J. Differential Equations 255 (2013) 812–839
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Journal of Differential Equations
www.elsevier.com/locate/jde
S-shaped and broken S-shaped bifurcation diagrams with
hysteresis for a multiparameter spruce budworm population
problem in one space dimension ✩
Shin-Hwa Wang a , Tzung-Shin Yeh b,∗
a
b
Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan, ROC
Department of Applied Mathematics, National University of Tainan, Tainan 700, Taiwan, ROC
a r t i c l e
i n f o
Article history:
Received 17 November 2012
Revised 7 May 2013
Available online 20 May 2013
MSC:
34B18
74G35
Keywords:
S-shaped bifurcation diagram
Broken S-shaped bifurcation diagram
Weak hysteresis
Strong hysteresis
Spruce budworm problem
Time map
a b s t r a c t
We study exact multiplicity and bifurcation diagrams of positive
solutions for a multiparameter spruce budworm population steadystate problem in one space dimension
⎧
⎨
⎩
u (x) + λ ru 1 −
u
q
u (−1) = u (1) = 0,
−
u2
1 + u2
= 0,
−1 < x < 1 ,
where u is the population density of the spruce budworm, q, r are
two positive dimensionless parameters, and λ > 0 is a bifurcation
parameter. Assume that either r η1 q and (q, r ) lies above the
2a3
,
a2 −1
3
√
r (a) = (a22a
, 1 < a < 3} or
+1)2
r η2 q for some constants η1 ≈ 0.0939 and η2 ≈ 0.0766. Then on
the (λ, u ∞ )-plane, we give a classification of three qualitatively
different bifurcation diagrams: an S-shaped curve, a broken Sshaped curve, and a monotone increasing curve. Our results settle
rigorously a long-standing open problem in Ludwig, Aronson and
Weinberger [Spatial patterning of the spruce budworm, J. Math.
Biol. 8 (1979) 217–258].
© 2013 Elsevier Inc. All rights reserved.
curve Γ1 = {(q, r ): q(a) =
✩
This work is partially supported by the National Science Council of the Republic of China under the grant numbers NSC
97-2115-M-024-001 and NSC 98-2115-M-024-002.
Corresponding author. Fax: +886 6 3017131.
E-mail addresses: [email protected] (S.-H. Wang), [email protected] (T.-S. Yeh).
*
0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.jde.2013.05.004
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
813
2
Fig. 1. (a) S-shaped bifurcation diagram S̄ of (1.1) with weak hysteresis. λ̂ = π4r < λ∗ < λ∗ and u λ∗ ∞ < u λ∗ ∞ < β3 . (b) Stabe
and unstable branches and the hysteresis loop DABCD for S-shaped bifurcation diagram S̄.
1. Introduction
We study exact multiplicity and bifurcation diagrams of positive solutions for the spruce budworm
population steady-state problem in one space dimension
u (x) + λ f (u ) = 0, −1 < x < 1,
u (−1) = u (1) = 0,
(1.1)
where u is the population density of the spruce budworm, f (u ) = ug (u ) is the growth rate,
g (u ) = r 1 −
u
q
−
u
1 + u2
(1.2)
is the growth rate per capita, q, r are two positive dimensionless parameters, and λ > 0 is a bifurcation
parameter. On the right-hand side of (1.2), the first term is the per capita birth rate and the second
term is the per capita death rate, both in terms of the scaled variables. This is the one-dimensional
steady-state case of a famous population problem in mathematical biology, which has been extensively studied by many authors, see e.g. Ludwig et al. [7,8], Murray [10,11], Jiang and Shi [4], and Shi
and Shivaji [12].
The spruce budworm is a very destructive native insect that lives in the spruce and fir forests
of Northeastern United States and Canada. Normally the spruce budworm exists in low numbers in
these forests, kept in check by the predators (primarily birds). However, every 40 years or so there is
an outbreak of these insects and their numbers can defoliate and damage most of the spruce and fir
trees in a forest in about 4 years. The trees (if they are not killed) can replace their foliage in about
7 to 10 years, and their life span in the absence of budworms is 100–150 years. The budworm is
capable of a five-fold increase in density per year (under ideal conditions of food and weather), and
the budworm can increase its density several hundred fold in a few years during outbreaks, see Ludwig
et al. [8, pp. 315 and 325]. Outbreaks can last for several years or they may collapse after only 1 or
2 years, see Maclauchlan et al. [9, p. 352]. As a consequence, the dynamics of the forest is reversed
and living conditions deteriorate, but for a while the budworm density remains relatively high before
it returns to low numbers again, see Figs. 1 and 2. This hysteresis effect is due to nonlinearity f (u )
in (1.1) reflecting the role of its predators. Notice that, as far as the budworm dynamics are concerned,
the forest variables may be treated as constants, see Ludwig et al. [7]. Also since the birds do not
feed exclusively on budworms, their numbers are for the most part independent of the budworm
population, see Strogatz [13] and Yodzis [14].
We define the bifurcation diagram of (1.1)
S̄ =
λ, u λ ∞ : λ > 0 and u λ is a positive solution of (1.1) .
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S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
2
Fig. 2. (a) Broken S-shaped bifurcation diagram S̄ of (1.1) with strong hysteresis. λ̂ = π4r < λ∗ and β1 <
(b) Stable and unstable branches of broken S-shaped bifurcation diagram S̄.
γ < u λ∗ ∞ < β3 .
(I) We say that the bifurcation diagram S̄ is an S-shaped curve on the (λ, u ∞ )-plane if S̄ consists
of a continuous curve with exactly two turning points at some points (λ∗ , u λ∗ ∞ ) and (λ∗ , u λ∗ ∞ )
such that
(i) λ∗ < λ∗ and u λ∗ ∞ < u λ∗ ∞ ,
(ii) at (λ∗ , u λ∗ ∞ ) the bifurcation diagram S̄ turns to the left,
(iii) at (λ∗ , u λ∗ ∞ ) the bifurcation diagram S̄ turns to the right.
See Fig. 1(a). Notice that, in Fig. 1, A = (λ∗ , u λ∗ ∞ ) and C = (λ∗ , u λ∗ ∞ ) represent threshold points.
A (stable) solution u λ satisfying 0 < u λ ∞ < u λ∗ ∞ represents a low endemic state; while a (stable)
solution u λ satisfying u λ ∞ > u λ∗ ∞ represents an outbreak state. The interpretation of Fig. 1(a)
2
for spruce budworm population problem (1.1) is that there are three critical values λ̂ = π4r < λ∗ < λ∗
such that (i) for 0 < λ λ̂ no population can persist, (ii) for λ̂ < λ < λ∗ a population can persist at
a low endemic density, (iii) for λ∗ λ λ∗ a population can persist at a low endemic density or a
large outbreak density, (iv) for λ > λ∗ a population can persist at a large outbreak density.
(II) We say that the bifurcation diagram S̄ is a broken S-shaped curve on the (λ, u ∞ )-plane if S̄
consists of two connected components such that
(i) the upper branch of S̄ has exactly one turning point at some point (λ∗ , u λ∗ ∞ ) where the curve
turns to the right,
(ii) the lower branch of S̄ is a monotone increasing curve.
See Fig. 2(a). A (stable) solution u λ satisfying 0 < u λ ∞ < β1 represents a low endemic state; while a
(stable) solution u λ satisfying u λ ∞ > u λ∗ ∞ represents an outbreak state. The upper stable branch
“collapses” at the threshold point C = (λ∗ , u λ∗ ∞ ) when λ decreases across λ∗ . The interpretation of
2
Fig. 2(a) for spruce budworm population problem (1.1) is that there are two critical values λ̂ = π4r < λ∗
such that (i) for 0 < λ λ̂ no population can persist, (ii) for λ̂ < λ < λ∗ a population can persist at
a low endemic density, (iii) for λ λ∗ a population can persist at a low endemic density or a large
outbreak density.
This paper is motivated by Ludwig et al. [7] and Ludwig et al. [8]. Ludwig et al. [8] first sought
to model the outbreak of the spruce budworm, by using the qualitative theory of ordinary differential equations and catastrophe theory. They modeled the budworm population dynamics (without
diffusion) to be governed by the equation
dN
dT
N
−B
= rN N 1 −
KN
N2
A2
+ N2
,
(1.3)
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
815
where N is budworm density. Subsequently, assuming that the dispersal of the budworm is purely diffusive so that the dispersal is therefore modeled by adding a diffusion term to (1.3), Ludwig et al. [7]
studied the diffusing budworm population dynamics governed by the equation
N2
∂N
∂2N
N
−B 2
= d 2 + rN N 1 −
∂T
KN
∂X
A + N2
(1.4)
in spatial one dimension. (Note that,
for the sake
of simplicity, in Ludwig et al.[7], the habitat is taken
as the infinite strip {( X , Y ): − 2L rd < X < 2L rd , −∞ < Y < ∞} of width rd L and the budworm
N
N
N
density is assumed to be independent of the Y coordinate.) Eqs. (1.3) and (1.4) contain parameters
which describe the foliage density and the interaction between the budworm and its predator. On the
right-hand side of (1.4), in the first term d > 0 is the diffusion (dispersion) coefficient characterizing
the rate of the spatial dispersion of the budworm population, the second term r N N (1 − N / K N ) represents logistic growth, where r N is the linear birth rate of the budworm and K N is the carrying capacity
which is related to the density of the foliage (food) available on trees, the third term B N 2 /( A 2 + N 2 )
represents predation of Holling type III (sigmoidal) functional response generated by birds, where B is
a positive constant which represents the predation rate of the birds and A is the budworm population
when the predation rate is at half of the maximum. More precisely, A is a measure of the threshold
where the predation is ‘switched on’. See Holling [2]. These real-world models in (1.3) and (1.4) are
simply the logistic model with one additional term which is designed to incorporate the effects of
predation. See Ludwig et al. [7, p. 230].
Let
w=
N
A
,
t̃ = r N T ,
x̃ =
rN
d
r=
X,
rN A
B
,
q=
KN
A
.
(1.5)
Then problem (1.4) takes the form
∂2 w
∂w
w
1 w2
−
=
+
w
1
−
,
2
q
r 1 + w2
∂ x̃
∂ t̃
− L /2 < x̃ < L /2, t̃ > 0.
(1.6)
Assume that the habitat − L /2 x̃ L /2 is surrounded by a totally hostile, outer environment. That
is, Eq. (1.6) holds in the strip |x̃| < L /2 and
w (− L /2, t̃ ) = w ( L /2, t̃ ) = 0,
Let v (x, t ) = w (x̃, t̃ ) with x =
2x̃
,
L
t̃ > 0.
(1.7)
t = ( 2L )2 t̃, and let
λ=
1
r
2
L
2
.
