Positive solutions to boundary value problems with nonlinear boundary conditions

Nonlinear Analysis 75 (2012) 417–432

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Positive solutions to boundary value problems with nonlinear boundary conditions Christopher S. Goodrich ∗ Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA

article

abstract

info

∆∆ σ In this paper, we consider the boundary value problem  2 y (t ) = −λf (t , y (t )) subject to the boundary conditions y(a) = φ(y) and y σ (b) = 0. In this setting, φ : Crd

Article history: Received 2 June 2011 Accepted 20 August 2011 Communicated by Ravi Agarwal

a, σ 2 (b) T , R → R is a continuous functional, which represents a nonlinear nonlocal boundary condition. By imposing sufficient structure on φ and the nonlinearity f , we deduce the existence of at least one positive solution to this problem. The novelty in our setting lies in the fact that φ may be strictly nonpositive for some y > 0. Our results are achieved by appealing to the Krasnosel’ski˘ı fixed point theorem. We conclude with several examples illustrating our results and the generalizations that they afford. © 2011 Elsevier Ltd. All rights reserved.



MSC: primary 34B09 34B10 34B15 34B18 34N05 secondary 26E70 39A05 47H10





Keywords: Time scales Second-order boundary value problem Nonlocal boundary condition Cone Positive solution

1. Introduction In this paper we are interested in the boundary value problem (BVP) y∆∆ (t ) = −λf (t , yσ (t )) ,

(1.1)

y(a) = φ(y),

(1.2)

y σ 2 (b) = 0,





where φ : Crd

a, σ 2 (b)

(1.3)

, R → R is a continuous functional, which represents a nonlocal boundary condition, and T   λ > 0 is a constant. We also assume that f : a, σ 2 (b) T × R → [0, +∞) is continuous, where T is an arbitrary time scale; note that the notation FT , where F is set, is defined by FT := F ∩ T. Observe that in case T = R or T = Z, (1.1) reduces, 





respectively, to an ordinary differential or difference equation. We remark that for the sake of simplicity in the sequel, we shall generally assume that the nonlinearity f has the form f (t , y(t )) := a(t )g (y(t )),

∗

Tel.: +1 402 472 3731. E-mail address: [email protected].

0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.08.044

(1.4)

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C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

where a(t ) and g (y) are continuous and nonnegative, though we shall give one result where we remove assumption (1.4); in any case, we additionally shall impose some further assumptions on these functions later. The novel contribution of this work in the context of the time scales literature is to allow φ to be nonpositive for some y ≥ 0 and, moreover, to allow this in the case when φ is not a linear multipoint condition. As a consequence of these ideas, we extend the available results on this type of problem and, moreover, due to the generality of the time scale T, we achieve this extension in both the continuous and discrete settings. We explain later in this section the specific ways in which our results improve and generalize certain results in the literature. We are motivated to study problem (1.1)–(1.3) on a general time scale because our results then apply in not only the setting of ordinary differential equations but also the setting of difference equations, q-difference equations, and even more exotic time scales. The study of dynamic equations has become a very vibrant area of research, following Hilger’s initial work on the subject [1]. This is due in part to the fact that dynamic equations provide a useful framework for mathematical modeling. It is also due to the fact that by casting problem (1.1)–(1.3) in the time scales setting we obtain results on all time scales rather than only in the setting of ordinary differential equations. In fact, so far as we are aware, the results of this paper are completely new on all time scales, including the important, interesting, and nontrivial cases T = R and T = Z. Prior to describing the novel idea employed in this work and the resultant existence theorems, let us place problem (1.1)–(1.3) in a meaningful context. In particular, the existence of positive solutions to nonlinear BVPs has been extensively studied over the past several decades. An important paper in this study was produced by Erbe and Wang [2]. Their general technique has been used successfully on a variety of time scales. In addition, there are numerous works on multipoint or more generally nonlocal boundary value problems, where, once again, these investigations have occurred in a variety of settings, especially for ordinary differential equations and difference equations — see, [3–21], for example, and the references therein. Recently, the present author has provided some extensions of the nonlocal boundary value problem to both the discrete and continuous fractional calculus setting — see [22–26]. Inasmuch as problem (1.1)–(1.3) is concerned, there have been several recent papers that have dealt with either nonlocal linear or nonlinear boundary conditions. A particularly noteworthy work is a recent paper by Infante and Webb [27], which provided for a unified treatment of nonlocal boundary value problems for second-order ordinary differential equations with linear nonlocal boundary conditions. Other recent papers by Webb [28] and Graef and Webb [29] also explore nonlocal linear boundary conditions. Of particular note in these works is the fact that the nonlocal terms may be negative for some positive functions. That is to say, if ψ(y) is a functional representing the nonlocality, then ψ(y) may be negative for some y ≥ 0. This leads to the interesting question of whether or not such BVPs can have positive solutions. As [27–29] and others show, the existence of at least one positive solution is possible given relatively mild conditions on the nonlinearity and the nonlocalities. On the other hand, in the case of nonlinear nonlocal boundary conditions, relevant works include [30–35] and certain of the references therein. Many of these works require that the nonlocality be monotone. In addition, due to the use of upper and lower solution techniques in many of these works, positivity of the solution, if a solution even exists, cannot be guaranteed. Moreover, existence of a solution relies on exhibiting a suitable upper and lower solution pair. Because of the lack of guarantee of positivity, these works do not extend in any natural way the theory of linear nonlocal conditions described in the preceding paragraph. Now, it is true that both Kong and Kong [32] and Infante [35] consider the existence of a positive solution, but in [32] it is assumed that the nonlocal terms are monotone, whereas in [35] it is assumed that the Borel measures associated to the Stieltjes integrals, which represent the boundary conditions, are positive. Consequently, neither of these works provides any direct extension of the results in [27]. Indeed, in [27] the key contribution is that the Stieltjes measures representing the nonlocal boundary conditions can be signed. So, in summary, while these works and others address the issue of nonlinear nonlocal boundary conditions, they often assume monotonicity of the boundary terms and, in addition, never make a significant and direct connection with the linear theory developed in [27–29] and other recent works. In the present work, we attempt to clarify certain of these relationships in the special setting of multipoint-type boundary conditions and, at the same time, provide generalizations and, to an extent that shall be described in the sequel, improvements over the works cited in the preceding paragraph. In particular, we shall provide the following improvements and generalizations. 1. As mentioned above, previous work on BVPs with nonlinear nonlocal boundary conditions often assume some sort of monotonicity of the nonlocality (cf., [30,31,33,34]). In our work, we do not make any sort of monotonicity assumption on the functionals defining the boundary conditions. Thus, we effectively remove this constraint. On the other hand, we do impose a growth condition, which, incidentally, is no stricter than that appearing, say, in [27]. So, at face value, we believe that our conditions are no worse than those imposed by other authors in the nonlinear boundary condition setting. Moreover, since we deduce existence of a positive solution and remove the monotonicity of the boundary conditions, we feel that, in this sense at least, our conditions are improvements over certain existing ones in the literature. 2. Regarding the existence of a positive solution, we feel that this is an important contribution since in our context, the nonlocal term need not be positive for each positive continuous function. Yet we are still able to deduce the existence of at least one positive solution. So, this provides a real improvement over some of the existing results on this sort of problem. Indeed, [30,31,33,34] do not explicitly consider the existence of a positive solution. While [32,35] do, our results, as already explained, provided a degree of improvement and generalization. 3. It is important to mention that many of the previous works on nonlinear boundary conditions are in the higherdimensional cases (e.g., fourth-order problems) or with boundary conditions other than conjugate-type. However, our

