Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems

Nonlinear Analysis 71 (2009) 1369–1378

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Multiple positive solutions of resonant and non-resonant nonlocal boundary value problemsI J.R.L. Webb a , M. Zima b,∗ a

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

b

Institute of Mathematics, University of Rzeszów, 35-310 Rzeszów, Poland

article

a b s t r a c t

info

Article history: Received 30 October 2008 Accepted 1 December 2008

We study the existence of positive solutions for equations of the form

−u00 (t ) = f (t , u(t )),

a.e. t ∈ (0, 1),

when f (t , u)+ω2 u ≥ 0 for u ≥ 0, for some constant ω > 0, and for the perturbed equation

MSC: 34B18 34B10

−u00 (t ) + ω2 u(t ) = h(t , u(t )),

a.e. t ∈ (0, 1),

when h(t , u) ≥ 0, subject to various non-local boundary conditions. In particular, we establish existence and multiplicity of positive solutions for non-perturbed boundary value problems at resonance by considering equivalent non-resonant perturbed problems with the same boundary conditions. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Nonlocal boundary conditions Fixed point index Positive solution Resonance

1. Introduction In the paper, we are interested in the existence of positive solutions of some non-local boundary value problems (BVPs) for equations, either of the form

− u00 (t ) = f (t , u(t )),

a.e. t ∈ (0, 1),

(1.1)

or, for some constant ω > 0,

− u00 (t ) + ω2 u(t ) = h(t , u(t )),

a.e. t ∈ (0, 1),

(1.2)

subject to one of the following non-local boundary conditions (BCs) u0 (0) = 0,

u(1) = β[u],

(1.3)

u(0) = 0,

u(1) = β[u],

(1.4)

u(0) = 0,

u (1) = β[u],

(1.5)

0

where β[u] is a linear functional on C [0, 1], that is, is given by a Riemann-Stieltjes integral

β[u] =

1

Z

u(s) dB(s) 0

I A support from TODEQ, project number MTK-CT-2005-030042, is gratefully acknowledged.

∗

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

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.12.010

(1.6)

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J.R.L. Webb, M. Zima / Nonlinear Analysis 71 (2009) 1369–1378

with B a function of bounded variation, that is, dB is a signed measure. Some kind of positivity on the functional β[u] is needed in order to have positive solutions, but we do not suppose that β[u] ≥ 0 for all u ≥ 0. By making the change of variable τ = 1 − t our discussion also covers the BCs u(0) = β[u],

u0 (1) = 0,

u(0) = β[u],

u(1) = 0,

u0 (0) + β[u] = 0,

u(1) = 0.

Examples of this type of BCs are the so-called multipoint BCs where β[u] := i=1 βi u(ηi ), 0 < η1 < η2 < · · · < ηm < 1, which have been extensively studied in recent years when all βi are positive or have the same sign (see for example [1–5]). For (1.1) we are particularly interested in cases where f (t , u) is not positive for all positive u but is such that f (t , u) + ω2 u ≥ 0 for u ≥ 0 for some constant ω > 0. One important motivation is that the original problem (1.1) with one of the BCs may be at resonance, that is, λ = 0 is an eigenvalue of the linear problem −u00 = λu with the given BCs. In such a case we can consider the equivalent problem

Pm

− u00 + ω2 u = h(t , u) := f (t , u) + ω2 u,

(1.7)

with the same BCs which, typically, is a non-resonant problem. For example, the BVP (1.1), (1.4) is at resonance when R1 P β[t ] := 0 t dB(t ) = 1 which, in the multipoint case, reads m i=1 βi ηi = 1. For problems at resonance, it can be necessary to have f changing sign in order to have positive solutions, see Theorem 3.3. A similar technique was used by Han in [6], where the following three-point BVP at resonance u00 (t ) = f (t , u(t )), u (0) = 0,

t ∈ (0, 1),

u(1) = u(η),

0

was studied via the equivalent BVP u00 (t ) + β 2 u(t ) = f (t , u(t )) + β 2 u(t ), u0 (0) = 0,

t ∈ (0, 1),

u(1) = u(η).

