟󶀵󶀵
≥𝑏𝑏𝑏
𝑌𝑌1
(88)
Lemma 19. For all 𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 and ‖𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 one has
𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼.
Proof. Using the fact that 𝛿𝛿 𝛿𝛿𝛿𝛿 𝛿𝛿, we have
𝛼𝛼 (𝑄𝑄𝑄𝑄) = (𝑄𝑄𝑄𝑄)(𝑇𝑇𝑇 𝑇𝑇)
≥
𝛿𝛿
‖𝑄𝑄𝑄𝑄‖
𝑇𝑇𝑇𝑇𝑇
𝛿𝛿𝛿𝛿
>
𝑇𝑇𝑇𝑇𝑇
(89)
≥𝑏𝑏𝑏
Applying the Leggett-Williams theorem (eorem 3), the
proof of eorem 16 is complete.
Discrete Dynamics in Nature and Society
11
Example 20. Let 𝕋𝕋 𝕋 𝕋𝕋𝕋𝕋𝕋𝕋 𝕋𝕋𝕋𝕋𝕋 𝕋 𝕋𝕋𝕋 𝕋𝕋𝕋𝕋 𝕋 𝕋𝕋𝕋𝕋𝕋𝕋ℕ0 },
where ℕ0 denotes the set of all nonnegative integers. Consider the following 𝑝𝑝-Laplacian functional dynamic equation
on the time scale 𝕋𝕋:
∇
2
󶁢󶁢Φ𝑝𝑝 󶀢󶀢𝑢𝑢Δ (𝑡𝑡)󶀲󶀲󶀲󶀲 + 󶀡󶀡𝑢𝑢1 + 𝑢𝑢2 󶀱󶀱 =0,
𝜓𝜓 (𝑡𝑡) ≡ 0,
𝑡𝑡 𝑡 (0,1)𝕋𝕋 ,
3
𝑡𝑡 𝑡 󶁤󶁤− ,0󶁴󶁴 ,
4 𝕋𝕋
1
𝑢𝑢 (0) − 𝐵𝐵0 󶀤󶀤𝑢𝑢Δ 󶀤󶀤 󶀴󶀴󶀴󶀴 =0,
4
(90)
𝑢𝑢Δ (1) =0,
where 𝑇𝑇 𝑇𝑇, 𝑝𝑝 𝑝𝑝𝑝𝑝, 𝑞𝑞 𝑞𝑞, 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎0 (𝑠𝑠𝑠𝑠 𝑠𝑠, 𝑤𝑤𝑤𝑤𝑤𝑤 𝑤
[0,1]𝕋𝕋 → [−3/4,1]𝕋𝕋 with 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤, 𝑟𝑟 𝑟𝑟𝑟𝑟, 𝜂𝜂 𝜂𝜂𝜂𝜂,
𝑙𝑙 𝑙𝑙𝑙𝑙, 𝑚𝑚 𝑚𝑚, and 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓2 , 𝑓𝑓𝑓𝑓𝑓1 , 𝑢𝑢2 )=(𝑢𝑢1 + 𝑢𝑢2 )2 .
We deduce that 𝑌𝑌1 =[0,3/4)𝕋𝕋 , 𝑌𝑌2 =[3/4,1]𝕋𝕋 . It is easy
to see that hypotheses (C1)–(C5) are veri�ed. On the other
hand, notice that lim𝑥𝑥 𝑥𝑥+ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑝𝑝𝑝𝑝 =0 < 𝑙𝑙𝑝𝑝𝑝𝑝 and
lim𝑥𝑥 𝑥 𝑥 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑝𝑝𝑝𝑝 )= +∞ > 𝑚𝑚𝑝𝑝𝑝𝑝 . us, hypotheses
(C6)–(C8) are obviously satis�ed. en, by eorem 16,
problem (90) has at least three positive solutions of the form
󶀂󶀂 𝑢𝑢𝑖𝑖 (𝑡𝑡) ,
𝑢𝑢 (𝑡𝑡) = 󶀊󶀊
𝜓𝜓 (𝑡𝑡) ,
󶀚󶀚
Acknowledgments
𝑡𝑡 𝑡 [0,1]𝕋𝕋 , 𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖
3
𝑡𝑡 𝑡 󶁤󶁤− ,0󶁴󶁴 .
4 𝕋𝕋
(91)
is work was supported by FEDER funds through COMPETE—Operational Programme Factors of Competitiveness
(Programa Operacional Factores de Competitividade)—by
Portuguese funds through the Center for Research and
Development in Mathematics and Applications (University
of Aveiro), and the Portuguese Foundation for Science and
Technology (FCT—Fundação para a Ciência e a Tecnologia),
within project PEst-C/MAT/UI4106/2011 with COMPETE
no. FCOMP-01-0124-FEDER-022690. e authors were also
supported by the project New Explorations in Control eory
rough Advanced Research (NECTAR) co�nanced by FCT,
Portugal and the Centre National de la Recherche Scienti�que
et Technique (CNRST), Morocco.
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