C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

419

results, we believe, can be easily extended to these cases, though we do not state so in this work. Therefore, while we only provide explicit results in the slightly simpler setting of the second-order BVP, we believe that results extend in a standard and obvious way to the higher-order cases treated by certain previous works. 4. We also provide extensions in the general eigenvalue setting. While [32] does treat this case, other works do not, and so, this provides a certain degree of novelty as well. 5. Finally, within the context of the existence of a positive solution, we feel that one of our contributions in the sequel deserves special mention. As has already been stated above, recent papers by Infante and Webb [27], Webb [28], and Graef and Webb [29], as well as other papers referenced therein, go into great detail regarding existence of positive solutions in case the nonlocalities are linear (even if the nonlocalities are not always positive for y nonnegative). While the methods that we present in this work do not precisely subsume [27–29] and others, we do demonstrate a strong connection between some of our results (cf., Theorem 3.10 and Corollary 3.11) and the results by Graef, Infante, and Webb, for example. Indeed, we show (cf., Example 4.3) that nonlocal terms of our variety that are asymptotically related to linear nonlocal terms of the type studied in the previously mentioned works are compatible with the existence of positive solutions. We shall make precise what we mean by saying that a nonlocal term of our variety is asymptotically related to a nonlocal term of the type found in [27] (cf., Corollary 3.11, for example), but let us demonstrate briefly what we mean by this, and the implications of this result. So, for instance, suppose that the nonlocal functional φ appearing in (1.2) has the form

φ(y) := −2



1 2

y (t0 ) −

where t0 , t1 ∈ a, σ 2 (b) defined by



H (y1 , y2 ) := −2





T

1 2

1 5



y (t1 ) e−y(t0 )−y(t1 ) +



1 2

y (t0 ) −

1 5



y (t1 ) ,

(1.5)

are chosen appropriately. If we consider the right-hand side of (1.5) as a function H : R2+ → R

y1 −

1 5

 y2

e−y1 −y2 +



1 2

y1 −

then a standard exercise, which we omit, shows that as

1 5



 y2

,

(1.6)

y21 + y22 → +∞ we find that

1 1 y1 − y2 , 2 5 in the sense that for each ϵ > 0 given there is N0 > 0 sufficiently large such that H (y1 , y2 ) →

(1.7)

     H (y1 , y2 ) − 1 y1 − 1 y2  < ϵ (1.8)   2 5  whenever ‖(y1 , y2 )‖ := y21 + y22 ≥ N0 . Thus, (1.7) implies that for (y1 , y2 ) ∈ R2+ sufficiently large in norm, (1.6) reduces essentially to the linear part in (1.6). What we shall show in the sequel is that this is, in fact, sufficient to get the existence of at least one positive solution to problem (1.1)–(1.3). We shall accomplish this by adjoining to our new results the theory developed by Infante and Webb in [27]. Essentially, as in this example, we shall show that provided for y very large φ is like a linear functional to which the theory of Infante and Webb applies, then our problem still admits a positive solution. In any case, we shall make this precise in Section 3, and we shall also give a specific example of these results in Section 4. In summary, we illustrate at least one very specific link between our theorems and those appearing in [27] and similar works, a connection that to the best of our knowledge other works do not provide. With the preceding outline in mind, the plan of this paper is as follows. We begin by presenting some well-known preliminary results that we shall use in the sequel. Then in Section 3 we present a variety of existence results in the case of a general eigenvalue problem. Our techniques are cone theoretic in nature, and our ideas here are loosely related to some recent ideas developed by Wang [36] in the context of singular BVPs for second-order ordinary differential equations with periodic boundary conditions. In this case, we show that for λ > 0 sufficiently large, problem (1.1)–(1.3) admits at least one positive solution. Finally, we conclude this work with three numerical examples, which explicate the implications of our results. In particular, to the best of the author’s knowledge, these examples cannot be treated by any results in the literature, and so, indicate how our theorems provide new results and significant generalizations. Moreover, because our results apply to any time scale, we get new results in several important cases such as ordinary differential equations (when T = R), difference equations (when T = Z), and q-difference equations (when T = qN0 ). 2. Preliminaries We begin by noting that a fixed point of the operator T : Crd

(Ty)(t ) :=

σ 2 (b) − t φ(y) + λ σ 2 ( b) − a

σ (b)

∫ a

a, σ 2 (b)



G(t , s)a(s)g (yσ (s)) ∆s,

 T

    , R → Crd a, σ 2 (b) T , R , defined by (2.1)

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C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

is a solution to problem (1.1)–(1.3), where G(t , s) is the Green’s function associated to the second-order conjugate delta BVP — that is,

   (t − a) σ 2 (b) − σ (s)    , σ 2 (b) − a  G(t , s) := 2 (σ (s) − a) σ (b) − t    , σ 2 (b) − a

t ≤s (2.2) t ≥ σ (s),

as may be found in, say, [37, Corollary 4.76]. Indeed, we note that

(Ty)∆∆ (t ) = −a(t )g (yσ (t )) = −f (t , yσ (t )) ,

(2.3)

whereas

(Ty)(a) = φ(y)

(2.4)

  (Ty) σ 2 (b) = 0.