Then problem (1.6), (1.7) takes the form
⎧
∂2v
v
v2
⎨ ∂v
−
, −1 < x < 1 , t > 0 ,
= 2 + λ rv 1 −
q
∂x
1 + v2
⎩ ∂t
v (−1, t ) = v (1, t ) = 0,
t > 0.
(1.8)
Let u (x) denote a positive steady-state population density of (1.8). Then u (x) satisfies (1.1) with
2
u
f (u ) = ru (1 − uq ) − 1+
. So by (1.5), for (1.1), roughly speaking, r measures the foliage density while
u2
q depends upon the properties of the budworm and the predators, but not upon forest conditions, see
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S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
Fig. 3. Classified graphs of growth rate per capita g (u ) = r (1 −
(q, r )-parameter plane P ∗
√
√
= (3 3, 3 3/8).
u
)
q
−
u
1+u 2
on (0, ∞), drawn on the first quadrant of
Ludwig et al. [7, p. 218]. Appropriate values for the parameters q and r of (1.1) have been estimated
by Ludwig et al. [8, Table 1], first using basic ecological knowledge (Level II-general quantitative information in Ludwig et al. [8]) and then via a refinement thereof from extensive field studies of the
forest of New Brunswick, Canada (Level III-empirical quantitative information in Ludwig et al. [8]),
see Ludwig et al. [8, pp. 316–330] for details. From basic ecological knowledge (Level II-general quantitative information in Ludwig et al. [8]), parameter q ranges from 50 to 300 and parameter r will
range from a minimum near 0 (for an infant forest) to a maximum of 1.07 to 3.84 (for a mature
forest). The more refined studies from the extensive field study of the forest lead to the parameters q = 302 and r ranging from a minimum near 0 to a maximum of 0.994, see Ludwig et al.
[8, Table 1].
In Fig. 3, we revise Ludwig et al. [7, Fig. 5.4] and Ludwig et al. [8, Fig. 2] and divide the first
quadrant of (q, r )-parameter plane into the disjoint union of the three curves Γ0 , Γ1 , Γ2 and four
regions R 1 , R 2 , R 3 , R 4 defined as follows:
Γ0 = (q, r ): r = q/8 = 0.125q > 0 ,
√
2a3
2a3
,
Γ1 = (q, r ): q(a) = 2
, r (a) = 2
,
1
<
a
<
3
a −1
(a + 1)2
√
2a3
2a3
Γ2 = (q, r ): q(a) = 2
, r (a) = 2
,
3
<
a
<
∞
,
a −1
(a + 1)2
R 1 = (q, r ): 0 < r < q/8 and (q, r ) lies above the curve Γ1 ,
(1.9)
R 2 = (q, r ): 0 < r < q/8, and (q, r ) lies between curves Γ1 and Γ2 ,
R 3 = (q, r ): 0 < r < q/8, and (q, r ) lies below the curve Γ2 ,
R 4 = (q, r ): r > q/8 > 0 .
(1.10)
It is well known that curves Γ1 and Γ2 are continuous and strictly decreasing on the
)-plane,
√ (q, r√
∗
and the curve of the set Cl(Γ1 ∪ Γ
√2 ) (the closure of Γ1 ∪ Γ2 ) has a cusp point P ≡ (3 3, 3 3/8) ∈
Cl(Γ1 ∪Γ2 )∩Γ0 when setting a = 3 in (1.9) and (1.10), see Ludwig et al. [7, Fig. 5.4 and pp. 234–235].
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
817
We thus write on √
curves Γ1 and Γ2 respectively, the functions
√ r1 (q) and r2 (q) with (q, r i (q)) ∈ Γi ,
i = 1, 2 for q > 3 3, which satisfy r1 (q) > r2 (q) for q > 3 3, see Ludwig et al. [7, Fig. 5.4].
Note that, for (q(a), r (a)) ∈ Γ1 , lima→1+ q(a) = ∞ and lima→1+ r (a) = 12 , and for (q(a), r (a)) ∈ Γ2 ,
lima→∞ q(a) = ∞ and lima→∞ r (a) = 0.
Accordingly, Ludwig et al. [7] studied exact multiplicity of positive solutions and shapes of bifurcation diagrams S̄ of spruce budworm population problem (1.1) for various parameters q, r > 0.
In particular, they chose parameter q = 302 and several various values of parameter r = 2, 0.7, 0.2,
0.015, 0.01, see Ludwig et al. [7, Fig. 6.8 and p. 241]. But their arguments are mostly heuristic except for (q, r ) ∈ Γ0 ∪ Γ2 ∪ R 3 ∪ R 4 ∪ R̄ 2 such that S̄ are monotone increasing curves, see below for
the definition of the set R̄ 2 . The exact multiplicity of positive solutions and the shape of bifurcation
diagram S̄ of (1.1) and the n-dimensional problem of (1.1) (n 2) have remained mostly open since
1979, see Jiang and Shi [4, p. 44] and Shi and Shivaji [12, p. 824]. One of the main difficulties is
that the growth rate per capita g (u ) = r (1 − uq ) − 1+uu 2 could be a decreasing–increasing–decreasing
function on (0, ∞) for (q, r ) ∈ R 1 ∪ R 2 ∪ Γ1 , see Shi and Shivaji [12, p. 824] and Fig. 3. Our results in
Theorems 2.1–2.3 stated below prove rigorously this long-standing open problem for (1.1).
In this paper we mainly study exact multiplicity of positive solutions and shapes of bifurcation diagrams S̄ of (1.1) for parameters q, r > 0. We first follow the analysis in Ludwig et al. [8, pp. 316–320]
to find numbers of positive zeros of g (u ) = r (1 − uq ) − 1+uu 2 . We then give a classification of growth
rate per capita g (u ) on the first quadrant of (q, r )-parameter plane according to the monotonicity
of g (u ).
It is easy to see that
g (u ) = r 1 −
u
q
−
u
1 + u2
=
1
1 + u2
r 1−
u 1 + u2 − u
q
(1.11)
satisfies g (0) = r > 0 and g (0) = −1 − r /q < 0, and limu →∞ g (u ) = −∞. In addition g (u ) has at
most three positive zeros and has at least one positive zero. According to Jiang and Shi [4], we classify
all growth rate patterns according to the monotonicity of the growth rate per capita g (u ) on [0, ∞)
in (1.11):
1. g (u ) is of logistic type, if g (u ) is strictly decreasing;
2. g (u ) is of hysteresis type, if g (u ) changes from decreasing to increasing then to decreasing again
when u increases.
In the hysteresis case, if g (u ) has three distinct positive zeros, then it is strong hysteresis, otherwise
it is weak hysteresis. We have that:
(i) If (q, r ) ∈ Γ0 ∪ R 4 , then g is of logistic type. In Γ0 ∪ R 4 Eq. (1.3) has a unique positive equilibrium
point at some β1 which is stable, and Γ0 ∪ R 4 is a monostable region for (1.3). We have that
g (0) = r > 0 and g (u ) < 0 on (0, β1 ) except possibly at some value β0 ∈ (0, β1 ) when (q, r ) ∈ Γ0 ,
and g (β1 ) = 0 and g (u ) < 0 on (β1 , ∞). Thus the bifurcation diagram S̄ of (1.1) is a monotone
increasing curve since f (u ) − u f (u ) = −u 2 g (u ) > 0 on (0, β1 ) except possibly at some value
β0 ∈ (0, β1 ) when (q, r ) ∈ Γ0 ; we omit the details of the proof.
(ii) If (q, r ) ∈ Γ2 ∪ R 3 , then g is of weak hysteresis type. Notice that:
(a) In Γ2 Eq. (1.3) has exactly two positive equilibrium points at some β1 < β3 , for which β1 is
stable and β3 is unstable. We have that g (0) = r > 0, g (u ) < 0 on (0, β1 ), g (β1 ) = g (β3 ) = 0,
and g (u ) < 0 on (β1 , β3 ) ∪ (β3 , ∞). Thus the bifurcation diagram S̄ of (1.1) is a monotone
increasing curve since f (u ) − u f (u ) = −u 2 g (u ) > 0 on (0, β1 ); we omit the details of the
proof.
(b) In R 3 Eq. (1.3) has a unique positive equilibrium at some β1 which is stable, and R 3 is a
monostable region for (1.3). We have that g (0) = r > 0, g (u ) < 0 on (0, β1 ), g (β1 ) = 0, and
g (u ) < 0 on (β1 , ∞). Thus the bifurcation diagram S̄ of (1.1) is a monotone increasing curve
since f (u ) − u f (u ) = −u 2 g (u ) > 0 on (0, β1 ); we omit the details of the proof.
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S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
(iii) If (q, r ) ∈ Γ1 ∪ R 2 , then g is of hysteresis type. (In particular, g is of strong hysteresis type if
(q, r ) ∈ R 2 .) Notice that:
(a) In Γ1 Eq. (1.3) has exactly two positive equilibrium points at some β1 < β3 , for which β1 is
unstable and β3 is stable. We have that g (0) = r > 0, g (u ) > 0 on (0, β1 ) ∪ (β1 , β3 ), g (β1 ) =
g (β3 ) = 0 and g (u ) < 0 on (β3 , ∞). We define γ = β1 in this subcase.
(b) In R 2 Eq. (1.3) has exactly three positive equilibrium points at some β1 < β2 < β3 , for which
β1 and β3 are stable and β2 is unstable. Notice that R 2 is called the bistable region for
, g (β1 ) = g (β2 ) = g (β3 ) = 0
(1.3). We have that g (0) = r > 0, g (u ) > 0 on (0, β1 ) ∪ (β2 , β3 )√
and g (u ) < 0 on (β1 , β2 ) ∪ (β3 , ∞). In addition, for fixed q > 3 3, there exists r̄2 = r̄2 (q) ∈
(r2 (q), r1 (q)) such that
β3
f (u ) du > 0 for r̄2 (q) < r < r1 (q),
β1
β3
f (u ) du = 0 for r = r̄2 (q),
β1
β3
f (u ) du < 0 for r2 (q) < r < r̄2 (q),
(1.12)
β1
see Ludwig et al. [7]. Notice that
number
γ ∈ (β2 , β3 ) such that
γ
β3
β1 f (u ) du
> 0 for r̄2 (q) < r < r1 (q), then there exists a
β1 f (u ) du = 0. We thus define the curve
√
Γ¯2 = (q, r ): (q, r ) ∈ R 2 , q > 3 3 and r = r̄2 (q)
and the regions
R̂ 2 = (q, r ): 0 < r < q/8, and (q, r ) lies between curves Γ1 and Γ¯2 ,
R̄ 2 = (q, r ): 0 < r < q/8, and (q, r ) lies on curve Γ¯2 or between curves Γ2 and Γ¯2 .
(So R 2 = R̂ 2 ∪ R̄ 2 with R̄ 2 ⊃ Γ¯2 .)
