Positive solutions of some nonlinear BVPs involving singularities and integral BCs

In this paper we discuss the existence of positive solutions of some 
nonlocal boundary value problems subject to integral boundary 
conditions and where the involved nonlinearity might be singular.


1.
Introduction. In this paper we establish new results on the existence of positive solutions of the following nonlocal boundary value problem (BVP) − u ′′ (t) = g(t)h(u(t)), t ∈ (0, 1), (1) where h is a non-negative function, allowed to be singular, with the nonlocal boundary conditions (BCs) We shall take α[u] to be a positive functional given by Multi-point and integral BCs are widely studied objects, see for example [4,8,10,11,12,21,22,24,27,28,29,30,31] and the reference therein. One motivation for studying (1)- (2) is that this type of BVP arises in some heat flow problems. For example the special case α[u] = 0 can be seen as a model for a heated bar of length 1 with a thermostat, where a controller at t = 1 adds or removes heat according to the temperature detected by a sensor at t = η. This case has been extensively studied by Infante and Webb [9,25,26], who were motivated by previous work of Guidotti and Merino [3]. The case of a nonzero term α[u], 100 GENNARO INFANTE studied by Infante and Webb in [10], has also a physical interpretation, for example when α[u] = αu(ξ), the four point BCs can be seen as a model for a heated bar, this time with two controllers, and two sensors at t = ξ and t = η.
Here we prove the existence of multiple positive solutions of (1)-(2) under suitable conditions.
As an application we also establish new results for second order differential equations of the form λu ′′ (t) + g(t)h(u(t)) = 0, t ∈ (0, 1).
Our approach is to rewrite the BVP (1)-(2) as a perturbed Hammerstein integral equation of the form where α[u] is a positive functional, γ is a positive continuous function and h is a positive function that is allowed to have singularities. This type of integral equation, with h non singular, has been studied recently by Infante and Webb [10,30], whereas the Hammerstein case, corresponding to γ(t) ≡ 0, but this time with a singular h, has been investigated by Lan [17]. The methods employed here rely on fixed point index theory, in particular we make use of results and ideas from the papers [10,17].
We mention that, with the same technique, one may study the case of a more general f (t, u) rather than g(t)h(u). Here for brevity and clarity, we refrain from doing do so.
2. Positive solutions of perturbed Hammerstein integral equations with singularities. We are interested in finding positive solutions of the integral equation We shall achieve this by seeking fixed points of an auxiliary operatorT in the cone of continuous function a type of cone first used by Krasnosel'skiȋ, see e.g. [13], and D. Guo, see e.g. [2]. Our main assumptions on h, g, k, α, γ satisfy are the following: and almost every s ∈ [0, 1].
where dA is a Lebesgue-Stieltjes measure with is well defined.
The above assumptions will enable us to use the well-known fixed point index for compact maps (see for example [1] or [2]) on the cone (7), with c = min{c 1 , c 2 }.
In order to use the results of [10], we extend h to all of [0, ∞) in a similar way to that of Lan [17]. We defineh(u) : and consider the operator We now look for fixed points ofT in C(r 1 , r 2 ) to find solutions of (6). First of all we note thatT : K → K is compact, that is,T is continuous and maps bounded sets in precompact sets.
Let q : We make use of the open set The set V ρ was introduced in [10] and is equal to the set called Ω ρ/c in [14]. One advantage of using V ρ is that it makes clearer that choosing c as large as possible yields a weaker condition to be satisfied by the nonlinearityh in Lemma 2.
The following Lemmas are special cases of Lemmas 2.5 and 2.7 of [10]. The first gives conditions which imply that the fixed point index is 0.
The second result implies that the index is 1.
Lemma 3. Suppose Γ < 1 and assume that there exists ρ > 0 such that u =T u for u ∈ ∂K ρ and Then we have i K (T , K ρ ) = 1.
Note that in Lemma 2.7 of [10] one needs to control the growth ofh on a larger set, here this is not needed, due to the fact that K contains only positive functions.
The above results valid forT allow us to give the following new result on existence of multiple positive solutions for Eq. (6).

Remark 1.
It is possible to state results for three or more positive solutions by similar arguments, we refer the reader to [14] for the type of results that may be stated.
3. Positive solutions of the BVP (1)- (2). We now consider the BVP with boundary conditions The solution of −u ′′ = y under these BCs can be written By a solution of the BVP (13)- (14) we mean a solution u ∈ C[0, 1] of the corresponding integral equation Note that k(t, s) in (15) is the kernel for the special case u ′ (0) = 0, studied by Infante and Webb in [9]. Here we discuss the case β + η > 1, that leads to the existence of positive solutions. Upper bounds Note that γ = β + η, In [9] it was shown that when β + η > 1 one may take Lower bounds Note γ(t) is a decreasing function of t in [0, 1] and min t∈[0,1] γ(t) = β + η − 1. A simple calculation shows that k(t, s) ≥ β + η − 1 for t ∈ [0, 1]. This leads to Hence we work on the cone with c as in (17). We state a result for the existence of one positive solution, of course there are more general results, analogous to Theorem 2 and Remark (1). (17), m be as in (12) and M as in (10) and Γ < 1. Then the BVP (13), (14) has at least one positive solution, if either (H 1 ) or (H 2 ) of Theorem 2 hold.

Theorem 3. Let c be as in
with BCs In this case g ≡ 1 and we may take A 0 = 0 and dA(s) the Dirac measure of weight α > 0 at ξ. Optimal values of the constants m and M , for the special case u ′ (0) = 0, were given by Webb in [25] as follows: Here we may take α 0 = α since, for u ∈ ∂V ρ , we have α[u] = αu(ξ) ≥ αρ.
With these BCs (21) reads We have that i K (T , K ρ ) = 1 if Γ < 1 and So we need Since K(s) = αk(ξ, s), by direct calculation one gets So (23) reads Note that all the numbers in (22) and (25) can be calculated.