First order differential systems with a nonlinear boundary condition via the method of solution-regions

In this article we extend the known theory of solution regions to encompass nonlinear boundary conditions. We both provide results for new boundary conditions and recover some known results for the linear case.


Introduction
In this paper, we study systems of differential equations with nonlinear boundary conditions of the form: where f : [0, T] × R n → R n is a Carathéodory function and L : R 2n × C(I, R n ) → R n is continuous.
To our knowledge, few results can be found in the literature in the case where n > 1 and L is nonlinear or does not depend only on the values at the boundary, u(0) and u(T). In [17] and [18], this problem was treated with linear integral boundary conditions. Results were obtained in the case where f satisfies a growth condition or a condition of contraction type.
The problem (1.1) in the case of a single differential equation (n = 1) and L nonlinear of the form L(x, y, u) = L(x, y) was studied by [1,2,4,15]. Existence results were obtained with the method of upper and lower solutions and under monotonicity conditions imposed on L.
In a recent work of Frigon [7], the concept of solution-region was introduced for the first time in order to obtain results concerning the existence and multiplicity of solutions of the system of differential equations (1.1) in the particular case of the initial condition This method of solution-region was extended by Tojo in [20] to treat more general linear boundary conditions such as This new method generalizes various means of obtaining existence and multiplicity of solutions of differential problems, such as the methods of upper and lower solutions [1,2,4,9,15,16], strict upper and lower solutions [13,19], and solution-tubes [3,6,8,10]. Furthermore, the method is closely related to that of Gaines and Mawhin concerning what they called bound sets [11,12]. The theory of bound sets was developed for the case where f is a continuous map and their existence results are obtained for bound sets not depending on t.
In this paper, we improve the works [7,20] by studying systems of differential equations with nonlinear boundary conditions such as (1.1). The results obtained recover, in most cases, the results in [7,20] for the case of linear boundary conditions. We also sharpen the definition and requirements for a set to be considered a solution-region (see Definition 3.1) for the case of nonlinear boundary conditions when compared to the linear one present in the literature (cf. [7,20]). It is worthwhile to mention that no monotonicity or growth conditions will be imposed on f and L.
The paper is structured as follows. In Sect. 2 we deal with some preliminaries that we will use afterwards. Section 3 is concerned with the main results of this work and contains detailed applications that render already known results, thus showing the generality of our approach. Finally, in Sect. 4, we draw some conclusions regarding future work.

Preliminaries
Throughout this paper, we consider R n with the Euclidean norm · , C( , R n ), the space of continuous functions endowed with supremum norm · ∞ , and L 1 (I, R) is the space of Lebesgue integrable functions endowed with the usual norm · 1 , where I = [0, T] and is some set. We consider also the Sobolev space W 1,1 (I, R n ) and, for J ⊂ I, the following set of locally absolutely continuous maps: We recall the notion of locally Carathéodory functions introduced in [7].
It is well known that a completely continuous operator is associated to a Carathéodory map.

Lemma 2.2 ([5])
Let g : I × R n → R n be a Carathéodory function and N g : Then, N g is continuous and completely continuous.
The following comparison result will be useful to ensure that solutions are in a given set. In order to present the problem of study, we recall the notion of an admissible region introduced in [7].

Definition 2.4 ([7])
We say that a set R ⊂ I × R n is an admissible region if there exist two continuous maps h : I × R n → R and p = (p 1 , p 2 ) : I × R n → I × R n satisfying the following conditions: x) ≤ 0} is bounded and, for every t ∈ I, R t = {x ∈ R n : (t, x) ∈ R} = ∅; (H2) the map h has partial derivatives at (t, x) for almost every t and every x with (t, x) ∈ R c := (I × R n )\R, and ∂h ∂t , ∇ x h are locally Carathéodory maps on R c ; (H3) p is bounded and such that p(t, x) = (t, x) for every (t, x) ∈ R and ∇ x h(t, x), p 2 (t, x)x < 0 for a.e. t and every x with (t, x) ∈ R c .
We call (h, p) an admissible pair associated to R.
In [20], a weaker notion of admissible region was considered where, in condition (H3), the inequality is taken to be non-strict. Definition 2. 5 We say that a set R ⊂ I × R n is a weak admissible region if there exist two continuous maps h : I × R n → R and p = (p 1 , p 2 ) : I × R n → I × R n satisfying (H1) and (H2) of Definition 2.4 and (H3)' p is bounded and such that p(t, x) = (t, x) for every (t, x) ∈ R and We call (h, p) a weak admissible pair associated to R.

Solution-regions and nonlinear boundary conditions
We consider systems of first order differential equations of the form: subject to boundary conditions of the form u ∈ B where B denotes one of the following boundary conditions: where f : [0, T] × R n → R n is a Carathéodory function and L : R 2n × C([0, T], R n ) → R n is continuous but not necessarily linear. The boundary conditions (3.2) and (3.3) generalize those considered in [7]. We will be interested in giving conditions ensuring the existence of a solution of (3.1) in a suitable weak admissible region. To this end, we introduce the notion of solution-regions of (3.1). Definition 3.1 A set R ⊂ I × R n is called a solution-region of (3.1) if it is a weak admissible region with an associated weak admissible pair (h, p) satisfying the following conditions: (i) for almost every t and every x with and one of the following conditions: Now we show that the existence of a solution-region ensures the existence of a solution of (3.1). Theorem 3.2 Let f : I × R n → R n be a Carathéodory function. Assume that there exists a solution-region R of (3.1). Then, problem (3.1), (3.2) has a solution u ∈ W 1,1 (I, R n ) such that (t, u(t)) ∈ R for every t ∈ I.
Proof Let (h, p) be a weak admissible pair associated to the solution-region R. For λ ∈ [0, 1], we consider the following family of problems: Observe that such a function c exists since f is Carathéodory and p is a bounded map. Let us consider the operators I 0 : where N f R is defined in (2.1). Since L is continuous and f is Carathéodory, using (H3)' of Definition 2.5, we deduce from Lemma 2.2 that I is continuous and completely continuous.
We claim that the fixed points of I are solutions of (3.5 λ ). Indeed, if u = I(λ, u), In particular, one has, for t = 0, and Thus, u is a solution of (3.5 λ ).
We present an example of application of the previous theorem in which there are no monotonicity assumptions or growth conditions imposed on the right-hand side term of (3.1) or in (3.2). Example 3. 3 We consider the following system of differential equations: We consider the closed and bounded set It is easy to verify that R is an admissible region with the associated admissible pair (h, p).
As a corollary of the previous theorem, we obtain an existence result established in [7] for the initial value problem.

