Stability theory and existence of solution to a multi-point boundary value problem of fractional differential equations

The aims and objectives of this manuscript are concerned with the investigation of some appropriate conditions to establish existence theory of solutions to a class of nonlinear four-point boundary value problem (BVP) corresponding to fractional order differential equations (FODEs) provided as $$\begin{aligned} \left\{ \begin{aligned} ^{c}&{\mathscr {D}}^{\omega }y(t)={\mathcal {F}}\left(t, y(t), {^{c}{\mathscr {D}}^{\omega -1}y(t)}\right),1<\omega \le 2,\,\, t\in {\mathbf{J }}=[0, 1],\\&y(0)=\zeta y(\alpha ), \,\, y(1)=\xi y(\beta ),\,\xi ,\ \zeta ,\ \alpha ,\,\beta \in (0,1), \end{aligned}\right. \end{aligned}$$cDωy(t)=Ft,y(t),cDω-1y(t),1<ω≤2,t∈J=[0,1],y(0)=ζy(α),y(1)=ξy(β),ξ,ζ,α,β∈(0,1),where $${^{c}{\mathscr {D}}^{\omega }}$$cDω is Caputo’s fractional derivative of order q and $${\mathcal {F}}\in ({\mathbf {J}}\times {\mathbf {R}} \times {\mathbf {R}} ,{\mathbf {R}})$$F∈(J×R×R,R) may be nonlinear. The required conditions are obtained by using classical results of functional analysis and fixed point theory. Further, we establish some adequate conditions for the Ulam–Hyers stability and generalized Ulam–Hyers stability for the solutions to the considered BVP of nonlinear FODEs. We include a proper problem to illustrate our established results.


Introduction
FODEs are used to model complex phenomena of physically significant problems arising from indifferent areas such as physics, engineering and other applied disciplines, and there is a broad set of applications. Due to the large numbers of applications of FODEs in sciences and engineering, plenty of research papers have been written in this area. As a result, the theory of FODEs has emerged as an important area of investigation in recent years [1,2]. Valuable contribution has been done dealing with the qualitative theory and numerical analysis of solutions to initial and boundary value problems (BVPs) of nonlinear FODEs. The researchers have given much attention on qualitative theory of solutions for mentioned FODEs [3][4][5][6][7][8][9][10][11][12], and references therein. Since BVPs arise in various disciplines of physics, engineering and in dynamics, etc., from applications point of view, here we refer some famous BVPs of differential equations which are the wave equation, like the computation of the normal modes, the Sturm-Liouville problems and Dirichlet problem, etc., see [13][14][15][16][17]. For usability purposes, a BVP should be well posed which implies that a unique solution exists corresponding to the input which depends continuously on the input . In thermal sciences BVPs have significant applications, for instance, to find the temperature at all points of an iron bar with one end kept at lowest energy level and the other end at the freezing point of water. Due to these importance applications, researchers studied BVPs of both classical and arbitrary order differential equations from different aspects. One of the important aspects which has been greatly developed and well explored by different researchers is known as existence theory. The respective aspects have been explored for BVPs of FODEs, see for some detail [18][19][20][21][22].
In the last few years, nonlinear BVPs of FODEs were investigated increasingly. For instance, in [22], Benchora and his co-authors investigated the following anti-periodic BVP given by In last few decades another important aspect which has greatly investigated for the solutions of differential, integral and functional equations is known as stability analysis. The mentioned aspects is very important from numerical and optimization point of view. This is due to the fact that most of the nonlinear problems of fractional calculus and applied analysis are quite difficult to solve for actual solution. In such a situation, one need approximate solutions which are near the actual solution of the corresponding problem. In the mentioned situation, stability of the solutions is necessary. Researchers investigated different kinds of stability to differential, integral and functional equations like exponential, Mittag-Leffler and Lyapunov stability, for detail see [23][24][25]. Recently, some authors explored the another form of stability known as Ulam-Hyers and generalized Ulam-Hyers stability for the solutions of FODEs, see [24,[26][27][28][29]. The aforesaid stability has been very well studied for initial value problems and simple two-point BVPs of linear and nonlinear FODEs, see [30][31][32]. The concerned stability is very rarely investigated for the three-or more point BVPs of FODEs. Here we remark that nonlocal BVPs of FODEs are of key importance for engineers, physics, etc. The stable solutions of the aforesaid problems help us in understanding the phenomenon which has the differential equations. Therefore, inspired from the aforementioned work and importance, here we investigate the aforesaid analysis to a four-point BVP of nonlinear FODEs suggested as where c is the Caputo derivative of order ∈ (1, 2] ,  ∈ ( × × , ) is continuous function and the parameters satisfy 0 ≤ , ≤ 1 , ∈ (0, 1) such that considered problem. The stability results are useful consequences of the existence theory and can be obtained by applying classical functional analysis. We include an example to illustrate our results.