(2.5)

and

Moreover, for convenience in the sequel we put

σ 2 (b) − t , (2.6) σ 2 (b) − a   where α : a, σ 2 (b) → [0, 1]. In fact, we get the following, obvious proposition, which will nonetheless be important. α(t ) :=

Proposition 2.1. Let α be defined as in (2.6). Then α is strictly decreasing and satisfies α(a) = 1 and α σ 2 (b) = 0.





Moreover, we recall that the Green’s function in (2.2) satisfies the following important property — see [38,39], for example. Namely, there exists a constant γ ∈ (0, 1) such that min [ t∈

(

2 a+σ 2 (b) 3 a+σ (b) , 4 4

)

]

G(t , s) ≥ γ

max

t ∈[a,σ 2 (b)]T

G(t , s) = γ G (σ (s), s) .

(2.7)

T

We shall use this fact in Sections 3 and 4 of this work. For additional background information on time scales, the reader can consult either of the excellent references by Bohner and Peterson [37,40]. We next recall as a preliminary lemma Krasnosel’ski˘ı’s fixed point theorem (see [41]). Lemma 2.2. 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 .





Let us conclude with one remark about notation. Remark 2.3. We use a to represent the left-hand endpoint of the interval a, σ 2 (b) T on which we study dynamic equation (1.1). On the other hand, when we wish to refer to the function a(t ) appearing in (1.4), we shall always include the argument so as to distinguish it from the number a.





3. Existence theorems for eigenvalue problem (1.1)–(1.3) In this section we consider the case in which λ > 0 is a parameter that may vary. In particular, we shall show that there exist a range of eigenvalues such that problem (1.1)–(1.3) admits a positive solution. We make the assumption that g (y) is unbounded and sublinear at +∞. In addition, we need to make some assumptions on the behavior of H. To this end, let us introduce the following conditions. H1: Let H : [0, +∞) → R be a real-valued, continuous function. We have that

φ(y) := H (y (ξ0 )) . H2: The number ξ0 from condition (H1) satisfies

 ξ0 ∈



a + σ 2 (b) 3 a + σ 2 (b)



4

,

.

4 T

C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

421

H3: There is a constant N0 > 0 such that H (y) > 0, whenever y ≥ N0 — that is, H is eventually strictly positive. H4: There is a constant c ∈ (0, 1) such that the inequality

|H (y)| ≤ cy holds eventually. H5: The function a(t ) is nonnegative, continuous, and not zero identically. H6: We find that limy→∞ g (y) = +∞. g (y) y

H7: We find that limy→∞

= 0.

Let us interrupt for a remark. Remark 3.1. It easy enough to modify condition (H2) to allow ξ0 to be an element of other proper subsets of a, σ 2 (b) T than the one given in condition (H2). We choose this subset only for definiteness throughout the remainder of this work. But an examination of the proofs in the sequel reveals that this interval could easily be changed. Therefore, this condition is not quite as restrictive as it would seem.





For notational convenience in the sequel, we introduce the notation ΩR for the open, bounded, convex set

ΩR := {y ∈ B : ‖y‖ < R} , where R > 0 and B is the Banach space defined by



B := y :

a, σ 2 (b)



 T

 → R : y is rd-continuous ,

when equipped with the usual supremum norm, ‖ · ‖. In addition, we need the cone K ⊆ B defined by

K :=

  

y ∈ B : y ≥ 0 and

min [

 

t∈

(

)

2 a+σ 2 (b) 3 a+σ (b) , 4 4

y(t ) ≥ γ0 ‖y‖

]

  

.

  T

The constant γ0 appearing in the definition of K is defined by

  γ0 := min γ , γ ∗ , where γ is the number from (2.7), and the number γ ∗ ∈ (0, 1) is defined by min [ t∈

γ := ∗

(

)

2 a+σ 2 (b) 3 a+σ (b) , 4 4

α(t )

]

   =α ρ

T

‖α‖

3 a + σ 2 (b) 4

  ;

here ρ is the backwards jump operator (see [37]). Note that γ0 ∈ (0, 1), evidently. We prove our first existence theorem, Theorem 3.3, by first showing that T maps an appropriate subset of K into K . Essentially, we restrict T to a subset of K containing y with suitably large norm so as to guarantee that the function H remains positive. In particular, we consider the set

K \ Ω N0 , γ0

N

which ensures that each y ∈ K \ Ω N0 satisfies ‖y‖ ≥ γ 0 . Importantly, then, we get that the minimum of any such y is at 0 γ 0

least N0 on an appropriate compact subset of a, σ 2 (b) T . This is the content of Lemma 3.2, which we state and prove now; from this lemma, we then deduce the first existence result, which is Theorem 3.3. We will give a specific example of the use of this result in Section 4.





Lemma 3.2. Let T be the operator defined in (2.1) and suppose that (H1)–(H3) and (H5) hold. We then find that T : K \ Ω N0 → γ0

K. Proof. Let y ∈ K \ Ω N0 be given. Then we note that γ0

y(t ) ≥ γ0 ‖y‖ ≥ γ0 ·

min [ ] 2 a+σ 2 (b) 3(a+σ (b)) t∈ , 4 4 T

N0

γ0

= N0 .

(3.1)

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C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

Thus, (3.1) implies that H (y (ξ0 )) > 0, provided that

 ξ0 ∈

a + σ 2 (b) 3 a + σ 2 (b)



,

4

 ,

4 T

which holds by condition (H2). Therefore, it follows that

(Ty)(t ) ≥ 0,   for each t ∈ a, σ 2 (b) T .

(3.2)

In addition, (3.1)–(3.2) straightforwardly imply that min [ t∈

(

2 a+σ 2 (b) 3 a+σ (b) , 4 4

)

]

(Ty)(t ) ≥ γ0 ‖Ty‖.

(3.3)

T

Putting (3.2) and (3.3) together, we conclude that Ty ∈ K — that is, T : K \ Ω N0 → K ,

(3.4)

γ0

as claimed.



