Third order boundary value problems with nonlocal boundary conditions

Nonlinear Analysis 71 (2009) 1542–1551

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Third order boundary value problems with nonlocal boundary conditions John R. Graef a,∗ , J.R.L. Webb b a

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA

b

Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK

article

a b s t r a c t

info

Article history: Received 29 September 2008 Accepted 14 December 2008

We treat the existence of multiple positive solutions for a third order boundary value problem with nonlocal boundary conditions. In our work, each of the three boundary conditions is a linear functional on C [0, 1], that is, given by a Stieltjes integral, but is not necessarily a positive functional. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Boundary value problems Hammerstein integral operator Nonlocal boundary conditions Third order problems

1. Introduction A recent paper [1] discussed the existence or nonexistence of at least one positive solution for the nonlinear third order equation u000 (t ) = g (t )f (u(t )),

0 < t < 1,

(1.1)

with the nonlocal boundary conditions (BCs) u(0) = 0,

u0 (p) = 0,

1

Z

w(t )u00 (t )dt = 0.

(1.2)

q

Here q > p > 1/2 are constants, w : [q, 1] → [0, ∞) is continuous and nondecreasing, and w(t ) > 0 for t > q. The results in [1] were extended to higher order equations in [2]. In this paper, we wish to treat the existence of multiple positive solutions for a similar but more general problem, with three nonlocal BCs. We do this by first studying the boundary value problem (BVP) u000 (t ) = g (t )f (t , u(t )),

u(0) = 0,

u0 (p) = 0,

u00 (1) = λ[u00 ],

R1

(1.3)

where p ≥ 1/2 and λ[v] = 0 v(t )dΛ(t ) is a linear functional on C [0, 1] given by a Riemann–Stieltjes integral with Λ a suitable function of bounded variation. Since λ can include both sums and integrals, this is a more general setup than in (1.2). Moreover, we do not require the signed measure dΛ to be supported on a subset of [p, 1] nor do we require the sign restriction on w . BC (1.2) restricts the solution to having a change of concavity between q and 1, but our BC does not. Other recent results in the study of nonlocal boundary value problems can be found in the works of Chu and Zhou [3], Du et al. [4, 5], Eggensperger and Kosmatov [6], Hao et al. [7], Infante and Webb [8], and Webb and Infante [9]. For recent contributions

∗

Corresponding author. E-mail addresses: [email protected] (J.R. Graef), [email protected] (J.R.L. Webb).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.12.047

J.R. Graef, J.R.L. Webb / Nonlinear Analysis 71 (2009) 1542–1551

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to the study of multi-point problems, see, for example, the papers of Anderson [10], Du et al. [11], Li et al. [12], Liu and Zhao [13], Maroun [14], and Zhou [15]. Once we have studied the boundary value problem (1.3), we can call on recent results of Webb and Infante [16] to study a considerably more general set of nonlocal BCs, namely, u(0) = α[u],

u0 (p) = 0,

u00 (1) + β[u] = λ[u00 ]

(1.4)

where α[u] and β[u] are given by Stieltjes integrals with signed measures. One motivation for the problems studied here is that of the beam equation u(4) (t ) = g (t )f (u(t )),

0 < t < 1,

(1.5)

with the nonlocal boundary conditions u(0) = u0 (0) = u00 (p) =

Z

1

w(s)u000 (s)ds = 0.

(1.6)

q

This models the deflection of a beam that is clamped at the endpoint t = 0, has a sensor placed at the point t = p, and has a distributed feedback controller at t = 1. Problem (1.5) and (1.6) can be viewed as a generalization of the problem u(4) (t ) = g (t )f (u(t )),

0 < t < 1,

(1.7)

u(0) = u (0) = u (1) = u (1) = 0, 0

00

000

(1.8)

which models a beam that is clamped at t = 0 and is free at t = 1 (a cantilever). To see that this is the case, consider the R1 boundary condition (1.3) with λ[v] = v(1) − ω[v] where ω[v] = q w(t )v(t )dt (i.e., condition (1.2)) and with w(t ) ≡ 1, namely, u(0) = u0 (0) = u00 (p) = 0,

u00 (q) − u00 (1) = 0.

(1.9)

The condition u (q) = u (1) implies there exists r ∈ (q, 1) such that u (r ) = 0. As p → 1, we have q, r → 1, and the condition u00 (q) − u00 (1) = 0 ‘‘tends to’’ the condition u000 (1) = 0, that is, we obtain condition (1.8). Notice that if we apply similar reasoning to the boundary condition (1.9) letting q → 1, we obtain the three-point condition 00

00

000

u(0) = u0 (0) = u00 (p) = u000 (1) = 0.