In Theorems 2.2 and 2.3 stated below we prove that the bifurcation diagram S̄ of (1.1) is a broken
S-shaped curve on the (λ, u ∞ )-plane and a bistable structure for the bifurcation diagram S̄
exists when (q, r ) ∈ Γ1 ∪ R 2 and satisfies r̄2 (q) < r η2 q for some constant η2 ≈ 0.0766 defined
in (2.3) below, see Fig. 2(a).
(iv) If (q, r ) ∈ R 1 , then g is of weak hysteresis type. In R 1 Eq. (1.3) has a unique positive equilibrium
at some β3 which is stable, and R 1 is a monostable region for (1.3). We have that g (0) = r > 0,
g (u ) changes from decreasing to increasing then to decreasing on [0, β3 ), g (u ) > 0 on (0, β3 ),
g (β3 ) = 0, and g (u ) < 0 on (β3 , ∞). In Theorem 2.1 stated below we prove that the bifurcation
diagram S̄ of (1.1) is an S-shaped curve on the (λ, u ∞ )-plane and a bistable structure for the
bifurcation diagram S̄ also exists when (q, r ) ∈ R 1 and satisfies r η1 q for some constant η1 ≈
0.0939 defined in (2.1) below, see Fig. 1(a). Indeed the S-shaped bifurcation diagram S̄ implies a
hysteresis loop even though the weak hysteresis nonlinearity g is positive until the zero at the
“carrying capacity” u = β3 . Hence this is a hysteresis induced by the diffusion, see Jiang and Shi
[4, p. 45].
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
819
Fig. 4. Graphs of the regions R ∗1 and R ∗2 , the curves Γ1 , Γ2 and Γ¯2 , the lines r = η1 q ≈ 0.0939q and r = η2 q ≈ 0.0766q, and the
√
√
points P 1 = (302, r1 (302)) ≈ (302, 0.502), P 2 = (302, r̄2 (302)) ≈ (302, 0.0175). P ∗ = (3 3, 3 3/8). This figure was obtained by
using Mathematica 7.0.
Remark 1. We notice that, for appropriate values for the parameter values q and r chosen in Ludwig et
al. [7], the points (q, r ) = (302, 2) ∈ R 1 , (q, r ) = (302, 0.7) ∈ R 1 , (q, r ) = (302, 0.2) ∈ R̂ 2 ⊂ R 2 , (q, r ) =
(302, 0.015) ∈ R̄ 2 ⊂ R 2 , and (q, r ) = (302, 0.01) ∈ R 3 , see Fig. 4.
The paper is organized as follows. Section 2 contains statements of the main results: Theorems 2.1–2.4, in which Theorem 2.4 follows from Theorems 2.1–2.3. Section 3 contains lemmas needed
to prove Theorems 2.1–2.3. Section 4 contains the proofs of Theorems 2.1 and 2.2.
2. Main results
The main results in this paper are Theorems 2.1–2.4. By applying Theorems 2.1–2.3, assuming that
either r η1 q and (q, r ) lies above the curve Γ1 or r η2 q, we give a classification of three qualitatively different bifurcation diagrams S̄ of (1.1) on the (λ, u ∞ )-plane: an S-shaped curve, a broken
S-shaped curve, and a monotone increasing curve. Hence we are able to determine the exact multiplicity of positive solutions by the values of q, r and λ. In Theorem 2.4 with parameter q = 302,
which follows from Theorems 2.1–2.3, we show that, on the (λ, u ∞ )-plane, the bifurcation diagram
S̄ evolves from a monotone increasing curve with a unique low endemic state, to a broken S-shaped
curve with strong hysteresis, then to an S-shaped curve with weak hysteresis as the evolution parameter r varies from 0+ to 302η1 ≈ 28.358.
We first define the numbers η1 and η2 as follows:
(i) Let
(0.0939 ≈) η1 be the unique positive intersection value of the two curves
η = I (u ) and η = K (u ) for u > 0,
where functions
(2.1)
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S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
I (u ) ≡
3u 2 − 1
(1 + u 2 )3
K (u ) ≡
and
2
1+
u2
2
1 + u2
−
3
u2
ln 1 + u 2
,
(2.2)
see Fig. 9 below.
(ii) Let
(0.0766 ≈) η2 be the unique positive intersection value of the two curves
η = I (u ) and η = M (u ) for u > 0,
(2.3)
where function
M (u ) ≡
3
u+
u3
u
1 + u2
− 2 tan−1 u ,
(2.4)
see Fig. 9 below.
Let u λ be a positive solution of (1.1) with
α ≡ u λ ∞ > 0.
Theorem 2.1. (See Figs. 1(a), 3 and 4.) Consider (1.1). Suppose
(q, r ) ∈ R ∗1 ≡ (q, r ): (q, r ) ∈ R 1 and r η1 q .
Then
lim λ(α ) = λ̂ ≡
α →0+
π2
4r
<
π2
2
,
lim λ(α ) = ∞,
(2.5)
α →β3−
and the bifurcation diagram S̄ is an S-shaped curve on the (λ, u ∞ )-plane. More precisely, S̄ consists of a
continuous curve with exactly two turning points at some points (λ∗ , u λ∗ ∞ ) and (λ∗ , u λ∗ ∞ ) such that
λ̂ < λ∗ < λ∗ < ∞ and u λ∗ ∞ > u λ∗ ∞ . Problem (1.1) has:
(i) exactly three positive solutions w λ , u λ , v λ with w λ < u λ < v λ for λ∗ < λ < λ∗ ,
(ii) exactly two positive solutions w λ , u λ with w λ < u λ for λ = λ∗ and exactly two positive solutions u λ , v λ
with u λ < v λ for λ = λ∗ ,
(iii) exactly one positive solution w λ for λ̂ < λ < λ∗ and exactly one positive solution v λ for λ > λ∗ ,
(iv) no positive solution for 0 < λ λ̂.
Furthermore,
lim w λ ∞ = 0
λ→(λ̂)+
and
lim v λ ∞ = β3 .
(2.6)
λ→∞
Remark 2. (See Figs. 3 and 4.) It is known that R 1 is a monostable region for the budworm equation without diffusion du
= f (u ). The existence of S-shaped bifurcation diagrams of S̄ for (1.1) (with
dt
diffusion) in region R ∗1 ⊂ R 1 exhibits diffusion-induced bistability and hysteresis.
In the next theorem for (q, r ) ∈ R 2 , we recall the number
γ ∈ (β2 , β3 ) satisfying
γ
β1 f (u ) du
= 0.
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
821
Theorem 2.2. (See Figs. 2(a), 3 and 4.) Consider (1.1). Suppose
(q, r ) ∈ R ∗2 ≡ (q, r ): (q, r ) ∈ R 2 and r̄2 (q) < r η2 q .
Then
lim λ(α ) = λ̂ ≡
α →0+
π2
4r
2π 2
> √ ,
3 3
lim λ(α ) = lim λ(α ) = lim λ(α ) = ∞,
α →β1−
α →γ +
α →β3−
(2.7)
and the bifurcation diagram S̄ is a broken S-shaped curve on the (λ, u ∞ )-plane. More precisely, S̄ consists of
two connected components such that the upper branch of S̄ has exactly one turning point (λ∗ , u λ∗ ∞ ), with
λ∗ > λ̂ and γ < u λ∗ ∞ < β3 , where the curve turns to the right, and the lower branch of S̄ is a monotone
increasing curve starting at (λ̂, 0). Problem (1.1) has:
(i) exactly three positive solutions w λ , u λ , v λ with w λ < u λ < v λ for λ > λ∗ ,
(ii) exactly two positive solutions w λ , u λ with w λ < u λ for λ = λ∗ ,
(iii) exactly one positive solution w λ for λ̂ < λ < λ∗ ,
(iv) no positive solution for 0 < λ λ̂.
Furthermore,
lim w λ ∞ = 0,
λ→(λ̂)+
lim w λ ∞ = β1 ,
lim u λ ∞ = γ ,
λ→∞
λ→∞
and
lim v λ ∞ = β3 .
λ→∞
(2.8)
Theorem 2.3 for (q, r ) ∈ Γ1 and r η2 q is quite similar to Theorem 2.2. The proof of Theorem 2.3
follows by Lemma 3.3 and a slight modification of the proof of Theorem 2.2, and hence we omit it.
Theorem 2.3. (See Figs. 2(a), 3 and 4.) Consider (1.1). Suppose (q, r ) ∈ Γ1 and r η2 q. Then
π2
2
> lim λ(α ) = λ̂ ≡
α →0+
π2
4r
2π 2
> √ ,
3 3
lim λ(α ) = lim λ(α ) = lim λ(α ) = ∞,
α →β1−
α →β1+
α →β3−
and the bifurcation diagram S̄ is a broken S-shaped curve on the (λ, u ∞ )-plane. More precisely, S̄ consists of
two connected components such that the upper branch of S̄ has exactly one turning point (λ∗ , u λ∗ ∞ ), with
λ∗ > λ̂ and β1 < u λ∗ ∞ < β3 , where the curve turns to the right, and the lower branch of S̄ is a monotone
increasing curve starting at (λ̂, 0). Problem (1.1) has:
(i) exactly three positive solutions w λ , u λ , v λ with w λ < u λ < v λ for λ > λ∗ ,
(ii) exactly two positive solutions w λ , u λ with w λ < u λ for λ = λ∗ ,
(iii) exactly one positive solution w λ for λ̂ < λ < λ∗ ,
(iv) no positive solution for 0 < λ λ̂.
Furthermore,
lim w λ ∞ = 0,
λ→(λ̂)+
lim w λ ∞ = β1 = lim u λ ∞ ,
λ→∞
λ→∞
and
lim v λ ∞ = β3 .
λ→∞
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S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
Fig. 5. Numerical simulation of global bifurcation diagrams S̄ of (1.1) with q = 302 and varying r > 0. (i) A monotone increasing
curve with a unique positive solution for 0 < r r̂ (and λ > λ̂). (ii) A broken S-shaped curve for r̂ < r r ∗ . (iii) An S-shaped
curve for r ∗ < r < r̃. (iv) A monotone increasing curve with exactly one (cusp type) degenerate positive solution for r = r̃.
(v) A monotone increasing curve for r > r̃.
In the next remark we recall some stability results of positive solutions w λ , u λ , v λ of (1.1) with
w λ < u λ < v λ (whenever w λ , u λ , v λ exist) for varying λ > λ̂, when they are viewed as equilibrium
solutions to (1.8).
Remark 3. (See Figs. 1 and 2.) In Theorem 2.1 for (q, r ) ∈ R ∗1 and Theorem 2.2 for (q, r ) ∈ R ∗2 for
budworm population steady-state problem (1.1), about the stability of the positive solutions w λ , u λ ,
v λ with w λ < u λ < v λ (whenever w λ , u λ , v λ exist) for varying λ > λ̂, Ludwig et al. [7, Figs. 6.6 and
6.7 and Appendix B] proved that w λ and v λ are stable and u λ is unstable.