Corollary 3.4
Let R ⊂ I × R n be a weak admissible region, r ∈ R n and f : I × R n → R n be a Carathéodory function. Assume that there exists (h, p) a weak admissible pair of R satisfying (i) of Definition 3.1 and h(0, r) ≤ 0. Then, (3.1) has a solution u ∈ W 1,1 (I, R n ) such that (t, u(t)) ∈ R for every t ∈ I, and u(0) = r.
Here is another corollary of Theorem 3.2 in which a solution of (3.1), (3.2) with the periodic boundary condition is obtained.
Observe that, for L(x, y, u) = xy, the last corollary does not recover the results in [7] for the periodic problem because of condition (ii). The next theorem will show that (ii) of Corollary 3.7 is not necessary if R is an admissible region or if the inequality in Definition 3.1(i) is strict (see Definition 3.1(ii)'(b)). Theorem 3.9 Let f : I × R n → R n be a Carathéodory function. Assume that there exists R, a solution-region of (3.1) with the boundary condition (3.3). Then, the problem (3.1), (3.3) has a solution u ∈ W 1,1 (I, R n ) such that (t, u(t)) ∈ R for every t ∈ I.
Proof Let (h, p) be a weak admissible pair associated to the solution-region R. For λ ∈ [0, 1], we consider the following family of problems: Observe that, integrating the equation in (3.16 λ ) between 0 and T, we have that which, combined with the boundary conditions in (3.16 λ ), yields Let us consider the operator P : [0, 1] × C(I, R n ) → C(I, R n ) defined by where N f R is defined in (2.1). As in the proof of the previous theorem, we deduce that P is continuous and completely continuous.
We claim that the fixed points of P are solutions of (3.16 λ ). Indeed, if u = P(λ, u), then, for every t ∈ I, In particular, for t = 0 and for t = T, we have Thus, 20) and (3.18) holds. From (3.18) and (3.19), we deduce that, for almost every t ∈ I, Thus, u is a solution of (3.16 λ ).
Let M and m be as in (3.8) and ( If k > m > 1 + p 2 ∞ ≥ m 0 + 1, by (3.17), Taking norms on both sides of this equality and combining it with (3.7) and (3.10), we obtain which is a contradiction, so u(t) = k ≤ m < M for every t ∈ I.
By (3.17) and (3.20), we have that either u(0) ≤ m or m < u(0) = u(T) . Taking this into account, it follows from Lemma 2.3 that u(t) ≤ m for every t ∈ I since u cannot be constant and greater than m. Hence, u ∞ < M for any solution u of (3.16 λ ). Observe that We claim that x = P 0 (λ, x) for every (λ, x) ∈ [0, 1] × ∂B R n (0, M). Indeed, for λ = 0 this fact is evident and, if λ ∈ (0, 1], assume that there exists x ∈ R n such that x = M and x = P 0 (λ, x). By (3.6), (3.7), (3.8), and (3.17), it satisfies So, taking norms on both sides, which is a contradiction. The homotopy property of the fixed point index implies that If h(0, u(0)) > 0, we deduce from Definition 3.1(ii)'(a) that h(0, u(0)) ≤ h(T, u(T)). Moreover, by Definition 3.1(ii)'(b), this last inequality is strict or the inequality (3.25) is strict on a subset of positive measure. This prevents the existence of k > 0 such that h(t, u(t)) = k for every t ∈ I. Thus, Lemma 2.3 implies that (t, u(t)) ∈ R for every t ∈ I. Therefore, u is a solution of (3.1), (3.3) since p = Id on R and L R (u(0), u(T), u) = L(u(0), u(T), u).

Conclusions
In this work, we have extended the known theory of solution regions to encompass nonlinear boundary conditions and, at the same time, we have recovered some known results for the linear case from our generalization. This was achieved by refining the definition of solution-region for the case of a weak admissible region and nonlinear boundary conditions. Nonetheless, some of the requirements in the definition (see Definition 3.1(ii)'(b)) arise from the fact that in the proof of Theorem 3.9 (see last paragraph) the properties of a weak admissible region are not enough to conclude, something which was indeed the case when solutions regions where defined from admissible regions (cf. [7,Theorem 5.1]). This raises the question as to whether or why consider weak admissible regions altogether. This laxer definition appeared in [20] in order to prove the following result. This way, weak admissible regions are characterized in a simple topological way, which simplifies the application of the theory. In [20], it was already conjectured that, in fact, all weak admissible regions are admissible regions for an adequate admissible pair. Unfortunately, to the best of our knowledge, there is no proof of that fact as of today. Such a proof would simplify the theory since we could profit from a simple characterization and, at the same time, a simpler definition of solution regions, so it is a result to look forward to.