Upper and Lower Solutions for BVPs on the Half-line with Variable Coefficient and Derivative Depending Nonlinearity

This paper is concerned with a second-order nonlinear boundary value problem with a derivative depending nonlinearity and posed on the positive half-line. The derivative operator is time dependent. Upon a priori estimates and under a Nagumo growth condition, the Schauder’s fixed point theorem combined with the method of upper and lower solutions on unbounded domains are used to prove existence of solutions. A uniqueness theorem is also obtained and some examples of application illustrate the obtained results.


Introduction
In this work, we are concerned with the existence of solutions to the following boundary value problem x ′′ (t) − k 2 (t)x(t) + q(t)f (t, x(t), x ′ (t)) = 0, t > 0, x(0) = 0, x(+∞) = 0, ( where q ∈ C(0, +∞) ∩ L 1 (0, ∞) while the nonlinearity f : I × R × R −→ R and the coefficient k : I → (0, ∞) are continuous.Here I = (0, +∞) refers to the positive half-line.Since BVPs on infinite intervals arise in many applications from physics, chemistry and biology, there has been so much work devoted to the investigation of positive solutions for such BVPs in the last couple of years (see e.g., [2,3,6,16] and the references therein) where superlinear or sublinear nonlinearities are considered.The positivity of solutions is motivated by the fact that the unknown x may refer to a density, a temperature or the concentration of a product.For instance, the linear operator of derivation −x ′′ + cx ′ + λx (c, λ > 0), which may be rewritten in reduced form as −x ′′ + k 2 x, stems from epidemiology and combustion theory and models the propagation of the wave front of a reaction-diffusion equation (see e.g., [6]).Methods used to investigate these problems range from the upper and lower solution techniques [15,17] to the fixed point theory in weighted Banach spaces and the index fixed point theory on cones of some Banach spaces [1,5,16,18].
When k is constant, the existence of solutions to problem (1.1) was established in [14] using the Tychonoff's fixed point theorem.It was also studied by Djebali et al in [7,8,9] where multiplicity results have been also given.In [17], B. Yan et al have used the upper and lower solution techniques to obtain some existence results when f is allowed to have a singularity at x = 0 and may change sign.
In the general case when the constant k is replaced by a bounded function k = k(t), the problem was recently investigated by Ma and Zhu in [13].The nonlinearity f ∈ C(R + ×R + , R) is assumed to satisfy a sublinear polynomial growth condition.The authors of [13] proved that if the parameter λ is less that some λ 0 , then the following problem x ′′ (t) − k 2 (t)x(t) + λq(t)f (t, x(t)) = 0, t > 0, x(0) = 0, x(+∞) = 0, has a positive solution; a fixed point theorem in a cone of a Banach space has been employed.Their investigation relies heavily on estimates of the corresponding Green's function.In [11], the authors first applied fixed point index theory in cones of Banach spaces to prove existence results when f = f (t, x) is positive and may exhibit a singularity at the origin with respect to the solution; then they used the Schauder's fixed point theorem together with the method of upper and lower solutions to prove existence of solutions when f is not necessarily positive.
Using an upper and lower solution method on infinity intervals, the aim of this paper is to investigate the more general problem where the nonlinearity f = f (t, x, y) is derivative depending.In [12], Lian et al. used unbounded upper and lower solutions on noncompact intervals to prove an existence result for a class of BVPs.In [10], the authors considered problem (1.1) with q = 1 and used topological degree theory combined with the existence of C 1 B upper and lower solutions to prove existence of solutions on bounded intervals.Solutions are then extended to the positive half-line by means of sequential arguments.In the present paper, we complement these existence theorems via a direct approach.
This paper is organized as follows.Some preliminaries and definitions are given in Section 2. Then we will enunciate our assumptions in Section 3 and present a modified problem.In Section 4, bounded and unbounded upper and lower solutions will be established for problem (1.1) which allow us to prove correspondingly two existence results under a Nagumo type growth condition.The truncated problem is first studied.The proofs rely on suitable a priori estimates.A uniqueness result is also given in Section 5. Finally, we give two examples of application to illustrate our existence and uniqueness results.

Auxiliary Lemmas
Let us first enunciate an assumption regarding the function k: (H 0 ) the function k : I → [0, ∞) is bounded and continuous and where In order to construct a Green's function of the corresponding linear problem, it is necessary to know a fundamental system of solutions.The following auxiliary results are brought from [13].
Lemma 2.1.Assume that k is bounded and continuous.Then the Cauchy problem has a unique solution φ 1 defined on [0, +∞).Moreover φ 1 is nondecreasing and unbounded.
Lemma 2.2.(See also [1], Thm.7) Assume that k is bounded and continuous.Then the problem has a unique solution φ 2 defined on [0, +∞) with If further (H 0 ) holds, then Lemma 2.3.Assume that (H 0 ) holds.Then there exists M > 0 such that Lemma 2.4.Assume (H 0 ) holds.Then for any function is equivalent to the integral equation where • Moreover, we have Lemma 2.6.For any t ∈ (0, ∞), we have Proof.Since {φ 1 , φ 2 } is the fundamental system, we have that φ Then our claim follows from the sign and the monotonicity of φ 1 , φ 2 .
Lemma 2.7.We have Now we define what we mean by lower and upper solutions.