Theorem 3.3. Suppose that conditions (H1)–(H7) are satisfied. Then there exists a number λ0 > 0 sufficiently large such that problem (1.1)–(1.3) has at least one positive solution for all λ > λ0 . Proof. Lemma 3.2 shows that T : K \ Ω N0 → K . Moreover, T is a completely continuous operator, which is a standard γ0

fact, whose proof, therefore, we omit. (Recall here that H is continuous, by assumption.) Now, condition (H6) implies that we may find a number r1 > 0 such that g (y) ≥ η1 ,

(3.5)

whenever y > r1 , where η1 satisfies

η1

∫ 

(

2 a+σ 2 (b) 3 a+σ (b) , 4 4

)

γ0 G(σ (s), s)a(s) ∆s = 1,



(3.6)

T

say. Now, by condition (H4), there is a number, say ζ0 ≥ 0, such that H satisfies the inequality in (H4) for all y ≥ ζ0 . So, given r1 , define r1∗ by



∗

r1 := max

N0 ζ0

r1



, , . γ0 γ0 γ0

(3.7)

Then for y ∈ ∂ Ωr ∗ ∩ K , we find that 1

min [

(

2 a+σ 2 (b) 3 a+σ (b) , 4 4

)

y(t ) ≥ γ0 ‖y‖ ≥ max {r1 , N0 , ζ0 } .

]

(3.8)

T

In particular, (3.5)–(3.8) imply that whenever y ∈ ∂ Ωr ∗ ∩ K , it follows that 1

y ∈ K \ Ω N0 ; γ0

that is, we find that

K \ Ω N0 ⊇ ∂ Ωr1∗ ∩ K . γ0

Now, for notational simplicity in inequality (3.9) below, let us put E := E ̸= ∅. Then, for y ∈ ∂ Ωr ∗ ∩ K , we have 1

(Ty)(t ) ≥ γ0 ‖α‖H (y (ξ0 )) + λ

∫

σ (b)

∫a

G(t , s)a(s)g (yσ (s)) ∆s

γ0 G(σ (s), s)a(s)g (yσ (s)) ∆s ∫ ≥ γ0 ‖α‖H (y (ξ0 )) + λη1 γ0 G(σ (s), s)a(s) ∆s ≥ γ0 ‖α‖H (y (ξ0 )) + λ

E

E



a+σ 2 (b) 4

3 a+σ 2 (b) , ( 4 )



, where we assume that T

C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

≥ λη1

∫

423

γ0 G(σ (s), s)a(s) ∆s E

= λ.

(3.9)

Note that in (3.9) we have used the fact that H (y (ξ0 )) > 0 since y ∈ K \ Ω N0 . We also note that minE y(t ) ≥ γ0 · γ = r1 . 0 γ r1

0

Furthermore, in (3.9) we see that by choosing λ > 0 sufficiently large (here λ := λ r1∗ depends on the choice of r1∗ , of course), we get

 

‖Ty‖ ≥ ‖y‖.

(3.10)

In particular, there exists a number λ0 > 0 such that for all λ > λ0 , inequality (3.10) holds for all y ∈ ∂ Ωr ∗ ∩ K . Thus, T is 1 a cone expansion on ∂ Ωr ∗ ∩ K . 1 On the other hand, condition (H7) implies that there is r2 > 0 such that g (y) ≤ ϵ1 y,

(3.11)

whenever y > r2 , where ϵ1 := ϵ1 (c , λ) > 0 satisfies

ϵ1 λ

σ (b)

∫

G(σ (s), s)a(s) ∆s < 1 − c ,

(3.12)

a

c is the number from condition (H4), and λ > λ0 , with λ0 as in the previous part of this proof. Now, condition (H6) implies that g (y) is unbounded on [0, +∞). Therefore, we may find a number r2∗ > max r1∗ , r2





(3.13)

such that g (y) ≤ g r2∗ ,

 

(3.14)

for all y ∈ 0, r2 . Consequently, for each y ∈ ∂ Ωr ∗ ∩ K , we have



∗



2

‖Ty‖ ≤ H (y (ξ0 )) + λ

σ (b)

∫

G(σ (s), s)a(s)g (yσ (s)) ∆s

a

  < cr2∗ + λg r2∗

σ (b)

∫

G(σ (s), s)a(s) ∆s

a

≤ cr2∗ + r2∗ ϵ1 λ

σ (b)

∫

G(σ (s), s)a(s) ∆s

a

< cr2∗ + (1 − c )r2∗ = ‖y‖,

(3.15)

whence T is a cone compression on ∂ Ωr ∗ ∩ K . Note that we have used in (3.15) above the fact that mint ∈E y(t ) ≥ ζ0 so that, 2 in particular, H (y (ξ0 )) ≤ cy (ξ0 ).   In summary, we may invoke Lemma 2.2 to deduce the existence of a fixed point y0 ∈ Ω r ∗ \ Ωr ∗ ∩ K . Moreover, y0 is a 2

1

solution of problem (1.1)–(1.3). Finally, that y0 is a positive solution follows from the proof of Lemma 3.2, for y0 ∈ K \ Ω N0 , γ0

by construction, and so, y0 (t ) = (Ty0 ) (t ) is nonnegative (and not zero identically). In fact, the preceding proof shows that y0 satisfies the bound 0<

N0

γ0

≤ r1∗ ≤ ‖y0 ‖ ≤ r2∗ .

And this completes the proof.

(3.16)



Remark 3.4. Although not stated in the theorem, the proof of Theorem 3.3 reveals that we can give explicitly a range of eigenvalues for which Theorem 3.3 applies. Indeed, if we put

 r1 := inf x ≥ 0 : g (y) ≥

[∫

γ0 G(σ (s), s)a(s) ∆s E

then by putting

λ0 := max



r1

,

N0 ζ0

,

γ0 γ0 γ0



,

] −1

 , for all y ∈ [x, +∞) ,

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C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

where each of r1 , N0 , ζ0 , and γ0 can be explicitly computed, we find that the set of admissible eigenvalues is (λ0 , +∞). In particular, if ζ0 = 0 (i.e., if the inequality in (H4) holds for all y ≥ 0), then the interval reduces to

    r1 N0 , , +∞ . λ ∈ max γ0 γ0 We now provide a slight generalization of Theorem 3.3. In particular, it is possible to generalize the function H to one that has as its domain Rn+ in lieu of [0, +∞). In this setting, boundary condition (1.2) would involve finitely many values of y. Therefore, we make the following assumptions. Note that by the notation Rn+ we mean the set

Rn+ := x := (x1 , . . . , xn ) ∈ Rn : xi ≥ 0, for each 1 ≤ i ≤ n .