(1.10)

Such boundary conditions were considered, for example, by Graef et al. [17–19]. Now, if in problem (1.5) and (1.6) we write v(t ) = u0 (t ), then (1.5) and (1.6) can be written as

v 000 (t ) = g (t )f

t

Z

 v(s)ds ,

0 < t < 1,

(1.11)

0

with the nonlocal boundary conditions

v(0) = v 0 (p) =

Z

1

w(s)v 00 (s)ds = 0,

(1.12)

q

which we see has the form of the problems under consideration in this paper. 2. The problem with one nonlocal BC A standard method to find positive solutions of the BVP u000 (t ) = g (t )f (t , u(t )), u(0) = 0,

u0 (p) = 0,

0 < t < 1, u00 (1) = λ[u00 ],

(2.1)

is to seek fixed points of the corresponding Hammerstein integral operator S0 u(t ) :=

1

Z

G(t , s)g (s)f (s, u(s))ds

(2.2)

0

where G is the Green function, provided this Green function has suitable positivity properties. If G is non-negative, one can work in the cone P = {u ∈ C [0, 1] : u ≥ 0} of non-negative functions in the space C [0, 1] of continuous functions endowed with the usual supremum norm. To obtain multiple positive solutions, it has proved to be convenient to work in a smaller cone than P, namely, for some subinterval [a, b] of [0, 1], K0 := {u ∈ P : min u(t ) ≥ c0 kuk}, t ∈[a,b]

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where c0 > 0 is a constant. The cone K0 is of a well-known type, apparently first used by Krasnosel’ski˘ı, and may be found in Section 45.4 of [20], and by D. Guo, see for example [21]; it has also been used by many other authors in the study of multiple solutions of BVPs. To use this cone, some additional positivity property of the Green function is needed. Lan and Webb [22] gave a framework which has been refined in [16,23] and which fits the use of this cone rather well. For examples of recent contributions to the study of weakly singular problems (singular in t), we refer the reader to the papers of Hao et al. [7], Lan [24], Lan and Webb [22], Li [25], for problems singular in u we refer to Liu et al. [26], and Staněk [27]. Here we study weakly singular problems in (2.1), (2.2); we only impose Carathéodory conditions on f (see (H4 )), and the term g can have singularities at arbitrary points of [0, 1]. The rather weak conditions imposed on G and g in (2.2) are the following ones.

(H1 ) The kernel G is measurable, non-negative, and for every τ ∈ [0, 1] satisfies lim |G(t , s) − G(τ , s)| = 0

for almost every s ∈ [0, 1].

t →τ

(H2 ) There exist a subinterval [a, b] ⊆ [0, 1], a measurable function Φ , and a constant c0 ∈ (0, 1] such that G(t , s) ≤ Φ (s)

for t ∈ [0, 1] and almost every s ∈ [0, 1],

G(t , s) ≥ c0 Φ (s) for t ∈ [a, b] and almost every s ∈ [0, 1].

Rb (H3 ) g Φ ∈ L1 [0, 1], g ≥ 0 almost everywhere, and a Φ (s)g (s)ds > 0. (H4 ) The nonlinear term f : [0, 1] × R+ → [0, ∞) satisfies Carathéodory conditions, that is, f (·, u) is measurable for each fixed u ∈ R+ , f (t , ·) is continuous for almost every t ∈ [0, 1], and for each r > 0, there exists φr ∈ L∞ [0, 1] such that 0 ≤ f (t , u) ≤ φr (t )

for all u ∈ [0, r ] and almost all t ∈ [0, 1].

It is often convenient to establish the following type of inequality

(H02 ) c (t )Φ (s) ≤ G(t , s) ≤ Φ (s), for 0 ≤ t , s ≤ 1, which proves (H2 ) when c (t ) ≥ c0 > 0 on [a, b], and we shall do this here. Condition (H2 ) is the key one to enable use of the cone K0 . It was shown in [16,23] that very many BCs fit this framework, for example, separated BCs, m-point BCs, and other nonlocal BCs. Firstly, we find the Green function for the basic problem under study and show that it satisfies these conditions. R1 Recall that λ[v] = 0 v(t )dΛ(t ) is a linear functional on C [0, 1], namely, a Riemann–Stieltjes integral with Λ a function of bounded variation, but we do not suppose it is a positive functional. A typical example, which covers multi-point BCs and integral BCs as special cases, is

λ[v] =

m X

αi v(ηi ) +

1

Z

α(t )v(t )dt 0

i=1

where ηi are distinct points in [0, 1] and α is continuous. In this case,

Λ(s) :=

s

Z

X

dΛ(t ) = 0

αi +

{i:ηi ≤s}

s

Z

α(t )dt . 0

Theorem 2.1. Suppose λ[1] 6= 1. Then the unique solution of the BVP u000 (t ) = y(t ),

u(0) = 0,

u0 (p) = 0,

u00 (1) = λ[u00 ],

is given by u(t ) = (tp − t 2 /2)

