Nonlinear differential equations with perturbed Dirichlet integral boundary conditions

This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation of the Dirichlet conditions by considering a homogeneous Dirichlet-type condition at one extreme of the interval and an integral operator on the other one. We obtain the expression of the Green’s function related to the linear part of the equation and characterize its constant sign. Such a property will be fundamental to deduce the existence of solutions of the nonlinear problem. The results hold from fixed point theory applied to related operators defined on suitable cones.


Introduction
This paper is devoted to the study of the existence of solutions of the following family of nonlinear second order ordinary differential equations: coupled to the following integral boundary conditions: where γ < π 2 and λ ∈ R.
To this end, we will distinguish the cases γ = 0, γ > 0 and γ < 0. We will analyze each of them and give optimal sufficient conditions on γ , λ and f that allow us to ensure the existence of a solution of the considered problem.
This kind of problems model the behavior of an harmonic oscillator, subject to an external force f , which is fixed at the left extreme of the interval and has some mechanism at the right one, that controls the displacement according to the feedback from devices measuring the displacements along parts of the oscillator. Integral boundary conditions have been considered in much work in the literature; see, for instance, [6,9,12] (for second and fourth order ordinary differential equations) or [1,5,7,8] (for fractional equations) and the references therein.
The paper is organized as follows: in Sect. 2, we study the linear part of problem (1)-(2), where we obtain the explicit expression of the related Green's function and calculate the exact values of γ and λ for which the Green's function has constant sign. In the next section we prove the existence of positive solutions for the nonlinear problem (1)- (2). Such solutions are given as the fixed points of a related integral operator defined on a suitable cone. At the end of this section we show two examples where the applicability of the obtained results is pointed out.
The following concept will be fundamental in order to deduce our existence results.
Definition 1 Let X be a Banach space. A subset K ⊂ X is a cone if: • K is closed, We will use the celebrated expansion/contraction theorem of Krasnosels'kiȋ [10].
Theorem 2 (Krasnosels'kiȋ) Let X be a Banach space and K ⊂ X a cone in X. Let 1 , 2 ⊂ X open bounded such that 0 ∈ 1 ⊂ 1 ⊂ 2 and T : K ∩ ( 2 \ 1 ) → K a compact operator that satisfies one of the following properties: Then T has a fixed point at K ∩ ( 2 \ 1 ).

Linear part: Green's function
In this section we obtain the expression of the related Green's function of the linear part of problem (1)-(2) and deduce some important properties that will be fundamental to obtain the existence of positive solutions of the nonlinear problem. To this end, we consider separately three cases depending on the sign of the real parameter γ . To do this, we will follow a treatment of a similar problem studied in [5] for fractional equations.

Case γ = 0
In this subsection, we obtain the expression of the Green's function related to the linear problem coupled to the boundary conditions (2).

Theorem 3
Let λ = 2 and σ ∈ C([0, 1]), then problem (3), (2) has a unique solution u ∈ C 2 ([0, 1]), which is given by the following expression: Proof By using the fundamental theorem of integral calculus to Eq. (3), together with Fubini's theorem, we arrive at the following expression: Since u(0) = 0, we deduce that c 2 = 0. Now, the boundary condition at t = 1 implies that As a consequence By denoting A = 1 0 u(s) ds, we have Thus, Then Substituting this value in (5), we obtain the expression of the solution u as follows: In the sequel, we will state two lemmas related to the properties of the Green's function that will be useful to prove the existence of a positive solution of the nonlinear problem (1)-(2) with γ = 0.
Lemma 4 Let G be the Green's function related to problem (3), (2), given by Eq. (4). Then, for all λ = 2, the following properties are fulfilled: 5. The result is immediately deduced from the following equality: 6. Since G(t, s) is linear on t, for all s ∈ [0, 1] fixed, G(t, s) attains its maximum and minimum at t = 0, t = s or at t = 1. From (7) it is clear that G(s, s) > 0 for all s ∈ (0, 1) and λ ≤ 0 and, moreover, if λ > 2, we have G(s, s) > 0 for s ∈ (0, 1) close enough to 0. Thus, this property is fulfilled. 8. From Property 6., we know that G(t, s) > 0 for all (t, s) ∈ (0, 1) × (0, 1). As in Property 6., we know that the maximum values will be attained at G(s, s) and/or G(1, s). Now, since the proof is concluded.
In the sequel, we deduce two sharp inequalities for the positiveness of the Green's function.

Lemma 5
Let λ ∈ (0, 2) and G(t, s) be the Green's function related to problem (3), (2), given by Eq. (4). Then the following properties hold: Proof For t = 0 t = 1, s = 0 or s = 1 the inequalities follow immediately from Properties 1. and 4. in Lemma 4. Now let t, s be such that 0 < t ≤ s < 1. In this case As a consequence, we have Consider now the case 0 < s ≤ t < 1, using s ≥ ts, we conclude that Moreover, On the other hand, since h(t, s) is a nonnegative, continuous and linear function with respect to t, we deduce From Property 4. in Lemma 4, we conclude that the inequalities (8) are satisfied.

