On Ulam Stability and Multiplicity Results to a Nonlinear Coupled System with Integral Boundary Conditions

: This manuscript is devoted to establishing existence theory of solutions to a nonlinear coupled system of fractional order differential equations (FODEs) under integral boundary conditions (IBCs). For uniqueness and existence we use the Perov-type ﬁxed point theorem. Further, to investigate multiplicity results of the concerned problem, we utilize Krasnoselskii’s ﬁxed-point theorems of cone type and its various forms. Stability analysis is an important aspect of existence theory as well as required during numerical simulations and optimization of FODEs. Therefore by using techniques of functional analysis, we establish conditions for Hyers-Ulam (HU) stability results for the solution of the proposed problem. The whole analysis is justiﬁed by providing suitable examples to illustrate our established results.


Introduction
Fractional order differential equations (FODEs) emerge in the scientific demonstration of numerous frameworks and different fields of science such as physics, chemistry , economics, polymer rheology, aerodynamics, electrodynamics of complicated medium, blood flow phenomena, biophysics, etc. (see [1][2][3][4][5]). Recently, many authors have studied FODEs from different aspects, one is the numerical and scientific techniques for finding solutions and the other is the theoretical perspective of uniqueness and existence of solutions. The interest of the researchers in the investigation of FODEs lies in the incontrovertible fact that fractional-order models (FOM) are found to be highly realistic and practical, compared to the integer order models. Because there are additional degrees of opportunity in the FOM, in consequence, the subject of FODEs is gaining more attention from researchers. Another facet of research, which has been completely studied for integer order differential equations is devoted to uniqueness and existence of solutions to boundary value problems (BVPs). The mentioned aspect has been very well studied for FODEs, we refer the readers [6][7][8][9][10]. Uniqueness and existence results of solutions to multi-point BVPs have been studied via classical fixed point theorems such as the Schauder fixed point theorem and the Banach contraction principle, see [11][12][13][14][15][16][17].
FODEs under integral boundary conditions (IBCs) have been investigated very well because these type of equations are increasingly used in fluid-mechanics and dynamical problems. Jankowski [18] studied the ordinary differential equation under IBCs given by      y (ϑ) = F(ϑ, y(ϑ)), ϑ ∈ [0, T], T > 0, where F ∈ C([0, T] × R, R) and δ = 1 or −1. He developed a sufficient condition for iterative approximate solutions to the above problem.
During the last few decades another part of research, which has been considered for FODEs and got much attention from the researchers is stability analysis. Numerous forms of stabilities have been studied in literature which are Mittag-Leffer stability, exponential stability, Lyapunov stability etc., we refer [21][22][23].
The Ulam stability was first presented by Ulam in 1940 and then brilliantly explained by Hyers in 1941. For more information about HU stability, we refer [24,25]. The HU stability results were generalized and extended by many researchers for FODEs under IBCs. In 1978, Jung studied the said stability for ODEs. Oblaz, Benchohra, etc., have studied the said stability for FODEs but their investigation was limited to initial value problems, we refer to [26][27][28]. To the best of our information and knowledge, the HU stability has been very rarely studied for coupled system of FODEs under IBCs. Therefore in this article we investigate HU stability to the considered problem. Here we remark that we also provide some necessary results for nonexistence of solution. Finally a series of examples are provided to support our analysis.