Z 1

1+

0

Λ(s)  y(s)ds − t 1 − λ[1]

p

Z

(p − s)y(s)ds + 0

0

Proof. Integrating the given differential equation once, we have u00 (t ) = u00 (0) + Y (t ),

where Y (t ) :=

t

Z

y(s)ds. 0

Therefore,

λ[u00 ] = u00 (0)λ[1] + λ[Y ]. Using the BC at 1 gives u00 (0) + Y (1) = u00 (0)λ[1] + λ[Y ],

that is u00 (0) =

t

Z

λ[Y ] − Y (1) . 1 − λ[1]

(t − s)2 2

y(s)ds.

(2.3)

J.R. Graef, J.R.L. Webb / Nonlinear Analysis 71 (2009) 1542–1551

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Integrating again, we obtain u0 (t ) = u0 (0) + tu00 (0) +

Z

t

(t − s)y(s)ds

0

from which, using the BC u0 (p) = 0, we obtain u0 (0) = −pu00 (0) −

p

Z

(p − s)y(s)ds. 0

We then have u(t ) = tu0 (0) + (t 2 /2)u00 (0) +

(t − s)2

t

Z

2

0

= (tp − t /2) 2

Y (1) − λ[Y ]

p

Z

(p − s)y(s)ds +

−t

1 − λ[1]

y(s)ds t

Z

(t − s)2

0

0

2

y(s)ds.

(2.4)

Since

λ[Y ] =

1

Z 0

t

Z



y(s)ds dΛ(t ) =

Z

0

1

(Λ(1) − Λ(s)) y(s)ds

0

= Λ(1)Y (1) −

1

Z

Λ(s) y(s)ds 0

and λ[1] = Λ(1), (2.4) gives (2.3).



As we are interested in the existence of positive solutions, we will suppose that λ[1] < 1. We will now verify the key step, namely, to show that (H02 ) is satisfied. We do this under n sufficiently general conditions on λ so as to include the case studied in [1]. Let H denote the Heaviside function H (t ) :=

1 0

if t ≥ 0, if t < 0.

Theorem 2.2. Suppose that 1/2 ≤ p ≤ 1 and λ[1] < 1, and let Gλ (t , s) be the Green function

Λ(s) Gλ (t , s) := (tp − t /2) 1 + 1 − λ[1] 2



Λ(s)



− t (p − s)H (p − s) +

(t − s)2 2

H (t − s).

(2.5)

(s−p)

Suppose that Λ(s) ≥ 0 for s ≤ p and 1−λ[1] ≥ − 1−p for s > p. Then Gλ satisfies (H02 ), that is, for 0 ≤ t ≤ 1, 0 ≤ s ≤ 1, we have

   2 p p2 Λ(s)   + ,  2 2 1 − λ[1] Gλ (t , s) ≤ Φ (s) :=    p2 Λ(s) s2   + , 2 2 1 − λ[1]

if s ≥ p, if s < p,

and, with c (t ) = (2tp − t 2 )/p2 , we have Gλ (t , s) ≥ c (t )Φ (s). Proof. The upper bounds are obtained by finding maxt ∈[0,1] {Gλ (t , s)} for each fixed s. We first consider a fixed s > p, and Λ(s) −s write Q (s) := 1 + 1−λ[1] so that Q (s) ≥ 11− > 0. We have p

(∂/∂ t )Gλ (t , s) = (p − t )Q (s) + (t − s)H (t − s), which is positive for t < p. For p < t < s, the derivative is negative, while for p ≤ s ≤ t ≤ 1 the derivative becomes s−p Q (s) zero at t = tc := 1−Q (s) . The derivative is positive for t > tc , so there is a local minimum at t = tc . The point tc satisfies −s s < tc ≤ 1 if and only if Q (s) < 1 and Q (s) ≤ 11− . Therefore, under our hypothesis, the maximum of Gλ (t , s) for this fixed p s and t ∈ [s, 1] occurs when t = s, and the value of Gλ (t , s) there is smaller than the value for t = p. This gives the upper half of the expression for Φ . When s ≤ p, using the fact that Λ(s) > 0, we have for t > s,

 Λ(s) Gλ (t , s) = (tp − t /2) 1 + − t (p − s) + (t − s)2 /2 1 − λ[1] Λ(s) Λ(s) = (tp − t 2 /2) + s2 /2 ≤ (p2 /2) + s2 /2. 1 − λ[1] 1 − λ[1] 2



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J.R. Graef, J.R.L. Webb / Nonlinear Analysis 71 (2009) 1542–1551

When s ≤ p and t ≤ s, we have

Λ(s) Gλ (t , s) = (tp − t /2) 1 + 1 − λ[1] 2



= (ts − t 2 /2) + (tp − t 2 /2)



− t (p − s)

Λ(s) 1 − λ[1]

≤ s2 /2 + (p2 /2)

Λ(s) 1 − λ[1]

.

For the lower bounds, we consider t fixed. When t < p we have to show:

2  2 Λ(s) + s2 /2 ≥ c (t ) s2 + p2 1−λ[ for 0 ≤ s ≤ t; 1]   2  2 p Λ(s) Λ(s) (ii) (ts − t 2 /2) + (tp − t 2 /2) 1−λ[1] ≥ c (t ) s2 + 2 1−λ[1] for t ≤ s ≤ p;    2  p p2 Λ(s) Λ(s) (iii) (tp − t 2 /2) 1 + 1−λ[1] ≥ c (t ) 2 + 2 1−λ[1] for p ≤ s ≤ 1. (i) (tp − t 2 /2)



Λ(s) 1−λ[1]



Recall that Λ(s) ≥ 0 for s ≤ p, that is, for cases (i) and (ii). Firstly, (i) holds since c (t ) = (tp − t 2 /2)/(p2 /2) ≤ 1. Secondly, (ii) is satisfied if (ts − t 2 /2) ≥ c (t )s2 /2, that is, if c (t ) t

≤

2 s

−

t

for t ≤ s ≤ p.

s2

The expression on the right side of the inequality is decreasing in s, so this inequality holds if and only if

c (t ) t

≤

2 p

−

t p2

,

which is true by the definition of c. Case (iii) is an equality. For t > p, each of the three similar inequalities is either the same as one of the above or the left side has an extra non-negative term (t − s)2 /2, so the inequalities clearly hold.  (s−p)

Λ(s)

R1

Remark 2.3. (1) The condition 1−λ[1] ≥ − 1−p is satisfied if λ[v] = v(1) − ω[v] where ω[v] = p w(t )v(t )dt and w > 0 Rs Λ(s) −Ω (s) is a nondecreasing function. In this case, writing Ω (s) := p w(t )dt, we have 1−λ[1] = Ω (1) for s < 1 and the condition Ω (s)

then becomes Ω (1) < 1−p which holds because w is nondecreasing. This shows that our case strictly includes the situation covered in [1]. (2) The function c in this result is the same function used in [1], where it was found by another method. (3) Even if Λ(s) ≥ 0 for all s (which we do not assume), λ need not be a positive functional. For example, if λ[v] = R1 R1 Rs (2π s) ≥ 0, but λ[t ] = 0 t sin(2π t )dt = −1/(2π ). v(t ) sin(2π t )dt, then Λ(s) = 0 sin(2π t )dt = 1−cos 0 2π s −p

Solutions of BVP (2.1) are fixed points of S0 u(t ) :=

1

Z

Gλ (t , s)g (s)f (s, u(s))ds. 0

Since we have shown that Gλ satisfies the required hypotheses to fit the framework, we could immediately give some results on existence of multiple positive solutions for problem (2.1) under suitable conditions on f . However, as this problem is a special case of more general problems to which we now turn, we defer stating such results. 3. General nonlocal BCs Webb and Infante [16] have shown that existence of positive solutions of nonlocal BVPs with two nonlocal boundary terms α[u], β[u] given by Stieltjes integrals with signed measures, can be studied, in a unified way, via a perturbed Hammerstein integral equation of the type u(t ) = γ (t )α[u] + δ(t )β[u] +

1

Z

G(t , s)g (s)f (s, u(s))ds 0

=: γ (t )α[u] + δ(t )β[u] + S0 u(t ) =: Tu(t ).

(3.1)

Here G(t , s) is the Green function for the corresponding, but simpler, problem with no nonlocal terms. The Green function can then be found for the problem with two nonlocal terms, without a lengthy calculation. The theory of [16] employs the following hypotheses. Suppose that (H1 )–(H4 ) listed above hold for the kernel G in (3.1). The following conditions are required on the remaining terms occurring in (3.1):

(H5 ) A, B are of bounded variation and GA (s), GB (s) ≥ 0 for a.e. s, where Z 1 Z 1 GA (s) := G(t , s)dA(t ) and GB (s) := G(t , s)dB(t ). 0

0

(H6 ) γ ∈ C [0, 1], γ (t ) ≥ 0, 0 ≤ α[γ ] < 1, β[γ ] ≥ 0. There exists c1 ∈ (0, 1] such that γ (t ) ≥ c1 kγ k for t ∈ [a, b], that is, γ is strictly positive on [a, b].

J.R. Graef, J.R.L. Webb / Nonlinear Analysis 71 (2009) 1542–1551

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(H7 ) δ ∈ C [0, 1], δ(t ) ≥ 0, 0 ≤ β[δ] < 1, α[δ] ≥ 0. There exists c2 ∈ (0, 1] such that δ(t ) ≥ c2 kδk for t ∈ [a, b]. (H8 ) D := (1 − α[γ ])(1 − β[δ]) − α[δ]β[γ ] > 0. It is shown in [16] that solutions of (3.1) are fixed points of the map S given by

γ (t ) h

Su(t ) =

D

+

1

Z

(1 − β[δ])

GA (s)g (s)f (s, u(s))ds + α[δ]

D

(1 − α[γ ])

1

Z

GB (s)g (s)f (s, u(s))ds + β[γ ]

G(t , s)g (s)f (s, u(s))ds :=

+

1

Z

GA (s)g (s)f (s, u(s))ds

i

0

0

1

Z

i

GB (s)g (s)f (s, u(s))ds 0

0

δ(t ) h

1

Z

1

Z

GS (t , s)g (s)f (s, u(s))ds.

(3.2)

0

0

The kernel GS is the Green function corresponding to the underlying BVP. Moreover, as shown in [16], GS satisfies hypotheses (H1 ), (H2 ). The idea in [16], which allows sign changing measures dA, dB in these problems, is to work in the cone K = {u ∈ P , min u(t ) ≥ c kuk, α[u] ≥ 0, β[u] ≥ 0},

(3.3)

t ∈[a,b]

where c = min{c0 , c1 , c2 }. It is then shown that T : K → K and S : P → K and are compact operators. In our case, the BVP has three nonlocal BCs, namely u000 (t ) = g (t )f (t , u(t )), u(0) = α[u],

t ∈ (0, 1),

u0 (p) = 0,

(3.4)

u00 (1) + β[u] = λ[u00 ].

Solutions of this BVP are fixed points of Tu(t ) = γ (t )α[u] + δ(t )β[u] + S0 u(t ), where S0 u(t ) =

R1 0

Gλ (t , s)g (s)f (s, u(s))ds and

γ (t ) = 0, γ (0) = 1, γ 0 (p) = 0, γ 00 (1) = 0, 000 0 00 δ (t ) = 0, δ(0) = 0, δ (p) = 0, δ (1) = −1. 2 2 Hence, we have γ (t ) ≡ 1, δ(t ) = p t − t /2 = p c (t )/2. Thus, c1 (t ) = 1, c2 (t ) = c (t ) in (H6 ) and (H7 ), with c (t ) as in Theorem 2.2. Hence, c = mint ∈[a,b] c (t ) in (3.3), where [a, b] can be chosen arbitrarily in (0, 1). For (H8 ) we need 000

0 ≤ α[1] < 1,



(1 − α[1]) 1 − GA (s) =

0 ≤ β[c ] < 2/p2 , 2

p

2



β[c ] −

p

2

2

α[c ] ≥ 0,

α[c ]β[1] > 0,

1

Z

β[1] ≥ 0,

Gλ (t , s)dA(t ) ≥ 0,

GB (s) =

(3.6) 1

Z

0

(3.5)

Gλ (t , s)dB(t ) ≥ 0.

(3.7)

0

We have

Z 1 Λ(s) p2 (t − s)2 GA (s) = α[c ] + α[c ] − α[t ](p − s)H (p − s) + dA(t ). 2 (1 − λ[1]) 2 2 s Therefore, using the assumptions on Λ given in Theorem 2.2, and noting that Z 1 p2 α[c ] − (p − s)α[t ] = (st − t 2 /2)dA(t ), p2

2

0

we see that GA (s) ≥ 0 if 1

Z

( t − s) 2 2

s

dA(t ) ≥ 0

for s ≥ p,

and s

Z

(st − t 2 /2)dA(t ) + 0

Z s

1

s2 2

dA(t ) ≥ 0

for s < p.

The same conditions are to be satisfied by the signed measure dB(t ). This is always satisfied when dA, dB are positive measures. An example of a sign changing measure that satisfies these conditions is given by dA(t ) = (1/4 + cos(2π t ))dt . In this case, α[1] = 1/4 < 1 and α[c ] =

(3p−1)π 2 −6 12p2 π 2

> 0.

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J.R. Graef, J.R.L. Webb / Nonlinear Analysis 71 (2009) 1542–1551

4. Existence and nonexistence results We now state some existence results which follow from the application of fixed point index theory and results of [16, 23]. We use the following constants: mS :=



Z sup

t ∈[0,1]

1

GS (t , s)g (s)ds

−1

,

MS = MS (a, b) :=

0



b

Z

GS (t , s)g (s)ds

inf

t ∈[a,b]

−1

.