Case γ > 0
In this subsection, we will obtain the expression of the Green's function related to the problem coupled to the boundary conditions (2).
. Then problem (9), (2) has a unique solution u ∈ C 2 ([0, 1]), which is given by Here, if m = kπ , k ∈ N odd, and, if m = kπ , for some k ∈ N odd, Proof It is immediate to verify that the spectrum of problem (9), (2) is given by the following pairs on the plane (m, λ): and w be defined as the unique solution of is a solution of (9), (2). In [3] there has been constructed and algorithm to calculate the exact expression of the Green's function related to any nth order differential equation, with constant coefficients coupled to arbitrary two-point linear boundary conditions. Such algorithm has been developed in a Mathematica package that is available in [4]. So, using this package, we obtain It is immediate to verify that Thus, Denoting A = Substituting this value in (11) we obtain the following expression: The uniqueness of the Green's function is deduced from the uniqueness of functions v and w. The expressions of the Green's function for m = kπ , with k ∈ N odd, follow by taking the limit of the expressions of G 1 m and G 2 m when m goes to kπ . By direct calculations, it is immediate to verify that such function satisfies the properties of the Green's function of problem (9), (2) with m = kπ , k ∈ N odd.
In the same way as in the case γ = 0, we will now state different results about the properties of the Green's function that we have just obtained. As before, we will try to give conditions that allow us to ensure that the Green's function attains a constant sign. To this end, we will use the following property, which is a direct consequence of the Sturm comparison theorem Remark 7 Since w(t) = sin(πt), is a nontrivial solution of w (t) + π 2 w(t) = 0, and it vanishes in Z, from the classical Sturm comparison result, we see that for γ < π 2 any nontrivial solution of the problem v (t) + γ v(t) = 0 vanishes at most once in the interval [0,1]. Therefore, the distance between two zeros of any nontrivial solution must be greater than 1. (9)  It is easy to check that if m ∈ (0, 2π) then function r m has a unique maximum at s = 1 2 and, since r m (0) = 0 = r m (1), we deduce that r m (s) > 0 for s ∈ (0, 1). Therefore for all s ∈ (0, 1), m ∈ (0, 2π) and λ > 0. 5. Assume now that 0 ≤ λ < m sin m 1-cos m , m ∈ (0, π]. Using Properties 1. and 4., we know that, for each s ∈ (0, 1), G m (0, s) = 0 and G m (1, s) ≥ 0. In addition,

Lemma 8 Let G m be the Green's function related to problem
Therefore, G m (t, s) is increasing and positive to the right of t = 0 for every s ∈ (0, 1).
• Now suppose t 0 = s. In this case, since ∂ 2 ∂t 2 G m (t, s)| t=s -exists and is finite, we can extend G m (·, s) as a C 2 function on the interval [0, s], which brings us back to the previous case. By direct computation, we have f m (0) = 1 for all λ ∈ R and m > 0, m = 2kπ , k = 1, 2, . . . . In consequence, G m (t, t) > 0 in a small enough neighborhood of (0, 0). Now we deduce the following stronger condition on the Green's function.

As a consequence,
from which the result follows.

Case γ < 0
Next, we will give the expression of the Green's function of the problem coupled to the boundary conditions (2). In this section we will omit most of the proofs because they are analogous to those presented in previous cases.
First, we will state a lemma that will be useful for the calculation of this function.
Proof In a similar way to Theorem 6 and using Lemma 10, we construct the Green's function taking into account that we can express the solution as where w(t) = sinh mt sinh m is the unique solution of the following problem: We will now enunciate some properties of G m . The proofs are analogous to those that have been presented in the two previous cases.

Nonlinear problem
This section is devoted to proving the existence of positive solutions on (0, 1) of problem (1)-(2). We will assume the following regularity condition for the nonlinear part of the equation: As in the previous section, we will distinguish three different cases depending on the sign of the parameter γ . The results hold from the application of the Krasnosel'skiȋ fixed point Theorem 2 to the operator Here, G γ corresponds to the function G, given by (4), if γ = 0, G m given by (10) if γ = m 2 > 0, and G m given by (16) if γ = -m 2 < 0.
As we have proved in previous section, we know that the fixed points of operator T γ coincide with the solutions of problem (1)- (2).
Now, we denote and In the sequel, we introduce the cone K γ ⊂ X, depending on the sign of the real parameter γ .
If γ = 0: If γ > 0: If γ < 0: Here h 1 and C 1 are given in Lemma 9, and h 2 and C 2 are given in Lemma 13. So, we arrive at the following existence result.
Proof Consider, to start with, the case γ = m 2 > 0.
Let us first see that T : K γ → K γ is a compact operator. Since G m and f are continuous and nonnegative in their domain of definition, we have T γ u ∈ C([0, 1]) and T γ u(t) ≥ 0, for all t ∈ [0, 1].
Let u ∈ K γ , using the properties stated in Lemmas 8 and 9, we have, for all t ∈ [0, 1], Furthermore, the continuity of the functions G m and f guarantees the continuity of the operator T : Let us now verify that the image by T γ of a bounded set is relatively compact. To this end, we will use the Arzelà-Ascoli theorem.
Let u ∈ K γ be such that u = ρ 2 , then using the definition of f and the above inequality we have     Finally, the first part of Theorem 2 implies that there is at least one positive solution of problem (1)-(2), with γ = m 2 > 0, u ∈ K γ , such that ρ 1 ≤ u ≤ ρ 2 .
From this, we deduce the following inequalities: