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 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 ≥ 0.
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 a 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 many works in the literature, see for instance, [5,8,11] (for second and fourth order Ordinary Differential Equations) or [1,4,6,7] (for Fractional equations) and references therein.
The paper is organized as follows: in Section 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 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 • K + K ⊂ K, λK ⊂ K for all λ ≥ 0 and K ∩ (−K) = {0}.

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 [4] 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 where Proof. By using the Fundamental Theorem of Integral Calculus to the equation (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 that

Thus,
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.
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 expression (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 that s ≥ ts, we conclude that Moreover, On the other hand, since h(t, s) is a non-negative, 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).

Theorem 6
Let λ = m sin m 1−cos m , m > 0, m = 2 k π, k = 1, 2, . . ., and σ ∈ C([0, 1]). Then problem (9), (2) has a unique solution u ∈ C 2 ([0, 1]), which is given by Here, if m = k π, k ∈ N odd, and, if m = k π, for some k ∈ N odd, 2λ sin(πks) cos(πkt) + sin(πkt)(λ(− cos(πks)) − πk sin(πks) + λ) 2πkλ and Proof. It is immediate to verify that the spectrum of problem (9), (2) is given by the following pairs on the plane (m, λ): Consider the case m = k π, k ∈ N odd, and let v be the unique solution of and w be defined as the unique solution of Using [3] we have that It is immediate to verify that Thus, Denoting A = 1 0 u(s)ds, we deduce from the previous expression that As a consequence, it follows that 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 = k π, with k ∈ N odd, follow by tacking the limit of the expressions of G 1 m and G 2 m when m goes to k π. 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 = k π, 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  1]. Therefore, the distance between two zeros of any non-trivial solution must be greater than 1.
Suppose that there exists t 0 ∈ (0, 1) such that G m (t 0 , s) = 0. We distinguish the following cases: • If t 0 ∈ [0, s), since G m (0, s) = 0, we would have two zeros that distances less than 1, which is not possible.
• Now suppose t 0 = s, in this case we can continuously extend G m (t, s) to the interval [0, s], which brings us back to the previous case.

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 made 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.

Lemma 12 Let G m be the Green's function associated with problem (13), (2), given by expression (15). Then for all λ = m sinh m
cosh m−1 , m > 0, the following properties hold:

Nonlinear Problem
This section is devoted to prove 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's fixed point Theorem 2 to the operator T γ : C([0, 1]) → C([0, 1]) defined as Here, G γ corresponds to function G, given by (4), if γ = 0, G m given by (10) if γ = m 2 > 0, and G m given by (15) 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 In the sequel, we introduce the cone K γ ⊂ X, depending on the sign of the real parameter γ.
So, we arrive at the following existence result.
Theorem 14 Let us consider problem (1)- (2), and let ∆ : (−∞, π 2 ) → R be the function defined as (see Figure 1): Suppose further that (f ) holds and one of the two following conditions is fulfilled: (i) (sublinear case) f 0 = ∞ and f ∞ = 0. So, for all γ < π 2 and 0 < λ < ∆(γ) there is a positive solution of problem Proof. Consider, in a first moment, the case γ = m 2 > 0. Let's first see that T : K γ → K γ is a compact operator. Since G m and f are continuous and non-negative in their domain of definition, then 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 that, 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 Ω ⊂ K γ bounded, that is, there exists M ∈ R, M > 0 such that ||u|| ≤ M , for all u ∈ Ω. Let's define Then for all u ∈ Ω and t ∈ [0, 1], we have that As a consequence, ||T γ u|| ≤ N, On the other hand, for each u ∈ Ω and t ∈ [0, 1] we have The regularity of the Green's function allows us to ensure that N ∈ R, N > 0. Thus, for all t 1 , t 2 ∈ [0, 1], t 1 < t 2 , the following inequality is fulfilled Thus, by virtue of the Arzelà-Ascoli Theorem, we deduce that the set T γ (Ω) is relatively compact, i.e., T : K γ → K γ is a compact operator.
Taking limits, it is easy to check that f 0 = ∞ and f ∞ = 0, i.e., we are in the sublinear case. Since f fulfills (f ), we are in the hypotheses of the case (i) Theorem 14, so we can ensure the existence of a positive solution for problem (1)- ( 2).
Again, taking limits we see that f 0 = 0 and f ∞ = ∞, that is, we are in the superlinear case. As f verifies (f ), the second part of Theorem 14 guarantees the existence of a positive solution of problem (1)- (2).