(4.1)

a

We also use some results which depend on the behavior of f (u)/u near 0 and near ∞ and were proved in [23] for integral operators of the form S with kernels satisfying the required weak conditions (H1 )–(H3 ). This allows us to give some sharp results. It uses comparison with the linear operator LS defined by LS u(t ) :=

1

Z

GS (t , s)g (s)u(s)ds 0

with GS as in (3.2). It is shown in [23] that, under our hypotheses, the radius of the spectrum r (LS ) is an eigenvalue of LS with eigenfunction ϕ in P. Since LS : P → K , it follows that ϕ ∈ K . We write µ1 := 1/r (LS ) and call µ1 the principal characteristic value of LS ; it is often called the principal eigenvalue of the corresponding differential equation. We use the following notation. f (u) := sup f (t , u),

f (u) := inf f (t , u);

f = lim sup f (u)/u,

f0 = lim inf f (u)/u;

t ∈[0,1]

0

t ∈[0,1]

u→0+

u→0+

f

∞

= lim sup f (u)/u, u→∞

f∞ = lim inf f (u)/u. u→∞

Theorem 4.1. Suppose λ satisfies the conditions in Theorem 2.2 and that α, β satisfy (3.5)–(3.7). Then BVP (3.4) has at least one positive solution u ∈ K if one of the following conditions holds.

(S1 ) 0 ≤ f 0 < µ1 and µ1 < f∞ ≤ ∞. (S2 ) µ1 < f0 ≤ ∞ and 0 ≤ f ∞ < µ1 . BVP (3.4) has at least two positive solutions in K if one of the following conditions holds.

(D1 ) There exists ρ > 0 such that 0 ≤ f 0 < µ1 , 0≤f

∞

f (t , u) > ρ MS

for t ∈ [a, b] and u ∈ [ρ, ρ/c ],

< µ1 .

(D2 ) There exists ρ > 0 such that µ1 < f0 ≤ ∞, µ1 < f∞ ≤ ∞.

f (t , u) < ρ mS

for t ∈ [0, 1] and u ∈ [0, ρ],

BVP (3.4) has at least three positive solutions in K if either (T1 ) or (T2 ) holds.

(T1 ) There exist 0 < ρ1 < c ρ2 < ∞, such that 0 ≤ f 0 < µ1 , f (t , u) < mS ρ2

f (t , u) > ρ1 MS

for t ∈ [a, b] and u ∈ [ρ1 , ρ1 /c ],

for t ∈ [0, 1] and u ∈ [0, ρ2 ],

µ1 < f∞ ≤ ∞.

(T2 ) There exist 0 < ρ1 < ρ2 < ∞, such that µ1 < f0 ≤ ∞, f (t , u) < ρ1 mS for t ∈ [0, 1] and u ∈ [0, ρ1 ], f (t , u) > ρ2 MS for t ∈ [a, b] and u ∈ [ρ2 , ρ2 /c ], 0 ≤ f ∞ < µ1 . Note that ‘positive’ here means a nonzero element of K . Instead of using the sharp conditions such as f 0 < µ1 , f∞ > µ1 , we could use the stronger conditions f 0 < mS , f∞ > MS ; similarly for the conditions with reversed inequality signs. It was shown in [23] that one always has mS ≤ µ1 ≤ MS and the inequalities are strict if the corresponding eigenfunction is not constant. In fact, 1/mS = kLS k shows that mS ≤ µ1 . The index results involving µ1 were proved in [23] where an extra hypothesis that was called (UPE ) was used for one result. However, a result of Nussbaum, Lemma 2 on page 226 of [28], shows that Theorem 3.7 of [23] is valid without that hypothesis, so the index result holds without (UPE ). We now give a nonexistence result which shows that the above result on existence of one solution is sharp. We will use a concept due to Krasnosel’ski˘ı [29,20].

J.R. Graef, J.R.L. Webb / Nonlinear Analysis 71 (2009) 1542–1551

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Definition 4.2. We say that a bounded linear operator L is u0 -positive on the cone P, if there exists u0 ∈ P \ {0}, such that for every u ∈ P \ {0} there are positive constants k1 (u), k2 (u) such that k1 (u)u0 (t ) ≤ Lu(t ) ≤ k2 (u)u0 (t ),

for every t ∈ [0, 1].