With this choice of sign, it is necessary that β ∈ (0, π2 ) in order to have a positive Green’s function. Our choice of sign imposes no size restriction on ω and is well suited to the consideration of positive solutions. The existence of one positive solution was shown in [6] under stronger conditions than we have; in [7] a result of the same type as one of ours is obtained. The method of [7] does not apply to the other BCs we study because the property u(t ) ≥ c kuk for all t ∈ [0, 1], to be satisfied by solutions, is used by Han, but this property does not hold for solutions of problems with the BC u(0) = 0. Our method proves the existence of multiple positive solutions, under suitable conditions on f . Moreover, our method yields a sharp result on existence of one positive solution. Multi-point boundary value problems at resonance have recently also been studied in [8,9,3,10–13]. In particular, Kosmatov [11] and Liu and Zhao [13] considered the existence of solutions for the BVP at resonance making use of the coincidence degree theory due to Mawhin (see, for example, [14]). Infante and Zima [3] proved the existence of positive solutions for the BVP at resonance by using the Leggett-Williams norm-type theorem for coincidences from [15]. It is to be noted that our method can prove the existence of several positive solutions for the resonance case, but the methods of Infante–Zima [3] only allow them to prove the existence of one positive solution. Resonance problems for local BCs have also been treated using the Lyapunov Schmidt procedure and the connectivity properties of solutions, for example [16]. 2. Existence results for integral equations A standard approach to studying positive solutions of a BVP such as

− u00 (t ) + ω2 u(t ) = h(t , u(t )),

t ∈ (0, 1),

(2.1)

with some BCs, when h(t , u) ≥ 0 for u ≥ 0, is to find the corresponding Green’s function G and seek solutions as fixed points of the integral operator Su(t ) :=

1

Z

G(t , s)h(s, u(s)) ds

(2.2)

0

in the cone P = {u ∈ C [0, 1] : u ≥ 0} of nonnegative functions in the space C [0, 1] of continuous functions endowed with the usual supremum norm. We will apply this approach using some results obtained by Webb and Infante [17], for precisely this type of problem. More general BCs are studied in [18]. The Green’s function for our BVPs

− u00 + ω2 u = h(t , u),

(2.3)

J.R.L. Webb, M. Zima / Nonlinear Analysis 71 (2009) 1369–1378

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with each of BCs (1.3)–(1.5), is found via the perturbed problem 1

Z

Tu(t ) = γ (t )β[u] +

G0 (t , s)h(s, u(s)) ds,

(2.4)

0

where G0 is the easily found Green’s function for the problem when β[u] ≡ 0 and γ is a solution of

−γ 00 + ω2 γ = 0, under (1.3)–(1.5), respectively, with β[u] replaced by 1. From this form, the Green’s function for each of our BVPs, the kernel in (2.2), is shown in [17] to be G(t , s) :=

γ (t ) 1 − β[γ ]

G(s) + G0 (t , s),

(2.5)

R1

where G(s) := 0 G0 (t , s) dB(t ). For an integral equation such as (2.2) with G as in (2.5) the following conditions on h and G0 are assumed to hold, as in [17].

(C1 ) G0 ≥ 0 is measurable, and for every τ ∈ [0, 1] we have lim |G0 (t , s) − G0 (τ , s)| = 0

t →τ

for a.e. s ∈ [0, 1].

(C2 ) There exists a subinterval [a, b] ⊆ [0, 1], a function Φ0 ∈ L∞ , with (c0 = c0 (a, b)) such that

Rb a

Φ0 (s)ds > 0, and a constant c0 ∈ (0, 1]

G0 (t , s) ≤ Φ0 (s) for t ∈ [0, 1] and almost every s ∈ [0, 1], G0 (t , s) ≥ c0 Φ0 (s)

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

R1 (C3 ) B is of bounded variation and G(s) := 0 G0 (t , s) dB(t ) ≥ 0 for a.e. s ∈ [0, 1]. (C4 ) γ ∈ C [0, 1], γ (t ) ≥ 0, 0 ≤ β[γ ] < 1 and there exists c1 ∈ (0, 1] (c1 = c1 (a, b)) such that γ (t ) ≥ c1 kγ k for t ∈ [a, b]. (C5 ) h : [0, 1] × R+ → R+ satisfies Carathéodory conditions, that is, h(·, u) is measurable for each fixed u ∈ R+ and h(t , ·) is continuous for almost every t ∈ [0, 1], and for each r > 0, there exists φr ∈ L∞ [0, 1] such that 0 ≤ h(t , u) ≤ φr (t ) for all u ∈ [0, r ] and almost all t ∈ [0, 1]. It is often convenient to prove the following type of inequality:

(C20 ) c0 (t )Φ0 (s) ≤ G0 (t , s) ≤ Φ0 (s), for 0 ≤ t , s ≤ 1, which establishes the inequality in (C2 ) when c0 (t ) ≥ c0 > 0 on [a, b]. Note that the sign condition in (C3 ) holds automatically for (positive) measures. As (C3 ), (C4 ) are integral (or sum) conditions, not pointwise ones, we can allow sign changing measures dB. The condition (C1 ) ensures that G(s) is well defined for a.e. s. γ (t ) Since γ (t ) ≥ kγ k kγ k for t ∈ [0, 1], the inequality in (C4 ) holds for arbitrary [a, b] ⊂ (0, 1) if γ (t ) > 0 for t ∈ (0, 1). Let P = {u ∈ C [0, 1] : u(t ) ≥ 0 for each t ∈ [0, 1]} be the standard cone of nonnegative continuous functions. To prove existence of multiple positive solutions, it is convenient to work in the following smaller cone, first used in [17]. It is a refinement of a well-known type of cone used by Krasnosel’ski˘ı, for example [19], and by many authors in the study of positive solutions. Let min u(t ) ≥ c kuk, β[u] ≥ 0},

K = {u ∈ P :

for c = min{c0 , c1 }.

t ∈[a,b]

(2.6)

This cone is used to avoid assuming that β[u] ≥ 0 for all u ∈ P and thus allows sign changing measures. Theorem 2.1 ([17]). Under the hypotheses (C1 )–(C5 ), T : K → K , S : P → K and both mappings are compact. Our results use the theory of fixed point index. One shows that the index on certain sets is 1 and on other sets is 0. Some of the results employ the constants m and M = M (a, b) defined by 1 m

1

Z

G(t , s)ds,

= sup

t ∈[0,1]

0

b

Z

1 M ( a, b )

G(t , s)ds.

= inf

t ∈[a,b]

(2.7)

a

Other results use comparison with the linear operator Lu(t ) :=

γ (t ) 1 − β[γ ]

Z 0

1

G(s)u(s) ds +

1

Z

G0 (t , s)u(s) ds. 0

(2.8)

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By the theory of [20], L has a principal eigenvalue r (L) (radius of its spectrum) with positive eigenfunction in K . Let µ1 = 1/r (L) be the principal characteristic value of L. For a function g that satisfies Carathéodory conditions, we use the following notations: g (u) := sup g (t , u), t ∈[0,1]

g = lim sup g (u)/u, 0

g (u) := inf g (t , u), t ∈[0,1]

g

∞

= lim sup g (u)/u, u→∞

u→0+

g0 = lim inf g (u)/u, u→0+

g∞ = lim inf g (u)/u. u→∞

For r > 0 we also set g 0,r = sup{g (t , u)/r : t ∈ [0, 1], u ∈ [0, r ]} and gr ,r /c = inf{g (t , u)/r : t ∈ [a, b], u ∈ [r , r /c ]}. We now have the following theorem on existence of multiple positive solutions for the following equation (fixed points of S), when h ≥ 0 satisfies (C5 ), which is based on fixed point index results of [20], see Theorems 4.1 and 4.2 of [17]. u(t ) =

1

Z

G(t , s)h(s, u(s)) ds,

t ∈ [0, 1].

(2.9)

0

Theorem 2.2. Assume that (C1 )–(C5 ) hold and also that, whenever we have the condition µ1 < h∞ , we suppose (C2 ) holds for an arbitrary [a, b] ⊂ (0, 1). Then Eq. (2.9) has a positive solution u ∈ K if one of the following conditions holds.

(S1 ) 0 ≤ h0 < µ1 and µ1 < h∞ ≤ ∞. (S2 ) µ1 < h0 ≤ ∞ and 0 ≤ h∞ < µ1 . Eq. (2.9) has at least two positive solutions in K if one of the following conditions holds.

(D1 ) 0 ≤ h0 < µ1 , hr ,r /c > M for some r > 0, and 0 ≤ h∞ < µ1 . (D2 ) µ1 < h0 ≤ ∞, h0,r < m for some r > 0, and µ1 < h∞ ≤ ∞. Eq. (2.9) has at least three positive solutions in K if either (T1 ) or (T2 ) holds. (T1 ) There exist 0 < r1 < cr2 < ∞, such that 0 ≤ h0 < µ1 ,

hr1 ,r1 /c > M ,

h0,r2 < m,

µ1 < h∞ ≤ ∞.

h r 2 ,r 2 / c > M ,

0 ≤ h∞ < µ1 .

