Investigation of a Class of Implicit Anti-Periodic Boundary Value Problems

This research is devoted to studying a class of implicit fractional order differential equations (FODEs) under anti-periodic boundary conditions (APBCs). With the help of classical fixed point theory due to Schauder and Banach, we derive some adequate results about the existence of at least one solution. Moreover, this manuscript discusses the concept of stability results including Ulam-Hyers (HU) stability, generalized Hyers-Ulam (GHU) stability, Hyers-Ulam Rassias (HUR) stability, and generalized Hyers-UlamRassias (GHUR)stability. Finally, we give three examples to illustrate our results.


Introduction
In previous years the area of FODEs has been considered a powerful procedure for solving practical problems that arise in several fields such as biological science, control theory, heat conduction, viscoelasticity, chemical physics, economics, ecology, aerodynamics [1,2,3,4,5,6], etc. A comprehensive study in the form of a book has been given in 1999 about FODEs and their applications, we provide reference as [7].
In present time, the study of nonlinear differential and integral equations have received much attention from mathematicians due to its worldwide applications in several fields of engineering and technologies. Since using integer order differential operators for modeling various dynamical systems, the hereditary process and memory description cannot be well explained in many situations. Therefore researchers brilliantly have applied the fractional differential operators to describe memory and hereditary precoces in more accurate way than integer order derivatives. This fact motivated researchers to take interest in FODEs. So far we know considerable amount of work has been done in this area. The said area has been investigated from different direction including qualitative theory, stability theory, optimization and numerical simulations. Abundant of work in this regard can be founded about existence theory of solutions, we refer some as [8,9,10,11,12]. On the other hand the area devoted to establish procedure for numerical solutions has been investigated very well. Therefore for this purposes plenty of research papers have been formed in literature which address very good investigations, for instance (we give references as [13,14,15,16,17,18]). Since it is necessary for numerical procedure to be stable to produce good results which are highly acceptable in applications. Therefore another aspect has been considered which is known as stability analysis. Various kinds of stability like exponential, Mittag-Leffler and Lyapunov type have been investigated for differential equations of integer order. In last few years the mentioned stabilities have been upgraded for linear and nonlinear FODEs and their systems, see detail as [19,20,21]. Establishing these stabilities for nonlinear systems have merits and de-merits in constructions. Some of them need a pre-defined Lyapunov function which often is very difficult and time consuming to construct on trail basis. on other hand the exponential and Mittag-Lefller stability involving exponential functions which often create difficulties in treating during numerical analysis of problems. In this regard another kind stability has been given proper attention by the mathematicians known as HU stability. Ulam in 1940 was the first man who pointed this stability during a talk. After that in 1940 Hyers very nicely explained for functional equations, for detail we refer [22,23]. Onward the said stability was further modified to more general form by other researchers for functional equations, ordinary differential equations. Some very fruitful results were formed in this regard which can be traced in [24,25,26,27,28], etc. In last two decades the said stability theory has been considered very well for FODEs and their systems, see [29,30,31,32,33,34].
Inspired from the above mentioned work, in this research article we are considered the following class of antiperiodic boundary value problem (ABVP) in implicit nature c D δ w(t) = h(t, w(t), c D δ w(t)), t ∈ J τ > 0, 2 < δ 3, where 0 < r < 1, 1 < s < 2, J = [0, τ] and h : J × R 2 → R is continous. We investigate qualitative theory as well as different kinds of stability including HU, GHU , HUR and GHUR stability for the considered problem. For qualitative theory we utilize classical fixed point theorem due to Schauder and Banach while for the stability theory nonlinear functional analysis is used. In last, this work is strengthened by providing examples and short conclusion.

Preliminaries
The space M = C 3 (J , R) is a Banach space with respect to the norm defined by Definition 2.1. [35] Integral of a function w ∈ L(J , R + ) with fractional order δ > 0 is defined by provided that integral on the right exists.

