Monotone iterative technique and Ulam-Hyers stability analysis for nonlinear fractional order differential equations with integral boundary value conditions

In this manuscript, the monotone iterative scheme has been extended to the nature of solution to boundary value problem of fractional differential equation that consist integral boundary conditions. In this concern, some sufficient conditions are developed in this manuscript. On the base of sufficient conditions, the monotone iterative scheme combined with lower and upper solution method for the existence, uniqueness, error estimates and various view plots of the extremal solutions to boundary value problem of nonlinear fractional differential equations have been studied. The obtain results have clarified the nature of the extremal solutions. Further, the Ulam–Hyers and Ulam–Hyers–Rassias stability have been investigated for the considered problem. Two illustrative examples of the BVP of the nonlinear fractional differential equations have been provided to justify our contribution. 2010 Mathematics Subject Classifications: 26A33, 34A08, 34A45


Introduction
In the last few decades, the class of nonlinear fractional differential equations (NFDEs) has been attracted the attentions of researchers very well.The area devoted to study NFDEs has attracted many researcher in almost every field of science.We refer some papers in [1][2][3][4][5][6][7][8] about the applications.Various aspects like existence theory for the aforesaid equations has been considered in plenty of research papers, see [9-12, 17, 18] for existence and uniqueness of solutions to NFDEs.
The iterative technique with the method of upper and lower solutions has gained much attention of most of the researcher in last few decades.It is very useful tools for the existence and approximation of solutions to the initial and boundary value problems.This scheme is well studied for the intial value problems, we refer to see the work in [13][14][15][16][19][20][21][22][23][24].However, The study of the scheme is on intial stage for integral boundary value problem.A few authors have its study for NFDEs with boundary conditions.for instance, Khan [25] have considered it for the following class of FDEs C D p z(t) + h(t, z(t)) = 0, 1 < p < 2, z ′ (t)| t=0 = 0, z(1) = δz(t), 4 where δ, t ∈ (0, 1).The aforementioned method has also been applied to the study of a coupled system of the NFDEs with three point boundary conditions by kamal [26] and his co-author.
Recently, the investigation of stability is also a key research area for the the development of the fractional calculus.It an important field of fractional calculus.In this concern many researchers have introduced numerous type of stability including exponential stability, Lyapunov stability, Ulam type stability analysis etc.Among them, the Ulam type stability has attracted the attention of most of the authors.The Ulam type stability has been well investigated for classical differential equations.In last few years, for initial value problems of FDEs, the Ulam type stability has been well investigated in many articles for detail see [27][28][29][30][31]. But, as far we know, its study is very limited for BVPs of FDEs.Therefore inspired from the aforesaid work, we study the following NFDEs with integral boundary value conditions as where C D p denotes Caputo derivative, Q : In comparison to the work [10,11,25,26], we contribute the improvement of iterative scheme together with upper and lower solution to study existence, uniqueness, error estimates and plotting of the iterative extremal solutions of nonlinear fractional differential equations with integral boundary conditions (1).In addition, we investigate stability analysis for the solution of nonlinear fractional differential equations which not established in the work of [10,11,25,26] .Two illustrative examples of problem (1) have been considered to demonstrate our existence theory.Behavior of the upper and lower solutions of proposed examples are plotted as Figure 1 and 2 via using Matlab which have not been done in the work of cited above.
Definition 1.The Riemann-Liouville integral of fractional order δ ∈ R + of a function φ is defined by provided the right side exits.Definition 2. [3] The Caputo fractional order derivative of order p of function φ is defined by provided that integral on the right is pointwise defined on (0, ∞), where n = ⌊p⌋+1 and ⌊p⌋ denotes the greatest integer which is less than or equal to the real number p. Definition 3. [33] Let V = C[0, 1] be the Banach space endowed with z = max s∈[0,1] |z(s)| which satisfies the partially ordering and let U = [z 1 , z 2 ] with z 1 ≤ z 2 be a set such that U ⊂ V , Then the operator T : U → V is known as increasing function if for each z 1 , z 2 ∈ U and z 1 ≤ z 2 gives T z 1 ≤ T z 2 .The operator T is known as decreasing function if for each z 1 , z 2 ∈ U and z 1 ≤ z 2 gives T z 1 ≥ T z 2 .Definition 4. [33] Let (I − T )z 1 ≤ 0, then z 1 ∈ U is known as a lower solution of (I −T )z = 0 and (I −T )z 2 ≥ 0, then z 2 ∈ U is known as an upper solution of (I −T )z = 0. Lemma 1. [18] Let W be Banach Space with satisfying partially order and z n , y n ∈ U so that z n ≤ y n , n ∈ Z + .If z n → z and y n → y, then z ≤ y.

Assume that
is the solution of fractional differential equation Lemma 3. [32] Let p > 0, then in view of Definition 24 and Lemma 2, the solution of the solution of fractional differential equation is given by

Iterative Solutions
This portion of the manuscript is devoted to existence and uniqueness of extremal solutions to the considered BVP of the nonlinear FDE (1).Further, we study the iterative technique coupled with method of the lower and upper solutions for the approximation and error estimates to the extremal solution of (1).We also investigate the various type stabilities like Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for the considered BVP.First, we convert our considered problem into the integral equation.
has a solution defined by where R(s, γ) is defined by Proof.We may apply a well known Lemma 3 to reduce the equation ( 6) to equivalent integral form By considering the given conditions z(0 Let say Then equation (3.5) implies Clearly, for all s, γ ∈ [0, 1], we have R(s, γ) ≥ 0.
Lemma 5.If we consider the Lemma 4 and condition (C 1 ) with Q ∈ C([0, 1], R), then we see that the integral equation is the solution of problem (1), where R(s, γ) is provided in the equation (8).
Lemma 6.For Green's function R(s, γ), we have that , it follows that max We define the operator T : U → U by Then, in view of ( 11) and ( 12), we obtain the operator equation Let consider the condition (C 2 ), then for z 1 , z 2 ∈ U with z 1 ≤ z 2 , we obtain that is, T is nondecreasing.We can assume another condition which is (C 4 ) For minimal and maximal solutions α, β ∈ U of the equation ( 13), the inequality α ≤ β on [0, 1] is obvious .
Lemma 7.Under the conditions (C 1 ) − (C 4 ) the iterative solutions of operator equation (13) is the monotonic sequence which converges to the solution of equation (11).
Proof.We consider that the conditions (C 1 ) − (C 4 ) hold, then we show that T is continuous.For this, let z 1 , z 2 ∈ U , then Thus T is continuous.It is easy to prove that T is also uniformly bounded and equicontinuous .Thank to the Arzela Ascoli theorem's statement: "If M be a family (finite or infinite) of an equi-continuous, uniformly bounded real valued functions z on an interval [0, 1] .Then M contains a uniformly convergent sequence of functions z n , converging to a function z in U as n → ∞, where U denotes the space of all continuous bounded functions on [0, 1].Thus any sequence in M contains a uniformly bounded convergent subsequence on [0, 1] and consequently M has a compact closure in U ".We obtain T is compact.Further, we may choose z 0 = α.then, in the view of C 4 , we obtain Similarly, , where z 2 = T z 1 .
In same way, we obtain the bounded monotone sequence {z n }, that is where z n = T z n−1 .Then, there will be z ∈ U such that z n → z as n → ∞.Hence, we obtain that z = T z.But z is solution of the equation (11) given by Let consider the condition (C 3 ), then for z, y ∈ U with z ≤ y, we have the inequality Hence in view of ( 15) and ( 16), we get Therfore, we can obtain (17) where m, n are positive integers and ∆ < 1.Now if n → ∞, then we obtain that z m+n − z n → 0. Hence {z n } is Cauchy sequence in U .Thus z * (s) = lim n→∞ z n (s).Therefore, we obtain that T z * = z * .Let m → ∞ in the equation ( 17), then estimate of error for the minimal solution is provided by , where e 1 = z 1 − z 0 .
Remark.If we consider that z 0 = β, then we have that which converges to the solution of the equation (11).Hence, we obtain corresponding estimate of error for the maximal solutions which is provided by e In the view of Lemma 7, we can fix the iterative schemes for NFDE (1) as Theorem 1.Under the assumptions (C 1 ), (C 2 ) and (C 3 ) with ∆ < 1, there exist unique lower and upper solutions to the problem (1).
Proof.Proof is similar as given in [33].
Definition 5. If, we have c K ∈ R + and for every ε > 0 such that for any solution there exists v ∈ AC 2 [0, 1] is a unique solution of equation (1) such that Then BVP (1) is said to be Ulam-Hyers stable.
is the solution of inequality (19) whenever, there exists the function ψ ∈ C([0, 1], R) (dependent on z), so that Theorem 2. If consider the conditions (C 2 ), (C 3 ) coupled with δ < 1, then the Ulam-Hyers stability results for the solutions of BVP (1) are obtained which further implies that the solutions of BVP (1) are generalized Ulam-Hyers stable.
is the solution of inequality (20) whenever there exists the function By Using the Remark 2, Let z ∈ AC 2 [0, 1] be any solution of BVP (1) such that Then, thank to Lemma 4, we have Theorem 3. If consider the conditions (C 2 ), (C 3 ) coupled with δ < 1, then the Ulam-Hyers-Rassias stability results for the solutions of BVP (1) are obtained which further implies that the solutions of BVP (1) are generalized Ulam-Hyers-Rassias stable.

Examples
In the concern section, we provide two examples of concern problem (1).We find iterative solution for the extremal solution of the considered examples.We also provide error estimates for the extremal solutions.Furthermore, stability analysis and error estimates for the solution of examples of the problem (1) is also our concern.In addition, we provide various plots view of every example of problem (1). Where Let say n = 4 is large enough.Then, approximate minimal and maximal solutions are by Let B = 0.00001 and the lower and upper solutions of BVP (1) are z 0 = −0.2, and z * 0 = 0.2 respectively.Then we have ∆ = 2B (2−δ)Γ(p) = 0.000010030 < 1.Therefore, the error estimates are: The maximal and minimal solutions of the Example (1) are plotted in the Figure 1.
Clearly, the condition ∆ < 1 is sufficient for various kinds of Ulam stabilities for the extremal solution of the Example (1).The stability of extremal solutions is also obvious from plots in the given Figure 1.
Where Q (s, z(s)) = t z(s)+ exp(z(s).Let say n = 4 is large enough.Then, approximate minimal and maximal solutions are given by z(s) = z 4 (s) =

Conclusion
We have successfully investigated sufficient conditions for maximal and minimal solutions using monotone iterative schemes together with upper and lower solutions method.We also provided maximum error estimates for the solutions.The obtained results were verified by plotting the graph of the illustrative problem.and academic environment.

Figure 1 A
Figure 1 A line graph of the extremal solutions of the Example 1.

Figure. 2
Figure. 2 A line graph of the extremal solutions of the Example 2.