Theorem 2.4 with parameter values q = 302 and 0 < r 302η1 ≈ 28.358 essentially follows from
Theorems 2.1–2.3. Our theoretic results in Theorem 2.4 on bifurcation diagram S̄ prove rigorously
simulation results in Ludwig et al. [7, Fig. 6.8] for specific values r = 2, 0.7, 0.2, 0.015, 0.01.
Theorem 2.4. (See Figs. 4 and 5.) Consider (1.1) with q = 302 and let r̂ ≡ r̄2 (q = 302) ≈ 0.0175 and r ∗ ≡
r1 (q = 302) ≈ 0.502. Then the following assertions (i)–(iii) hold:
(i) (See Fig. 5(i).) If 0 < r r̂ (that is, (q = 302, r ) ∈ R 3 ∪ Γ2 ∪ R̄ 2 ), then
140.994 ≈
π2
4r̂
lim λ(α ) = λ̂ =
α →0+
π2
4r
< ∞, lim λ(α ) = ∞,
α →β1−
and the bifurcation diagram S̄ is a monotone increasing curve on the (λ, u ∞ )-plane.
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
823
(ii) (See Fig. 5(ii).) If r̂ < r r ∗ (that is, (q = 302, r ) ∈ R ∗2 ∪ Γ1 ), then
4.918 ≈
π2
4r ∗
lim λ(α ) = λ̂ =
α →0+
π2
4r
<
π2
4r̂
≈ 140.994,
lim λ(α ) = lim λ(α ) = lim λ(α ) = ∞,
α →β1−
α →γ +
α →β3−
and the bifurcation diagram S̄ is a broken S-shaped curve on the (λ, u ∞ )-plane. Moreover, all the results
in Theorems 2.2 and 2.3 hold.
(iii) (See Fig. 5(iii).) If r ∗ < r 302η1 ≈ 28.358 (that is, (q = 302, r ) ∈ R ∗1 ), then
lim λ(α ) = λ̂ =
α →0+
π2
4r
<
π2
4r ∗
≈ 4.918 and
lim λ(α ) = ∞,
α →β3−
and the bifurcation diagram S̄ is an S-shaped curve on the (λ, u ∞ )-plane. Moreover, all the results in
Theorem 2.1 hold.
Remark 4. (See Fig. 4.) By Theorems 2.1–2.3, we are not only able to determine precisely the shapes
of the bifurcation diagram S̄ of (1.1) for parameters q = 302 and 0 < r 0.994 from field studies
(Level III-empirical quantitative information in Ludwig et al. [8]), but also to determine precisely
the shapes of the bifurcation diagram S̄ for (q, r ) belonging to the rectangle R̃ ≡ {(q, r ): 50 q 300 and 0 < r 3.84} which are obtained based on basic ecological knowledge (Level II-general quantitative information in Ludwig et al. [8]), since 50η1 ≈ 4.695 > 3.84.
Finally, we give next simulation results on the global bifurcation diagrams S̄ of (1.1) with q = 302
and varying r > r ∗ ≈ 0.502.
Remark 5. (Cf. Theorem 2.4(i)–(iii).) For q = 302, numerical simulations show that when evolution
parameter r increases across some critical value r̃ ≈ 33.61 (> 28.358 ≈ 302η1 ), there exists a cusp
bifurcation from a S-shaped curve to a monotone increasing curve on the (λ, u ∞ )-plane. More precisely, by numerical simulations, the following assertions (iii ), (iv) and (v) hold:
(iii ) (See Fig. 5(iii).) For r ∗ < r < r̃, the bifurcation diagram S̄ is an S-shaped curve on the
(λ, u ∞ )-plane.
(iv) (See Fig. 5(iv).) For r = r̃, the bifurcation diagram S̄ is a monotone increasing curve on the
(λ, u ∞ )-plane. Moreover, problem (1.1) has exactly one (cusp type) degenerate positive solution u λ̃ with λ̃ = ( T (u λ̃ ∞ ))2 ≈ 0.0749.
(v) (See Fig. 5(v).) For r > r̃, the bifurcation diagram S̄ is a monotone increasing curve on the
(λ, u ∞ )-plane. Moreover, all positive solutions u λ of (1.1) are nondegenerate.
3. Lemmas
To prove Theorems 2.1–2.3, we need the following three lemmas: Lemmas 3.1–3.3 which are of
independent interest. We first consider the bifurcation diagram of positive solutions of the Dirichlet
problem
u (x) + λ f̂ u (x) = 0, −1 < x < 1,
u (−1) = u (1) = 0,
(3.1)
where f̂ ∈ C [0, ∞) ∩ C 2 (0, ∞) is a general nonlinearity, and λ > 0 is a bifurcation parameter. We
define
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u
F̂ (u ) =
u
f̂ (t ) dt ,
θ̂(u ) = 2 F̂ (u ) − u f̂ (u ),
t f̂ (t ) dt − u 2 f̂ (u )
Ĥ (u ) = 3
and
0
(3.2)
0
and assume that function f̂ satisfies hypotheses (H1a) and (H2a) below.
(H1a) There exists a positive number β̂ such that f̂ (0) = f̂ (β̂) = 0, f̂ (u ) > 0 on (0, β̂), and f̂ (u ) < 0
on (β̂, ∞).
(H2a) There exist positive numbers B̂ 1 < Ĉ 1 < B̂ 2 < Ĉ 2 < β̂ such that θ̂ (Ĉ 1 ) = θ̂ (Ĉ 2 ) = 0,
⎧
⎨ = 0 for u = B̂ 1 and u = B̂ 2 ,
θ̂ (u ) = −u f̂ (u ) > 0 on (0, B̂ 1 ) ∪ ( B̂ 2 , β̂),
⎩
< 0 on ( B̂ 1 , B̂ 2 ),
(3.3)
Ĥ ( B̂ 2 ) 0, and 2 Ĥ ( B̂ 1 ) − B̂ 21 θ̂ ( B̂ 1 ) 2 Ĥ (Ĉ 2 ).
To prove Theorem 2.1, we need the key Lemma 3.1. We prove Lemma 3.1 by applying the timemapping method (quadrature method) which was used by Ludwig et al. [7]. The time map formula
which we apply to study problem (3.1) with f satisfying (H1a) and (H2a), takes the form as follows:
α
1
T (α ) ≡ √
2
0
1
[ F̂ (α ) − F̂ (u )]1/2
du =
√
λ for 0 < α < β̂;
(3.4)
see Laetsch [6] for the derivation of the time map formula√T (α ) for problem (3.1). So positive solutions u λ of (3.1) correspond to u λ ∞ = α and T (α ) = λ. Thus, studying the exact number√of
positive solutions of (3.1) is equivalent to studying the number of roots of the equation T (α ) = λ
on (0, β̂).
Lemma 3.1. Consider (3.1). Suppose f̂ ∈ C [0, ∞) ∩ C 2 (0, ∞) satisfies (H1a) and (H2a), then
π
lim T (α ) = √
∈ [0, ∞)
2 m0
α →0+
and
lim T (α ) = ∞,
α →β̂ −
(3.5)
where 0 < limu →0+ f̂ (u )/u ≡ m0 ∞. In addition, T (α ) has exactly two positive critical points, at some
α ∗ < α∗ , on (0, β̂), such that T (α ∗ ) is a local maximum on (0, β̂) and T (α∗ ) is a local minimum on (0, β̂).
In the proof of Lemma 3.1, we apply arguments similar to those in the proof of Hung and Wang
[3, Lemma 3.2].
Proof of Lemma 3.1. Eq. (3.5) follows by (H1a) and Laetsch [6, Theorems 2.6, 2.9 and 2.10]. By (3.4)
for T (α ), we compute that
1
T (α ) = √
2 2α
α
0
θ̂(α ) − θ̂(u )
[ F̂ (α ) − F̂ (u )]3/2
du .
(3.6)
It is clear that θ̂(0) = 0 and θ̂ (0) = f̂ (0) = 0. In addition, by (H1a) and since 0 < Ĉ 1 < β̂ , we
obtain that θ̂(β̂) = 2 F̂ (β̂) − β̂ f̂ (β̂) = 2 F̂ (β̂) > 2 F̂ (Ĉ 1 ) > 2 F̂ (Ĉ 1 ) − Ĉ 1 f̂ (Ĉ 1 ) = θ̂ (Ĉ 1 ). Thus, by (H2a),
there exists a positive number Ê ∈ (Ĉ 2 , β̂) such that θ̂ ( Ê ) = θ̂ (Ĉ 1 ) and
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
825
Fig. 6. Three possible graphs of θ̂ (u ). (i) θ̂ ( B̂ 2 ) 0. (ii) θ̂ ( B̂ 2 ) > 0 > θ̂ (Ĉ 2 ). (iii) θ̂( B̂ 2 ) > θ̂ (Ĉ 2 ) 0.
⎧ ⎨ θ̂ (Ĉ 1 ) = θ̂ (Ĉ 2 ) = 0,
θ̂ (u ) > 0 on (0, Ĉ 1 ) ∪ (Ĉ 2 , β̂),
⎩ θ̂ (u ) < 0 on (Ĉ 1 , Ĉ 2 ).
(3.7)
The graph of θ̂(u ) can be depicted in Fig. 6. Thus, we obtain that
⎧
⎨ = 0 for u = Ĉ 1 and u = Ĉ 2 ,
Ĥ (u ) = u f̂ (u ) − u 2 f̂ (u ) = u θ̂ (u ) > 0 on (0, Ĉ 1 ) ∪ (Ĉ 2 , β̂),
⎩
< 0 on (Ĉ 1 , Ĉ 2 ).
(3.8)
First, we prove that T ( B̂ 2 ) < 0. There are two Cases (A) and (B) to be studied as follows:
Case (A). (See Fig. 6(i).) Suppose θ̂( B̂ 2 ) 0. So θ̂(u ) > θ̂( B̂ 2 ) for all u ∈ (0, B̂ 2 ), and hence
T ( B̂ 2 ) < 0 by (3.6).
Case (B). (See Fig. 6(ii)–(iii).) Suppose θ̂( B̂ 2 ) > 0. In this case, by applying arguments similar to
those in the proof of Hung and Wang [3, Lemma 3.2], we are able to prove T ( B̂ 2 ) < 0. We omit the
proof.
By the graphs of θ̂(u ) in Fig. 6(i)–(iii), we obtain that θ̂(u ) < θ̂(α ) for all α ∈ (0, Ĉ 1 ], u ∈ (0, α ).