H1a: Let H : Rn+ → R be a real-valued function and assume that

φ(y) := H (y (ξ1 ) , y (ξ2 ) , . . . , y (ξn )) . Moreover, assume that there is a number N0 > 0 such that H (y1 , . . . , yn ) > 0 whenever (y1 , . . . , yn ) ∈ Rn+ \ B (0; N0 ), where B (x; r ) represents the ball of radius r > 0 centered at x ∈ Rn . H2a: Assume that ξi satisfies condition (H2), for each i. H4a: There is a finite sequence {ci }ni=1 ⊆ [0, 1) such that 0≤

n −

ci < 1,

i=1

and H satisfies

|H (y1 , . . . , yn )| ≤

n −

ci yi ,

i =1

eventually. It is obvious that Lemma 3.2 still holds in this setting; thus, we do not repeat its proof. On the other hand, Theorem 3.3 can be easily modified to deduce an analogous result in this slightly more general setting. Theorem 3.5. Suppose that conditions (H1a), (H2a), (H3), (H4a), and (H5)–(H7) are satisfied. Then there exists a number λ0 > 0 sufficiently large such that problem (1.1)–(1.3) has at least one positive solution for all λ > λ0 . Proof. The first part of the proof of Theorem 3.3 proceeds unchanged, for we can again select r1∗ sufficiently large and then λ sufficiently large so that y ∈ K \ Ω N0 , γ0

which implies that H (y (ξ1 ) , . . . , y (ξn )) > 0. Therefore, we will not repeat that part of the argument. On the other hand, since by condition (H4a) n −

0≤

ci < 1,

i=1

we may select a number δ0 ∈ (0, 1) such that n −

ci < δ0 < 1

(3.17)

i=1

holds. Then condition (H7) implies the existence of a number r2 > 0 such that g (y) ≤ ϵ1 y,

(3.18)

whenever y > r2 , where ϵ1 := (δ0 , λ) > 0 satisfies

ϵ1 λ

∫ a

σ (b)

G(σ (s), s)a(s) ∆s < 1 − δ0 .

(3.19)

C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

425

As before, we can then select a number r2∗ satisfying r2∗ > max r1∗ , r2 ,





(3.20)

such that inequality (3.14) still holds, and so, for y ∈ ∂ Ωr ∗ ∩ K we find that 2

‖Ty‖ ≤ |H (y (ξ1 ) , . . . , y (ξn ))| + λ

∫

σ (b)

G(σ (s), s)a(s)g (yσ (s)) ∆s

a

≤

n −

ci y (ξi ) + λg r2

 ∗

G(σ (s), s)a(s) ∆s

a

i=1

≤ r2∗

σ (b)

∫

n −

ci + r2∗ ϵ1 λ

σ (b)

∫

G(σ (s), s)a(s)y(s) ∆s

a

i=1

< δ0 r2∗ + (1 − δ0 ) r2∗ ≤ ‖y‖,

(3.21)

whence we again have that T is a cone compression on ∂ Ωr ∗ ∩ K . 2





Consequently, we conclude that T has a fixed point, say y0 , with y0 ∈ K ∩ Ω r ∗ \ Ωr ∗ , which is a positive solution of 2

problem (1.1)–(1.3). And this completes the proof.

1



Remark 3.6. As in with the case of Theorem 3.3, we point out that condition (H2a) can actually be weakened to require merely that there is an index i0 such that ξi0 satisfies condition (H2). A correspondingly more general version of Theorem 3.5 may then be given. Remark 3.7. We point out that if condition (H2a) is weakened as in Remark 3.6, then we need condition (H4) to hold for all y ≥ 0. In any case, we omit this statement, too. We now present our final existence result in this section. This result is rather different in structure and is built upon some of the recent ideas of Infante and Webb [27]. In particular, of the results we have presented thus far, none can be applied to a function H : R2+ → R having the form H (y1 , y2 ) := −2



1 2

y1 −

1 5

 y2

e−y1 −y2 +



1 2

y1 −

1 5

 y2

.

Indeed, for H as in (3.22), there is no N0 such that H (y1 , y2 ) ≥ 0 whenever

(3.22)



y21 + y22 ≥ N0 . But note that H has a special

structure of the form H (y1 , y2 ) =



1 2

y1 −

1 5

 y2



 −2e−y1 −y2 + 1 .

(3.23)

In particular, if we put L(y1 , y2 ) :=

1 2

y1 −

1 5

y2 ,

(3.24)

where L : R2+ → R, then H (y1 , y2 ) = L(y1 , y2 ) −2e−y1 −y2 + 1 ,





(3.25)

where

− 2e−y1 −y2 + 1

(3.26)

is eventually nonnegative. Since L is a linear function, we may ensure its nonnegativity by using a cone recently introduced in [27]. By doing this, we shall be able to incorporate a function such as given in (3.22) into our theory. So, to set up our next existence theorem, let us introduce a new set of conditions. They are the following.





H1b: Let H , L : Rn+ ∩ y ∈ Rn+ : L(y) ≥ 0 → R be real-valued functions, where L is linear, and assume that

φ(y) := H (y (ξ1 ) , y (ξ2 ) , . . . , y (ξn )) + ψ(y),    where ψ : Crd a, σ 2 (b) T , R → R is a linear functional having the form ψ(y) = L (y (ξ1 ) , . . . , y (ξn )) .

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C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

In particular, assume that L has the multipoint form L (y1 , . . . , yn ) :=

n −

βj yj ,

j=1

 n

where the finite sequence βj j=1 ⊆ R satisfies both n −

  βj α ξj ≥ 0

j =1

and n −

  βj G ξj , s ≥ 0,

j =1

for each s ∈ a, σ 2 (b) T . Finally, under the assumption that L (y1 , . . . , yn ) ≥ 0, there is a number N0 > 0 such that





H (y1 , . . . , yn ) + L (y1 , . . . , yn ) > 0 whenever ‖y‖ :=



y21 + · · · + y2n > N0 .

