Positive solutions of some higher order nonlocal boundary value problems, Electron

We show how a unified method, due to Webb and Infante, of tackling many nonlocal boundary value problems, can be applied to nonlocal versions of some recently studied higher order boundary value problems. In particular, we give some explicit examples and calculate the constants that are required by the theory.

The existence of at least one positive solution was shown for λ in certain intervals, defined in terms of the behaviour of f (u)/u as u → 0 + and as u → ∞, by using the wellknown Krasnosel'skiȋ fixed point theorems of cone compression and cone expansion.This methodology was previously used, for example, in [15,23] on some fourth order and third order problems respectively with some other BCs.For some other work on similar BVPs see, for example, [1,3,4,5], and for some higher order systems, see [7].
In the present paper, we discuss the higher order equation (1.1), under weaker assumptions on f and g, with nonlocal versions of the BCs written above.We obtain results on the existence of one positive solution by utilizing results of Webb and Lan [21] involving comparison with the principal eigenvalue of a related linear problem and show that these results can be sharp.We also show that, in some cases, a previous theory using other constants does not apply.
In particular, we give some explicit examples and calculate some constants named m, M(a, b) that are commonly used on such problems.It is interesting to note that, in the case g(t) ≡ 1, for an arbitrary n we can compute explicitly the constant m and the optimal value of M(a, b); that is, we determine a, b so that M(a, b) is minimal for each of the three local problems.
We then use the theory worked out by Webb and Infante in [18,19] to study the same equation with some quite general nonlocal BCs.For example, corresponding to (1.2) we can treat the BCs where α is a linear functional on , a Stieltjes integral.We similarly obtain sharp existence results for this case.This case covers multi-point BCs, where α[u] = m i=1 a i u(η i ), and also integral BCs, where α[u] = 1 0 a(s)u(s) ds, in a single framework.
Nonlocal BCs have been extensively studied in recent years, see the survey article [14] for many references.Nonlocal BCs defined by Stieltjes integrals were studied in [8,9,10,13] where it was assumed that the measure dA is positive.In contrast, the method of [18,19,20] does not require that α[u] ≥ 0 for all u ≥ 0; that is, we can allow dA to be a signed measure.In particular, in the multi-point case we can have coefficients a i of both signs.We give examples where we explicitly calculate the constants required by the theory.