So by (3.6), T (α ) > 0 for α ∈ (0, Ĉ 1 ]. By this and since T ( B̂ 2 ) < 0, we obtain that T (α ) has at least
one critical point, a local maximum on (Ĉ 1 , B̂ 2 ). By a slight modification of the proof of Korman [5,
Theorem 2.4(i)], we obtain that T (α ) has at most one critical point, a local maximum on (Ĉ 1 , B̂ 2 ). So
T (α ) has exactly one critical point, a local maximum at some α ∗ , on (0, B̂ 2 ).
Next, we prove that, T (α ) has exactly one critical point, a local minimum on ( B̂ 2 , β̂).
(I) (See Fig. 6(i)–(iii).) For any α ∈ ( B̂ 2 , Ĉ 2 ], (3.8) implies Ĥ (α ) < Ĥ ( B̂ 2 ) 0. Moreover, if θ̂(α ) 0,
then θ̂ (u ) > θ̂(α ) for all u ∈ (0, α ) by θ̂(0) = 0 and (3.7). By (3.6), we obtain that T (α ) < 0 for
α ∈ ( B̂ 2 , Ĉ 2 ]. If θ̂ (α ) > 0, then T (α ) < 0 for α ∈ ( B̂ 2 , Ĉ 2 ] since Ĥ (α ) < 0 and by applying arguments
similar to those in the proof of Hung and Wang [3, Lemma 3.2]. We omit the proof.
(II) (See Fig. 6(i)–(iii).) For any α ∈ [ Ê , β̂), by the graphs of θ̂(u ), we obtain that θ̂(Ĉ 1 ) θ̂(α ) and
θ̂(u ) < θ̂ (α ) for all u ∈ (0, Ĉ 1 ) ∪ (Ĉ 1 , α ). By (3.6), we obtain that T (α ) > 0 for α ∈ [ Ê , β̂).
By above parts (I) and (II), we obtain that T (α ) has at least one critical point, a local minimum
on (Ĉ 2 , Ê ). We finally prove that T (α ) has exactly one critical point, a local minimum at some α∗ , on
(Ĉ 2 , Ê ).
(III) Consider α ∈ (Ĉ 2 , Ê ). By (3.6), we compute that
1
T (α ) = √
2 2α 2
α
0
− 32 [θ̂ (α ) − θ̂(u )][α f̂ (α ) − u f̂ (u )] + [ F̂ (α ) − F̂ (u )][α θ̂ (α ) − u θ̂ (u )]
[ F̂ (α ) − F̂ (u )]5/2
du
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S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
Fig. 7. Three possible graphs of φ(u ) with
α ∈ (Ĉ 2 , Ê ). (i) φ(α ) φ( B̂ 1 ). (ii) 0 < φ(α ) < φ( B̂ 1 ). (iii) φ(α ) 0.
and
2
1
T (α ) + T (α ) = √
α
2 2α 2
2 2α
0
α
1
√
α
3
[θ̂ (
2
α ) − θ̂ (u )]2 + [ F̂ (α ) − F̂ (u )][φ(α ) − φ(u )]
du
[ F̂ (α ) − F̂ (u )]5/2
φ(α ) − φ(u )
[ F̂ (α ) − F̂ (u )]3/2
2
0
du ,
(3.9)
where φ(u ) = u θ̂ (u ) − θ̂(u ). It is clear that φ(0) = 0, φ(Ĉ 1 ) = −θ̂ (Ĉ 1 ) < 0, and
⎧
⎨ = 0 for u = B̂ 1 and u = B̂ 2 ,
φ (u ) = u θ̂ (u ) > 0 on (0, B̂ 1 ) ∪ ( B̂ 2 , β̂),
⎩
< 0 on ( B̂ 1 , B̂ 2 ),
(3.10)
since θ̂(0) = θ̂ (0) = 0 and by (3.3) and (3.7). The graph of φ(u ) can be depicted in Fig. 7.
Now for any α ∈ (Ĉ 2 , Ê ), we next prove
T (α ) +
2 T (α ) > 0.
α
(3.11)
There are three Cases (i)–(iii) to be studied as follows:
Case (i). (See Fig. 7(i).) Suppose φ(α ) φ( B̂ 1 ). Then φ(u ) < φ(α ) for all u ∈ (0, B̂ 1 ) ∪ ( B̂ 1 , α ) since
φ(0) = 0, φ(Ĉ 1 ) < 0 and by (3.10). So (3.11) holds for any α ∈ (Ĉ 2 , Ê ) with φ(α ) φ( B̂ 1 ) by (3.9).
Case (ii). (See Fig. 7(ii).) Suppose 0 < φ(α ) < φ( B̂ 1 ). Since φ(0) = 0, φ(Ĉ 1 ) < 0, f̂ (u ) > 0 on (0, β̂),
and by (3.10), there exist two positive numbers ᾱ , α̃ with ᾱ < B̂ 1 < α̃ < B̂ 2 (< α ) such that
⎧
⎨ = 0 for u = ᾱ and u = α̃ ,
φ(α ) − φ(u ) > 0 on (0, ᾱ ) ∪ (α̃ , α ),
⎩
< 0 on (ᾱ , α̃ )
(3.12)
and
α
F̂ (α ) − F̂ (u ) =
f̂ (t ) dt
u
> F̂ (α ) − F̂ (α̃ ) on (0, α̃ ),
< F̂ (α ) − F̂ (α̃ ) on (α̃ , α ).
(3.13)
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
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By (3.9), (3.12) and (3.13), we obtain that
2
1
T (α ) + T (α ) √
α
2 2α 2
ᾱ
φ(α ) − φ(u )
[ F̂ (α ) −
0
α
1
+ √
2 2α 2
α̃
α̃
1
= √
φ(α ) − φ(u )
[ F̂ (α ) − F̂ (u )]3/2
ᾱ
α
2 [ F̂ (
α ) − F̂ (α̃
)]3/2
α̃
ᾱ
φ(α ) − φ(u )
[ F̂ (α ) − F̂ (u )]3/2
du
du
1
du + √
2 2α 2
[ F̂ (α ) − F̂ (α̃ )]3/2
1
2 2α
du + √
2 2α 2
F̂ (u )]3/2
φ(α ) − φ(u )
> √
2 2α 2
1
α
α̃
φ(α ) − φ(u )
[ F̂ (α ) − F̂ (α̃ )]3/2
du
φ(α ) − φ(u ) du .
ᾱ
It follows that
2
1
T (α ) + T (α ) > √
(α − ᾱ )φ(α ) −
α
2 2α 2 [ F̂ (α ) − F̂ (α̃ )]3/2
1
= √
2 2α 2 [ F̂ (α ) − F̂ (α̃ )]3/2
2 2α 2 [ F̂ (α ) − F̂ (α̃ )]3/2
φ(u ) du
ᾱ
α
φ(u ) du
ᾱ
α
1
= √
u =α
u φ(u )u =ᾱ −
α
u φ (u ) du .
(3.14)
ᾱ
Since Ĉ 2 ∈ ( B̂ 2 , α ), θ̂ (Ĉ 2 ) = 0 and by (3.8) and (3.10), we obtain that
α
Ĉ 2
Ĉ 2
u φ (u ) du =
u φ (u ) du >
ᾱ
ᾱ
u 2 θ̂ (u ) du
ᾱ
u =Ĉ
= u θ̂ (u )u =ᾱ2 − 2
2 Ĉ 2
u θ̂ (u ) du
ᾱ
2 = −ᾱ θ̂ (ᾱ ) − 2 Ĥ (Ĉ 2 ) + 2 Ĥ (ᾱ ).
We compute that (2 Ĥ (u ) − u 2 θ̂ (u )) = −u 2 θ̂ (u ), and hence 2 Ĥ (u ) − u 2 θ̂ (u ) is a strictly decreasing
function on (0, B̂ 1 ] by (3.3). Thus
α
ᾱ
u φ (u ) du > 2 Ĥ ( B̂ 1 ) − 2 Ĥ (Ĉ 2 ) − B̂ 21 θ̂ ( B̂ 1 ) 0
(3.15)
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ᾱ < B̂ 1 and the assumption that 2 Ĥ ( B̂ 1 ) − B̂ 21 θ̂ ( B̂ 1 ) 2 Ĥ (Ĉ 2 ). Hence (3.14) and (3.15) imply that
2
1
T (α ) + T (α ) > √
2
α
2 2α [ F̂ (α ) − F̂ (α̃ )]3/2
α
u φ (u ) du > 0
ᾱ
for any α ∈ (Ĉ 2 , Ê ) with 0 < φ(α ) < φ( B̂ 1 ).
Case (iii). (See Fig. 7(iii).) Suppose φ(α ) 0. In this case, by applying arguments similar to those
in the proof of Hung and Wang [3, Lemma 3.2], we are able to prove T (α ) + α2 T (α ) > 0 for any
α ∈ (Ĉ 2 , Ê ) with φ(α ) 0. We omit the proof.
By above, we conclude that T (α ) + α2 T (α ) > 0 for any α ∈ (Ĉ 2 , Ê ). So, if α∗ is a critical point of
T (α ) on (Ĉ 2 , Ê ), then T (α∗ ) > 0, and hence T (α∗ ) must be a local minimum. So T (α ) has exactly
one critical point, a local minimum at some α∗ , on (Ĉ 2 , Ê ).
2
The proof of Lemma 3.1 is complete.
To prove Theorem 2.2, we need the key Lemma 3.2. We assume that function f̂ satisfies hypotheses (H1b), (H2b) and (H3) below.
(H1b) There exist positive numbers β̂1 < β̂2 < β̂ such that f̂ (0) = f̂ (β̂1 ) = f̂ (β̂2 ) = f̂ (β̂) = 0, f̂ (u ) > 0
on (0, β̂1 ) ∪ (β̂2 , β̂), and f̂ (u ) < 0 on (β̂1 , β̂2 ) ∪ (β̂, ∞).
(H2b) There exist positive numbers B̂ 1 < Ĉ 1 < Ê 1 B̂ 2 < Ĉ 2 < Ê 2 < β̂ such that θ̂( Ê 1 ) = θ̂( Ê 2 ) =
θ̂ (Ĉ 1 ) = θ̂ (Ĉ 2 ) = 0 and (3.3) holds. Also, θ̂ ( B̂ 1 ) − B̂ 1 θ̂ ( B̂ 1 ) − θ̂ (Ĉ 2 ) 0.
(H3) There exists a positive number
γ̂ ∈ (β̂2 , β̂) satisfies
γ̂
β̂1
f̂ (u ) du = 0.
The time map formula which we apply to study problem (3.1) with f satisfying (H1b), (H2b) and
(H3), takes the form as follows:
1
T (α ) = √
2
α
0
1
[ F̂ (α ) −
F̂ (u )]1/2
du =
√
λ for α ∈ (0, β̂1 ) ∪ (γ̂ , β̂).
