On nonlinear boundary conditions satisfying certain asymptotic behavior

Nonlinear Analysis 76 (2013) 58–67

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On nonlinear boundary conditions satisfying certain asymptotic behavior Christopher S. Goodrich ∗,1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA

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Article history: Received 7 May 2012 Accepted 21 July 2012 Communicated by S. Carl MSC: primary 34B09 34B10 34B15 34B18 47H07

abstract In this paper we consider a second-order boundary value problem of the form y′′ (t ) = −λa(t )g (y(t )), y(0) = ϕ(y), y(1) = 0. We demonstrate that if the nonlinear functional ϕ(y) satisfies certain asymptotic behavior, then the problem will possess at least one positive solution. These results generalize and improve on some recent results in the literature on boundary value problems with nonlocal, nonlinear boundary conditions, and we provide some examples, which illustrate these generalizations and improvements. © 2012 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear boundary condition Nonlocal boundary condition Positive solution Cone Eigenvalue

1. Introduction In this paper we consider a class of boundary value problems (BVP) with nonlocal boundary conditions (BC) of the form y′′ (t ) = −λa(t )g (y(t )) y(0) = ϕ(y)

(1.1)

y(1) = 0, where ϕ : C ([0, 1]) → R is a nonlinear functional and λ ∈ R is an eigenvalue; throughout we assume that ϕ is continuous. We consider in the sequel conditions which guarantee the existence of at least one positive solution to problem (1.1). Let us begin by straightaway explaining the novel approach that we take to problem (1.1); we shall then briefly describe the relationship of our results to those in the existing literature. First of all, we work in the cone defined by





K := y ∈ C ([0, 1]) : y(t ) ≥ 0, min y(t ) ≥ γ0 ∥y∥, L1 (y) ≥ 0 , t ∈[a,b]

(1.2)

where L1 : C ([0, 1]) → R is a linear functional to be specified shortly, γ0 ∈ (0, 1) is a constant to be stated later, (a, b) b (0, 1) is a fixed subinterval of [0, 1], and ∥ · ∥ is the usual max norm on C ([0, 1]). Note that K is a type of cone

∗

Tel.: +1 402 393 1190. E-mail addresses: [email protected], [email protected].

1 Also at: Department of Mathematics, Creighton Preparatory School, Omaha, NE 68114, USA. 0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.07.023

C.S. Goodrich / Nonlinear Analysis 76 (2013) 58–67

59

first introduced by Webb and Infante [1]. We also note that the linear functional L1 may be very general here, being as it is realized as L1 (y) :=

 [0,1]

y(s) dβ1 (s)

(1.3)

with β1 ∈ NBV ([0, 1]). Moreover, due to the structure of K , we can allow the Stieltjes measures associated with L1 to be signed. With this structure in place, we then make two novel assumptions in the context of problem (1.1). In particular, we first assume that ϕ satisfies the limit superior condition lim sup ∥y∥→+∞ y∈K

|ϕ(y)| < ρ, ∥y∥

(1.4)

for some ρ ∈ [0, 1). We then assume that the linear functional in K , namely L1 , has the property that

ϕ(y) ≥ L1 (y),

(1.5)

