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Existence and stability results for nonlocal boundary value problems of fractional order
Boundary Value Problems volume 2022, Article number: 25 (2022)
Abstract
In this paper, we prove the existence and uniqueness of solutions for the nonlocal boundary value problem (BVP) using Caputo fractional derivative (CFD). We derive Green’s function and give some estimation for it to derive our main results. The main principles applied to investigate our results are based on the Banach contraction fixed point theorem and Schauder fixed point approach. We dwell in detail on some results concerning the HyersUlam (HU) type and generalized HU (gHU) type stability also for problem we are considering. We justify our results with an illustrative example.
1 Introduction
Fractional differential equations (FDEs) refer to the generalization of integerorder differential equations. They appear in various scientific and engineering fields, for instance, the mathematical modeling of different dynamical processes in epidemiology [1–4], chemistry, physics [5], psychology [6], biology, signal and image processing, control theory, ecology [7], etc. As a result, the concept of noninteger order differential equations is continuously getting more consequences and relevancy. For instances and details, see [8–11] and the given references therein. After all, even though the BVPs theory for nonlinear and nonclassical differential equations is still in its infancy, many directions of this literature need to be expanded further.
The topic of multipoint nonlocal BVPs has been raised by various researchers (we refer [12–15]). The multipoint boundary constraints arise in various problems of physics, fluid mechanics, and wave propagation (we refer to [16, 17] for interest). For instance, in controllers, the multipoint boundary constraints may be found such that the controllers at the endpoints spread or add energy with related sensors placed at middlelevel positions. In the same line, the thirdorder differential equations in which differentiation of acceleration is involved are called jerk equations, where the time derivative of acceleration occurs (see details in [18, 19]). The said equations are important for engineers and physicists, and they try to plan the vehicles in a way that jerks may be minimal. The thirdorder differential equations are special cases of FDEs with orders between 2 and 3. The fractional order goes through three, the considered equation possibly corresponds to the jerk equation.
The results related to investigating the existence and uniqueness of solutions for nonlinear multipoint BVPs have been studied by a number of researchers. For example, authors in [20, 21], and references therein have investigated various classes of BVPs of FDEs. Authors [22] have proved the existence of a solution for noninteger order BVP with nonlocal multipoint boundary constraints using Schaefer’s and Krasnoselskii’s fixed point theorems. On the other hand, stability results are important to be investigated in most cases to demonstrate the authenticity and validity of numerical algorithms, methods, and procedures. In this regard, valuable work has been done in the last many decades. Various concepts of stability have been introduced, including exponential, MittagLeffler, and Lyapunov types. The stability aspects are very important from the optimization and numerical point of view. So far, we know, a huge amount of work has been done [23–27]. Another version of stability introduced by Ulam and explained by Hyers attracted the attention of researchers very well. The said version is easy and understandable for approximate solutions. For functional equations, this kind of stability has been investigated very well (see details in a few articles as [28–30]). The mentioned stability has been very well studied for ordinary differential problems. In the last few decades, this aspect has been given more attention and investigated for different classes of initial value problems of FDEs (see [31–34]). However, for simple BVPs, the concerned stability has also been well studied. However, in the case of nonlocal BVPs, it is very rarely investigated, especially for jerklike problems. In this regard, we refer to [35–38].
Inspired by the above work, we will extend the results of the following problem [39] to fractional order as
Here \(\beta \in ({c,d})\), \(k \in \mathbb{R}\), \(\Lambda \in C([{c,d}]\times \mathbb{R},\mathbb{R})\), and \(\Lambda (t,0)\neq 0\). In our work, we intend to extend the above (1.1) and (1.2) by taking the Caputo fractional order derivative instead of ordinary (we refer the reader to see [40] for definitions and basic consequences on noninteger order calculus) in the place of the classical operator \(y'''\). Here, we prove the existence and uniqueness of the solution for the following noninteger order BVP of FDEs as
where \(2< \zeta \le 3\), by assuming that \(\Lambda :[{c,d}]\times \mathbb{R} \to \mathbb{R}\) is a continuous mapping and follows a uniform Lipschitz inequality with respect to y on \([{c,d}]\times \mathbb{R}\), such that there exists a constant \(L>0\), where for every \((t,y),(t,z) \in [{c,d}]\times \mathbb{R}\), we have
If \((dc)^{2} \neq k(c\beta )^{2}\) with \(c \neq \beta \), \(c\neq d\) and \((dc)\) is as small such that
Under the said condition, a unique solution to Problem (1.3) and (1.4) exists. This result will be investigated using fixed point techniques. After that, we will also evaluate the proposed stability analysis for our considered Problem (1.3) and (1.4). Also, HU and gHU stability results are developed for the considered problem using some sophisticated procedure of nonlinear analysis.
The remaining article is arranged as follows: In Sect. 2, we recall some basic definitions and results. In Sect. 3, we compute fractional Green’s function. In Sect. 4, some results about Green’s function and its estimation are given. Section 5 is devoted to existing results. In Sect. 6, we elaborate on stability results. Also, we have given an example to demonstrate our results in Sect. 7. The conclusion is given in Sect. 7.