H2b: Assume that ξi satisfies condition (H2), for each i. H4b: There is a finite sequence {ci }ni=1 ⊆ (0, 1) such that n −

0≤

ci < 1,

i=1

and H + L satisfies

|H (y1 , . . . , yn ) + L (y1 , . . . , yn )| ≤

n −

ci yi ,

i=1

eventually. Remark 3.8. Evidently, condition (H4b) places a restriction on the quantity max

j∈{1,2,...,n}

  βj  ,

though we do not explicitly write down a restriction of this sort. In addition, we require a different cone, which we label as K ∗ below. In particular, the cone K ∗ is defined by

K ∗ :=

  

y ∈ B : y ≥ 0,

 

min [ t∈

(

2 a+σ 2 (b) 3 a+σ (b) , 4 4

  

)

]

y(t ) ≥ γ0 ‖y‖, ψ(y) ≥ 0

,

(3.27)

 

T

which is obviously adapted from [27], for example. We first show that the operator T leaves a certain subset of K ∗ invariant. We then use this result together with Lemma 2.2 to deduce that problem (1.1)–(1.3) has at least one positive solution. Lemma 3.9. Let T be the operator defined in (2.1). Suppose that conditions (H1b), (H2b), (H3), and (H5) hold. Then T : K ∗ \ Ω N0 → K ∗ . γ0

Proof. Let y ∈ K ∗ \ Ω N0 . Observe that as y ∈ K ∗ , we may assume that γ0

ψ(y) ≥ 0.

(3.28)

Inequality (3.28) is crucial, and it justifies the assumption in condition (H1b) that ψ(y) (and thus L (y1 , . . . , yn )) is, in fact, nonnegative. So, we first claim that (Ty)(t ) ≥ 0, for each t. To this end, note that a calculation identical to (3.1) implies that y(t ) ≥ N0 , provided that t satisfies

 t ∈

a + σ 2 (b) 3 a + σ 2 (b)



4

,

 .

4

(3.29)

T

Consequently, from condition (H1b), we deduce that φ(y) > 0 for all such y. From this, it is clear that (Ty)(t ) ≥ 0, as claimed.

C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

427

We next argue that min [ t∈

(

)

2 a+σ 2 (b) 3 a+σ (b) , 4 4

(Ty)(t ) ≥ γ0 ‖Ty‖.

]

(3.30)

T

But, once again, a calculation identical to (3.3) reveals that this, too, is true. (We again use the positivity of H + L here.) Finally, we must show that ψ(Ty) ≥ 0. So, note that σ (b)

 ∫ ψ(Ty) = ψ α(t )φ(y) +

G(t , s)a(s)g (yσ (s)) ∆s



a

= ψ (α(t )φ(y)) + ψ

σ (b)

∫

G(t , s)a(s)g (yσ (s)) ∆s



a

=

n −

n −   βj βj α ξj φ(y) + j=1

j =1

= φ(y)

n −

  βj α ξj +

σ (b)

∫

a

j =1





a

 n σ (b) −

∫

G ξj , s a(s)g (yσ (s)) ∆s

βj G ξj , s 

 

a(s)g (yσ (s)) ∆s

j=1

≥ 0,

(3.31)

where to get the final inequality we have used the conditions, again from (H1b), that both n −

  βj α ξj ≥ 0

(3.32)

  βj G ξj , s ≥ 0,

(3.33)

j=1

and n − j=1

for each s ∈ a, σ 2 (b) T , hold. Moreover, we have used the fact that φ(y) ≥ 0 since y ∈ K ∗ \ Ω N0 . Thus, from (3.31), we





conclude that ψ(Ty) ≥ 0. Consequently, we see that T : K ∗ \ Ω N0 → K ∗ . And this completes the proof. γ0

γ0



Theorem 3.10. Suppose that conditions (H1b), (H2b), (H3), (H4b), and (H5)–(H7) hold. Then there exists a number λ0 > 0 sufficiently large such that for all λ > λ0 , problem (1.1)–(1.3) has at least one positive solution. Proof. Essentially this proceeds like the proofs previously given in this section. Therefore, we shall merely outline the differences here. In particular, Lemma 3.9 shows that T : K ∗ \ Ω N0 → K ∗ . Moreover, as before, T can be shown to be a completely γ0

continuous operator. Then, just as in the proof of Theorem 3.3, we can use condition (H6) to find that

(Ty)(t ) ≥ λη1

∫

γ0 G(σ (s), s)a(s) ∆s,

(3.34)

E

whenever y ∈ ∂ Ωr ∗ ∩ K ∗ . It then follows that 1

‖Ty‖ ≥ ‖y‖,

(3.35)

at least for λ sufficiently large. Evidently, in the preceding work we use the crucial fact that φ(y) > 0 whenever y ∈ K \Ω N0 . ∗

On the other hand, by using conditions (H4b) and (H7), we can get, by way of (3.21), that

‖Ty‖ ≤ ‖y‖,

γ0

(3.36)

whenever y ∈ ∂ Ωr ∗ ∩ K . 2 Therefore, upon putting (3.35) and (3.36) together, we deduce by way of Lemma 2.2 that problem (1.1)–(1.3) has at least one positive solution. And this completes the proof.  ∗

As remarked earlier, Theorem 3.10 applies to many nonlocalities to which the preceding theorems of this section do not. We shall provide in Section 4 an explicit numerical example of Theorem 3.10. We conclude by remarking as a corollary that Theorem 3.10 applies in a special case. This corollary records the fact that if one of our nonlinear functionals is asymptotically

428

C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

similar to a linear function of the type in [27], then problem (1.1)–(1.3) can exhibit at least one positive solution, at least for sufficiently large values of the eigenvalue λ. Corollary 3.11. Suppose that the conditions of Theorem 3.10 hold, but in addition assume that φ(y) has the special form

φ(y) = L (y (ξ1 ) , . . . , y (ξn )) [H (y (ξ1 ) , . . . , y (ξn )) + 1] , where H (y1 , . . . , yn ) → 0 as ‖(y1 , . . . , yn )‖ → +∞. Then there exists a number λ0 > 0 sufficiently large such that for all λ > λ0 , problem (1.1)–(1.3) has at least one positive solution. 4. Examples We give now three numerical examples illustrating how certain of the preceding theorems may be satisfied in nontrivial settings. We limit ourselves to explicit examples of Theorems 3.3, 3.5 and 3.10. We remark as in Section 1 that, to the best of our knowledge, none of these examples could be handled by any of the existing results (cf., Remarks 4.4 and 4.5). Indeed, as our examples will illustrate, we do not require the boundary terms to be monotone, and we get the existence of positive solutions rather than just the existence of a nontrivial solution. Furthermore, we note that while we provide all examples on the time scale T = R, seeing as this time scale has particular importance in both pure and applied mathematics, it is obvious how our examples could be modified to treat other important time scales such as the difference equations case (T = Z) or the q-difference equations case (T = qN0 ), for example. Example 4.1. Let us give first an example illustrating the use of Theorem 3.3. We shall consider this example on the time scale T = R with 0 ≤ t ≤ 1. So, to this end, let g : [0, +∞) → [0, +∞) be defined by g (y) := ln(ln(y + 3)),

