Positive solutions of BVPs on the half-line involving functional BCs

We study the existence of positive solutions on the half-line of a second order ordinary differential equation subject to functional boundary conditions. Our approach relies on a combination between the fixed point index for operators on compact intervals, a fixed point result for operators on noncompact sets, and some comparison results for principal and nonprincipal solutions of suitable auxiliary linear equations.


Introduction
In this manuscript we discuss the existence of multiple non-negative solutions of the boundary value problem (BVP) where p, f are continuous functions on their domains, α > 0, β ≥ 0 and B is a suitable functional with support in [0, R]. The functional formulation of the boundary conditions (BCs) covers, as special cases, the interesting setting of (not necessarily linear) multi-point and integral BCs; there exists a wide literature on this topic, we refer the reader to the recent paper [11] and references therein.
The approach that we use to solve the BVP (1.1), in the line of the papers [7,8,10,19,20], consists in considering two auxiliary BVPs separately, the first one on the compact interval [0, R], where B has support and f is nonnegative, and the second one on the half-line [R, ∞), where f is allowed to change its sign. Unlike the above cited articles, in which the problem of gluing the solutions is solved with some continuity arguments and an analysis in the phase space, here both the auxiliary problems have the same slope condition in the junction point (namely the condition u ′ (R) = 0), which simplifies the arguments. This kind of decomposition is some sort of an analogue of one employed by Boucherif and Precup [2] utilized for equations with nonlocal initial conditions, where the associated nonlinear integral operator is decomposed into two parts, one of Fredholm-type (that takes into account the functional conditions) and another one of Volterra-type.
We make the following assumptions on the terms that occur in (1.1).
As we will show in Section 3 (see also Theorem 4.1), if the condition (1.6) is not satisfied, then our approach leads to the existence of a bounded non-negative solution of the differential equation in (1.1), namely a solution of the problem The problem of the existence and multiplicity of the solutions for the equation in (1.1), which are non-negative in the interval [0, R] and satisfy the functional BCs and the additional assumptions at u ′ (R) = 0, is considered in Section 2 and is solved by means of the classical fixed point index for compact maps. A BVP on [R, +∞) is examined in Section 3, where 2 we deal with the existence of positive global solutions which have zero initial slope and are bounded or tend to zero at infinity. This second problem is solved by using a fixed point theorem for operators defined in a Frechét space, by a Schauder's linearization device, see [5,Theorem 1.3], and does not require the explicit form of the fixed point operator, but only some a-priori bounds. These estimates are obtained using some properties of principal and nonprincipal solutions of auxiliary second-order linear equations, see [15,Chapter 11] and [9].
Finally, the existence and multiplicity of solutions for the BVP (1.1) and (1.7) is obtained in Section 4, thanks to the fact that the problem in [R, +∞) has at least a solution for every initial value u(R) sufficiently small. An example completes the paper.
2. An auxiliary BVP on the compact interval [0, R] In this Section we investigate the existence of multiple positive solutions of the BVP First of all we recall some results regarding the linear BVP It is known, see for example [18], that the Green's function k for the BVP (2.2) is given by and satisfies the inequality (see [18,Lemma 2.1]) Note that the constant function 1 α solves the BVP We associate to the BVP (2.1) the perturbed Hammerstein integral equation We seek fixed points of the operator T in a suitable cone of the space of continuous functions C[0, R], endowed with the usual norm w := max{|w(t)|, t ∈ [0, R]}.
We recall that a cone K in a Banach space X is a closed convex set such that λ x ∈ K for x ∈ K and λ ≥ 0 and K ∩ (−K) = {0}. In the following Proposition we recall the main properties of the classical fixed point index for compact maps, for more details see [1,12].
In what follows the closure and the boundary of subsets of a cone K are understood to be relative to K.
has the following properties: The assumptions above allow us to work in the cone a type of cone firstly used by Krasnosel'skiȋ, see [16], and D. Guo, see e.g. [12]. In (2.5) , with the restriction a > 0 when β = 0. Note also that the constant function equal to r ≥ 0 (that we denote, with abuse of notation r) belongs to K, Regarding the functional H we assume that • H : K → [0, +∞) is continuous and maps bounded sets in bounded sets.

4
With these ingredients it is routine to show that T leaves K invariant and is compact. We make use of the following open bounded set (relative to K) We now employ some local upper and lower estimates for the functional H, in the spirit of [13,14]. We begin with a condition which implies that the index is 1.
there exists ρ > 0, such that the following algebraic inequality holds: We prove that µ u = T u for every µ ≥ 1 and u ∈ ∂K ρ . In fact, if this does not happen, there exist µ ≥ 1 Then we obtain, for t ∈ [0, R], Taking the supremum for t ∈ [0, R] in (2.7) and using the inequality (2.6) we obtain µρ < ρ, a contradiction that proves the result.
Now we give a condition which implies that the index is 0 on the set K ρ .