2 Elementary results
Here, we recall some basic results of fractional integral and derivative found in [40].
Definition 2.1
The fractional integral of order \(\zeta >0\), for an absolutely continuous function \(y:(0,\infty )\to R\) is defined as
provided the integral converges at the right sides over \((0, \infty )\).
Definition 2.2
The Caputo fractional derivative of order \(\zeta >0\), for an absolutely continuous function \(y\in C^{n}[c, d]\) is defined as
provided that the right side is point wise defined on \((0,\infty )\).
Lemma 2.1
([40])
Let \(\zeta > 0\) and if y is absolutely continuous function, then we have
for some constants \(C_{i} \in \mathbb{R}\), \(i=0,1,2,\ldots,n1\).
3 Computation and estimation of Green’s function
Now, let us establish Green’s function for the following twopoint BVP
with \(c \neq \beta \), \(c\neq d\) and afterwards, supposing that the solution of the following threepoint BVP
can be stated as follows
where \(\lambda _{0}\), \(\lambda _{1}\), and \(\lambda _{2}\) are constants that will be specified later. We will estimate Green’s function for (3.3) and (3.4), respectively.
Proposition 3.1
If \(h:[{c,d}] \to \mathbb{R}\) is continuous mapping, then BVP (3.1) and (3.2) has a unique solution given by
which can be expressed in compact form as
where
Proof
It is well known that Problem (3.1) and (3.2) is similar to solving the integral equation
where \(c_{1}\), \(c_{2}\), and \(c_{3}\) are some real constants. Using boundary conditions given in (1.4), we can obtain
Thus, we get
□
The unique result exists from the assumption that the completely homogeneous BVP has only the trivial solution. So Proposition 3.1 has been proved.
Proposition 3.2
Let \(h:[{c,d}] \to \mathbb{R}\) be a continuous mapping, if \(k(c\beta )^{2} \neq (cd)^{2}\) and \(c\neq \beta \), \(c\neq d\), then BVP (3.3) and (3.4) has a unique solution given by
The solution can be written further as
where
Proof
Let
where \(\lambda _{0}\), \(\lambda _{1}\), \(\lambda _{2}\) are constants that will be determined using boundary conditions given in (3.4) and
Therefore, to compute \(\lambda _{0}\), \(\lambda _{1}\), \(\lambda _{2}\), we proceed as
We get
or
Solving the system, we get the corresponding values as
Therefore, the final solution becomes
Now we derive the proof of the uniqueness. Let z be also a solution to (3.3) and (3.4), that is
Let \(\Omega (t)=z(t)y(t)\), \(t\in [{c, d}]\). Due to linearity property of the Caputo noninteger order derivative, we have
Therefore, \(\Omega (t)=c_{1}+c_{2}t+c_{3}t^{2}\), where \(c_{1}\), \(c_{2}\), and \(c_{3}\) are constants that will be computed later. We have
or
We get the following homogeneous system
with determinant
Therefore, the homogeneous system contains only the trivial solution, and hence \(\Omega (t)\equiv 0\), \(t\in [{c,d}]\) or \(y(t)\equiv z(t)\), \(t\in [{c, d}]\). Thus, the proof is completed. □
4 Green’s function estimations
Proposition 4.1
Let \(R(t,s)\) be Green’s function given in Proposition 3.1, then
Proof
We deduce the proof as
□
The next result is also important in our study.
Proposition 4.2
Green’s function \(G(t, s)\) given in Proposition 3.2satisfies the following inequality
for \(t\in [{c, d}]\).
Proof
Here we derive the proof as
□
5 Existence results for the solution
For further correspondence in this work, we define the following:
For further analysis, we also denote the Banach space by \(X=C[c, d]\) under the norm \(\y\=\sup_{t\in [c, d]}y(t)\). Hence we define the operator \(T:X\rightarrow X\) by
The following hypothesis needed also to be held:
 \((H_{1})\):

Let \(\Lambda :[{c,d}]\times \mathbb{R} \to \mathbb{R}\) is continuous and follows the uniform Lipschitz inequality for y on \([{c, d}]\times \mathbb{R}\), such that there exists a constant L, for all \((t,y),(t,z) \in [{c,d}] \times \mathbb{R}\),
$$ \bigl\vert \Lambda (t,y)\Lambda (t,z) \bigr\vert \le L \vert yz \vert .$$
Theorem 5.1
Under the hypothesis \((H_{1})\) and if the condition \((dc)^{2} \neq k(c\beta )^{2}\), with \(c \neq \beta \) holds and \(dc\) is sufficiently small, such that
then there exists a unique solution to (1.3) and (1.4).
