Existence of solutions for some nonlinear problems with boundary value conditions

In this paper we study the existence of solutions for nonlinear boundary value problems ({\phi}(u' ))' = f(t,u,u'), l(u,u')=0 where l(u,u') =0 denotes the Dirichlet or mixed conditions on [0, T], {\phi} is a bounded, singular or classic homeomorphism such that {\phi}(0)=0, f(t,x,y) is a continuous function, and T a positive real number. All the contemplated boundary value problems are reduced to finding a fixed point for one operator defined on a space of functions, and Schauder fixed point theorem or Leray-Schauder degree are used.


Introduction
The purpose of this article is to obtain some existence results for nonlinear boundary value problems of the form (ϕ(u )) = f (t, u, u ) l(u, u ) = 0, (1.1) where l(u, u ) = 0 denotes the Dirichlet or mixed boundary conditions on the interval [0, T ], ϕ is a bounded, singular or classic homeomorphism such that ϕ(0) = 0, f : [0, T ] × R × R → R is a continuous function, and T a positive real number.
The main purpose of this section is an extension of the results obtained in the previous theorem. For this, we use topological methods based upon Leray-Schauder degree [10] and more general properties of the function f . In Section 4, we use the fixed point theorem of Schauder to show the existence of at least one solution for boundary value problems of the type (ϕ(u )) = f (t, u, u ) u(T ) = u(0) = u (T ).
where ϕ : (−a, a) → R (we call it singular). We call solution of this problem any function u : [0, T ] → R of class C 1 such that max [0,T ] u (t) < a, satisfying the boundary conditions and the function ϕ(u ) is continuously differentiable and (ϕ(u (t))) = f (t, u(t), u (t)) for all t ∈ [0, T ]. In Section 5, for u(T ) = u (0) = u (T ) boundary conditions and classic homeomorphisms (ϕ : R → R), we investigate the existence of at least one solution using Leray-Schauder degree, where a solution of this problem is any function u : [0, T ] → R of class C 1 such that ϕ(u ) is continuously differentiable, which satisfies the boundary conditions and (ϕ(u (t))) = f (t, u(t), u (t)) for all t ∈ [0, T ]. Such problems do not seem to have been studied in the literature. In the present paper generally we follow the ideas of Bereanu and Mawhin [1,2,3,4,5,6].

Notation and preliminaries
We first introduce some notation. For fixed T , we denote the usual norm in L 1 = L 1 ([0, T ] , R) for · L 1 . For C = C([0, T ] , R) we indicate the Banach space of all continuous functions from [0, T ] into R witch the norm · ∞ , C 1 = C 1 ([0, T ] , R) denote the Banach space of continuously differentiable functions from [0, T ] into R endowed witch the usual norm u 1 = u ∞ + u ∞ and for C 1 0 we designate the closed subspace of C 1 defined by C 1 0 = u ∈ C 1 : u(T ) = 0 = u(0) . We introduce the following applications: the Nemytskii operator N f : the following continuous linear applications: For u ∈ C, we write The following lemma is an adaptation of a result of [4] to the case of a homeomorphism which is not defined everywhere. We present here the demonstration for better understanding of the development of our research.
Moreover, the function Q ϕ : B → R is continuous and sends bounded sets into bounded sets.
Proof. Let h ∈ B. We define the continuous application Using the injectivity of ϕ −1 we deduce that r = s. Let us now show the existence. Because ϕ −1 is strictly monotone and ϕ −1 (0) = 0, we have that It follows that there exists s ∈ [h m , h M ] such that G h (s) = 0. Consequently for each h ∈ B, the equation (2.2) has a unique solution. Thus, we define the function On the other hand, because h ∈ B, we have that Therefore, the function Q ϕ sends bounded sets into bounded sets.
Finally, we show that Q ϕ is continuous on B. Let (h n ) n ⊂ C be a sequence such that h n → h in C. Since the function Q ϕ sends bounded sets into bounded sets, then (Q ϕ (h n )) n is bounded. Hence, (Q ϕ (h n )) n is relatively compact. Without loss of generality, passing if necessary to a subsequence, we can assume that where for each n ∈ N we obtain Using the dominated convergence theorem, we deduce that T 0 ϕ −1 (h(t) − a)dt = 0, so we have that Q ϕ (h) = a. Hence, the function Q ϕ is continuous.
The following extended homotopy invariance property of the Leray-Schauder degree, can be found in [9].

Dirichlet problems with bounded homeomorphisms
In this section we are interested in Dirichlet boundary value problems of the type Let Clearly Ω is an open set in [0, 1] × C 1 0 , and is nonempty because {0} × C 1 0 ⊂ Ω. Using Lemma 2.1, we can define the operator M : Here ϕ −1 with an abuse of notation is understood as the operator . It is clear that ϕ −1 is continuous and sends bounded sets into bounded sets. When the boundary conditions are periodic or Neumann, an operator has been considered by Bereanu and Mawhin [5].
The following lemma plays a pivotal role to study the solutions of the problem (3.4).
where the continuity of M (λ, u) and (M (λ, u)) follows from the continuity of the applications H and N f .
On the other hand using Lemma 2.1, we have Therefore M (Ω) ⊂ C 1 0 and M is well defined. The continuity of M follows by the continuity of the operators which compose it M . Now suppose that (λ, u) ∈ Ω is such that M (λ, u) = u. It follows from (3.5) that Applying ϕ to both of its members and differentiating again, we deduce that for all t ∈ [0, T ]. Thus, u satisfies problem (3.4). This completes the proof.
Remark 3.2. Note that the reciprocal of Lemma 3.1 is not true because we can not guarantee that λH(N f (u) ∞ < a/2 for u solution of (3.4) In our main result, we need the following lemma to obtain the required a priori bounds for the possibles fixed points of M .
Proof. Let λ = 0 and (λ, u) ∈ Ω be such that M (λ, u) = u. Using Lemma 3.1, we have that u is solution of (3.4), which implies that where for all t ∈ [0, T ], we obtain On the other hand, because ϕ is a homeomorphism such that Using the integration by parts formula, the boundary conditions and the fact that Since λ ∈ (0, 1] and u is solution of (3.4), we have that T 0 f (t, u(t), u (t))n(u(t))dt ≤ 0, and hence On the other hand, since that Q ϕ (λH(N f (u))) ∈ Im(λH(N f (u))), we get Using again the boundary conditions, we have that Finally, if u = M (0, u), then u = 0, so the proof is complete.
Let ρ, κ ∈ R be such that h L 1 < κ < a/2, ρ > L + LT and consider the set On the other hand using an argument similar to the one introduced in the proof of Lemma 6 in [5], it is not difficult to see that M : V → C 1 0 is well defined, completely continuous and