(T2 ) There exist 0 < r1 < r2 < ∞, such that µ1 < h0 ≤ ∞,

h0,r1 < m,

Note that in [17,20] an extra assumption (UPE ) was used in one part of this result, but, by a result of Nussbaum (Lemma 2 on page 226 of [21]), Theorem 3.7 of [20] is valid without needing (UPE ) so (UPE ) is not needed. It is known, [20], that one always has m ≤ µ1 ≤ M and the inequalities are strict if the eigenfunction corresponding to µ1 is not constant. It is possible to give a result for an arbitrary finite number of positive solutions by extending, in a routine manner, the list of conditions given above. Of course, the conditions on h become ever more restrictive. We illustrate the meaning of some of these conditions when applied to the resonance case in Figs. 1 and 2, later in the paper. 3. Resonance case We now establish the existence of multiple positive solutions for the Eq. (1.1) at resonance via our results for the perturbed equation (1.7). For this purpose, first consider the equation

− γ000 (t ) = 0

(3.1)

under one of the local boundary conditions:

γ00 (0) = 0,

γ0 (1) = 1,

(3.2)

γ0 (0) = 0,

γ0 (1) = 1,

(3.3)

γ0 (0) = 0,

γ0 (1) = 1.

(3.4)

0

Clearly the solutions are γ0 (t ) = 1 for (3.1) and (3.2) and γ0 (t ) = t for (3.1)–(3.3) and (3.1)–(3.4), respectively. Now the BVPs (1.1)–(1.3), (1.1)–(1.4), (1.1)–(1.5) are at resonance if β[γ0 ] = 1 while the equivalent perturbed problems (1.7)–(1.3), (1.7)–(1.4), (1.7)–(1.5) are not at resonance if 0 ≤ β[γ ] < 1, where γ is a solution of

−γ 00 + ω2 γ = 0, under the respective BCs, as described in Section 2.

J.R.L. Webb, M. Zima / Nonlinear Analysis 71 (2009) 1369–1378

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Fig. 1. Case (Da). A has co-ordinates (r1 , Mr1 − ω2 r1 ), B has co-ordinates (r1 /c , Mr1 − ω2 r1 /c ).

Fig. 2. Case (Db). C has co-ordinates (0, mr1 ), D has co-ordinates (r1 , mr1 − ω2 r1 ).

Theorem 3.1. Suppose that f (t , u) + ω2 u ≥ 0 for all u ≥ 0, that β[γ0 ] = 1, and 0 ≤ β[γ ] < 1. Moreover, suppose that (C3 ) holds. Then the resonance problem (1.1) has at least one positive solution if either

(Sa) f 0 < 0 and f∞ > 0, or (Sb) f0 > 0 and f ∞ < 0. Let c be as in the definition of K . The resonance problem (1.1) has at least two positive solutions if either of the following conditions hold. (Da) f 0 < 0, and f ∞ < 0, and there exists r1 > 0 such that inft ∈[a,b] f (t , u) + ω2 u > r1 M for r1 ≤ u ≤ r1 /c. (Db) f0 > 0, and f∞ > 0, and there exists r1 > 0 such that supt ∈[0,1] f (t , u) + ω2 u < r1 m for 0 ≤ u ≤ r1 . Proof. The Eq. (1.7) has at least one positive solution under condition (S1 ) or (S2 ) of Theorem 2.2. Since 0 is an eigenvalue of (1.1) we have µ1 = ω2 for (1.7). Since f 0 = h0 + ω2 , with similar expressions for the other terms, clearly (S1 ), that is, 0 ≤ h0 < µ1 and µ1 < h∞ ≤ ∞ holds for h if and only if (Sa) holds, and (S2 ) holds for h when (Sb) holds for f . Similarly (Da) and (Db) imply that (D1 ) and (D2 ) hold for h.  Remark 3.2. We can similarly write a result for three positive solutions using the conditions (T1 ), (T2 ) of Theorem 2.2. Our method proves the existence of multiple positive solutions but the methods of Infante–Zima [3] only allow them to prove existence of one positive solution. Figs. 1 and 2 illustrate the meaning of conditions (Da) and (Db) in the case when f depends only on u. In these figures, r0 is to be regarded as very small and R is to be regarded as very large. The graph of f should not lie in the hashed regions. The cases (Sa), (Sb) of Theorem 3.1 say that f (u) has opposite signs at 0 and ∞. For the BVP (1.1)-(1.3), when β[u] arises from a positive measure, change of sign is close to a necessary condition for positive solutions to exist. We have the following simple result.