(4.1)

a : [0, 1] → [0, +∞) be defined by a(t ) := 2e3t

(4.2)

 9  and H : [0, +∞) → − 10 e, +∞ be defined by  1   sin(π y), 0≤y≤1   2  1 1y H (y) := − e 10 (y − 1), 1 ≤ y ≤ 10  10    ln(y − 9) − 9 e, 10 ≤ y < +∞. 10   3 Let us also fix ξ0 = 5 . Note then that ξ0 ∈ 41 , 34 .

(4.3)

Now, notice that lim

y→∞

g (y) y

= ∞,

(4.4)

as is easy to verify. Moreover, clearly g (y) → +∞ as y → +∞. So, conditions (H6) and (H7) are verified. It is easy to see 9

that H is eventually positive — namely, for y ≥ 9 + e 10 e =: N0 . As the other conditions, (H1)–(H5), are obviously satisfied, we find that we may apply Theorem 3.3. In particular, then, there exists a number λ0 > 0 such that the ordinary differential equation y′′ (t ) = −2λe3t ln(ln(y(t ) + 3))

(4.5)

subject to the boundary conditions

   1 3  sin πy ,    2 5    1 1 3  3 y y(0) = − e 10 5 y −1 ,  10      5    ln y 3 − 9 − 9 e, 5

10

  0≤y 1≤y

3

5 3

≤1

≤ 10 5  3 10 ≤ y < +∞

(4.6)

5

and y(1) = 0,

(4.7)

has at least  one positive  solution for all λ > λ0 . Once again, we note that H is nonpositive for some y > 0 — in particular, 9

for y ∈ 1, 9 + e 10 e .

C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

429

Example 4.2. We next give an example of Theorem 3.5. As before, our time scale shall be T = R, and we set a = 0 and b = 1. So, suppose that H has the form H (y1 , y2 ) :=

1 20

y1 −

1 5

y1 y2 e−y2 ,

(4.8)

where H : R2+ → R; evidently, H is continuous on R2+ . Note that H (y1 , y2 ) > 0 whenever 1 5

 y1

1

− y2 e−y2

4



> 0,

(4.9)

and it is obvious that (4.9) is true provided that y1 > 0 and y2 > ln (4y2 ), which can be numerically approximated to reveal that y2 > 2.154, say, must hold. Consequently, whenever (y1 , y2 ) ∈ R2+ satisfies



y21 + y22 > 3,

(4.10)

say, we find that H (y1 , y2 ) > 0. In particular, condition (H1a) holds. Now, so that condition (H2a) is satisfied, let us set ξ1 := 25 and ξ2 := 35 . Finally, we check that condition (H4a) holds. To see that it does, merely notice that

   1 1 |H (y1 , y2 )| =  y1 − y1 y2 e−y2  20 5 ≤

1 20 1

|y1 | +

1 5

|y1 | |y2 | e−y2

1

y1 y2 e−y2 5 1 ≤ y1 + y1 e−1 20 10 1 1 y1 + y1 ≤ 20 20 1 = y1 + 0y2 . 10

=

20 1

y1 +

(4.11)

Note that in (4.11) we have appealed to the elementary fact that sup

y2 ∈[0,+∞)

y2 e−y2 = e−1

to deduce that y2 e−y2 ≤ e−1 ,

(4.12)

for all y2 ∈ [0, +∞). Consequently, (4.11) implies the condition (H4a) holds with, say, c1 := Now consider the ordinary differential equation

  − y′′ = λ t 2 + 2t ln (y(t ) + 1)

1 10

and c2 := 0. (4.13)

subject to the boundary conditions y(0) =

1 20

  y

2

5

1

− y 5

    2

5

y

3

5

−y

e

  3 5

(4.14)

and y(1) = 0.

(4.15)

It is easy to check that the function g (y) := ln(y + 1) satisfies both conditions (H6) and (H7). Obviously, a(t ) := t + 2t satisfies condition (H5). Therefore, it is clear that each of the necessary conditions holds, and so, Theorem 3.5 implies that problem (4.13)–(4.15) has at least one positive solution for all λ > λ0 , where λ0 > 0 is selected sufficiently large. As in Example 4.1, we note that this holds in spite of the fact that H (y1 , y2 ) is negative for some y1 , y2 > 0. 2

Example 4.3. We conclude this section and this paper with an example of Theorem 3.10. Recall, as discussed earlier, that this theorem provides a more direct link between the results of [27] and the results of the present work. As in each of the preceding examples, we shall assume that the time scale here is T = R and that a = 0 and b = 1.

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C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

So, let us define the functional φ by way of a modification of (3.22). In particular, we assume that

φ(y) := −2



 

1 10

1

y

−

2

 

1 20

3

y



    −y 12 −y 34

+

e

4

 

1 10

y

1 2



  1 3 − y . 20 4  

(4.16)

=:ψ(y)

From (4.16), it is clear that we have set ξ1 := and ξ2 := so that condition (H2b) holds. Now, from (4.16), we can identify the real-valued functions L and H as 1 2

L(y1 , y2 ) :=

1 10

y1 −

1 20

3 4

y2

(4.17)

and H (y1 , y2 ) := −2



1 10

y1 −



1 20

y2

e−y1 −y2 ,

(4.18)

respectively, where L, H : R2+ → R. Then we note that condition (H4b) holds, seeing as

       1 1 1 1 −y1 −y2  |L(y1 , y2 ) + H (y1 , y2 )| ≤ −2 y1 − y2 e + y1 − y2  10 20 10 20     1 1   1 1  ≤ 2e−y1 −y2  y1 − y2  +  y1 − y2  10 20 10 20     1    1 1 1 ≤ 2  y1 − y2  +  y1 − y2  10 20 10 20 ≤ 2· =

3 10

1 10

1

y1 + 2 ·

y1 +

3 20

20

y2 +

1 10

y1 +

1 20

y2

y2 .