General Assumptions and a Modified Problem
We first posit some assumptions: (H 1 ) There exist α ≤ β lower and upper solutions of problem (1.1) respectively. .
(H 3 ) There exist continuous functions ψ : where D β α is defined by and for any y ∈ R and t ∈ (0, ∞), we have Now, define the Banach space The following compactness criterion will be needed (see [4], p. 62).
Given two continuous functions α and β such that α ≤ β, we define the truncated function f by where Proof.We prove that x(t) ≤ β(t), ∀ t ∈ I. Suppose, on the contrary that Moreover, by definition of an upper solution, we have the successive estimates: To check that the last right-hand term is nonnegative, we distinguish between two cases: (a) In case (H 4 ) holds, consider the sub-cases: Our claim is then proved leading to a contradiction.Similarly, we can prove that x(t) ≥ α(t) for every t ∈ [0, ∞).
Proof.Since solving problem (3.4) amounts to proving existence of a fixed point for T , let us consider the operator T : X −→ X defined by (a) T : X −→ X is well defined.Let x ∈ X.From (3.1) and (3.3), we get where From the monotonicity of φ 1 and φ ′ 1 together with Lemma 2.6, we obtain that For s ≤ t, we have Hence for any ε > 0, there exists N > 0 such that for t ≥ N , we have For s ≥ t, we have It follows that lim t→∞ (T x) ′ (t) = 0. EJQTDE, 2011 No. 14, p. 8 (b) T : X −→ X is continuous.Let (x n ) n∈N be a sequence converging to some limit x in X; then there exists r > 0 such that ||x|| ≤ r and ||x n || ≤ r.
Let H r = max 0≤t≤r h(t).We have and From continuity of f , (4.2) and the Lebesgue dominated convergence theorem, the last term goes to 0 as n → ∞.
(c) T : X −→ X is compact.Let B be any bounded subset of X and let x ∈ B. Then there exists r > 0 such that ||x|| ≤ r.First, notice that as above we have EJQTDE, 2011 No. 14, p. 9 Now, given T > 0 and t 0 , t 1 ∈ [0, T ], we have the estimates By (3.1), the continuity of the Green's function and the Lebesgue dominated convergence theorem, we get lim In addition, (3.1) and the continuity of φ 1 imply that lim Hence the right-hand term goes to 0 as |t 1 − t 0 | → 0.Moreover, for t 0 ≤ t 1 , the following estimates hold true EJQTDE, 2011 No. 14, p. 10 Consequently, each of the four terms above tends to 0 as |t 1 − t 0 | tends to 0, proving that T x is almost equicontinuous.
To prove equiconvergence, we first notice that lim t→∞ T x(t) = 0. Moreover from lim t→∞ φ 2 (t) = 0 and ∞ 0 q(s)(H r ψ(s) + 1)ds < ∞, for any ε > 0, there exists N > 0 such that for t ≥ N, the following estimates hold true: Furthermore, for any ε > 0, there exists N > 0 such that for t ≥ N EJQTDE, 2011 No. 14, p. 11 and As a consequence, for t ≥ N , we obtain the estimates Thus we have proved equiconvergence of T ending the proof that T is completely continuous.Finally, by the Leray-Schauder fixed point theorem, we deduce that T has at least a fixed point x, solution of problem (3.4).

The Original Problem
Theorem 4.2.Assume that either Assumptions (H 0 )−(H 4 ) or (H 0 )−(H 3 ) and (H 4 ) ′ hold.Then problem (1.1) has at least one solution x having the representation where G(t, s) is the Green's function defined in (2.3).
Then for t ≥ γ, we have which is a contradiction.Hence there exists Case 3.There exists an interval [t 0 , t 1 ] ⊂ [0, ∞) such that either For the sake of brevity, we only consider the first case.Using the fact that we get t 0 q(s)ψ(s))ds.
Remark 4.2.The condition h(s) ≥ 1 in (H 3 ) is not essential; in fact it is sufficient to suppose h(s) ≥ h 0 for some h 0 > 0. Indeed, in this case h(s) h 0 ≥ 1 and then we have to write in the above estimates:

So we have just to modify (4.3) by
Our second existence result is Theorem 4.3.Assume that all conditions of Theorem 4.2 are satisfied but (H 2 ) replaced by Then problem (1.1) has at least one solution x having the representation Then the proof runs parallel to the proof of Theorem 4.2 with R replaced by R.However, in Case 3 of the proof of Theorem 4.2, we have the following EJQTDE, 2011 No. 14, p. 14 estimates instead: Finally, we complete the proof using (4.4).

A Uniqueness Result
The following result complements Theorems 4.2 and 4.3.

Lemma 3 . 1 .
Let M ⊆ X.Then M is relatively compact in X if the following conditions hold (a) M is uniformly bounded in X, EJQTDE, 2011 No. 14, p. 5 (b) the functions belonging to M and the functions belonging to {u : u(t) = x ′ (t), x ∈ M } are locally equicontinuous on [0, +∞), (c) the functions belonging to M and the functions belonging to {u :

Remark 4 . 1 .
Assumption (H 4 ) is essential in Proposition 4.1.Such an hypothesis is missing to complete the proof of Theorem 3.1 in[12].