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Theorem 3.3. Let f be continuous. Suppose that dB is a (positive) measure and that Then the boundary value problem

− u00 (t ) = f (u(t )),

u0 (0) = 0,

R1 0

h(t )dB(t ) > 0 if h(t ) > 0 on (0, 1).

u(1) = β[u],

(3.5)

when β[1] = 1 (the resonance case), has a positive solution if and only if there exists (a constant) x > 0 such that f (x) = 0. Proof. Sufficiency: If f (x) = 0 for some x > 0, then u(t ) ≡ x is a positive solution. Necessity: Suppose there exists a positive solution u, u(t ) > 0 for t ∈ (0, 1), and that f (x) 6= 0 for all x > 0. Then either f (x) > 0 or else f (x) < 0 for all x > 0. If f (x) > 0 for all x > 0 then u0 is strictly decreasing and u0 (0) = 0 implies u is strictly R1 decreasing. Then we have u(t ) > u(1) for t ∈ [0, 1) and hence β[u] = 0 u(t )dB(t ) > u(1)β[1] = u(1), contradicting the boundary condition. Similarly, if f (x) < 0 for all x > 0, then u0 and u are strictly increasing. Then u(t ) < u(1) for t ∈ [0, 1) R1 and hence β[u] = 0 u(t )dB(t ) < u(1)β[1] = u(1), a contradiction.  This result includes the multipoint BCs u0 (0) = 0, u(1) = i=1 βi u(ηi ), when all coefficients βi are positive. Our theorems cover nonlinearities that can depend explicitly on t, and a more general case which allows some coefficients of both signs.

Pm

4. The BCs u0 (0) = 0, u(1) = 0 In order to discuss the BVP

− u00 + ω2 u = h(t , u),

u0 (0) = 0,

u(1) = β[u],

(4.1)

for a linear functional β we first need some results about the local problem with β[u] ≡ 0. The well-known Green’s function for

− u00 + ω2 u = h(t , u),

u0 (0) = 0,

u(1) = 0,

(4.2)

is G0 (t , s) :=

cosh(ωs) sinh(ω(1 − t )), ω cosh(ω) cosh(ωt ) sinh(ω(1 − s)),

s ≤ t, s > t.



1

(4.3)

Observe that G0 (t , s) ≥ 0 for t, s ∈ [0, 1]. Moreover, (C20 ) holds with

Φ0 (s) =

cosh(ωs) sinh(ω(1 − s))

ω cosh(ω)

,

s ∈ [0, 1],

and c0 (t ) = min



cosh(ωt ) cosh(ω)

,

sinh(ω(1 − t )) sinh(ω)



,

hence (C2 ) holds for an arbitrary [a, b] ⊂ (0, 1). The perturbed Hammerstein integral operator corresponding to (4.1) is Tu(t ) = γ (t )β[u] +

1

Z

G0 (t , s)h(s, u(s)) ds

0

where

−γ 00 +ω2 γ = 0,

γ 0 (0) = 0,

γ (1) = 1,

so that γ (t ) =

cosh(ωt ) cosh(ω)

.

Clearly, γ (t ) ≥ 0 and γ (t ) ≥ c1 (t )kγ k for c1 (t ) = γ (t ) so (C4 ) is satisfied on any interval [a, b] ⊂ [0, 1). Thus we can use any interval [a, b] ⊂ [0, 1) and take c = mint ∈[a,b] c0 (t ) in the definition of the cone K . The Green’s function G(t , s) corresponding to (4.1) is now given by (2.5). Theorem 2.2 now applies at once. All we need to do in specific problems is exhibit the values of the constants that occur. Example 4.1. We first consider the local problem (4.2) with Green’s function G0 given by (4.3). In this case we can give a formula for some of the constants. By a calculation, we have m0 = ω2

cosh(ω) cosh(ω) − 1

.