Axillary Results
In the current section, we review some fundamental definitions and useful results of functional analysis, fractional calculus and fixed point theory (see reference [1,2,8,[29][30][31][32]). Here, first of all, we define the Banach space which is utilized throughout in this article.
As in [31], we define positive solution as follows.
where the integral is point-wise defined on (0, ∞).
Definition 4. [32,33] On the Banach space E defined afore, the mapping d : E × E → R n is called a generalized metric on E if ∀ x, y, and y, z ∈ E with y = x, z = y, z = y, then the following hold Further the pair (E, d) is called a generalized metric space.  Lemma 5. [32,33] Consider a Banach space E with cone C ⊆ E and y ⊂ C is relatively open set with 0 ∈ y and B : y → y be a completely continuous mapping. Then one of the following hold (A1) The mapping B has a fixed point in y (A2) There exist y ∈ ∂y and η ∈ (0, 1) with y = ηBy. Lemma 6. [33,34] Consider a cone C in the Banach space E and if A 1 and A 2 be two bounded open sets in E, such that 0 ∈ A 1 ⊂ A 1 ⊂ A 2 . Let B : C ∩ (A 2 A 1 ) → C be completely continuous operator and one of the following satisfied: Then B has at least one fixed point in C ∩ (A 2 A 1 ) .
This H(ϑ, s) has the given properties Now according to Lemma 7, we can write system (1) as follows Let B : E × E → E × E be the operator defined as Then the fixed point of operator B coincides with the solution of the coupled system (1).
Proof. To prove that B(C) ⊂ C, let (y, z) ∈ C, then by Lemma 8, we have B(y, z) ∈ C and from (F 4 ) and ∀ ϑ ∈ K, we obtain Also from (F 3 ), we obtain Thus from (16) and (17), we have Similarly, one can write that Hence we have B(y, z) ∈ C ⇒ B(C) ⊂ C. Next, like the proof of Theorem 1 of [35], and applying the Arzelà-Ascoli's theorem, it can be easily proven that B : C → C is completely continuous Then the system (1) has a unique positive solution (y, z) ∈ C.
Proof. Let us define a generalized metric d : Obviously (E × E, d) is a generalized complete metric space. Then for any (y, z), (ȳ,z) ∈ E × E and using property (F 3 ) we get Similarly we can show that Thus we have As ae(B) < 1, in the light of Lemma 4, system (1) has a unique positive solution.
Theorem 5. Under the conditions (C 1 ) − (C 3 ) and if the following assumptions hold  (18) and (19), then the the considered system (1) has at least one positive solution.
Proof. Proof can be obtained as proof of Theorem 4.
Thus the system (1) has at least two positive solutions.

Proof.
We left the proof out, as it similar to the proof of Theorem 6.
In same line for multiple solutions we give the following results.
Hence we conclude that there exist at least m positive solutions (y L , z L ), corresponding to coupled system (1) which satisfy u L ≤ (y L , z L ) ≤û L , L = 1, 2 . . . m.

Hyers-Ulam Stability
Definition 6. [30] Let B 1 , B 2 : E × E → E × E be the two operators. Then the system of operator equations is called the HU stability if we can find J i (i = 1, 2, 3, 4) > 0, with ae i (i = 1, 2) > 0 and for each solution (y * , z * ) ∈ E × E of the inequalities given by there exists a solution (y, z) ∈ E × E of system (28) such that ||y * − y|| E×E ≤ k 1 ae 1 + k 2 ae 2 , ||z * − z|| E×E ≤ k 3 ae 1 + k 4 ae 2 , Theorem 10. [30] Let B 1 , B 2 : E × E → E × E be the two operators such that and if the matrix converges to zero, then the fixed points corresponding to operator system (28) are HU-stable. Further, the given condition holds (M 11 ) under the continuity of ϕ i , i = 1, 2, there exist f i , H i ∈ C(0, 1), i = 1, 2 and (y, z), (y, z) such that In this section, we study HU stability for the solutions of our proposed system.
Theorem 11. Suppose that the assumption (M 11 ) along with condition that matrix is converging to zero. Then, the solutions of (1) are HU-stable.
Proof. Thanks to Theorem 2, we have From which we get Analogously one has such that Hence, we get the required results.

Example
To verify the aforesaid established analysis we provide some test problems here in the given sequel.

Non-Existence of Positive Solution
Here some conditions are developed under which the coupled system (1) with given IBCs has no solution.
Then there does not exist positive solution to BVPs (1).
To demonstrate the results of Theorems 12 and 13 respectively, we give the following example.
Example 5. Taking the given system of FODEs with given IBCs as

Conclusions
In the above research work we have successfully investigated a coupled system of nonlinear FODEs with IBCs for multiplicity results. Further, the aforesaid investigation has been strengthened by developing some conditions under which the solutions of the proposed system are HU-stable.