Preliminaries
Since the space C( , ) is a Banach space endowed with a norm ‖y‖ ∞ = sup{�y� ∶ t ∈ } , L 1 ( , ) for the space of Lebesgue integrable functions defined on which is a Banach space corresponding to the norm ‖y‖ L 1 = ∫ 1 0 �y(t)�dt . The space defined as endowed with a norm ‖y‖C = max{‖y‖ ∞ , ‖ c −1 y‖ ∞ } is a Banach space [22]. We provide some basic results and definitions.

Definition 2.1 ([1]). The integral with fractional order
> 0 for a function g ∈ L 1 ( , ) is recalled by ). The FODE with order > 0 given by has a unique solution provided as The next lemma plays an important role for converting BVPs to integral equations.

Lemma 2.5
For g ∈ C( , ), the linear fractional order BVP

Existence of at least one solution: main result
This part of the manuscript is devoted to study existence and uniqueness of solutions for the considered problem (1). For the required results, we use Schauder's fixed point theorem [33] and Banach contraction principle. Thanks to Lemma 2.5, the proposed problem (1) is equivalent to the given integral equation Let N ∶C( , ) →C( , ) be the operator defined as  Proof Applying ℐ on (2) and thanking to Lemma 2.4, we obtain The boundary condition y(0) = y( ) implies and the boundary condition y(1) = y( ) yields It follows from (4) that solution of the BVP (2) is then by solutions of the BVP (1) we mean fixed points of N. Further, we need the following result for onward analysis.
and again from the continuity of  and Lebesgue dominated convergence theorem, we get Hence, it follows that Finally, we prove that N maps D into an equi-continuous set of C (J, ) . Consider 1 < 2 ∈ and y ∈ D , then w h e r e p * = sup{p(t), t ∈ J}, * = sup t∈ ∫ 1 0 | (t, )|d . Then, the suggested FODE (1) has at least one solution with |y(t)| ≤ r on . Proof To prove the continuity of the operator N defined in (6), we consider a sequence {y n } such that y n → y in C ( , ) . Then, there exists r > 0 such that for ‖y n ‖C ≤ r, ‖y‖C ≤ r , we get due to the continuity of  and Lebesgue dominated convergence theorem implies that and From the continuity of , it follows that |(Ny) .

which implies that
From the relations (11) and (12), it follows that thus (10) implies that N is a contraction. Thanks to Banach contraction theorem, N has a unique fixed point. □

Generalized Ulam-Hyers stability of the solutions of BVP (1)
In this section, we prove necessary and sufficient conditions for the Ulam-Hyers (UHS) and generalized Ulam-Hyers stability (GUHS) of the solutions to considered BVP (1) of nonlinear FODEs. To come across the required result, we give the following auxiliary results needed onward.

Conclusion
In this paper we have considered a class of BVP of nonlinear FODEs. By using well-known Schauder's and Banach fixed point theorems, we have established some sufficient conditions for existence and uniqueness of solution to the considered problem. Since stability analysis is very important aspect of existence theory, we have also developed some results about Ulam-Hyers and generalized Ulam-Hyers type stability. For the demonstration of our analysis, we have given a suitable example. The established results generalize many results of the literature in two different ways. First of all we have taken four-point BVP of nonlinear FODEs instead of two-point BVP in our paper. Secondly, we have investigated two important forms of stability including Ulam-Hyers and generalized Ulam-Hyers type which have not been investigated for those BVP of FODEs in which nonlinear function depends on derivative term of dependent function (solution to be determined) to the best of our information. Further, our results generalize many results of the literature; we refer few of them as [34,35].