(4.19)

3 3 and c2 := 20 . Consequently, (4.19) shows that condition (H4b) holds with c1 := 10 We next check that the conditions of (H1b) hold. It is obvious that L has an admissible form. Moreover, since

  (H + L)(y1 , y2 ) = −2e−y1 −y2 + 1



1 10

y1 −

1 20

 y2

(4.20)

and since

− 2e−y1 −y2 + 1 > 0

(4.21)

whenever y1 + y2 ≥ ln 2, it follows that (H + L)(y1 , y2 ) ≥ 0 whenever, say, both y21 + y22 ≥ ln 2

(4.22)

and 1 10

y1 −

1 20

y2 ≥ 0

(4.23)

hold. Note that (4.23) may be assumed since by the construction of the cone K ∗ , we are looking for solutions only over those functions for which 1 10

  y

1 2

−

1 20

  y

3

4

≥0

(4.24)

holds and thus (4.23) is true. Furthermore, straightforward numerical calculations show that both 2 −

  βj α ξj ≥ 0

(4.25)

  βj G ξj , s ≥ 0

(4.26)

j =1

and 2 − j =1

C.S. Goodrich / Nonlinear Analysis 75 (2012) 417–432

hold, for each s ∈ [0, 1]. (Note that we choose β1 = holds. Now consider the ordinary differential equation

1 10

431

1 and β2 = − 20 here.) Consequently, we see that condition (H1b)

− y′′ = λet ln(y(t ) + 1)

(4.27)

together with the boundary conditions y(0) = −2



1 10

  y

1

−

2

1 20

  y

3

−y

e

4

  1 2

  −y 34

 +

1 10

  y

1

2

−

1 20

  y

3

(4.28)

4

and y(1) = 0.

(4.29)

Then it is clear that we may apply Theorem 3.10 to deduce that problem (4.27)–(4.29) has at least one positive solution for all λ > λ0 , for some λ0 > 0 sufficiently large. Remark 4.4. Observe that problem (4.27)–(4.29) of Example 4.3 does not fit into the framework captured by Theorem 3.5. This is because the function

(H + L)(y1 , y2 ) := −2



1 10

y1 −

1 20

 y2

e−y1 −y2 +



1 10

y1 −

1 20

 y2

is not eventually positive in the sense that there exists no N0 such that whenever

(4.30)



y21 + y22 ≥ N0 , we find that (H +

L)(y1 , y2 ) > 0. In fact, this is due to the existence of the linear part L(y1 , y2 ) =

1 10

y1 −

1 20

y2

(4.31)

in (4.30). However, by combining our ideas together with those introduced in [27], we are nonetheless able to deduce the existence of at least one positive solution to this problem. Remark 4.5. As remarked elsewhere, none of the preceding examples could be treated by any of the existing results. In particular, Example 4.3 could not be treated by the results of [27], and it could not even be handled by [35], for there it is assumed that the multipoint term is nonnegative for each nonnegative function y — a condition which clearly fails in Example 4.3 above. Nonetheless, it is closely related to these results. In particular, the nonlocality φ in this case is asymptotically similar to a linear functional which does fit the theory of [27]. Consequently, this example explicitly shows how a nonlinear functional that is asymptotically similar to a linear functional from [27] can, under other suitable conditions, imply the existence of at least one possible solution to problem (1.1)–(1.3). In fact, this is exactly the content of Corollary 3.11 stated earlier. References [1] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56. [2] L.H. Erbe, H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994) 743–748. [3] F.M. Atici, G. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002) 75–99. [4] D.R. Anderson, R.I. Avery, An even-order three-point boundary value problem on time scales, J. Math. Anal. Appl. 291 (2004) 514–525. [5] L. Bian, X. He, H. Sun, Multiple positive solutions of m-point BVPs for third-order p-Laplacian dynamic equations, Adv. Difference Equ. (2009) 12 pp. Art. ID 262857. [6] J.J. DaCunha, J.M. Davis, P.K. Singh, Existence results for singular three point boundary value problems on time scales, J. Math. Anal. Appl. 295 (2004) 378–391. [7] A. Dogan, J. Graef, L. Kong, Higher order semipositone multi-point boundary value problems on time scales, Comput. Math. Appl. 60 (2010) 23–35. [8] F. Geng, D. Zhu, Multiple results of p-Laplacian dynamic equations on time scales, Appl. Math. Comput. 193 (2007) 311–320. [9] C.S. Goodrich, Existence of a positive solution to a first-order p-Laplacian BVP on a time scale, Nonlinear Anal. 74 (2011) 1926–1936. [10] E.R. Kaufmann, Y.N. Raffoul, Positive solutions for a nonlinear functional dynamic equation on a time scale, Nonlinear Anal. 62 (2005) 1267–1276. [11] B. Liu, Z. Zhao, A note on multi-point boundary value problems, Nonlinear Anal. 67 (2007) 2680–2689. [12] Y. Liu, Three positive solutions of nonhomogeneous multi-point BVPs for second order p-Laplacian functional difference equations, J. Appl. Math. Comput. 29 (2009) 437–460. [13] Y. Sang, H. Su, Several existence theorems of nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales, J. Math. Anal. Appl. 340 (2008) 1012–1026. [14] Y.H. Su, W.T. Li, Triple positive solutions of m-point BVPs for p-Laplacian dynamic equations on time scales, Nonlinear Anal. 69 (2008) 3811–3820. [15] H.R. Sun, W.T. Li, Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales, J. Differential Equations 240 (2007) 217–248. [16] J.P. Sun, Twin positive solutions of nonlinear first-order boundary value problem on time scales, Nonlinear Anal. 68 (2008) 1754–1758. [17] J.P. Sun, W.T. Li, Existence of solutions to nonlinear first-order PBVPs on time scales, Nonlinear Anal. 67 (2007) 883–888. [18] J.P. Sun, W.T. Li, Positive solutions to nonlinear first-order PBVPs with parameter on time scales, Nonlinear Anal. 70 (2009) 1133–1145. [19] Y. Tian, W. Ge, Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales, Nonlinear Anal. 69 (2008) 2833–2842.

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