Lemma 2.3. [36]
For δ > 0, the given result holds there is a unique solution w ∈ M of the considered problem (1.1), such that Definition 2.5. [27] The fractional order ABVP (1.1) becomes GHU stable if there exist Ψ ∈ C((0, 1), R + ), Ψ(0) = 0, such that for any solutionw ∈ M of the relation (2.4), there is at most solution w ∈ M of the considered problem (1.1) such that Definition 2.6. [27] The fractional order ABVP (1.1) is called HUR stable corresponding to φ ∈ C((0, 1), R + ), with a constant C h > 0, such that for > 0 and for any solution w ∈ M of the relation there exists at most one solution w ∈ M of problem (1.1), such that there is C h ∈ R + , such that for any solutionw ∈ M of the relation (2.5) there at most one solution w ∈ M of problem (1.1), such that

Main work
is given by Proof. Let w be a solution of (3.1). Then by Lemma 2.3, we have Thank to the conditions due to boundary Plugging the values of c 0 , c 1 and c 2 in (3.4), one has the following solution where G (t, η) is the same Green's function given in (3.3).

Lemma 3.3.
The Green function G (t, η), given in (3.3) satisfies the given relations: (A 3 ) moreover for the Green's function one has the given result Proof. Hypothesis (A 1 ) and (A 2 ) are obvious.
Hence this complete the proof.
Here we note that for convince we use To go ahead we give the assumptions bellow for t ∈ J To convert the proposed problem into fixed point problem, we define an operator N : where β w (t) ∈ M , such that β w (t) = h(t, w(t), c D δ w(t)). (3.9) Using property (A 3 ) of Lemma 3.3 and the relation (3.9) in the relation (3.8), we obtain.
Which shows that N is uniformly bounded. To derive N is equicontinuous, let t 2 > t 1 ∈ J and consider Since at t 1 → t 2 , (3.10) tends to zero in the right hand side. Therefore, operator N is equicontinuous and hence it is uniformly continuous. Also it is easy to show that N(B) ⊂ B. Hence by Arzelá-Ascoli theorem N is completely continuous. Proof. We define a set E as The operator N :Ē → M as defined in (3.7) is completely continuous by Theorem 3.4. Take w ∈ E then by definition of the set E , one has by using (A 5 ) From which we have Hence the set E is bounded. So the operator N has at least one solution. Consequently the APBVP (1.1) has at least one solution. Proof. Here we shall use Banach contraction principle to prove the required result. Let w,w ∈ M , then for t ∈ J consider |Nw(t) − Nw(t)| = τ 0 G (t, η) h(η, w(η), β w (η)) − h(η,w(η), βw(η)) dη τ 0 |G (t, η)|| h(η, w(η), β w (η)) − h(η,w(η), βw(η))|dη.
On taking maximum of both sides and repeating the same fashion as in (3.9), we have Since ∆K h 1−L h < 1, therefore, the operator N is contraction. Thus by Banach contraction principle, we get that N has a unique fixed point. Consequently APBVP (1.1) has unique solution.

Stability Analysis
In this section, we provide stability results for the corresponding problem of previous section. Here we provide an assumption needed in further analysis. h(t, w(t), c D δ w(t)) + ψ(t), t ∈ J , 2 < δ 3,

Lemma 4.1. For the given APBVP
we have the following inequality Proof. Thank to Corollary 3.2 the solution of perturbed problem (4.1) is given by From which one has by using (i) of Remark 2.8

Theorem 4.2. Inview of hypothesis (A 5 ) and Lemma 4.1, the solution of the APBVP (1.1) is HU stable and consequently it is GHU stable if the condition
Proof. Letw ∈ M be unique solution of APBVP (1.1) and w ∈ M be any solution of the said problem, then consider with t ∈ J Upon simplification (4.3) yields .  Thus APBVP (1.1) is GHU stable.
Proof. Thank to Corollary 3.2 the solution of perturbed problem (4.1) is given by From which one has by using (i) of Remark 2.9 Proof. For the proof follow Lemma 4.1.

Examples
In this section, we prove suitable examples to illustrate our analysis. Hence we have K h = 1 75 , L h = 1 100 . On computation we have ∆ = 1.26098028. Now thank to Theorem 3.6, we see that Thus the APBVP (5.1) has at most one solution. Further by using Theorem 4.2, we observe that L h + ∆K h = 0.0168130 + .01 = 0.0268130704 < 1.
Hence the solution is HU stable. Further it is also GHU stable. For HUR stability we thank Theorem 4.5 by taking a nondecreasing function φ(t) = t for t ∈ (0, 1). One has C h = ∆(1−L h ) 1−(L h +∆K h ) = 3.25778 × 10 −2 . Hence we see that the results for unique solution w ∈ M and any solutionw ∈ M the following relation w −w M 1.7103 × 10 −2 t, for all t ∈ [0, 1] holds true. Hence the solution of (5.1) is HUR stable. Consequently it is obviously GHUR stable on using Theorem 4.6.