Existence and uniqueness results for fractional Navier boundary value problems

We establish the existence, uniqueness, and positivity for the fractional Navier boundary value problem: {Dα(Dβω)(t)=h(t,ω(t),Dβω(t)),0<t<1,ω(0)=ω(1)=Dβω(0)=Dβω(1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \textstyle\begin{cases} D^{\alpha }(D^{\beta }\omega )(t)=h(t,\omega (t),D^{\beta }\omega (t)), & 0< t< 1, \\ \omega (0)=\omega (1)=D^{\beta }\omega (0)=D^{\beta }\omega (1)=0, \end{cases}\displaystyle \end{aligned}$$ \end{document} where α,β∈(1,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha,\beta \in (1,2]$\end{document}, Dα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\alpha }$\end{document} and Dβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\beta }$\end{document} are the Riemann–Liouville fractional derivatives. The nonlinear real function h is supposed to be continuous on [0,1]×R×R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0,1]\times \mathbb{R\times R}$\end{document} and satisfy appropriate conditions. Our approach consists in reducing the problem to an operator equation and then applying known results. We provide an approximation of the solution. Our results extend those obtained in (Dang et al. in Numer. Algorithms 76(2):427–439, 2017) to the fractional setting.


Introduction
An elastic beam is an important element needed in structures like buildings, bridges, ships, and aircrafts. The deformations of the beam can be modeled (see, e.g., [2]) by the fourthorder Navier boundary value problem ⎧ ⎨ ⎩ ω (4) (t) = h(t, ω(t), ω (t)), 0 < t < 1, where h : [0, 1] × R × R → R is continuous. Aftabizadeh [3] studied problem (1.1) under the restriction that h is bounded on [0, 1] × R × R. By using a topological degree method he proved the existence and uniqueness of a solution. In [4] (see also [5]) the authors established the existence of a solution for problem (1.1) by means of the lower and upper solutions method. Differently from this method, Dang et al. [1] investigated problem (1.1) by reducing it to an operator equation and using some easily verified conditions. In [6] the authors studied the existence of a solution of a fourth-order differential equation boundary value problem by proving a new fixed point result based on a new distance structure called the extended Branciari b-distance.
Motivated by the novel approach presented in [1], our purpose is generalization of their results to the frame of fractional differentiation. More precisely, we address the question of existence and uniqueness of solutions of the following problem: where α, β ∈ (1, 2], D α and D β are the standard Riemann-Liouville differentiation, and the real function h is supposed to be continuous on [0, 1] × R × R and satisfying some appropriate conditions. For α = β = 2, we recover the results obtained in [1].
In the literature, various mathematical procedures have been considered by scientists through different research-oriented aspects of fractional differential equations. In particular, the fixed point theory has been used very extensively to find solutions of such equations. For instance, in [7] the authors studied the existence of solutions to nonlinear Volterra-Fredholm integral equations of certain types and to nonlinear fractional differential equations of the Caputo type by using the technique of a fixed point with numerical experiment in an extended b-metric space. On the other hand, in [8] the authors established some new fixed-point theorems, which extend and unify several existing results in the literature. As application of their results, they have proved the existence and uniqueness of solutions to some fractional and integer-order differential equations. In [9] the authors established the existence and uniqueness of solutions of boundary value problems for a nonlinear fractional differential equation by means of a fixed point problem for an integral operator. The conditions for the existence and uniqueness of a fixed point for an integral operator are derived via b-comparison functions on complete b-metric spaces. Our approach in the present study consists in applying the Banach fixed point theorem.
Our paper is organized as follows. In Sect. 2, we establish key inequalities on the Green operator functions. In Sect. 3, by reducing problem (1.2) to an operator equation we prove the existence, uniqueness, and positivity of a solution. We propose an approximation process of this solution. We provide some examples at the end of Sect. 3.

Preliminaries and lemmas
For the convenience of the reader, we recall some basic definitions and known results related to fractional calculus [10,11].

Definition 2.1
Let ω : (0, ∞) → R be a measurable function. The Riemann-Liouville fractional integral of order γ > 0 for ω is defined as where is the Euler gamma function.
where m is the smallest integer greater than or equal to γ .
where m is the smallest integer greater than or equal to γ .
Proof To make the argument complete and self-contained, we reproduce this short proof. By means of Lemma 2.3 we can equivalently reduce (2.1) to Substituting c 1 and c 2 into (2.4), we obtain (2.2).

Remark 3.2 Theorem 3.1 extends Theorem 1 in [1] to the fractional setting.
To establish the positivity of solution of problem (1.2), for M > 0, we denote

Corollary 3.3 Let h be a continuous function on
Then problem (1.2) admits a unique nonnegative continuous function ω satisfying (3.5)

Theorem 3.4 (Iterative method)
Under the assumptions of Theorem 3.1, consider the iterative process defined by The sequence (G β (G α ϕ k )) k≥0 converges uniformly to ω, the unique solution of problem (1.2), and we have where q := L 1 M α M β + L 2 M α < 1.