(3.16)
Lemma 3.2. Consider (3.1). Suppose f̂ ∈ C [0, ∞) ∩ C 2 (0, ∞) satisfies (H1b), (H2b) and (H3), then
π
lim T (α ) = √
∈ [0, ∞)
2 m0
α →0+
and
lim T (α ) = lim T (α ) = lim T (α ) = ∞,
α →β̂1−
α →γ̂ +
α →β̂ −
(3.17)
where 0 < limu →0+ f̂ (u )/u ≡ m0 ∞. In addition, T (α ) is strictly increasing on (0, β̂1 ) and T (α ) has exactly one positive critical point at some α∗ on (γ̂ , β̂), such that T (α∗ ) is a local minimum on (γ̂ , β̂).
Proof of Lemma 3.2. Eq. (3.17) follows by (H1b) and Laetsch [6, Theorems 2.6, 2.9 and 2.10], and by
applying the same arguments in the proofs of Addou and Wang [1, Lemma 5.3(iii) and (iv)].
It is clear that θ̂ (0) = θ̂ (0) = 0. By (H2b), we obtain that
⎧
⎨ θ̂( Ê 1 ) = θ̂( Ê 2 ) = 0,
θ̂(u ) > 0 on (0, Ê 1 ) ∪ ( Ê 2 , β̂),
⎩
θ̂(u ) < 0 on ( Ê 1 , Ê 2 )
(3.18)
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
829
Fig. 8. (i) Graph of θ̂ (u ). (ii) Graph of φ(u ).
and
⎧ ⎨ θ̂ (Ĉ 1 ) = θ̂ (Ĉ 2 ) = 0,
θ̂ (u ) > 0 on (0, Ĉ 1 ) ∪ (Ĉ 2 , β̂),
⎩ θ̂ (u ) < 0 on (Ĉ 1 , Ĉ 2 ).
(3.19)
The graph of θ̂(u ) can be depicted in Fig. 8(i). By (H1b) and (3.3), we obtain that f̂ (β̂1 ) < 0 and
f̂ (β̂2 ) > 0. These and by (H1b), we obtain that θ̂ (β̂1 ) = −β̂1 f̂ (β̂1 ) > 0 and θ̂ (β̂2 ) = −β̂2 f̂ (β̂2 ) < 0.
So β̂1 < Ĉ 1 by (3.19). This and by (3.6) and (3.19), we obtain that T (α ) > 0 for α ∈ (0, β̂1 ). Hence T (α )
is strictly increasing on (0, β̂1 ). In addition, by the result that limα →γ̂ + T (α ) = limα →β̂ − T (α ) = ∞, we
obtain that T (α ) has at least one critical point, a local minimum on (γ̂ , β̂). We then prove that T (α )
has exactly one critical point, a local minimum on (γ̂ , β̂). There are three Cases (A)–(C) to be studied
as follows:
Case (A). Suppose γ̂ < Ê 1 ( B̂ 2 ). Then by modifying the proof of Korman [5, Theorem 2.4(i)], we
obtain that T (α ) < 0 on (γ̂ , Ê 1 ). In addition, since θ̂(0) = θ̂ (0) = 0 and by (3.6), (3.18) and (3.19), we
obtain that T (α ) < 0 for α ∈ [ Ê 1 , Ĉ 2 ]. We then prove that T (α ) has exactly one critical point, a local minimum on (Ĉ 2 , β̂). By (H2b), we obtain that (3.10) holds. In addition, we know that φ(0) = 0,
φ(Ĉ 1 ) = −θ̂ (Ĉ 1 ) < 0 and φ(Ĉ 2 ) = −θ̂ (Ĉ 2 ) > 0. The graph of φ(u ) can be depicted in Fig. 8(ii). Hence
we obtain that φ(u ) is strictly increasing on (Ĉ 2 , β̂) and φ(u ) φ(Ĉ 2 ) for u ∈ (0, Ĉ 2 ) by the assumption in (H2b) that θ̂( B̂ 1 ) − B̂ 1 θ̂ ( B̂ 1 ) − θ̂(Ĉ 2 ) 0. So φ(u ) < φ(α ) for all α ∈ (Ĉ 2 , β̂), u ∈ (0, α ). By
(3.9), we obtain that T (α ) + (2/α ) T (α ) > 0 for α ∈ (Ĉ 2 , β̂). That is, if α∗ is a critical point of T (α )
on (Ĉ 2 , β̂), then T (α∗ ) > 0 and hence T (α∗ ) must be a local minimum. Thus T (α ) has exactly one
critical point, a local minimum, on (Ĉ 2 , β̂). So T (α ) has exactly one critical point at some α∗ , a local
minimum, on (γ̂ , β̂).
Case (B). Suppose Ê 1 γ̂ < Ĉ 2 . Then by applying the same arguments for Case (A), we obtain that
T (α ) < 0 for α ∈ (γ̂ , Ĉ 2 ] and T (α ) has exactly one critical point, a local minimum on (Ĉ 2 , β̂). So
T (α ) has exactly one critical point at some α∗ , a local minimum, on (γ̂ , β̂).
Case (C). Suppose Ĉ 2 γ̂ . Then by applying the same arguments for Case (A), we obtain that T (α )
has exactly one critical point at some α∗ , a local minimum, on (γ̂ , β̂).
We summarize above results and we obtain that T (α ) is strictly increasing on (0, β̂1 ) and T (α )
has exactly one positive critical point α∗ , on (γ̂ , β̂), such that T (α∗ ) is a local minimum on (γ̂ , β̂).
The proof of Lemma 3.2 is complete. 2
To prove Theorem 2.3, we need the key Lemma 3.3. We assume that function f̂ satisfies hypotheses (H1c) and (H2c) below.
(H1c) There exist positive numbers β̂1 < β̂ such that f̂ (0) = f̂ (β̂1 ) = f̂ (β̂) = 0, f̂ (u ) > 0 on (0, β̂1 ) ∪
(β̂1 , β̂), and f̂ (u ) < 0 on (β̂, ∞).
830
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(H2c) There exist positive numbers B̂ 1 < Ĉ 1 < Ê 1 B̂ 2 < Ĉ 2 < Ê 2 < β̂ such that θ̂( Ê 1 ) = θ̂( Ê 2 ) =
θ̂ (Ĉ 1 ) = θ̂ (Ĉ 2 ) = 0 and (3.3) holds. Also, θ̂ ( B̂ 1 ) − B̂ 1 θ̂ ( B̂ 1 ) − θ̂(Ĉ 2 ) 0.
The time map formula which we apply to study problem (3.1) with f satisfying (H1c) and (H2c),
takes the form as follows:
1
T (α ) = √
2
α
0
1
[ F̂ (α ) −
F̂ (u )]1/2
du =
√
λ for α ∈ (0, β̂1 ) ∪ (β̂1 , β̂).
Lemma 3.3. Consider (3.1). Suppose f̂ ∈ C [0, ∞) ∩ C 2 (0, ∞) satisfies (H1c) and (H2c), then
π
lim T (α ) = √
∈ [0, ∞)
2 m0
α →0+
lim T (α ) = lim T (α ) = lim T (α ) = ∞,
and
α →β̂1−
α →β̂1+
α →β̂ −
where 0 < limu →0+ f̂ (u )/u ≡ m0 ∞. In addition, T (α ) is strictly increasing on (0, β̂1 ) and T (α ) has exactly one positive critical point at some α∗ on (β̂1 , β̂), such that T (α∗ ) is a local minimum on (β̂1 , β̂).
The proof of Lemma 3.3 follows by a slight modification of the proof of Lemma 3.2; we omit it.
4. Proofs of Theorems 2.1 and 2.2
Proof of Theorem 2.1. We prove Theorem 2.1 mainly by applying Lemma 3.1. We shall show that,
2
u
for (q, r ) ∈ R ∗1 = {(q, r ): (q, r ) ∈ R 1 and r η1 q ≈ 0.0939q}, f (u ) = ug (u ) = ru (1 − uq ) − 1+
satisfies
u2
∗
(H1a) and (H2a). For (q, r ) ∈ R 1 , we know that there exists a number β3 > 0 such that
⎧
⎨ f (0) = f (β3 ) = 0,
f (u ) > 0 on (0, β3 ),
⎩
f (u ) < 0 on (β3 , ∞).
So, for (q, r ) ∈ R ∗1 , f satisfies (H1a).
We then compute that, for θ(u ) in (3.2) with f̂ = f (u ),
r
θ(u ) = 2F (u ) − u f (u ) =
3q
u3 − u −
u
1 + u2
+ 2 tan−1 u ,
(4.1)
and hence
θ (u ) = f (u ) − u f (u ) = u
2
r
q
+
1 − u2
(1 + u 2 )2
(4.2)
and
θ (u ) = −u f (u ) = 2u
r
q
+
1 − 3u 2
(1 + u 2 )3
.
By (2.2) and (4.3), it is clear that u = ũ is a positive zero of θ (u ) if and only if u = ũ satisfies
r
q
=
3u 2 − 1
(1 + u 2 )3
= I (u ).
(4.3)
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
Fig. 9. Graphs of functions
831
η = I (u ), η = J (u ), η = K (u ), η = M (u ), η = η1 ≈ 0.0939, and η = η2 ≈ 0.0766 on (0, 5].
We compute that
I (u ) =
12u (1 − u 2 )
(1 + u 2 )4
.
Thus
⎧
√
⎪
3 ) = 0,
⎨ I (0) = −1, I (1/ √
I (u ) < 0 on (0, 1/ 3 ),
⎪
√
⎩
I (u ) > 0 on (1/ 3, ∞)
and
⎧ ⎨ I (0) = I (1) = 0,
I (u ) > 0 on (0, 1),
⎩ I (u ) < 0 on (1, ∞).
In addition, it is clear that limu →∞ I (u ) = 0, and hence
max I (u ) = I (1) =
u ∈[0,∞)
1
4
> η1 ≈ 0.0939.
By the above analysis for I (u ), for 0 < r η1 q,√we obtain that the equation I (u ) = r /q ( η1 ) has
exactly two positive roots, say, B 1 , B 2 with 1/ 3 < B 1 < 1 < B 2 , see Fig. 9. Thus, for 0 < r η1 q,
θ (u ) = 2u [(r /q) − I (u )] has two positive zeros B 1 , B 2 such that
⎧ ⎨ θ ( B 1 ) = θ ( B 2 ) = 0,
θ (u ) > 0 on (0, B 1 ) ∪ ( B 2 , ∞),
⎩ θ (u ) < 0 on ( B 1 , B 2 ).
So, for 0 < r η1 q, θ (u ) satisfies (3.3).
By (4.2), it is clear that u = ũ is a positive zero of θ (u ) if and only if u = ũ satisfies
r
q
=
u2 − 1
(1 + u 2 )2
≡ J (u ).