Proof
Note that y is a solution to (1.3) and (1.4) if and only if it is a solution to (3.3) and (3.4) with \(h(t)\) replaced by \(\Lambda (t,y(t))\). However, (3.3) and (3.4) have a unique solution given by
where \(G(t,s)\) is specified in Proposition 3.2. Define the operator \(T:X \to X\) by
We will apply the Banach fixed point theorem to determine whether the operator T has a unique fixed point. Assume that \(y,z\in X\), then for \(t\in [c, d]\), we have
where we have utilized Proposition 4.1. It agrees that
where
Hence, we deduce that T is a contraction mapping on X, and from the Banach contraction mapping theorem, we receive the required result. □
For next result, the following hypothesis need to be hold:
 \((H_{2})\):

For \((i=1,2)\), let there exists \(\varphi _{i}: [c, d]\rightarrow R\) and \(y\in X\), such that
$$ \bigl\vert \Lambda \bigl(t, y(t)\bigr) \bigr\vert \leq \varphi _{1}(t)+ \varphi _{2}(t) \vert y \vert ,\quad t\in [c, d].$$Further putting \(\varphi ^{*}_{1}= \sup_{t\in [c, d]}\varphi _{1}(t)\) and \(\varphi ^{*}_{2}= \sup_{t\in [c, d]}\varphi _{2}(t)\).
Theorem 5.2
Under the hypothesis \((H_{2})\), if \(T:B\rightarrow B\) is a completely continuous operator, then T has at least one fixed point in \(B_{r}\).
Proof
Let X be a Banach space and \(B_{r}\subset X\) be a bounded closed convex subset.
Step 1: First, we are going to show that operator T is continuous.
Let \(\{y_{n}\}\) be a sequence, such that \(B_{r} = \{y \in X: \y\\leq r\}\), then, for \(t\in [c, d]\), we have
where Ω is given in (5.5).
then we have
Hence T is continuous.
Step 2: Next, to show that T is bounded means maps bounded sets to bounded sets on X. Let \(y\in B_{r}\), then for \(t\in [c, d]\), we have
Hence one has
Thus, T is a bounded operator.
Step 3: Now, we are going to show that T is equicontinuous. Let \(t_{1}\) and \(t_{2}\in [c, d]\), then
On further simplification, one has
We see that as \(t_{1}\rightarrow t_{2}\), then the righthand side tends to 0. As T is bounded and continuous on \(B_{r}\) therefore is uniformly continuous.
Hence we claim that
Thus, using Arzelá Arcoli theorem, one can say that operator T is relatively compact, bounded, and uniformly continuous. Thus, T is the completely continuous operator. Hence, T has at least one fixed point. Therefore, the considered problem has at least one solution. □
6 Stability results
Here we describe some stability results. The said stability results are based on the HU concept.
Remark 6.1
We consider a mapping ϕ independent of y, such that \(\phi (t)\leq \epsilon \), for every, \(t \in [c, d]\).
Theorem 6.1
The solution to the following perturbed problem
satisfies the following relation
where Ω has been given above in (5.5).
Proof
In view of Lemma 2.1, the solution is given by
Further, we have for \(t\in [c, d]\)
□
Theorem 6.2
In view of Theorem 6.1and assumption \((H_{1})\), the solution of Problem (1.3) and (1.4) is HU stable and in the same line will be gHU stable if and only if the condition \(\Omega <\frac{1}{L}\) holds.
Proof
Let y be any solution and x be the unique solution of Problem (1.3) and (1.4), such that \(x, y \in X\), then consider
Hence
where \(C_{L,\Omega } = \frac{\Omega }{1L\Omega }\).
Forth, if we have a nondecreasing function \(\psi : [c, d]\rightarrow R^{+}\), then
where \(\psi (0) = 0\). So, the solution to the considered problem is also gHU stable. □
7 Illustrative example
Here to support our findings, we present the following example.
Example 7.1
Consider the following nonlocal nonlinear BVP of FDEs as
We see from (7.1) that
and
So, Λ is a Lipschitz with respect to y on \([0, 1]\times \mathbb{R}\) with the Lipschitz constant \(L=1\).
Since \((dc)^{2}=1\ne \frac{3}{16}=k(c\beta )^{2}\) and
Now in view of Theorem 5.1, Problem (7.1) has a unique solution. The graph of the solution \(y(t)\) is displayed in Fig. 1. Note that the solution has been obtained here by the generalized differential transform method, which is a very effective tool to give semianalytical solutions for FDEs (see for details [41]). Also, the condition of HU stability and gHU is obvious. The graphical presentation of the approximate solution is given in Fig. 1.
8 Conclusion
We have given some sufficient conditions that demonstrate the existence and uniqueness of the solution for a noninteger order threepoint nonlocal BVP of FDEs. Some pertinent results regarding HU and gHU stability have been incorporated. Thanks to the fixed point approach and nonlinear functional analysis, these findings have been established. For validation of our results, an interesting example has also been given. From the mentioned discussion and results, we conclude that fixed point theory is a powerful tool to deal with nonlinear problems of FDEs corresponding to different initial and boundary conditions.
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Acknowledgements
The authors Kamal Shah and Thabet Abdeljawad would like to thank Prince Sultan University for paying the APC and for the support through the TAS research lab.
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Ertürk, V.S., Ali, A., Shah, K. et al. Existence and stability results for nonlocal boundary value problems of fractional order. Bound Value Probl 2022, 25 (2022). https://doi.org/10.1186/s13661022016060
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DOI: https://doi.org/10.1186/s13661022016060