Positive solutions with exponential decay for a singular semipositone Fisher-like equation

This paper is concerned with the existence of positive solutions to the boundary value problem:


Introduction
In this article, we deal with the existence of positive solutions to the boundary value problem (BVP for short): −u + cu + λu = f (t, u), t ∈ R, lim t→−∞ e k|t| u(t) = lim t→+∞ e l|t| u(t) = 0, where λ and c are positive constants, k, l ∈ R, and f : R × (0, +∞) → R is a continuous function. By a positive solution to the BVP (1), we mean a function u ∈ C 2 (R) satisfying the ordinary differential equation in (1) such that u(t) > 0 for all t ∈ R and lim t→−∞ e k|t| u(t) = lim t→+∞ e l|t| u(t) = 0. Notice that if the constants k and l are positive, then by the boundary conditions in (1) we mean that we look for solutions having an exponential decay at ±∞.
The positivity of the solution u is required here since the BVP (1) arises in the modeling of the propagation of wave fronts in combustion theory and epidemiology (see [2,9]), where u stands for a concentration or a density. The positive constants c and λ refer to the wave speed of the front and the removal rate, respectively. The case where the BVP (1) is autonomous, that is f (t, u) = f (u) with f having a prescribed form, corresponds to the classical Fisher's equation.
There are many papers in the literature considering the case of the BVP (1) posed on the half-line, see [1,[4][5][6][10][11][12] and references therein. However, to the best of authors' knowledge, there is no paper in the literature considering the singular semipositone case posed on the whole real line. Thus, the purpose of the present paper is to fill this gap.

Remark 1.2. Hypothesis (3) cover the case of the BVP
To see this, one has to take Notice that satisfying Hypothesis (3), the nonlinearity f may exhibit singular at u = 0. It is well known that the BVP (1) is called positone if q(t) = 0 for all t ∈ R , and semipositone if q(t 0 ) > 0 for some t 0 ∈ R. Study of existence of positive solutions for semipositone BVPs still attract the attention of many researchers (see for instance, [7,8] and references therein).
Our approach in this work is based on a fixed point formulation and we will use the Guo-Krasnoselskii's fixed point theorem. So, let us present some basic background related to this principle.
Let (E, ||.||) be a real Banach space. A nonempty closed convex subset C of E is said to be a cone in E if C ∩ (−C) = {0 E } and tC ⊂ C for all t ≥ 0.
Let Ω be a nonempty subset in E. A mapping A : Ω → E is said to be compact if it is continuous and A (Ω) is relatively compact in E.
The main tool of this work is the following Guo-Krasnoselskii's fixed point theorem.
is a compact mapping such that either: 1. T u ≤ u for u ∈ P ∩ ∂Ω 1 and T u ≥ u for u ∈ P ∩ ∂Ω 2 , or 2. T u ≥ u for u ∈ P ∩ ∂Ω 1 and T u ≤ u for u ∈ P ∩ ∂Ω 2 , Then T has at least one fixed point in P ∩ (Ω 2 \ Ω 1 ).
The paper is organized as follows. The next section is devoted to the fixed point formulation of the BVP (1). In Section 3, we present the main existence result of this paper. We end the paper by giving an illustrative example.

Fixed point formulation
We start this section by the following important lemma. It will propose a cone in a specific functional Banach space, favorable to the use of Theorem 1.1. Let G : R × R → R + be the function defined by Lemma 2.1. The function G has the following properties: (ii). For all t, τ, s ∈ R, the following inequality holds (iii). Let h : R −→ R + be a measurable function. If δh ∈ L 1 (R) then for all t ∈ R, the following inequality holds Case (a). τ, t ≥ 0. In this case, we have Case (b). τ, t ≤ 0. In this case, we have Case (c). τ ≤ 0, t ≥ 0. In this case, we have In this case, we have This completes the proof.
It is well known that the use of Theorem 1.1 requires the positivity. Since the nonlinearity f is not positive, we will make on the BVP (1) a translation v = u + φ where the associated modified problem has a positive nonlinearity. The following lemma provides such a function φ.

Lemma 2.2. Assume that Hypothesis (2) holds and let φ be the function defined by
Proof. For all t ≥ 0, we have Similarly, for all t ≤ 0 we have This completes the proof.
The functional framework in which we will solve the BVP (1) consists in the following Banach space E and the cone P given below and suggested by Lemma 2.1. In the remaining part of this paper, let E be the linear space defined by Equipped with the norm · , where for u ∈ E u = sup t∈R (p(t) |u(t)|), E becomes a Banach space. The subset P of E given by is a cone of E.
The following lemma is an adapted version to the case of the linear space E of Corduneanu's compactness criterion (see [3], p. 62). This lemma will be used to prove that the operator in the fixed point formulation of the BVP (1) is compact.  (2) and (3) hold k < −r 1 and l < r 2 . Then for all real numbers r and R with R > r > φ * there exists a compact operator T r,R : P r,R → P, where P r,R = P ∩ B(0, R \ (B(0, r)) such that if v is a fixed point of T r,R then u = v − φ is a positive solution to the BVP (1).