The optimal choice of interval [a, b], that is, the one for which M0 (a, b) is minimal, is of the form [0, b] and 1 M0 (0, b)

b

Z

G0 (0, s) ds,

= min 0

b

Z



G0 (b, s) ds = 0

b

Z

G0 (b, s) ds. 0

J.R.L. Webb, M. Zima / Nonlinear Analysis 71 (2009) 1369–1378

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Hence, the optimal choice is b = 1/2. Then we have M0 (0, 1/2) = ω2 (coth2 (ω/2) + 1). The constants m0 (ω), M0 (ω) are increasing function of ω and their limits as ω → 0+ are m0 (0) = 2, M0 (0) = 4, which coincide with the known (optimal) values for the BVP with ω = 0. If we write µ1 (ω) for the principal characteristic value of L0 , the linear operator defined by (2.8) when γ ≡ 0, then, from the differential equation (4.2), we clearly have

µ1 (ω) = µ1 (0) + ω2 = π 2 /4 + ω2 . As illustration, we have calculated the constants in some specific cases using the above formulae for m0 (ω), M0 (ω), µ1 (ω). The figures are rounded to 3 decimal places. (1) ω = 0.5. Then m0 ≈ 2.209, M0 ≈ 4.418, µ1 ≈ 2.717. (2) ω = 1. Then m0 ≈ 2.841, M0 ≈ 5.683, µ1 ≈ 3.467. (3) ω = 2. Then m0 ≈ 5.448, M0 ≈ 10.896, µ1 ≈ 6.467. Example 4.2. We consider β[u] = β

R1 0

u(t ) dt.

sinh(ω)

cosh(ω)−cosh(ωs)

Then we have β[γ ] = β ω cosh(ω) and G(s) = β ω2 cosh(ω) Then β[γ ] ≈ 0.4621 and by numerical calculation m ≈ 1.423,

µ1 ≈ 1.609,

. For a specific numerical example we take β = 1/2, ω = 1/2.

M (0, 0.648) ≈ 2.345.

We now deal with the resonance case for the problem

− u00 = f (t , u),

u0 (0) = 0,

This problem is at resonance if β[1] =

u(1) = β[u].

R1 0

dB(t ) = 1. The equivalent perturbed problem is

− u00 + ω2 u = h(t , u) := f (t , u) + ω2 u,

u0 (0) = 0,

which is not at resonance when 0 ≤ β[γ ] < 1, where γ (t ) = 1

Z 0≤ 0

cosh(ωt ) cosh(ω)

(4.4)

u(1) = β[u], cosh(ωt ) , cosh(ω)

(4.5)

that is,

dB(t ) < 1.

R1

This is automatically satisfied if dB is a positive measure with 0 dB(t ) = 1 since γ (t ) < 1 on [0, 1). In the case of a three-point BVP, that is, β[u] = β u(η), where η ∈ (0, 1), the resonance case is β = 1 and

β[γ ] = β cosh(ωη)/ cosh(ω) and 0 ≤ β[γ ] < 1 is clearly satisfied for β = 1. Pm Pm For the more general multipoint case when β[u] = i=1 βi u(ηi ), the problem (4.4) is at resonance when i=1 βi = 1. Pm For the perturbed problem (4.5) with i=1 βi = 1 and βi ≥ 0, i = 1, 2, . . . , m, we have 0≤

m X

βi cosh(ωηi ) < cosh(ω),

i =1

that is, 0 ≤ β[γ ] < 1 holds. We now give an explicit example to show that some negative coefficients βi are allowed in such multipoint problems. We consider a four-point BVP, that is,

− u00 + ω2 u = h(t , u),

u0 (0) = 0,

u(1) = β[u],

(4.6)

with

β[u] = β1 u(η1 ) + β2 u(η2 ),

for 0 < η1 < η2 < 1.

(4.7)

We will determine the restrictions to be placed on β1 , β2 and will see that both signs are allowed with appropriate cosh(ωt ) restrictions. For this BVP, γ (t ) = cosh(ω) and we need: 0 ≤ β[γ ] < 1

and

G(s) = β1 G0 (η1 , s) + β2 G0 (η2 , s) ≥ 0.

The first condition to be satisfied is 0 ≤ β1 cosh(ωη1 ) + β2 cosh(ωη2 ) < cosh(ω). The second condition has 3 parts, one for s < η1 , one for η1 ≤ s ≤ η2 , and one for s > η2 . The extreme cases simplify to

β1 cosh(ωη1 ) + β2 cosh(ωη2 ) ≥ 0, β1 sinh(ω(1 − η1 )) + β2 sinh(ω(1 − η2 )) ≥ 0.

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The middle case (which includes the two extreme cases) is

β1 cosh(ωs) sinh(ω(1 − η1 )) + β2 cosh(ωη2 ) sinh(ω(1 − s)) ≥ 0,

for η1 ≤ s ≤ η2 .

Since cosh(ωs) is increasing and sinh(ω(1 − s)) is decreasing, the region defined by all the half spaces corresponding to these lines is precisely the region for which the two extreme cases hold, i.e.

