Existence and Iterative Method for Some Riemann Fractional Nonlinear Boundary Value Problems

: In this paper, we prove the existence and uniqueness of solution for some Riemann–Liouville fractional nonlinear boundary value problems. The positivity of the solution and the monotony of iterations are also considered. Some examples are presented to illustrate the main results. Our results generalize those obtained by Wei et al (Existence and iterative method for some fourth order nonlinear boundary value problems. Appl. Math. Lett. 2019, 87, 101–107.) to the fractional setting.


Introduction
Forth-order boundary value problems, can be used to model the deformation of the elastic beam, which is considered to be one of the most used elements in structures such as bridges, buildings and aircraft (see, for instance, [1,2]).
In [1], Aftabizadeh considered Equation (1) together with the boundary conditions: where f : [0, 1] × R 2 → R is continuous. Under adequate conditions imposed on f he proved that problem (1)-(2) has a unique solution. To do this, he transforms Equation (1) into a second-order integro-differential equation and apply the Schauder's fixed point theorem.
In [4], by using the method of lower and upper solutions for a fourth-order equation and some restrictive conditions on f , Bai established an existence result to problem (1)- (2).
Following a different approach, they addressed the question of existence and uniqueness of positive continuous solution.
In [31], the authors considered the two-dimensional fractional Schrödinger equation (FSE) without potential for the slowly varying envelope ψ of the optical field and 1 < α ≤ 2. They transformed Equation (5) into a Dirac-Weyl-like equation, which is used to establish a link with light propagation in the honeycomb lattice (HCL). They discovered a very similar behavior-the conical diffraction. This similarity in behavior is broken if an additional potential is brought into system. Our paper is organized as follows. In Section 2, we establish some estimates on the Green's function and we prove appropriate inequalities on some integral operators involving the Green' function. In Section 3, under adequate conditions imposed on function f , we prove the existence and uniqueness of a solution of problem (4). Our approach is based on the Banach contraction principle. The positivity of the solution and the monotony of iterations are also considered. Some examples are given to illustrate our existence results.

Fractional Calculus
We recall in this section some basic definitions on fractional calculus (see [33][34][35][36]). Definition 1. The Riemann-Liouville fractional integral of order γ > 0 for a measurable function f : (0, ∞) → R is defined as provided that the right-hand side is pointwise defined on (0, ∞). Here Γ is the Euler Gamma function.

Definition 2.
The Riemann-Liouville fractional derivative of order γ > 0 for a measurable function f : provided that the right-hand side is pointwise defined on (0, ∞) . Here n = [γ] + 1, where [γ] denotes the integer part of γ. Please note that if γ = m ∈ N\{0}, then we obtain the classical derivative of order m.
where m is the smallest integer greather than or equal to γ and c i ∈ R (i = 1, ..., m) are arbitrary constants.
Proof. For the convenience of the reader, we provide the proof of property (ii) which plays an important role in the rest of the paper.

Estimates on the Green's Function
, then the boundary-value problem, has a unique solution where for x, t ∈ [0, 1] , G β (x, t) is called Green's function of boundary-value problem (6).
Proof. By means of Lemma 1, we can reduce equation The boundary condition v(0) = 0 implies that c 3 = 0, while the condition v(1) = 0, gives On the other hand, since v (1) = 0, we obtain Hence Therefore the unique solution of problem (6) is The proof is completed.
In the following, for some values of β we give the representation of the Green function G β (x, t) with the contours and the projections on some coordinate planes (see Figures 1-3). These details give an immediate idea of the behavior of these functions.     Proposition 1. Let 2 < β ≤ 3. The Green function G β (x, t) satisfies the following properties. (8) and (9) we have On the other hand, since t − x ≥ 0, we get (iii) Now, assume that 0 ≤ t ≤ x ≤ 1. Since it follows from (8) and (10) that Now, using the fact that we deduce from (13) that Similarly, using again (13) and (14), we obtain Throughout this paper, for 2 < β ≤ 3 and ϕ ∈ C([0, 1]), we denote by where G β (x, t) is given by (8).

Now, using Definition 1 and
where Observe that By combining (22) and (23), we obtain the second inequality in (16).

Existence and Uniqueness of a Solution
Theorem 1. Let f : [0, 1] × R 2 → R be a continuous function and assume that there exist numbers M, L 1 , L 2 ≥ 0 such that where G β ϕ is defined by (15) and I α is the Riemann-Liouville fractional integral operator given by Definition 1. We shall investigate problem (4) via the operator equation (25). Observe that if ϕ is a fixed point of the operator T, then by Lemma 1,(15) and Lemma 2, where q is defined in assumption (iii).
The the proof is completed.
Next, we present a particular case of Theorem 1. To this end, denote Corollary 1. Let f : [0, 1] × R 2 → R be a continuous function and assume that there exists numbers M, L 1 , L 2 ≥ 0 such that Then the boundary value problem (4) has a unique nonnegative solution u ∈ C([0, 1]) satisfying

Iterative Method and Examples
Consider the following iterative process.
Theorem 2. Assume that hypotheses of Theorem 1 are satisfied. The sequence (ϕ k ) k≥0 converges with the rate of geometric progression and we have where u is the exact solution of problem (4) and q is given in assumption (iii) in Theorem 1.
Proof. It is known by the Banach contracting mapping principle that the sequence (ϕ k ) k≥0 converges with the rate of geometric progression and we have where ϕ is the unique fixed point of the operator T in B[O, M].
Using this fact and Lemma 3, we obtain The proof is completed.
Proof. (i) We claim that for all k ∈ N, we have We proceed by induction. From hypothesis, the inequality is clear for k = 0. For a given k ∈ N, assume that ϕ k (x) ≤ ψ k (x).
Since the Green function is nonnegative, we deduce from (15) and Definition 1 that Combining this fact and that the function f (x, u, v) is nondecreasing in u and v, we obtain So our claim is proved. Using (35), (15) and Definition 1 we get inequality in (32) (ii) From Theorem 2, we know that the sequences (I α (G β ϕ k )) k≥0 and (I α (G β ψ k )) k≥0 converge to the unique solution u of problem (4).
We claim that the sequence (ϕ k ) k≥0 is nondecreasing. (15), Definition 1 and the monotony of the function f , we deduce that Hence the sequence (ϕ k ) k≥0 is nondecreasing. Therefore, by using again (15) and Definition 1, it follows that the sequence (I α (G β ϕ k )) k≥0 is nondecreasing.

Example 1.
Consider the following boundary value problem: It is easy to verify that M = 4 is an example of suitable choice.
By Proposition 2, (u k := I α (G β ϕ k )) k≥0 is a nonnegative increasing sequence which converges to the unique nonnegative solution u. Some iterations are depicted in Figure 5.