for all y ∈ K with mint ∈[a,b] y(t ) sufficiently large; note that mint ∈[a,b] y(t ) can be controlled by ∥y∥ since y ∈ K . Condition (1.5) allows us to gain control over the sign of ϕ(y), and it thus permits ϕ to be negative for some y with arbitrarily large norm. As we shall show in the examples in the sequel, conditions (1.4) and (1.5) collectively permit a wide variety of nonlinear functionals ϕ and thus are not unreasonably restrictive. Moreover, this technique of including a linear functional in K which does not appear in either BC and whose sole purpose is to gain some control over ϕ does not appear to be a tactic that has been attempted previously, and it permits us to provide, in many cases, more general results than currently exist in the literature and yet obtain these results only by a relatively slight modification of the proof structure that we introduced in [2,3]. Having now outlined the novelty of the results we present herein, we now explain briefly in what ways these results complement and extend other results in the recent literature on problem (1.1) and its relatives. In particular, we have recently provided in [2,4,3,5] an analysis of asymptotic conditions in a few different settings. As we shall detail shortly, the results here complement, generalize, and in some cases in the scalar setting improve on certain of those in [2,4,3,6–11]. More generally, we point out that nonlocal, nonlinear BVPs, of which problem (1.1) is a prototypical case, have seen a great deal of study lately. This is partly due to their interesting theory and partly due to their applications to various problems such as heat flow in a bar of finite length (cf. [6, Section 1]). In particular, papers by Infante [6], Infante and Pietramala [7,8], Kang and Wei [9], and Yang [10,11] have recently contributed to this area, and as already pointed out the present author has also given some recent contributions [2,4,3]; see also the series of papers by Yang [12–14] for some related ideas. On the other hand, the more general study of nonlocal BVPs has recently seen excellent progress, largely due to the important work of Graef et al. [15,16,1,17–19]. Such studies have also occurred in the context of fractional BVPs in both the discrete and the continuous setting as well as in the time scales setting—see, for instance, [20–24] and certain of the references therein. The results that we give here afford a few different improvements and generalizations over the existing literature. We specifically enumerate a couple of these. 1. We do not assume that the boundary condition at t = 0 has the form H ◦ L, an assumption made in a variety of contexts such as [4,3,5–11]; in this formulation the function H : [0, +∞) → [0, +∞) is a nonnegative continuous function and L is a linear functional. Consequently, the boundary conditions allowed here can be rather more general. Furthermore, we eliminate hypothesis (H4b) in [2]—at least in the case of the time scale T = R. Thus, in certain cases, our results here are an improvement of the results, which we presented in [2]. 2. We make no general assumption about the sign of ϕ(y). In other words, it is possible that for each M > 0 there exists y0 ∈ C ([0, 1]) satisfying ∥y0 ∥ > M such that ϕ(y0 ) < 0. In fact, this generalizes the techniques utilized in [2,4,3,5–11]. Indeed, due to the various assumptions there (primarily the composition assumption that ϕ = H ◦ L), the various nonlinear BCs in those works cannot have this property. That our nonlinear BCs can have this property makes problem (1.1) more interesting. In particular, in light of this and the preceding point we achieve some of the same generalities here with respect to our nonlinear boundary condition as Infante and Webb do in [1] with respect to their linear boundary condition. 2. Preliminaries In this section, we very briefly set up the basic mathematical tools with which we shall study problem (1.1). In particular, we study problem (1.1) by means of the operator T : C ([0, 1]) → C ([0, 1]) defined by

(Ty)(t ) := α(t )ϕ(y) +

1



G(t , s)f (s, y(s)) ds. 0

(2.1)

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C.S. Goodrich / Nonlinear Analysis 76 (2013) 58–67

It can be shown that a fixed point of T is a solution to problem (1.1). Note that in (2.1) the function G(t , s) is the Green’s function associated with the second-order conjugate BVP—that is, G(t , s) :=



t (1 − s), s(1 − t ),

t ≤s t ≥ s;

(2.2)

this may be found in [25, Corollary 4.76], for example. Moreover, the function α in (2.1) is defined by

α(t ) := 1 − t ,

(2.3)

where α : [0, 1] → [0, 1]. We recall that the Green’s function in (2.2) satisfies the following important property—see [26], for example. Namely, there exists a constant γ0 := γ0 (a, b) ∈ (0, 1) such that min G(t , s) ≥ γ0 max G(t , s) = γ0 G(s, s).

t ∈[a,b]

(2.4)

t ∈[0,1]

In particular, γ0 = min{a, 1 − b}. Note that throughout this section and the sequel, we assume that the numbers a and b are fixed such that (a, b) b (0, 1). Finally, we recall as a preliminary lemma Krasnosel’ski˘ı’s fixed point theorem (see [27]). Lemma 2.1. Let B be a Banach space and let K ⊆ B be a cone.Assume that  Ω1 and Ω2 are bounded, open sets contained in B such that 0 ∈ Ω1 and Ω 1 ⊆ Ω2 . Assume, further, that T : K ∩ Ω 2 \ Ω1 → K is a completely continuous operator. If either 1. ∥Ty∥ ≤ ∥y∥ for y ∈ K ∩ ∂ Ω1 and ∥Ty∥ ≥ ∥y∥ for y ∈ K ∩ ∂ Ω2 ; or 2. ∥Ty∥ ≥ ∥y∥ for y ∈ K ∩ ∂ Ω1 and ∥Ty∥ ≤ ∥y∥ for y ∈ K ∩ ∂ Ω2 ; then T has at least one fixed point in K ∩ Ω 2 \ Ω1 .