Theorem 4.3. Suppose that LS is u0 -positive for some u0 ∈ P \ {0}. Let µ1 = 1/r (LS ) be the principal characteristic value of LS . Suppose that one of the following conditions hold. (i) f (t , u) < µ1 u, for all u > 0 and almost all t ∈ [0, 1]. (ii) f (t , u) > µ1 u, for all u > 0 and almost all t ∈ [0, 1]. If (i) holds, then 0 is the unique fixed point of S in P. If (ii) holds, then 0 is the only possible fixed point of S in P. Proof. (i) If u = Su for some u ∈ P \ {0}, then, since G(t , s) > 0 for a.e. s, t ∈ (0, 1), we have 1

Z

u( t ) =

G(t , s)g (s)f (s, u(s))ds <

1

Z

G(t , s)g (s)µ1 u(s)ds = µ1 LS u(t ). 0

0

Since also ϕ = µ1 LS ϕ for an eigenfunction ϕ ∈ P, by Theorem 2.19 of [29] we must have u = αϕ for some α 6= 0. But then we would have u = µ1 LS u, which is a contradiction. (ii) The argument is exactly similar with a reversed inequality, but in this case f (t , 0) need not be 0.  To apply this result, we need to know when our operator is u0 -positive. Recall that we have shown that Gλ (t , s) ≥

Φ (s)c (t ) in Theorem 2.2.

Theorem 4.4. Suppose that Gλ (t , s) ≤ W (s)c (t ) where Wg ∈ L1 (0, 1). Then the linear operator L0 defined by L0 u(t ) := R1 Gλ (t , s)g (s)u(s)ds is c-positive on P. 0 Proof. We have c (t )

1

Z

Φ (s)g (s)u(s)ds ≤ 0

1

Z

Gλ (t , s)g (s)u(s)ds ≤ c (t ) 0

1

Z

W (s)g (s)u(s)ds, 0

so we can take (with c = mint ∈[a,b] c (t ) > 0) k1 (u) =

1

Z

Φ (s)g (s)u(s)ds ≥ 0

k2 (u) =

b

Z

Φ (s)g (s)c kukds > 0 a

1

Z

W (s)g (s)u(s)ds.  0

Theorem 4.5. Suppose that p > 1/2. If g and g Λ are integrable functions, then Gλ (t , s) ≤ W (s)c (t ) for a function W with Wg ∈ L1 (0, 1), so L0 is c-positive on P. Proof. We have



Gλ (t , s) = (tp − t 2 /2) 1 +

Λ(s) 1 − λ[1]



− t (p − s)H (p − s) +

(t − s)2 2

H (t − s).

Since c (t ) = 2(tp − t 2 /2)/p2 , it suffices to prove that (t − s)2 /2 ≤ C (tp − t 2 /2). In fact, (t − s)2 /2 ≤ t 2 /2 ≤ C (tp − t 2 /2) for C = 1/(2p − 1).  We illustrate the applicability of these results with some simple examples. Example 4.6. Consider the BVP, which is Example 1 in [1],

(1 + t ) u(1 + 3u) , t ∈ (0, 1), 10 (1 + u) u(0) = 0, u0 (2/3) = 0, u0 (3/4) = u0 (1)

u000 = ξ

(4.2)

where ξ > 0 is a parameter. Here, we have p = 2/3, g (t ) =

(1+t )

u(1+3u)

, f (u) = ξ (1+u) . The third BC can be written as R1 where λ[v] := v(1) − 3/4 v(t )dt. Then λ[1] = 3/4, 10

 Z s  − dt = −(s − 3/4), Λ(s) =  3/4 0,

3/4 < s < 1, s ≤ 3/4.

R1

3/4

u00 (t )dt = 0 or as u00 (1) = λ[u00 ]

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J.R. Graef, J.R.L. Webb / Nonlinear Analysis 71 (2009) 1542–1551

Hence, we have Gλ (t , s) = (2t /3 − t 2 /2)(1 + (3 − 4s)H (s − 3/4)) − t (2/3 − s)H (2/3 − s) +

1 2

(t − s)2 H (t − s).

By a numerical calculation, µ1 ≈ 66.643. In the present case we have f0 = f 0 = ξ , f∞ = f ∞ = 3ξ and ξ u ≤ f (u) ≤ 3ξ u for all u > 0. Combining the existence and nonexistence results gives the following conclusion. BVP (4.2) has at least one positive solution if 22.214 ≈ µ1 /3 < ξ < µ1 ≈ 66.643 and has no positive solution if

ξ < µ1 /3 ≈ 22.214 or if ξ > µ1 ≈ 66.643. Our result here is an improvement of the results of [1] since the constants here are sharp. (1+t ) u(1+3u) Note that we could have taken g˜ ≡ 1 and f˜ (t , u) = ξ 10 (1+u) . Then we have f˜0 = ξ /10, f˜ 0 = ξ /5, f˜∞ = 3ξ /10, and

f˜ ∞ = 3ξ /5. The principal characteristic value is, by a numerical calculation, µ ˜ 1 ≈ 10.963. This gives the result: BVP (4.2) has at least one positive solution if 36.54 ≈ 10µ ˜ 1 /3 < ξ < 5µ ˜ 1 ≈ 54.81 and has no positive solution if

