Existence Results for a Class of Fractional Differential Beam Type Equations

: Fractional differential beam type equations are considered. By using an efﬁcient approach, we prove the existence and uniqueness of continuous solutions. An iterative scheme for approximating the solution is given. Some examples are presented

For instance, in [2], Aftabizadeh considered the problem where h ∈ C [0, 1] × R 2 , R satisfying some appropriate conditions.By transforming problem (1) into a second-order boundary value problem, and applying known results, the author proved an existence results.Minhós et al. [12] proved the existence of a solution for where h ∈ C [0, 1] × R 4 , R satisfying a Nagumo-type condition.The proof is based on the degree theory.Dang and Ngo [13] studied the problem where They have proved that problem (2) admits a unique solution.Their method consists of reducing the problem to an operator equation, then proving the contraction of the operator.
Currently, many researchers from various fields have become interested in the topic of fractional calculus based on integrals and derivatives of fractional order.It has numerous applications in the widespread field of science and engineering (see, for instance [14][15][16][17][18] and references therein).Fractional calculus offers superior tools to cope with the time-dependent effects noticed compared to integer-order calculus, which forms the mathematical foundation of most mathematical systems.As a result, fractional calculus is crucial to model real-life problems and finding mathematical solutions is a great challenge.Since fractional differential equations are used to model real-life problems, many mathematical methods (numerical/analytical/exact) are being developed to obtain the solutions to fractional differential equations/models/systems.
In [19], the authors studied the problem Existence results of positive solutions are obtained for the above problem by means of lower and upper solution methods.
In [20], the author proved some existence results for , which need not to be a Caratheodory function.An approximating of the solution is also obtained.
The approach relies on the Schäuder fixed-point theorem.
In [21], by using the Schäuder fixed point theorem, the authors proved the existence of a unique positive solution to the problem More related existence results can be found in [22][23][24][25][26][27][28] and their references.
In this paper, motivated by the previous cited works, we are interested in the study of the following fractional beam type problem: where 2 Our goal is to address the existence and uniqueness of a solution for the above problem.The convergence of an iterative process to the unique solution is also proposed.The method consists at reducing the problem to an operator equation, then proving the contraction of the operator.
where m = γ is the ceiling function and a i ∈ R.

}, and we denote by
) and assume that for some A > 0 and L > 0, we have Then, problem (3) admits a unique solution ϑ ∈ Proof.Define T on C([0, 1], R) by where V α φ is defined by (4) and I β is given by Definition 1.
We claim that T maps B A into itself.Note that Tφ ∈ C([0, 1], R).
To study the positivity, we denote by and assume that for some A > 0 and L > 0, we have Theorem 2. (Iterative method) Under the same hypotheses of Theorem 1, let ϑ be the unique solution of problem (3) and consider the following iterative process: Then, where p := βL αΓ(α + β + 1) .

Conclusions
In this work, we presented a new approach to study a class of Riemann-Liouville fractional differential equations of beam type.Under suitable conditions, we have proved the existence and uniqueness of continuous solutions by reducing the problem to an operator equation, then proving the contraction of the operator.Some examples with numerical approximation are given.In future works, we can extend this problem to more fractional derivatives, such as the Hadamard fractional derivative, ψ-Hilfer and Quantum Fractional Derivatives.

Figure 3 .
Figure 3. Approximation of the solution regarding Example 3.