3. The main results and numerical examples In this section we shall state and prove our existence results. Afterwards we shall give some examples to explicate in what way these results improve on and generalize certain existing results in the literature. To carry out this program, as before, define the cone K ⊆ C ([0, 1]) by (1.2). As above, the constant γ0 , which appears in (1.2), is defined by

γ0 := min{a, 1 − b}, where we assume that 0 < a < b < 1 so that γ0 ∈ (0, 1). Note that the function α introduced in Section 2 satisfies min α(t ) ≥ γ0 ∥α∥ = γ0 .

t ∈[a,b]

(3.1)

Let us next state the structural assumptions, which we assume regarding problem (1.1). For our second existence result, we shall alter certain of these conditions slightly. H1: There exists

ρ ∈ [0, 1)

(3.2)

such that lim sup ∥y∥→+∞ y∈K

|ϕ(y)| < ρ. ∥y∥

(3.3)

H2: There is a linear functional L1 , which may be realized as L1 (y) :=

 [0,1]

y(s) dβ1 (s),

(3.4)

such that there exists a constant M0 > 0 for which

ϕ(y) ≥ L1 (y)

(3.5)

whenever y ∈ K satisfies mint ∈[a,b] y(t ) ≥ M0 . Here it is assumed that β1 ∈ NBV ([0, 1]). H3: We assume that the functions a : [0, 1] → [0, +∞) and g : R → [0, +∞) are continuous functions such that a is not identically zero on any subinterval of [0, 1]. H4: Assume that limy→+∞ g (y) = +∞. H5: Assume that limy→+∞ H6: It holds both that

 [0,1]

g (y) y

= 0.

α(t ) dβ1 (t ) ≥ 0

(3.6)

C.S. Goodrich / Nonlinear Analysis 76 (2013) 58–67

61

and that



G(t , s) dβ1 (t ) ≥ 0,

[0,1]

(3.7)

where the latter condition – i.e., (3.7) – holds for each s ∈ [0, 1]. Note that condition (H3) is assumed only for convenience in the sequel. Other works invoking cone theoretic arguments demonstrate clearly how we can easily eliminate assumption (H3)—e.g., see [28]. Moreover, note that due to condition (3.6) it holds that α ∈ K , whence K ̸= {0}. We now state a preliminary lemma followed by the statement and proof of our first existence result; we note that in the sequel, we use the following notation. Notation 3.1. The set ΩR ⊆ C ([0, 1]), for R > 0, is the open set

ΩR := {y ∈ C ([0, 1]) : ∥y∥ < R} .

(3.8)

Lemma 3.2. Suppose that each of conditions (H2) and (H6) holds. Then it holds that T : K \ Ω M0 → K . γ0

Proof. First of all, observe that since y ∈ K \ Ω M0 , we see that γ0

min y(t ) ≥ γ0 ∥y∥ ≥ γ0 ·

t ∈[a,b]

M0

γ0

= M0 .

(3.9)

Consequently, condition (H2) implies that ϕ(y) ≥ L1 (y) ≥ 0, for each y ∈ K \ Ω M0 , where the second inequality holds γ0

since y ∈ K . Thus, for all such y ∈ K \ Ω M0 it is evident that (Ty)(t ) ≥ 0, for each t ∈ [0, 1]. Furthermore, we see that γ0

t ∈[a,b]

t ∈[a,b]

1



min (Ty)(t ) ≥ min α(t )ϕ(y) + min

t ∈[a,b]

G(t , s)a(s)g (y(s)) ds 0

≥ γ0 max α(t )ϕ(y) + γ0 max t ∈[0,1]

t ∈[0,1]

1



G(t , s)a(s)g (y(s)) ds 0

≥ γ0 ∥Ty∥.