β1 cosh(ωη1 ) + β2 cosh(ωη2 ) ≥ 0, β1 sinh(ω(1 − η1 )) + β2 sinh(ω(1 − η2 )) ≥ 0. The total requirement is therefore 0 ≤ β1 sinh(ω(1 − η1 )) + β2 sinh(ω(1 − η2 ))

(4.8)

0 ≤ β1 cosh(ωη1 ) + β2 cosh(ωη2 ) < cosh(ω).

(4.9)

and

Put k1 = −

sinh(ω(1 − η1 )) sinh(ω(1 − η2 ))

,

k2 = −

cosh(ωη1 ) cosh(ωη2 )

,

and

k3 =

cosh(ω) cosh(ωη2 )

.

Then (4.8) and (4.9) define the region in the β1 , β2 -plane shown in Fig. 3. The dashed line β1 + β2 = 1 corresponds to the resonance case for the problem (4.4). It is clear that β1 and β2 of opposite signs are allowed and our region is much larger than the one defined for non-negative coefficients.

Fig. 3. The region defined by (4.8) and (4.9).

Example 4.3. We consider the four-point BC with β[u] = β1 u(η1 ) + β2 u(η2 ) and take as a specific numerical example ω = 1/2, η1 = 1/2, η2 = 3/4, β1 = 3/2, β2 = −1. Then we have β[γ ] ≈ 0.422, we check that G(s) ≥ 0 and by numerical calculations we find that m ≈ 1.074, M (0, 0.612) ≈ 1.539, µ1 ≈ 1.161. Example 4.4. As a second example we take ω = 1/2, η1 = 1/2, η2 = 3/4, β1 = 2, β2 = −1. Then we have β[γ ] ≈ 0.87945, we check that G(s) ≥ 0 and using some numerical calculations we find that m = M (0, 1) = µ1 = 1/4. This becomes clear when one realizes that this problem has a constant eigenfunction and the problem with ω = 0 is at resonance. 5. The BCs u(0) = 0, u(1) = β[u] As in Section 4, in order to discuss the BVP

− u00 + ω2 u = h(t , u),

u(0) = 0,

u(1) = β[u],

(5.1)

for a linear functional β we first establish some results about the local problem with β[u] ≡ 0. The Green’s function for

− u00 + ω2 u = h(t , u),

u(0) = 0,

u(1) = 0,

(5.2)

J.R.L. Webb, M. Zima / Nonlinear Analysis 71 (2009) 1369–1378

1377

is G0 (t , s) =

sinh(ωs) sinh(ω(1 − t )), ω sinh(ω) sinh(ωt ) sinh(ω(1 − s)), 1



s ≤ t, s > t.

(5.3)

This satisfies (C20 ) with

Φ0 (s) =

sinh(ωs) sinh(ω(1 − s))

ω sinh(ω)

,

c0 (t ) = min



sinh(ωt ) sinh(ω(1 − t )) sinh(ω)

,

sinh(ω)



.

cosh(ω/2)

sinh(ωt )

Moreover, γ (t ) = sinh(ω) . Then we have m = ω2 cosh(ω/2)−1 . Also we find that M (a, b) is minimized by taking b = 1 − a and for a < 1/2 we have 1

1

[sinh(ω) − cosh(aω)(sinh(aω) + sinh((1 − a)ω))]. ω2 sinh(ω) The optimal choice is then a = 1/4 and M ( a, 1 − a)

=

ω2 sinh(ω) . 4 cosh(ω/4) sinh3 (ω/4) The corresponding constant c0 = min{c0 (t ) : t ∈ [a, b]} is c0 (1/4, 3/4) ≈ 0.215. It is clear from the differential equation M (1/4, 3/4) =

(5.2) that we have

µ1 (ω) = µ1 (0) + ω2 = π 2 + ω2 . Example 5.1. For ω = 1/2, m ≈ 8.2085, µ1 ≈ 10.1196, and M (1/4, 3/4) ≈ 16.4169. For ω = 1, m ≈ 8.8354, µ1 ≈ 10.8696, and M (1/4, 3/4) ≈ 17.6708. In a similar way to the previous discussion, we can prove that for the BCs u(0) = 0, u(1) = β[u] with

β[u] = β1 u(η1 ) + β2 u(η2 ),

for 0 < η1 < η2 < 1,

we need 0 ≤ β1 sinh(ω(1 − η1 )) + β2 sinh(ω(1 − η2 ))

(5.4)

0 ≤ β1 sinh(ωη1 ) + β2 sinh(ωη2 ) < sinh(ω).