Preliminaries
We will study the problem with one of the BCs where α[u] is given by a Riemann-Stieltjes integral with A a function of bounded variation.This is quite natural because such a functional α[u] is a linear functional on C[0, 1], and it includes sums and integrals as special cases.
We do not suppose that α[u] ≥ 0 for all u ≥ 0 but we allow dA to be a signed measure.The condition α[u] ≥ 0 is only imposed on a positive solution u.The set-up of [19] allows us to study also more general BCs with two nonlocal terms, for example we could readily treat the BCs Using [20] we could similarly easily handle the case when any number of the BCs has a nonlocal part.We only treat the case of one nonlocal term here, firstly for simplicity, and secondly because, in our approach, the main effort has to be directed at the local problems (1.1)-(1.2),(1.1)-(1.3),(1.1)-(1.4);then the theory of [19,20] can be applied.
The problem is then reduced to calculating the constants that occur in the theory.This has been done in a number of papers.For second order equations with one nonlocal term with a variety of BCs see [18,19], for some some typical fourth order problems see [20,22], and for fourth order conjugate type BCs see [17].
We will apply the standard methodology of seeking solutions as fixed points of the integral operator where G is the Green's function corresponding to each BVP consisting of the equation (2.1) with the respective BCs (2.2), (2.3), (2.4).We use the well known cone where [a, b] is some subset of [0, 1] and c > 0. K 0 is a well-known type of cone, first used by Krasnosel'skiȋ, see e.g.[12] and D. Guo, see e.g.[6].
The following condition is a key one that allows use of the cone K 0 .We use the same label as [19] for convenience.In fact, we will determine functions Φ and c so that c(t)Φ(s) ≤ G(t, s) ≤ Φ(s) for s, t ∈ [0, 1].Since we find c satisfying c(t) > 0 for t ∈ (0, 1), the subinterval [a, b] can be chosen arbitrarily in (0, 1).
We shall assume throughout, and without further mention, that gΦ ∈ L 1 (so g can have pointwise singularities at arbitrary points of [0, 1]) and satisfies the non-degeneracy condition b a g(s)Φ(s) ds > 0. We also assume that f satisfies Carathéodory conditions.When α[u] ≡ 0 the Green's function is denoted G 0 (t, s).For each of the BCs above G 0 is easily found and is as follows, see for example [2]. where is the Heaviside function.
To treat a nonlocal BC such as we make use of the function γ defined to be the solution of the equation γ (n) (t) = 0 with the corresponding BCs We need the following 'positivity' assumptions on the 'boundary term', again using the same labels as in [19].
(C 5 ) A is a function of bounded variation, and Remark 2.1 Because (C 5 ), (C 7 ) are integral (or sum) conditions, not pointwise ones, we can allow some sign changing measures dA.
Under these conditions, it is shown in [18] that the Green's function G for each nonlocal problem is given by For the BCs given above the functions γ i satisfy the respective BCs (2.12) .
A major advantage of the technique developed in [18,19,20] is that it is only necessary to verify the key positivity assumption (C 2 ) for the simpler Green's function G 0 , corresponding to the problem with no nonlocal terms, obtaining a constant c 0 .It is then shown that (C 2 ) holds for the full Green's function G.Moreover, in each of the cases studied here we have c = c 0 .
We are able to allow sign changing measures by working in the following cone: It is shown in [18] that S : P → K and known fixed point index results can then be applied to S. We use connections with the related linear operator Then L is a compact linear operator in C[0, 1] and, by (C 2 ), the spectral radius r(L) of L satisfies r(L) > 0. By the Krein-Rutman theorem, L has an eigenfunction ϕ ∈ P \{0} corresponding to the principal eigenvalue r(L); we suppose that ϕ = 1.Since L maps P into K, we have ϕ ∈ K.We set µ 1 := 1/r(L), and call it the principal characteristic value of L; it is often called the principal eigenvalue of the corresponding BVP.
For r > 0 we define the following extended real numbers: We use the following constants: Fixed point index results given in [21] can be used in a standard way to get multiplicity results.The important point of [21] was to prove these results in sufficient generality and for non-symmetric kernels.
Theorem 2.1 Assume that, whenever we have the condition µ 1 < f ∞ , the condition (C 2 ) holds for an arbitrary [a, b] ⊂ (0, 1).Then S has at least one positive fixed point in K if one of the following conditions holds.
S has at least two positive fixed points in K if one of the following conditions holds.
The results using (S 1 ), (S 2 ) are sharp.Instead of using the sharp conditions such as f 0 > µ 1 , f ∞ < µ 1 in (S 2 ), the stronger conditions f 0 > M, f ∞ < m can be used; similarly for the conditions in (S 1 ).It was shown in [21] that one always has m ≤ µ 1 ≤ M and the inequalities are strict if the corresponding eigenfunction is not constant.
It is routine to extend the list of conditions in order to show the existence of four, five, or an arbitrary finite number of fixed points, under increasingly restrictive conditions on f .We do not write the obvious statements.
The restrictions on f are weaker if [a, b] is chosen so that M(a, b) is as small as possible: the height to be exceeded by the graph of f is less.Also, for a given [a, b] the restrictions on f are weaker when c is chosen as large as possible, since the length of interval on which f has to be large is reduced.Remark 2.2 Using conditions (S 1 ) and (S 2 ) with µ 1 can give existence results when using the stronger conditions with m and M (even the optimal M) might not apply.Example 3.1 below illustrates this fact.
In the case g(t) ≡ 1, for an arbitrary n and for each of the three local problems we can compute explicitly the constant m and the optimal value of M(a, b); that is, we determine a, b so that M(a, b) is minimal.
Positive solutions do not exist if the nonlinearity does not cross the 'principal eigenvalue' of the differential equation.This is a sharper nonexistence result than has been used in [2,15].To state the result we need a concept due to Krasnosel'skiȋ, [11,12].EJQTDE Spec.Ed.I, 2009 No. 29 Definition 2.1 We say that L is u 0 -positive on a cone K, if there exists u 0 ∈ K \ {0}, such that for every u ∈ K \ {0} there are positive constants k 1 (u), k 2 (u) such that It is shown in [16] that many nonlocal BVPs have corresponding linear operators that are u 0 -positive.The following nonexistence result is shown in [20] for quite general nonlocal problems, with a short proof.
If (i) holds then 0 is the unique fixed point of S in K.If (ii) holds and L S is u 0 -positive for some u 0 ∈ K \ {0} then 0 is the only possible fixed point of S in K.

The First Local BVP
We need to establish certain properties of the local problem, when α[u] ≡ 0. The Green's function for this problem is We will use the following properties.
Lemma 3.1 For n ≥ 3, The Green's function G(t, s) satisfies the inequalities His proof uses uses some inequalities without trying for optimal conditions (and has some misprints).We seek an optimal bound so we give our method for completeness.
Proof.We write We now want to show that G(t, s) The minimum on the right occurs when s = 1, so we need c 0 (t)) ≤ t(n − 1) A similar, but more complicated, argument shows that G 1 (t, s)/Φ(s) ≥ c 0 (t) for 0 ≤ s ≤ t when c 0 (t) ≤ t(n − 1) For the case g(t) ≡ 1, we now compute the constant m and the optimal value of M(a, b), that is, we determine a, b so that M(a, b) is minimal.
We have , and the maximum of this expression occurs For a < b we have by direct integration b a The sign of the derivative shows that this is an increasing function of t so the minimum occurs at t = a.Let The We can also compute these constants when g ≡ 1 but we cannot give explicit formulae.We do this is our first example below.
We reconsider Example 1 of [2] to show how our results sharpen the ones there.Similar types of nonlinear terms have been used before in examples; see for example, Example 5.1 of [23], Example 3.5 of [15] and Example 1 of [5].
There is at least one positive solution if 8λ < µ 1 and 49λ > µ 1 ; that is, there is a positive solution if λ ∈ (0.6709, 4.1094).
There does not exist a positive solution if either 8λ > µ 1 or 63λ < µ 1 ; that is, if λ < 0.5218 or λ > 4.1094 no positive solution exists.
This shows that our results improve those of [2] and can give sharp estimates.The result of [2] and one of [15] use constants called A, B which in our notation are defined to be [16] that 1/A ≥ µ 1 , so using these constants rather than µ 1 will always give a poorer estimate on λ.
If we had tried to use the more stringent conditions λf 0 < m and λf ∞ > M we would need 8λ < 9.863 and 49λ > 75.681 and there are no λ satisfying both inequalities, so the theory that uses the constant M is ineffective on this example.However, the EJQTDE Spec.Ed.I, 2009 No. 29 J. R. L. Webb point of using m, M is that they can be used together with the behaviour of f on bounded intervals, not solely on the behaviour of f (u)/u near 0 and ∞ alone.

The First Nonlocal BC
We now consider the nonlocal problem with the BCs where α[u] is a linear functional on C[0, 1] give by a Stieltjes integral α[u] = 1 0 u(s) dA(s) with A a function of bounded variation.
For this problem the function γ is, as seen above, γ(t) = t.The Green's function is where Since we have verified that G 0 satisfies the key condition (C 2 ), it follows from the form of γ and from [18,19] that G also satisfies these conditions with the same function c(t).The theory therefore is directly applicable.We give an example for the 4th-order equation with a 4-point BVP with coefficients of both signs.If µ 1 /3 < λ < µ 1 , that is, 5.857 < λ < 17.5707, then the problem has at least one positive solution.
This shows that the estimates are sharp.We could easily give other examples where two or more positive solutions exist using Theorem 2.1.

The Second BC
The second BVP we study is with the BCs The Green's function for the local problem, when α[u] ≡ 0 is The following properties hold.We omit this proof, it is readily shown using the same method as in Lemma 3.1; it is already shown in [2] (with a misprinted = sign).
We now compute the constants m and the smallest M(a, b) when g(t) ≡ 1.
We have