(3.10)

Finally, it holds that L1 (Ty) = L1



α(t )ϕ(y) +

1



G(t , s)a(s)g (y(s)) ds



0

 = [0,1]

= ϕ(y)

α(t )ϕ(y) dβ1 (t ) +  [0,1]

α(t ) dβ1 (t ) +



1

 [0,1] 1

 0

G(t , s)a(s)g (y(s)) ds dβ1 (t ) 0

 [0,1]

 G(t , s) dβ1 (t ) a(s)g (y(s)) ds

≥ 0,

(3.11)

 where to get the final inequality we use condition (H6). As this demonstrates that T complete.

K \ Ω M0 γ0

 ⊆ K , the proof is



Theorem 3.3. Suppose that each of conditions (H1)–(H6) holds. Then there exists λ0 > 0 such that problem (1.1) has at least one positive solution for each λ ∈ [λ0 , +∞).

 Proof. We have already shown that T

K \ Ω M0



γ0

⊆ K holds. Furthermore, that T is a completely continuous operator is

also standard here, and so, we omit the proof of this fact. Now, condition (H4) implies that there is a number r1 > 0 such that g (y) ≥ 1,

(3.12)

whenever y ≥ r1 . Choose the number r1∗ such that r1∗ := max



r1

,

M0

γ0 γ0



.

(3.13)

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C.S. Goodrich / Nonlinear Analysis 76 (2013) 58–67

Then for each y ∈ K satisfying ∥y∥ = r1∗ , we find that min y(t ) ≥ M0 .

(3.14)

t ∈[a,b]

Thus, ϕ(y) ≥ 0 for each y ∈ K ∩ ∂ Ωr ∗ . In particular, for any t0 ∈ [a, b] this implies that 1

(Ty)(t0 ) = α(t0 )ϕ(y) + λ

1



G (t0 , s) a(s)g (y(s)) ds 0

≥λ

b



G (t0 , s) a(s) ds.

(3.15)

a

So, by choosing

λ ≥ b a

r1∗ G (t0 , s) a(s) ds

,

(3.16)

we estimate

∥Ty∥ ≥ ∥y∥,

(3.17)

for each y ∈ K ∩ ∂ Ω

 λ∈

r∗ 1

and each fixed λ satisfying



r1∗

b a

G (t0 , s) a(s) ds

, +∞ .

(3.18)

On the other hand, select a number ε2 > 0 sufficiently small such that

ρ + ε2 < 1

(3.19)

holds. Then condition (H5) implies the existence of a number r2 > 0 such that g (y) ≤ η2 y,

(3.20)

whenever y ≥ r2 , where η2 > 0 satisfies

η2 λ

1



G(s, s)a(s) ds ≤ ε2 ;

(3.21)

0

here λ in (3.21) and the sequel is the number λ selected in (3.18) above. Moreover, select another number ε3 > 0 such that

ρ + ε2 + ε3 < 1.

(3.22)

Then condition (H1) implies the existence of a number r2 > 0 such that ∗

|ϕ(y)| < (ρ + ε3 ) ∥y∥

(3.23)

∗

whenever ∥y∥ ≥ r2 and y ∈ K . Finally, because g is assumed to be unbounded at +∞, it follows that there exists a number r2∗∗ such that g (y) ≤ g r2∗∗ ,





for each y ∈ 0, r2



∗∗

(3.24)



∗∗

. In particular, we may select r2 sufficiently large such that

r2∗∗ ≥ max 2r1∗ , r2 , r2∗ .





(3.25)

Then it holds that

∥Ty∥ ≤ ϕ(y) + λ

1



G(s, s)a(s)g (y(s)) ds 0

≤ (ρ + ε3 ) ∥y∥ + λ

1



G(s, s)a(s)g r2∗∗ ds





0

≤ (ρ + ε3 ) ∥y∥ + η2 λ∥y∥

1



G(s, s)a(s) ds 0

≤ (ε2 + ε3 + ρ) ∥y∥ < ∥y∥, for each y ∈ K ∩ ∂ Ωr ∗∗ . 2

(3.26)

C.S. Goodrich / Nonlinear Analysis 76 (2013) 58–67

63

Consequently, T is a cone expansion on K ∩ ∂ Ωr ∗ , whereas T is a cone compression on K ∩ ∂ Ωr ∗∗ . Thus, Lemma 2.1 1



implies the existence of a fixed point, y0 , of T with y0 ∈ K ∩ Ω r ∗∗ \ Ωr ∗ 2

1



2

⊆ K \ Ω M0 . And this completes the proof. γ0



We next state a second existence theorem. First, however, we introduce a different growth condition on g together with a somewhat stronger assumption regarding the lower boundedness of ϕ . H2a: There is a linear functional L2 , which may be realized as L2 (y) :=

 [0,1]

y(s) dβ2 (s),

(3.27)

such that

ϕ(y) ≥ L2 (y),

(3.28)

whenever y ∈ K . We assume here that β2 ∈ NBV ([0, 1]). H7: Assume that limy→0+

g (y) y

= +∞.