(5.5)

and

Example 5.2. We consider the four-point BC with β[u] = β1 u(η1 ) + β2 u(η2 ). We take as a specific numerical example ω = 1, η1 = 1/4, η2 = 3/4, β1 = 2, β2 = −1/2. Then we have β[γ ] = 4 sinh(1/4)−sinh(3/4) ≈ 0.080, G(s) = β1 G0 (η1 , s) − β2 G0 (η2 , s) ≥ 0, either by a direct calculation, or by checking that (5.4) 2 sinh(1) and (5.5) are satisfied. By numerical calculations we find that m ≈ 5.384, µ1 ≈ 6.888, and M (0.3, 0.972) ≈ 11.592, and the corresponding value of c is c (0.3, 0.972) ≈ 0.024. One can choose [a, b] = [1/4, 3/4] to get a larger value of c, namely c [1/4, 3/4] ≈ 0.215 at the expense of a larger value of M, M (1/4, 3/4) ≈ 13.085, or even [a, b] = [0.45, 0.55] with c (0.45, 0.55) ≈ 0.396 but with a much larger M (0.45, 0.55) ≈ 35.871. Which choice is appropriate depends on the behaviour of the given nonlinearity. Finally we will deal with the resonance case for the non-perturbed equation. The problem

− u00 = f (t , u),

u(0) = 0,

is at resonance when β[t ] :=

R1 0

u(1) = β[u],

(5.6)

t dB(t ) = 1 while the perturbed problem

− u00 + ω2 u = h(t , u) := f (t , u) + ω2 u, is non-resonant if 0 ≤ β[γ ] < 1 with γ (t ) = equivalent problem (5.7) is non-resonant if

u(0) = 0, sinh(ωt ) . sinh(ω)

u(1) = β[u],

(5.7)

For example, for the 3-point problem with β[u] = η u(η), the 1

sinh(ωη) < η sinh(ω). This holds for every η ∈ (0, 1), since sinh(x)/x is increasing . Pm on (0, ∞)P m Similarly, for a multipoint problem, when β[u] = i=1 βi u(ηi ), if i=1 βi ηi = 1 and βi ≥ 0, i = 1, 2, . . . , m, then m X

βi sinh(ωηi ) < sinh(ω),

i =1

that is, 0 ≤ β[γ ] < 1 holds for the perturbed problem.

(5.8)

1378

J.R.L. Webb, M. Zima / Nonlinear Analysis 71 (2009) 1369–1378

6. The BCs u(0) = 0, u0 (1) = β[u] We note that the local problem

− u00 + ω2 u = h(t , u),

u(0) = 0,

u(1) = 0,

(6.1)

is equivalent to (4.2), by writing τ = 1 − t. Therefore, in particular, the optimal interval [a, b], that is, the one for which M0 (a, b) is minimal, becomes of the form [b, 1]. Hence, the optimal choice is b = 1/2. The nonlocal non-perturbed problem

− u00 = f (t , u),

u(0) = 0,

is at resonance when β[t ] :=

R1 0

u0 (1) = β[u]

(6.2)

t dB(t ) = 1. The perturbed problem

− u00 + ω2 u = h(t , u) := f (t , u) + ω2 u, is non-resonant if 0 ≤ β[γ ] < 1 with γ (t ) =

u(0) = 0,

sinh(ωt ) . ω cosh(ω)

u0 (1) = β[u],

(6.3)

Let β[u] = η u(η). Then we have a non-resonance case for (6.3) if 1

sinh(ωη) < ηω cosh(ω).

(6.4)

This holds for every since sinh(x)/x < cosh(x) for x > 0. Pm η ∈ (0, 1),P m Let β[u] = i=1 βi u(ηi ). If i=1 βi ηi = 1 and βi ≥ 0, i = 1, 2, . . . , m, then m X

βi sinh(ωηi ) < ω cosh(ω),

i=1

that is, 0 ≤ β[γ ] < 1 holds. In case of four-point BVP, the total requirement is similarly shown to be 0 ≤ β1 cosh(ω(1 − η1 )) + β2 cosh(ω(1 − η2 )) and 0 ≤ β1 sinh(ωη1 ) + β2 sinh(ωη2 ) < ω cosh(ω). We omit further details since the procedure is the same as for the other sets of BCs. Acknowledgment We thank Dr G. Infante for drawing the figures. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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