Solvability of a boundary value problem with a Nagumo Condition

ABSTRACT The aim of this paper is to discuss the existence and localization of solutions for a generalized Emden-Fowler equation involving a conformable derivative and with a Dirichlet boundary condition. Our approach is based on the lower and upper solutions method and Schauder fixed-point theorem under a Nagumo condition.


Introduction
Recently, Khalil et al. introduced a conformable derivative based on a limit form as in the case of the classical derivative [1]. Since this derivative has properties similar to those of the classical one, it has attracted more attention.
The conformable derivative of order 0 < α < 1, of a function h : [a, ∞) → R is defined by If T a α h(t) exists on (a, b), b > a and lim t→a + T a α h(t) exists, then we define T a α h(a) = lim t→a + T a α h(t). Moreover, if h (n) exists, the conformable derivative of order n < α < n + 1 of h, is defined by where β = α − n ∈ (0, 1). Later, many papers considering boundary value problems with conformable derivative are presented [2,3]. Our interest will be focused upon the problem to prove the existence of solution for the following boundary value problem: where T a α denotes the conformable derivative of order α, 1 < α < 2, u is the unknown function and f : [a, b] × R 2 → R is a given function. To solve this problem, we use the method of upper and lower solutions together with Schauder's fixed-point theorem. Since the nonlinear term f depends on the derivative of the function u, we have to find a priory bound for the derivative of the solution. The method of upper and lower solutions is efficient since it gives both the existence and localization of the solution. For some problems applying this method for the study of the existence of solutions, we refer to [4][5][6][7][8][9][10].
Some particular cases of problem (P) may represent important problems such as the well-known Emden-Fowler equation with a Dirichlet condition of the form u + a(t)u σ (t) = 0, t ∈ (0, 1) , that appear in many branches of physics and engineering, for example in fluid dynamics, stellar dynamics, quantum mechanics . . . , see [10][11][12].
Let us recall some essential definitions on conformable derivative.
Let n < α < n + 1, and set β = α − n, for a function h : [a, ∞) → R, we denote by For the properties of the conformable derivative, we mention the following: Let n < α < n + 1 and h be an (n + 1)-differentiable at t > a, then we have and Next, we give a property on the extremum of a function that its conformable derivative exists:

Main results
Let AC[a, b] be the space of absolutely continuous func- The lower and upper solutions for problem (P) are defined as follows.
Next, we solve the corresponding linear problem.

Lemma 2.2: Assume that y ∈ C[a, b], then the linear problem
has a unique solution given by where Proof: Applying the integral I a α , to both sides of the differential equation (3), it yields Thanks to the boundary condition u(a) = 0, we get Since u(b) = 0, then Substituting c by its value in Equation (6) gives where the Green function G is given by Equation (5).

Proof:
Putting we can easily see that the function g 1 is decreasing and the function g 2 is increasing with respect to t, consequently Since the nonlinear term f depends on the derivative of the function u, we have to find a priory bound for the derivative of the solution, for this reason, the function f should satisfy the following Nagumo condition. Definition 2.4: (Nagumo condition) Let σ and σ be the lower and upper solutions of problem (P) such that is said to satisfy Nagumo condition on D if there exists a function H ∈ C(R + , (0, +∞)) such that Now we give the existence result for the nonlinear problem (P).

Theorem 2.5: Let σ and σ be the lower and upper solutions of problem (P) such that σ ≤ σ and assume that f (t, x, y) is continuous on D. Then the problem (P) has at least one solution u ∈ AC 2 ([a, b]) such that
Proof: Define the modified problem where the function F(t, x, y) is a modification of f (t, x, y) associated with the coupled of lower and upper solutions σ and σ : where R is a positive constant and large enough such that that exists from Equation (8). The proof of Theorem 2.5 will be done in three steps.
Step 1: Existence of solution of the modified problem (MP). From the definition of F, we see that it is continuous and bounded, i.e. |F(t, Let us prove that A( ) is uniformly bounded. For u ∈ and using Lemma 2.3, we get On the other hand, we have Hence, A( ) is equicontinuous. Thanks to Arzela-Ascoli's theorem we get that A is completely continuous. Moreover, by Schauder fixed-point theorem, we conclude that A has a fixed point u ∈ which is a solution of the modified problem (MP).
Step 2: Localization of the solutions. Let us prove that if u is a solution of the modified problem (MP), it satisfies If t 0 ∈ (a, b), then Proposition 1.1 implies T a α w(t 0 ) ≤ 0. Since σ is an upper solution for problem (P), then that leads to a contradiction. Now, if t 0 = a, we get taking into account that u(a) = 0 then σ (a) < 0, which contradicts the fact that σ is an upper solution of problem (P). By the same way, we get a contradiction if t 0 = b. Applying similar reasoning, we prove that σ (t) ≤ u(t), ∀ t ∈ [a, b].
Step 3: Priory bound for the derivative of the solution. Now Let us prove that if u is a solution of the modified problem (MP), then we have −R ≤ u (t) ≤ R, ∀ t ∈ a, b .
Let u be a solution of the modified problem (MP), then it satisfies from step 2 the inequalities σ (t) ≤ u(t) ≤ σ (t), ∀ t ∈ [a, b]. Suppose that there exists t 0 ∈ [a, b] such that u (t 0 ) > R. on the other side, by the Mean value theorem and conditions (2), there exists θ ∈ (a, b) such that u (θ ) = (u(b) − u(a))/(b − a) = 0. Since the function u is continuous then there exist two points t 1 , t 2 between t 0 and θ such that u (t 1 ) = r, u (t 2 ) = R and u (t) ≥ r, ∀ t ∈ I = [t 1 , t 2 ] (or [t 2 , t 1 ]). By the change of variable s = u (t) and in view of Equation (7), it yields