Theorem 3.4. Suppose that conditions (H1), (H2a), (H3), (H5)–(H7) are satisfied. Then problem (1.1) has at least one positive solution in the case λ = 1. Proof. As in the proof of Theorem 3.3, we focus here on proving that T is a cone compression and cone expansion on appropriate sets. Let us note that here T : K → K since the stronger condition (H2a) is in force. In any case, observe that by condition (H7), it follows that we may select r1 > 0 sufficiently small such that g (y) ≥ η1 y,

(3.29)

for each y ∈ (0, r1 ], where η1 satisfies

η1 γ02

b



G (s, s) a(s) ds ≥ 1.

(3.30)

a

As already mentioned, condition (H2a) implies that ϕ(y) ≥ L2 (y) ≥ 0, for each y ∈ K . Thus, for any t0 ∈ [a, b] it holds that

(Ty)(t0 ) ≥ γ0

b



G(s, s)a(s)g (y(s)) ds ≥ γ η ∥y∥ 2 0 1

a

b



G(s, s)a(s) ds ≥ ∥y∥,

(3.31)

a

for each y ∈ K ∩ ∂ Ωr1 . On the other hand, we consider cases. Let us first suppose that g is unbounded at +∞. Then we essentially give an argument exactly as in the previous proof but with λ = 1. Conversely, if g is bounded, then there is N ∈ R such that g (y) ≤ N, for each y ≥ 0. In this case, we may choose the r2∗∗ in (3.25) to satisfy

 r2 := max 2r1 , r2 , ∗∗

∗

N

1 0

G(s, s)a(s) ds

 ,

ε2

(3.32)

with ρ + ε2 + ε3 < 1 as before. Then proceeding as in the proof of Theorem 3.3, we estimate

∥Ty∥ ≤ (ρ + ε3 ) ∥y∥ + N

1



G(s, s)a(s) ds 0

≤ (ρ + ε3 ) ∥y∥ + ε2 r2

∗∗

< (ρ + ε2 + ε3 ) ∥y∥.

(3.33)

Hence, ∥Ty∥ ≤ ∥y∥ for each y ∈ K ∩ ∂ Ωr ∗∗ . 2 So, in either case we find that ∥Ty∥ ≤ ∥y∥ on K ∩ ∂ ΩR0 for some appropriate R0 > r1 . And this completes the proof.



Let us conclude by giving a few examples and some remarks on how these examples illustrate the ways in which our results both generalize and improve on certain of the existing results in the literature. Example 3.5. Consider the nonlinear functional

ϕ(y) :=

1 2

  y

2

5

−

1 3

 1−e

  3 −y 10

 y

3 10



.

(3.34)

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C.S. Goodrich / Nonlinear Analysis 76 (2013) 58–67

Moreover, take [a, b] := L1 (y) :=

1 2

  2

y

5

, 4 . Define the linear functional L1 by   1 3 − y . 1

 3

4

3

(3.35)

10

Observe that 1 1 |ϕ(y)| ≤ + < 1. ∥y∥ 2 3

0 ≤ lim sup ∥y∥→+∞ y∈K

(3.36)

Furthermore, we notice that 1

ϕ(y) ≥

2

  2

y



1

− y

5

3



3 10

= L1 (y) ≥ 0,

(3.37)

for each y ∈ K . Finally, we may numerically check that



1

α(t ) dβ1 (t ) =

[0,1]

15

≥0

(3.38)

and that



G(t , s) dβ1 (t ) ≥ 0,

[0,1]

(3.39)

for each s ∈ [0, 1]. Thus, (3.38)–(3.39) imply that condition (H6) holds. Consequently, in the case λ = 1 we may invoke Theorem 3.4 to deduce that problem (1.1) has at least one positive solution for each function f (t , y) satisfying conditions (H3), (H5), and (H7). Example 3.6. Next, putting [a, b] :=

 

ϕ(y) :=

1

L1 (y) :=

1

3

1

y

−y

− 3e

2

1 10

 , 34 , say, consider the nonlinear functional

3 10



  1 3



y(s) ds −

+ 1 4



1

1+e

10

  −2y 15

  y

1

5

.