(4.4)
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S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
We compute that
2u (3 − u 2 )
J (u ) =
(1 + u 2 )3
.
Then
⎧
⎨ J (0) = −1, J (1) = 0,
J (u ) < 0 on (0, 1),
⎩
J (u ) > 0 on (1, ∞)
and
√
⎧ 0,
⎨ J (0) = J ( 3 ) =√
J (u ) > 0 on (0, 3 ),
√
⎩ J (u ) < 0 on ( 3, ∞).
In addition, it is clear that limu →∞ J (u ) = 0, and hence
√
1
max J (u ) = J ( 3 ) =
u ∈[0,∞)
8
> η1 ≈ 0.0939.
By the above analysis for J (u ), for 0 < r η1 q, we √
obtain that the equation J (u ) = r /q ( η1 ) has
exactly two positive roots, say, C 1 , C 2 with 1 < C 1 < 3 < C 2 , see Fig. 9. Thus, for 0 < r η1 q,
θ (u ) = u 2
r
q
− J (u )
(4.5)
has two positive zeros C 1 , C 2 such that
⎧ ⎨ θ ( C 1 ) = θ ( C 2 ) = 0,
θ (u ) > 0 on (0, C 1 ) ∪ (C 2 , ∞),
⎩ θ (u ) < 0 on (C 1 , C 2 ).
(4.6)
To finish verifying (H2a) for f (u ), it suffices to prove that:
(I) B 1 < C 1 < B 2 < C 2 < β3 ,
(II) H ( B 2 ) 0,
(III) 2H ( B 1 ) − B 21 θ ( B 1 ) 2H (C 2 ),
which we prove as follows:
We compute that, for H (u ) in (3.2) with f̂ = f (u ),
u
t f (t ) dt − u 2 f (u ) =
H (u ) = 3
r
4q
u4 −
1
2
u2 −
0
u2
1 + u2
+
3
2
ln 1 + u 2 .
By (2.2) and (4.7), it is clear that u = ũ is a positive zero of H (u ) if and only if u = ũ satisfies
r
q
=
2
u2
1+
2
1 + u2
−
3
u2
ln 1 + u 2
= K (u ).
(4.7)
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
833
We compute that
K (u ) ≡
−4
u 5 (1 + u 2 )2
6u 2 + 9u 4 + u 6 − 6 1 + u 2
2
ln 1 + u 2 .
Let
K̃ (u ) ≡
u5
24
K (u ) = ln 1 + u 2 −
6u 2 + 9u 4 + u 6
6(1 + u 2 )2
,
(4.8)
and hence
K̃ (u ) =
u 5 (3 − u 2 )
3(1 + u 2 )3
.
Thus,
√
⎧ 0,
⎨ K̃ (0) = K̃ ( 3 ) =√
K̃ (u ) > 0 on (0, 3 ),
⎩ √
K̃ (u ) < 0 on ( 3, ∞).
Since K̃ (0) = 0 and limu →∞ K̃ (u ) = −∞, we obtain that there exists a positive number D̃ ≈ 2.396
such that
⎧
⎨ K̃ (0) = K̃ ( D̃ ) = 0,
K̃ (u ) > 0 on (0, D̃ ),
⎩
K̃ (u ) < 0 on ( D̃ , ∞).
By (4.8), we obtain that
⎧ ⎨ K ( D̃ ) = 0,
K (u ) > 0 on (0, D̃ ),
⎩ K (u ) < 0 on ( D̃ , ∞).
In addition, it is clear that limu →0+ K (u ) = −∞ and limu →∞ K (u ) = 0, and hence
0.104 ≈ max K (u ) = K ( D̃ ) > η1 ≈ 0.0939.
u ∈(0,∞)
By the above analysis for K (u ), for 0 < r η1 q, we obtain that the equation K (u ) = r /q ( η1 ) has
exactly two positive roots, say, D 1 , D 2 with D 1 < D̃ < D 2 , see Fig. 9. Thus, for 0 < r η1 q,
H (u ) =
u4 r
4
q
− K (u )
(4.9)
has two positive zeros D 1 , D 2 such that
⎧
⎨ H ( D 1 ) = H ( D 2 ) = 0,
H (u ) > 0 on (0, D 1 ) ∪ ( D 2 , ∞),
⎩
H (u ) < 0 on ( D 1 , D 2 ).
(4.10)
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S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
We compute that, for u > 5,
I (u ) − J (u ) =
u 2 (3 − u 2 )
(1 + u 2 )3
<0
(4.11)
and
J (u ) − K (u ) =
−u 2 + 6 ln(1 + u 2 )
−(6 + 9u 2 + u 4 )
6 2
<
+
ln
1
+
u
< 0.
u 2 (1 + u 2 )2
u4
u4
(4.12)
Thus,
(1) I (u ) < J (u ) < K (u ) for u > 5.
In addition, we know that
(2) I (0) = J (0) = −1 and limu →0+ K (u ) = −∞.
(3) limu →∞ I (u ) = limu →∞ J (u ) = limu →∞ K (u ) = 0.
(4) I (u ) has exactly one critical √
point, a local maximum, at 1, on (0, 5). J (u ) has exactly one critical
point, a local maximum, at 3, on (0, 5). K (u ) has exactly one critical point, a local maximum,
at D̃ ≈ 2.396, on (0, 5).
(5) Functions I (u ), J (u ) and K (u ) are all strictly decreasing on [5, ∞).
√
(6) The unique positive zeros of functions I (u ) − J (u ), I (u ) − K (u ), and J (u ) − K (u ) on (0, 5] are 3,
ξ1 ≈ 1.950, and D̃ ≈ 2.396, respectively. Moreover,
√
√
I ( 3 ) = J ( 3 ) = 1/8 = 0.125,
I (ξ1 ) = K (ξ1 ) = η1 ≈ 0.0939,
J ( D̃ ) = K ( D̃ ) ≈ 0.104
by (2.1). See Fig. 9 for graphs of functions I (u ), J (u ), and K (u ) on (0, 5].
By parts (1)–(6), we obtain that:
(a) for 0 < r < η1 q, B 1 < C 1 < D 1 < B 2 < C 2 < D 2 ;
(b) for r = η1 q, B 1 < C 1 < D 1 = B 2 < C 2 < D 2 ;
see Fig. 9. In addition, we know that, for (q, r ) ∈ R ∗1 , g (u ) changes from decreasing to increasing then
to decreasing on [0, β3 ). Thus, by θ (u ) = −u 2 g (u ) and (4.6), we obtain that C 2 < β3 . So we obtain
that, for (q, r ) ∈ R ∗1 , B 1 < C 1 < B 2 < C 2 < β3 . Moreover, since D 1 B 2 < D 2 and by (4.10), we obtain
that H ( B 2 ) 0.
Letting r /q = η , by (4.5) and (4.9), we obtain that
2H ( B 1 ) − B 21 θ ( B 1 ) =
=
=
=
since (4.11) and (4.12). Define
B 41 2
B 41 2
B 41 2
η − K ( B 1 ) − B 41 η − J ( B 1 )
B 41 I ( B 1 ) − K ( B 1 ) − B 41 I ( B 1 ) − J ( B 1 )
J (B1) − K (B1) −
− B 21 (6 + 15B 21 + 13B 41 )
2(1 +
B 21 )3
2
I (B1) − J (B1)
+ 3 ln 1 + B 21
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
N 1 (u ) ≡
835
−u 2 (6 + 15u 2 + 13u 4 )
+ 3 ln 1 + u 2
2(1 + u 2 )3
and compute that
N 1 (u ) =
6u 5 (u 2 − 1)
(1 + u 2 )4
.
We know that, for 0 < η η1 , B 1 (η) B 1 (η1 ) ≈ 0.651 < 1, and it is clear that N 1 (u ) is strictly decreasing on (0, B 1 (η1 )]. Thus, for 0 < r η1 q, 2H ( B 1 ) − B 21 θ ( B 1 ) = N 1 ( B 1 ) N 1 ( B 1 (η1 )) ≈ −0.0186.
Letting r /q = η and by (4.9), (4.12), we obtain that
−2H (C 2 ) = −
C 24 2
η − K (C 2 ) = −
C 24 2
J (C 2 ) − K (C 2 ) =
C 22 (6 + 9C 22 + C 24 )
2(1 +
C 22 )2
− 3 ln 1 + C 22 .
Define
N 2 (u ) ≡
u 2 (6 + 9u 2 + u 4 )
2(1 + u 2 )2
− 3 ln 1 + u 2
and compute that
N 2 (u ) =
u 5 ( u 2 − 3)
(1 + u 2 )3
.
√
We know that, for 0 < η η1 , C 2 (η) C 2 (η1 ) ≈ 2.642 > 3, and it is clear that N 2 (u ) is strictly
increasing on [C 2 (η1 ), ∞). Thus, for 0 < r η1 q, −2H (C 2 ) = N 2 (C 2 ) N 2 (C 2 (η1 )) ≈ 0.211. So for
0 < r η1 q, we obtain that 2H ( B 1 ) − B 21 θ ( B 1 ) − 2H (C 2 ) N 1 ( B 1 (η1 )) + N 2 (C 2 (η1 )) ≈ 0.193 > 0. So
by above analyses, for (q, r ) ∈ R ∗1 , f satisfies (H2a).
We summarize the preceding results and we conclude that, for (q, r ) ∈ R ∗1 , f satisfies (H1a) and
(H2a). By Lemma 3.1 and (3.4) and since r > 1/2, the results in (2.5) hold and the bifurcation diagram
S̄ is an S-shaped curve on the (λ, u ∞ )-plane. More precisely, S̄ consists of a continuous curve with
exactly two turning points at some points (λ∗ , u λ∗ ∞ ) and (λ∗ , u λ∗ ∞ ) such that 0 < λ∗ < λ∗ < ∞
and u λ∗ ∞ > u λ∗ ∞ .
In addition, since
f (u ) = ru 1 −
u
q
−
u2
1 + u2
u
≡ f 0 (u ) for u ∈ (0, β3 ),
< ru 1 −
q
by a comparison theorem of Laetsch [6, Theorem 2.3], we obtain that T f (α ) > T f 0 (α ) for α ∈ (0, β3 ).
(Note that, for T (α ) in (3.4), to make clearer for (3.1) the dependence on the nonlinearities f and f 0 ,
we write T f (α ) and T f 0 (α ), respectively.) Since
r
θ f 0 (u ) = f 0 (u ) − u f 0 (u ) = u 2 > 0 for u ∈ (0, β3 ),
q
and by (3.6), we obtain that T f 0 (α ) is strictly increasing on (0, β3 ). Thus by (3.5),
π
T f (α ) > T f 0 (α ) > lim T f 0 (α ) = √ = lim T f (α )
α →0+
α →0+
2 r
for α ∈ (0, β3 ),
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S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
2
and hence λ∗ > π4r = λ̂ by (3.4). Thus λ̂ < λ∗ < λ∗ < ∞. So we obtain immediately the exact multiplicity result and ordering results of the solutions in parts (i)–(iv). The proofs of results in (2.6) are
easy but tedious; we omit them.