Lemma 2.4. Assume that Hypotheses
Proof. Let r, R be real numbers such that R > r > φ * and let Φ be the function defined by where ω R and Ψ R are the functions given by Hypothesis (3) for ρ = R and notice that for all u ∈ P r,R and all t ∈ R, we have The proof is divided into three steps.
Step 1. In this step we prove the existence of the operator T r,R . To this aim let u ∈ P r,R , for all t ∈ R we have from Assertion (iii) in Lemma 2.1 and Hypothesis (3), Thus, let v be the function defined by Clealy, v is continuous on R and v(t) > 0 for all t ∈ R. Moreover, we have where J 1 (t) = t −∞ e −r1s (Φ (s) + q(s)) ds exp (r 2 |t| − r 1 t) and J 2 (t) = +∞ t e −r2s (Φ (s) + q(s)) ds exp (r 2 |t| − r 2 t) .
Since for t ≤ 0, and for t ≥ 0, we obtain from Hypotheses (2) and (3) that lim t→−∞ J 1 (t) = lim t→+∞ J 2 (t) = 0. Now, applying L'Hopital's rule, we obtain from Hypotheses (2) and (3) that Hence, we conclude that lim |t|→+∞ p(t)v(t) = 0 and v ∈ E. Assertion (ii) in Lemma 2.1 leads to for all t, τ ∈ R. Passing to the supremum on τ yields proving that v ∈ P and the operator T r,R : P r,R → P where for u ∈ P r,R , is well defined.
Step 2. In this step, we prove that the operator T r,R is compact. Let (u n ) be a sequence in P r,R such that lim n→∞ u n = u in E given by Hypothesis (3), then for all n ≥ 1 we have the Lebesgue dominated convergence theorem guarantees that lim n→∞ T r,R u n − T r,R u = 0. Hence, we have proved that T r,R is continuous. Also, for all u ∈ P r,R , we have Next, let t 1 , t 2 ∈ [α, β] ⊂ R, for all u ∈ P r,R then we have where for i = 1, 2, p i (t) = e −r2|t|+rit and C η,ζ = 2 sup t,s∈ [η,ζ] p(t)G(t, s).
Because p 1 , p 2 , and t → t 0 (Φ (s) + q(s)) ds are uniformly continuous on compact intervals, the above estimate proves that T r,R is equicontinuous on compact intervals. Furthermore, for all u ∈ T r,R and t ∈ R, we have By means of L'Hopital's rule, we obain from Hypotheses (2) and (3) that proving the equiconvergence of T r,R . Therefore, in view of Lemma 2.3, T r,R is relatively compact in E.
Step 3. In this step, we prove that if v ∈ P r,R is a fixed points of T r,R , then u = v − φ is a positive solution to the BVP (1).
Let v ∈ P r,R be a fixed point of T r,R and let u = v − φ. For all t ∈ R we have e −r1s f (s, u(s))ds + (r 2 ) 2 e r2t r 2 − r 1 +∞ t e −r2s f (s, u(s))ds − f (t, u(t)).
Since for t ≤ 0, and for t ≥ 0, Hypothesis (3) and L'Hopital's rule lead to Taking into account the conditions k < −r 1 and l < r 2 and Hypothesis (3), the L'Hopital's rule leads to Hence, we have proved that lim completing the proof of the lemma.

Main result
Before proving the main result of this paper, we first introduce the following notations. Let  (2) and (3) hold, k < −r 1 , l < r 2 , and (a). there exist a function α ∈ L 1 δ (R) and for all t ∈ R and u ∈ (0, R 1 ) ; Then, the BVP (1) admits a positive solution.
Proof. Without loss of generality, assume that R 1 < R 2 and let T = T R1,R2 be the operator given by Lemma 2.4. The following estimates hold, for all v ∈ P ∩ ∂B (0, R 1 ) and all t ∈ R, Passing to the supremum in the above estimates, we get For all v ∈ P ∩ ∂B (0, R 2 ) and s ∈ [−θ, θ], Assumption (b) and (4) lead to the following estimates Thus, it follows from Theorem 1.1 that T R1,R2 admits a fixed point v such that R 1 ≤ v ≤ R 2 . Then, by Lemma 2.4, u = v − φ is a positive solution to the BVP (1).

Corollary 3.1. Suppose that Hypotheses
Then, the BVP (1) admits a positive solution.

Example
In this example, we consider the case of the BVP (1) when with β < 0 and α, c > 0. We obtain from Theorem 3.1 the following corollary.
Then the BVP (1) within f given in (5), has a positive solution.
Proof. We have to show that all the assumptions of Theorem 3.1 are satisfied. We have leading to ) ≤ e −α|t| w β + c + 1 .