(3.40)

Put

3

  1

y

−

2

 

1 100

1

y

3

3 10

 + 1 4

y(s) ds −

11 100

  y

1

5

.

(3.41)

Now, note that 0 ≤ lim sup ∥y∥→+∞ y∈K

1 1 1 |ϕ(y)| ≤ + + < 1. ∥y∥ 3 20 10

(3.42)

On the other hand, observe that 1

ϕ(y) ≥

3

  y

1

2

−

 

1 100

1

y

3 10

 +

3

1 4

y(s) ds −

11 100

  y

1 5

= L1 (y) ≥ 0,

(3.43)

for each y ∈ K with ∥y∥ sufficiently large; note that to obtain estimate (3.43) we have used both the fact that −y

  1 3

− 3e

≥−

1 100

  y

1

3

,

(3.44)

and the fact that

−



1 10

1+e

−2y

  1 5

  y

1

5

≥−

11 100

  y

1

5

,

(3.45)

for each y ∈ K with ∥y∥ sufficiently large, where we have again used the fact that min 

1 3 t ∈ 10 ,4



y(t ) ≥ γ0 ∥y∥. Finally, as in

Example 3.5, we may numerically check that

 [0,1]

α(t ) dβ1 (t ) =

433 4000

>0

(3.46)

C.S. Goodrich / Nonlinear Analysis 76 (2013) 58–67

65

and that

 [0,1]

G(t , s) dβ1 (t ),

(3.47)

for each s ∈ [0, 1]. Thus, (3.46)–(3.47) imply that condition (H6) holds. Thus, we conclude from Theorem 3.3 that there exists λ0 > 0 sufficiently large such that for each λ ∈ [λ0 , +∞) and for each function f (t , y) satisfying conditions (H3)–(H5), problem (1.1) has at least one positive solution. Example 3.7. Finally, putting [a, b] :=

 , 34 , consider the nonlinear functional        1 1 2 2 1 1 ϕ(y) := sin y + y − y . 5

2

3



1 10

5

10

6

(3.48)

Define the linear functional L1 by L1 (y) := −

1 10

  y

1

6

2

 

3

5

+ y

2

−

 

2 3π

y

1 2

.

(3.49)

Note first that 0 ≤ lim sup ∥y∥→+∞ y∈K

2 1 |ϕ(y)| ≤ + < 1. ∥y∥ 3 10

(3.50)

On the other hand, it is clear that

ϕ(y) ≥ −

1 10

  y

1

6

2

 

3

5

+ y

2

−

2 3π

  y

1

2

= L1 (y) ≥ 0,

(3.51)

for each y ∈ K with ∥y∥ sufficiently large. Finally, as in each of Examples 3.5 and 3.6, we may numerically check that

 [0,1]

α(t ) dβ1 (t ) =

19π − 20 60π

>0

(3.52)

and that

 [0,1]

G(t , s) dβ1 (t ) ≥ 0,

(3.53)