The proof of Theorem 2.1 is now complete. 2
Proof of Theorem 2.2. We prove Theorem 2.2 mainly by applying Lemma 3.2. We shall show that,
2
u
for (q, r ) ∈ R ∗2 = {(q, r ): (q, r ) ∈ R 2 and r̄2 (q) < r η2 q ≈ 0.0766q}, f (u ) = ug (u ) = ru (1 − uq ) − 1+
u2
∗
satisfies (H1b), (H2b) and (H3). For (q, r ) ∈ R 2 , we know that there exist three positive numbers
β1 < β2 < β3 such that
⎧
⎨ f (0) = f (β1 ) = f (β2 ) = f (β3 ) = 0,
f (u ) > 0 on (0, β1 ) ∪ (β2 , β3 ),
⎩
f (u ) < 0 on (β1 , β2 ) ∪ (β3 , ∞).
(4.13)
So, for (q, r ) ∈ R ∗2 , f satisfies (H1b).
In the proof of Theorem 2.1, for 0 < r η2 q (< η1 q), we know that:
(A) θ (u ) has two positive zeros B 1 < B 2 satisfying (4.4).
(B) θ (u ) has two positive zeros C 1 < C 2 satisfying (4.6).
Moreover, by (2.4) and (4.1), it is clear that u = ũ is a positive zero of θ(u ) if and only if u = ũ
satisfies
r
q
=
3
u3
u+
u
1 + u2
− 2 tan−1 u = M (u ).
We compute that
M (u ) =
6
u 4 (1 + u 2 )2
3 1 + u2
2
tan−1 u − u 3 + 5u 2 + u 4
≡
18
u4
M̃ (u ),
where
M̃ (u ) = tan−1 u −
u (3 + 5u 2 + u 4 )
3(1 + u 2 )2
.
For M̃ (u ), we compute that
M̃ (u ) =
u 4 (3 − u 2 )
3(1 + u 2 )3
.
Thus
√
⎧ ⎨ M̃ (0) = M̃ ( 3 ) =√0,
0, 3 ),
⎩ M̃ (u ) > 0 on (√
M̃ (u ) < 0 on ( 3, ∞).
Since M̃ (0) = 0 and limu →∞ M̃ (u ) = −∞, we obtain that there exists a positive number Ẽ ≈ 2.726
such that
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
837
⎧
⎨ M̃ (0) = M̃ ( Ẽ ) = 0,
M̃ (u ) > 0 on (0, Ẽ ),
⎩
M̃ (u ) < 0 on ( Ẽ , ∞).
Thus, we obtain that
⎧ ⎨ M ( Ẽ ) = 0,
M (u ) > 0 on (0, Ẽ ),
⎩ M (u ) < 0 on ( Ẽ , ∞).
In addition, it is clear that limu →0+ M (u ) = −1 and limu →∞ M (u ) = 0, and hence
0.0905 ≈ max M (u ) = M ( Ẽ ) > η2 ≈ 0.0766.
u ∈(0,∞)
By the above analysis for M (u ), for 0 < r η2 q, we obtain that the equation M (u ) = r /q ( η2 ) has
exactly two positive roots, say E 1 , E 2 with E 1 < Ẽ < E 2 , see Fig. 9. Thus, for 0 < r η2 q,
θ(u ) =
u3 r
3
q
− M (u )
(4.14)
has two positive zeros E 1 , E 2 such that
⎧
⎨ θ( E 1 ) = θ( E 2 ) = 0,
θ(u ) > 0 on (0, E 1 ) ∪ ( E 2 , ∞),
⎩
θ(u ) < 0 on ( E 1 , E 2 ).
(4.15)
To finish verifying (H2b) for f (u ), it suffices to prove that:
(I) B 1 < C 1 < E 1 B 2 < C 2 < E 2 < β3 , and
(II) θ( B 1 ) − B 1 θ ( B 1 ) − θ(C 2 ) 0,
which we prove as follows:
We know that I (u ) < J (u ) for u > 5. In addition, we compute that, for u > 3π /2 ≈ 4.712,
J (u ) − M (u ) =
−2(3 + 5u 2 + u 4 )
6
3π − 2u
+ 3 tan−1 u <
< 0.
2
2
2
u (1 + u )
u
u3
(4.16)
Thus,
(1) I (u ) < J (u ) < M (u ) for u > 5.
In addition, we know that
(2) I (0) = J (0) = limu →0+ M (u ) = −1.
(3) limu →∞ I (u ) = limu →∞ J (u ) = limu →∞ M (u ) = 0.
(4) I (u ) has exactly one critical √
point, a local maximum, at 1, on (0, 5). J (u ) has exactly one critical
point, a local maximum, at 3, on (0, 5). M (u ) has exactly one critical point, a local maximum,
at Ẽ ≈ 2.726, on (0, 5).
(5) Functions I (u ), J (u ), and M (u ) are all strictly decreasing on [5, ∞).
838
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
(6) √
The unique positive zeros of functions I (u ) − J (u ), I (u ) − M (u ), and J (u ) − M (u ) on (0, 5] are
3, ξ2 ≈ 2.106 and Ẽ ≈ 2.726, respectively. Moreover,
√
√
I ( 3 ) = J ( 3 ) = 1/8 = 0.125,
I (ξ2 ) = M (ξ2 ) = η2 ≈ 0.0766,
J ( Ẽ ) = M ( Ẽ ) ≈ 0.0905
by (2.3). See Fig. 9 for graphs of functions I (u ), J (u ) and M (u ) on (0, 5].
By parts (1)–(6), we obtain that:
(a) for 0 < r < η2 q, B 1 < C 1 < E 1 < B 2 < C 2 < E 2 ;
(b) for r = η2 q, B 1 < C 1 < E 1 = B 2 < C 2 < E 2 ;
see Fig. 9. In addition, we know that, for (q, r ) ∈ R ∗2 , g (u ) changes from decreasing to increasing then
to decreasing on [0, β3 ). Thus, by θ (u ) = −u 2 g (u ) and (4.6), we obtain that C 2 < β3 . By (1.12), we
β
know that β 3 f (u ) du > 0. Then θ(β3 ) = 2F (β3 ) > 2F (β1 ) > 0 by (4.13). Thus E 2 < β3 by (4.15) and
1
β3 > C 2 > E 1 . So we obtain that, for (q, r ) ∈ R ∗2 , B 1 < C 1 < E 1 B 2 < C 2 < E 2 < β3 .
Letting r /q = η , by (4.5) and (4.14), we obtain that
B 31 θ( B 1 ) − B 1 θ ( B 1 ) =
3
B 31 =
3
B 31 =
3
I ( B 1 ) − M ( B 1 ) − B 31 I ( B 1 ) − J ( B 1 )
J (B1) − M(B1) −
−2B 1 (3 + 8B 21 + 9B 41 )
=
η − M ( B 1 ) − B 31 η − J ( B 1 )
3(1 + B 21 )3
2
3
B 31 I ( B 1 ) − J ( B 1 )
+ 2 tan−1 B 1
since (4.11) and (4.16). Define
N 3 (u ) ≡
−2u (3 + 8u 2 + 9u 4 )
+ 2 tan−1 u
3(1 + u 2 )3
and compute that
N 3 (u ) =
8u 4 (u 2 − 1)
(1 + u 2 )4
.
We know that, for 0 < η η2 , B 1 (η) B 1 (η2 ) ≈ 0.636 < 1, and it is clear that N 3 (u ) is strictly
decreasing on (0, B 1 (η2 )]. Thus, for 0 < r η2 q, θ( B 1 ) − B 1 θ ( B 1 ) = N 3 ( B 1 ) N 3 ( B 1 (η2 )) ≈ −0.0467.
Letting r /q = η and by (4.14), (4.16), we obtain that
−θ(C 2 ) = −
C 23 3
η − M (C 2 ) = −
C 23 3
J (C 2 ) − M (C 2 ) =
2C 2 (3 + 5C 22 + C 24 )
3(1 + C 22 )2
Define
N 4 (u ) ≡
2u (3 + 5u 2 + u 4 )
3(1 + u 2 )2
− 2 tan−1 u
− 2 tan−1 C 2 .
S.-H. Wang, T.-S. Yeh / J. Differential Equations 255 (2013) 812–839
839
and compute that
N 4 (u ) =
2u 4 (u 2 − 3)
3(1 + u 2 )3
.
√
We know that, for 0 < η η2 , C 2 (η) C 2 (η2 ) ≈ 3.096 > 3, and it is clear that N 4 (u ) is strictly
increasing on [C 2 (η2 ), ∞). Thus, for 0 < r η2 q, −θ(C 2 ) = N 4 (C 2 ) N 4 (C 2 (η2 )) ≈ 0.114. So for 0 <
r η2 q, we obtain that θ( B 1 ) − B 1 θ ( B 1 ) − θ(C 2 ) N 3 ( B 1 (η2 )) + N 4 (C 2 (η2 )) ≈ 0.0672 > 0. So by
above analyses, for (q, r ) ∈ R ∗2 , f satisfies (H2b).
√
β
By (1.12), we know that, for fixed q > 3 3 and r̄2 (q) < r < r1 (q), β 3 f (u ) du > 0. Then there exists
1
γ
a positive number γ ∈ (β2 , β3 ) satisfying β1 f (u ) du = 0 since f (u ) < 0 on (β1 , β2 ) and f (u ) > 0 on
(β2 , β3 ). So, for (q, r ) ∈ R ∗2 , f satisfies (H3).
We summarize the preceding results and we conclude that, for (q, r ) ∈ R ∗2 , f satisfies (H1b), (H2b)
√
and (H3). By Lemma 3.2 and (3.16) and since r < 3 3/8, the results in (2.7) hold and the bifurcation
diagram S̄ is a broken S-shaped curve on the (λ, u ∞ )-plane. More precisely, S̄ consists of two
connected components such that the upper branch of S̄ has exactly one turning point (λ∗ , u λ∗ ∞ ),
and γ < u λ∗ ∞ < β3 , where the curve turns to the right, and the lower branch of S̄ is a monotone
increasing curve starting at (λ̂, 0). In addition, we have λ̂ < λ∗ , see the proof of Theorem 2.1. So we
obtain immediately the exact multiplicity result and ordering results of the solutions in parts (i)–(iv).
The proofs of results in (2.8) are easy but tedious; we omit them.
The proof of Theorem 2.2 is now complete. 2
Acknowledgments
Most of the computation in this paper has been checked using the symbolic manipulator Mathematica 7.0. The authors appreciate the anonymous referee for giving valuable suggestions in the
manuscript.
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