ξ < 5µ ˜ 1 /3 ≈ 18.27 or if ξ > 10µ ˜ 1 ≈ 109.63. This illustrates how it is advantageous to allow the term g to be incorporated into the Green function. Example 4.7. Consider the BVP u000 = ξ u3/2 + u1/2 , u(0) = 0,

t ∈ (0, 1),

u (2/3) = 0, 0

u0 (3/4) = u0 (1),

(4.3)

where ξ > 0 is a parameter. Here we have g (t ) = 1, f (u) = ξ u3/2 + u1/2 , with the same BCs as before. In this case, µ1 ≈ 10.963, and m = 567/55 ≈ 10.309. Also f0 = ∞, f∞ = ∞ if ξ > 0. In this case, if ξ is not too large, there are two positive solutions. In fact, by a simple √ calculation, we see that for a given ξ > 0, the minimum of ξ r 1/2 + r −1/2 for r > 0 occurs when r = 1/ξ and equals 2 ξ . 2 Therefore, there exists r > 0 such that f (u) < m r for 0 ≤ u ≤ r if and only if ξ < m /4 ≈ 26.57. We may take r = 1/ξ in this case. Similarly, f (u) > µ1 u for all u > 0 if ξ > µ21 /4 ≈ 30.05, so there are no positive solutions in this case. References [1] J.R. Graef, B. Yang, Positive solutions of a third order nonlocal boundary value problem, Discrete Contin. Dyn. Syst. Ser. S 1 (2008) 89–97. [2] J.R. Graef, J. Henderson, B. Yang, Existence and nonexistence of positive solutions of an n-th order nonlocal boundary value problem, Proc. Dynam. Systems Appl. 5 (2008) 186–191. [3] J. Chu, Z. Zhou, Positive solutions and eigenvalues of nonlocal boundary-value problems, Electron. J. Differential Equations 2005 (86) (2005) 9. (electronic). [4] Z. Du, X. Lin, W. Ge, Nonlocal boundary value problem of higher order ordinary differential equations at resonance, Rocky Mountain J. Math. 36 (2006) 1471–1486. [5] Z.J. Du, X.J. Lin, W.G. Ge, Wei Gao, Solvability of a third-order nonlocal boundary value problem at resonance, Acta Math. Sinica (Chin. Ser.) 49 (2006) 87–94. [6] M. Eggensperger, N. Kosmatov, A higher-order nonlocal boundary value problem, Int. J. Appl. Math. Sci. 2 (2005) 196–206. [7] X. Hao, L. Liu, Y. Wu, Positive solutions for nonlinear nth-order singular nonlocal boundary value problems, Bound. Value Probl. (2007) 10. Art. ID 74517. [8] G. Infante, J.R.L. Webb, Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations, Proc. Edinb. Math. Soc. 49 (2006) 637–656. [9] J.R.L. Webb, G. Infante, Nonlocal boundary value problems of arbitrary order, J. London Math. Soc. (in press). [10] D. Anderson, Multiple positive solutions for a three-point boundary value problem, Math. Comput. Modelling 27 (1998) 49–57. [11] Z. Du, W. Liu, X. Lin, Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations, J. Math. Anal. Appl. 335 (2007) 1207–1218. [12] C. Li, L. Liu, Y. Yonghong, Multiple positive solutions for nonlinear second order m-point boundary value problems, J. Nonlinear Funct. Anal. Differ. Equ. 1 (2007) 121–135. [13] B. Liu, Z. Zhao, A note on multi-point boundary value problems, Nonlinear Anal. 67 (2007) 2680–2689. [14] M. Maroun, Positive solutions to an n-th order right focal boundary value problem, Electron. J. Qual. Theory Differ. Equ. (4) (2007) 17 (electronic). [15] Y. Zhou, The existence of positive solutions for four-point nonlinear boundary value problem, Int. J. Math. Anal. (Ruse) 1 (2007) 423–435. [16] J.R.L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc. 74 (2006) 673–693. [17] J.R. Graef, J. Henderson, P.J.Y. Wong, B. Yang, Three solutions of an nth order three-point focal type boundary value problem, Nonlinear Anal. 69 (2008) 3386–3404. [18] J.R. Graef, J. Henderson, B. Yang, Positive solutions of a nonlinear n-th order eigenvalue problem, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13B (Supplementary Volume) (2006) 39–48. [19] J.R. Graef, J. Henderson, B. Yang, Positive solutions of a nonlinear higher order boundary value problem, Electron. J. Differential Equations 2007 (45) (2007) 1–10.

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