for each s ∈ [0, 1]. Thus, (3.52)–(3.53) imply that condition (H6) holds. So, we may invoke Theorem 3.3 to deduce that for each λ > 0 sufficiently large, problem (1.1) has at least one positive solution for each function f (t , y) satisfying conditions (H3)–(H5). Remark 3.8. We should remark that the range of admissible eigenvalues in Theorem 3.3 is explicitly computable as is evident from the proof of Theorem 3.3. One may consult, for example, [2, Remark 3.4] for a more thorough explanation of this point. Remark 3.9. Observe that none of functionals ϕ in Examples 3.5–3.7 may be realized in the form ϕ(y) = (H ◦ L)(y) (cf. Section 1). In particular, this means that the approach in [4,3,6–11] is not applicable. Moreover, since the ϕ that we use here may be negative for y arbitrarily large in norm and since H is generally taken to be, at worst, eventually positive, those approaches fail for a second reason. It is even the case, moreover, that none of these examples can be treated by the techniques introduced in [2]. Indeed, in that work we produced results only in the case where the nonlinear functional ϕ was a sort of nonlinear n-point condition insofar as ϕ could be realized in the form ϕ(y) = H (y (ξ1 ) , . . . , y (ξn )) for some nonlinear, eventually nonnegative function H. Moreover, as remarked in Section 1, we also imposed some stricter growth conditions on the nonlinearity H. So, attempting to apply the results of [2] to the examples here runs afoul in three distinct ways. First of all, it need not be the case that ϕ is ever ‘‘eventually’’ positive in the sense introduced in [2]. Second, the growth condition utilized in [2] (cf. hypothesis (H4b) in [2]) may not be satisfied. Third, the functional ϕ may not be an n-point condition as Example 3.6 illustrates. Remark 3.10. We note that condition (H1) can be implied by a sort of asymptotic relatedness condition. Indeed, we suppose that there is a linear functional, say L2 (y) :=

 [0,1]

y(s) dβ2 (s),

(3.54)

66

C.S. Goodrich / Nonlinear Analysis 76 (2013) 58–67

where β2 ∈ NBV ([0, 1]), with the property that for each ε > 0 given there exists Mε ≥ 0 such that

|ϕ(y) − L2 (y)| < ε∥y∥,

(3.55)

whenever ∥y∥ > Mε . Here we assume that L2 satisfies |L2 (y)| ≤ C0 ∥y∥ for some C0 ∈ [0, 1). By then further restricting the cone K to those y such that L2 (y) ≥ 0, the proofs of each of Theorems 3.3 and 3.4 go through (essentially) without change. Moreover, this condition, namely (3.55), implies condition (1.4), for we write

|ϕ(y) − L2 (y)| + |L2 (y)| |ϕ(y) − L2 (y)| |ϕ(y)| ≤ ≤ + C0 . ∥y∥ ∥y∥ ∥y∥

(3.56)

But then condition (3.55) implies, for y ∈ K sufficiently large in norm, that

|ϕ(y)| < ε + C0 . ∥y∥

(3.57)

By the arbitrariness of ε > 0 in (3.57), we conclude that 0 ≤ lim sup ∥y∥→+∞ y∈K

|ϕ(y)| ≤ C0 < 1. ∥y∥

(3.58)

Thus, the asymptotic relatedness condition (3.55) implies the limit superior condition (1.4), though not necessarily conversely (cf. Remark 3.12). Remark 3.11. In spite of the greater generality of (1.4), the usefulness of the asymptotic relatedness condition is principally that the functionals ϕ in each of Examples 3.5 and 3.6 ‘‘look like’’ the linear functionals studied in [1], for example, in the sense that (3.55) holds for L2 (y) :=

1 y 2

2

− 13 y

5

3 10

and L2 (y) :=

1 y 3

1 2

+



3 10 3 4

y(s) ds −

1 y 10

1 5

in Examples 3.5 and 3.6,

respectively. And this, in any case, establishes a connection between the results stated herein and the results studied in [1] and related works, with said connection in particular being that if the nonlinear boundary condition ϕ(y) possesses the property that it ‘‘looks like’’ a linear function L2 in the sense of (3.55), then this may be sufficient for obtaining the existence of at least one positive solution of problem (1.1). Remark 3.12. As a final remark we note that the ϕ studied in this paper need not contain any linear term, and as Example 3.7 illustrates it need not be ‘‘eventually linear’’ in any sense either. For instance, if H : [0, +∞) → [0, +∞) is a function satisfying C2 y ≤ H (y) ≤ C1 y, for suitably chosen C1 , C2 > 0, then, for example, the functional

  

ϕ(y) := H y

1

4

1

 

4

3

− y

1

2

  

cos

y

1 3

(3.59)

  y)| − 14 y 31 and lim sup ∥y∥→+∞ |ϕ( ≤ C1 + 41 . Consequently, with a suitable ∥y∥ y∈K choice of the constants C1 and C2 the functional ϕ can fit within the framework discussed in this paper. satisfies ϕ(y) ≥ L1 (y) with L1 (y) := C2 y

1 4

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