Existence results
In this subsection, we present and prove our main result.
for all x, y ∈ R and t ∈ [0, T ]. If (λ, u) ∈ Ω is such that M (λ, u) = u, then Proof. Since ϕ is an increasing homomorphism we have that ϕ(y)y ≥ 0 for all y ∈ R. Using Lemma 3.3 with n(x) = x for all x ∈ R, we can obtain the conclusion of Corollary 3.6. The proof is achieved. Let us give now an application of Theorem 3.7 when f is unbounded.
x, y) = x − 2 and h(t) = 4. Using Theorem 3.7, we obtain that the problem has at least one solution if T < 1/8.
Composing with the function ϕ −1 , we obtain where c = ϕ(u(0)). Integrating from 0 to t ∈ [0, T ], we have that Using an argument similar to the introduced in Lemma 2.1, it follows that c = −Q ϕ (K (N f (u))). Hence, Let u ∈ C 1 be such that u = M (u). Then N f (u)))] dt = 0, therefore, we have that u(0) = u(T ). Differentiating (4.7), we obtain that In particular, Applying ϕ to both members and differentiating again, we deduce that   Next, we show that M (Λ) ⊂ C 1 is a compact set. Let (v n ) n be a sequence in M (Λ), and let (u n ) n be a sequence in Λ such that v n = M (u n ). Using (4.8), we have that there exists a constant L > 0 such that, for all n ∈ N, which implies that Hence the sequence (K(N f (u n )) − Q ϕ (K(N f (u n )))) n is bounded in C. Moreover, for t, t 1 ∈ [0, T ] and for all n ∈ N, we have that which implies that (K(N f (u n )) − Q ϕ (K(N f (u n )))) n is equicontinuous. Thus, by the Arzelà-Ascoli theorem there is a subsequence of (K(N f (u n )) − Q ϕ (K(N f (u n )))) n , which we call (K(N f (u n j )) − Q ϕ (K(N f (u n j )))) j , which is convergent in C. Using that that the sequence (M (u n j ) ) j is convergent in C. Then, passing to a subsequence if necessary, we obtain that (v n j ) j = (M (u n j )) j is convergent in C 1 . Finally, let (v n ) n be a sequence in M (Λ). Let (z n ) n ⊆ M (Λ) be such that lim n→∞ z n − v n 1 = 0.
Let (z n j ) j be a subsequence of (z n ) n such that converge to z. It follows that z ∈ M (Λ) and (v n j ) j converge to z. This concludes the proof.
The next result is based on Schauder's fixed point theorem. Proof. Let u ∈ C 1 . Then Moreover, Hence, Because the operator M is completely continuous and bounded, we can use Schauder's fixed point theorem to deduce the existence of at least one fixed point. This, in turn, implies that problem (4.6) has at least one solution. The proof is complete.

Problems with classic homeomorphisms and tree-point boundary conditions
We finally consider boundary value problems of the form where ϕ : R → R is a homeomorphism such that ϕ(0) = 0 and f : [0, T ] × R × R → R is a continuous function. We remember that an solution of this problem is any function u : [0, T ] → R of class C 1 such that ϕ(u ) is continuously differentiable, satisfying the boundary conditions and (ϕ(u (t))) = f (t, u(t), u (t)) for all t ∈ [0, T ]. Let us consider the operator Analogously to the section 3, here ϕ −1 is understood as the operator ϕ −1 : C → C defined for ϕ −1 (v)(t) = ϕ −1 (v(t)). It is clear that ϕ −1 is continuous and sends bounded sets into bounded sets.
Lemma 5.1. u ∈ C 1 is a solution of (5.9) if and only if u is a fixed point of the operator M 1 .
Proof. Let u ∈ C 1 , we have the following equivalences: .
Using an argument similar to the introduced in Lemma 4.2, it is easy to see that, M 1 : C 1 → C 1 is completely continuous.
In order to apply Leray-Schauder degree to the operator M 1 , we introduced a family of problems depending on a parameter λ. We remember that to each continuous function f : Using the same arguments as in the proof of Lemma 4.2 we show that the operator M is completely continuous. Moreover, using the same reasoning as above, the system (5.10) (see Lemma 5.1) is equivalent to the problem u = M (λ, u).

Existence results
In this subsection, we present and prove our main results. These results are inspired on works by Bereanu and Mawhin [5] and Manásevich and Mawhin [8]. We denote by deg B the Brouwer degree and for deg LS the Leray-Schauder degree, and define the mapping G : Assume that Ω is an open bounded set in C 1 such that the following conditions hold.

The equation
has no solution on ∂Ω∩R 2 , where we consider the natural identification (a, b) ≈ a + bt of R 2 with related functions in C 1 .

The Brouwer degree
Then problem (5.9) has a solution.
The following result gives a priori bounds for the possible solutions of (5.11), adapts a technique introduced by Ward [11]. Theorem 5.4. Assume that f satisfies the following conditions.
Theorem 5.5. Let f be continuous and satisfy condition (1) and (2) of Theorem 5.4. Assume that the following conditions hold for some ρ ≥ r(2 + T ).

The equation
G(a, b) = (0, 0), has no solution on ∂B ρ (0) ∩ R 2 , where we consider the natural identification (a, b) ≈ a + bt of R 2 with related functions in C 1 .

The Brouwer degree
Then problem (5.9) has a solution.