Lemma 2.3. Assume that
(I 0 ρ ) there exists ρ > 0 such that the following algebraic inequality holds: Then i K (T, K ρ ) is 0. 5 Proof. Note that the constant function 1 belongs to K. We prove that u = T u + λ1 for every u ∈ ∂K ρ and for every λ ≥ 0. If this is false, there exist u ∈ ∂K ρ and λ ≥ 0 such that Using the inequality (2.8) in (2.9) we obtain ρ > ρ, a contradiction that proves the result.
In view of the Lemmas above, we may state our result regarding the existence of one or more nontrivial solutions. Here, for brevity, we provide sufficient conditions for the existence of one, two or three solutions. It is possible to obtain more solutions, by adding more conditions of the same type, see for example [17].
Theorem 2.4. The BVP (2.1) has at least one non-negative solution u 1 , with ρ 1 < u 1 (R) < ρ 2 if either of the following conditions holds.
Proof. Assume condition (S 1 ) holds, then, by Lemma 2.3, we have i K (T, K ρ 1 ) = 0 and, by Lemma 2.2, i K (T, K ρ 2 ) = 1. By Proposition 2.1 we obtain a solution u 1 for the integral and therefore u ′ 1 ≥ 0 in [0, R], which, in turn, implies u 1 (R) = u 1 . Assume now that condition (S 3 ) holds, then we obtain in addition that i K (T, K ρ 3 ) = 0. By Proposition 2.1 we obtain the existence of a second solution u 2 of the integral equation (2.4) in K ρ 3 \ K ρ 2 . A similar argument as above holds for the monotonicity of u 2 .
The remaining cases are dealt with in a similar way.
3. An auxiliary BVP on the half-line [R, ∞] In this section we state sufficient conditions for the existence of solutions of the following BVP where u 0 > 0 is a given constant. The BVP (3.1), (3.2) involves both initial and asymptotic conditions, and a global condition (i.e., the positivity on the whole half-line). The continuability at infinity of solutions of (3.1) is not a simple problem, see for example [4].
For instance, the Emden-Fowler equation The problem (3.1), (3.2) has been consider in [9] for nonlinear equations with p-laplacian operator and nonlinear term f (t, u(t)) = b(t)F (u(t)). We address the reader to such a paper for a complete discussion on the issues related to the BVP (3.1), (3.2) and for a review of the existing literature on related problems. The approach used in [9] to solve the BVP was where f is a continuous function on J ×R and S is a subset of C 1 (J, R). Let g be a continuous function on J × R 2 such that and assume that there exist a closed convex subset Ω of C 1 (J, R) and a bounded closed subset S 1 of S ∩ Ω which make the problem uniquely solvable for all q ∈ Ω. Then the BVP (3.4) has at least one solution in Ω.
In view of this result, no topological properties of the fixed-point operator are needed to be checked, since they are a direct consequence of a-priori bounds for the solutions of the "linearized" problem (3.5).
We state here the existence results in the form which will be used in the next section, addressing to [9,Theorem 2] for the general result, in case of a factored nonlinearity. For reader's convenience we provide a short proof, focusing only on those points which require some adjustments due to the more general nonlinearity. We point out that the present results are obtained by using the Euler equation for some n > 1, then for every u 0 ∈ (0, d/2] the equation (3.1) has a solution u, satisfying Proof. The result follows from [9, Theorem 2 and Corollary 3], with some technical adjustments due to the actual general form of the nonlinearity. Indeed, it is sufficient to observe that, for every continuous function q : [R, ∞) → (0, d] fixed, the equations is a Sturm minorant of the linearized equation and (3.6) is a Sturm majorant, due to (3.8) (see for instance [15]). Since Thus, put We have S ∩ Ω = Ω and the set S 1 = T (Ω), where T is the operator which maps every q ∈ Ω into the unique solution of (3.10) satisfying the initial conditions u(R) = u 0 , u ′ (R) = 0, satisfies S 1 ⊂ S ∩ Ω = Ω and is bounded in C 1 [R, +∞) (for the detailed proof see [9, Theorem 2]). Then Proposition 3.1 can be applied, and the existence of a solution of (3.1) in the set S ∩ Ω is proved.  Proof. The proof is analogous to the second part of the proof of [9, Theorem 2], with obvious modifications due to the more general form of the nonlinear term here considered. In particular notice that (3.1) with conditions (1.3) gives the inequality for every positive solution u of (3.1) and all t ≥ R. Thus the arguments in the proof of [9, Theorem 2] apply also to the present case.
We conclude this Section pointing out that the case b 1 (t) ≥ 0 for t ≥ R is included in the previous results, and in this case Theorems 3.2, 3.5 have a more simple statement. Indeed, B − 1 = 0, and therefore (1.5) and (3.7) are trivially satisfied. Further, every solution of (3.1) is nonincreasing on the whole half-line.

The main result
Combining Theorems 2.4 and 3.2 or 2.4 and 3.5, we obtain an existence result for one or more solutions of the BVP (3.1) and (3.2), respectively. We limit ourself to state results for the existence of one or two solutions, for sake of simplicity. Clearly, as pointed out in Section 2, adding more conditions, with similar arguments it is possible to obtain sufficient conditions for the existence of three solutions (see Theorem 2.4) or more solutions. Notice that, from Theorem 2.4, the solutions u 1 , u 2 satisfy ρ 1 < u 1 (R) < ρ 2 < u 2 (R) < ρ 3 and therefore they are distinct solutions. Further, since solutions on [0, R] are increasing, In case also assumption (1.6) is satisfied, from the above Theorem we obtain sufficient conditions for the existence of solutions of the BVP (1.1).