On Coupled p-Laplacian Fractional Differential Equations with Nonlinear Boundary Conditions

This paper is related to the existence and uniqueness of solutions to a coupled system of fractional differential equations (FDEs) with nonlinear p-Laplacian operator by using fractional integral boundary conditions with nonlinear term and also to checking the Hyers-Ulam stability for the proposed problem. The functions involved in the proposed coupled system are continuous and satisfy certain growth conditions. By using topological degree theory some conditions are established which ensure the existence and uniqueness of solution to the proposed problem. Further, certain conditions are developed corresponding to Hyers-Ulam type stability for the positive solution of the considered coupled system of FDEs. Also, from applications point of view, we give an example.


Introduction
Due to high profile accuracy and usability, FDEs become an area of interest for various fields of scientists and mathematicians.In last few years, some physical phenomenons were described through FDEs and compared with integer order differential equations which have better results, that is why researchers of different areas have paid great attention to study FDEs.The applications of FDEs can be studied in several disciplines including aerodynamics, engineering, electrical circuits, plasma physics, chemical reaction design, turbulent filtration in porous media, and signal and image processing; for further details we refer to [1][2][3][4][5].
Nonlinear operators have vital roles in differential equations; one of the most important operators used in FDEs is the classical nonlinear -Laplacian operator, which is defined as For further details and applications of the nonlinear -Laplacian operators, reader should study [6].
Using classical fixed point theory needs strong conditions to establish conditions for existence and uniqueness of solutions to FDEs and therefore restrict the applicability to certain classes of FDEs and their systems.To relax the criteria degree theory plays excellent roles for the existence of solutions to FDEs and their systems.Various degree theories including Brouwer and Leray-Schauder were established to deal with the existence theory of differential equations.An important degree theory known as topological degree theory which was introduced by Stamova [15] and later on extended by Isaia [16] has been used to establish existence theory for solutions to nonlinear differential and integral equations.The mentioned method is called prior-estimate method which needs no compactness of the operator and relaxes much the condition for existence and uniqueness of solutions to differential and integral equations.Recently, the aforesaid degree theory has been applied to investigate certain classes of FDEs with boundary conditions; see [17][18][19].
In recent years another aspect of FDEs which has greatly attracted the attentions of researchers is devoted to the stability analysis of the mentioned equations.Stability analysis plays significant roles in the optimization and numerical analysis of the aforesaid equations.Different kinds of stability have been studied for fractional differential equations including exponential, Mittag-Leffler, and Lyapunov stabilities; see [15,20,21].An important stability was pointed out by Ulam [22], in 1940, which was formally introduced by Hyers [23], in 1941.The aforesaid stability has now been considered in many papers for classical differential equations; see [24][25][26].For instance, Urs [27] has investigated the Hyers-Ulam stability for the following coupled periodic BVPs given as The Hyers-Ulam stability has been investigated for certain FDEs with boundary and initial conditions; see [28][29][30].
In many situations, Lyapunov type stability and its investigation are very difficult and time-consuming for certain nonlinear fractional differential equations.This is due to the predefined Lyapunov function which is often very difficult to construct for FDEs.Therefore, Hyers-Ulam type stability plays important roles in such a situation.Inspired from the above-mentioned work, in this paper, we study a coupled system of FDEs with nonlinear p-Laplacian operator by using topological degree theory.Further, we also investigated some conditions for the Hyers-Ulam stability of the solution to the proposed problem.The proposed problem is given by where 2 <   < 3, 0 <   < 1, ,   ,   > 0,  Here, we remark that applying degree method to deal with existence and uniqueness and to find conditions for Hyers-Ulam stability to a coupled system of FDEs with -Laplacian operator has not been investigated properly to the best of our knowledge.Therefore thanks to the coincidence degree theory and nonlinear functional analysis greatly developed by Deimling [31], we establish necessary and sufficient conditions for existence and uniqueness as well as for Hyers-Ulam stability corresponding to the aforementioned problem considered by us.We also demonstrate our result through expressive example.

Axillary Results
Here we recall some special definitions, theorems, and Hyers-Ulam stability results from the literature [1][2][3][4] which have important applications throughout this paper.
Definition 1.The integral with fractional order  > 0 of Riemann-Liouville type is defined for the function F as provided that the integral on the right converges pointwise on (0, ∞).
Definition 2. The derivative with fractional order  > 0 of Caputo type is defined for the function F as where  = [] + 1, [] is the integer part of  such that the integral on the right converges pointwise on (0, ∞).
Let L be the space of all continuous functions  : [0, 1] → R endowed with a norm sup ∈[0,1] {|()| :  ∈ [0, 1]} which is obviously a Banach space.Then the product space denoted by L = L 1 × L 2 under the norms ‖(, V)‖ = ‖‖+‖V‖ is also a Banach space which will be used throughout this paper.For the coincidence degree theory and nonlinear functional analysis, we recall the following definitions which can be traced in [15,16,31] as follows.
Definition 6. Assume that  :  → U is bounded and continuous mapping such that  ⊂ U. Then  is a -Lipschitz, where  ≥ 0 such that Then  is called strict -contraction under the condition  < 1.
The condition  < 1 yields that  is a strict contraction.
Under the conditions G ⊂ L is bounded for  > 0 and G ⊂ ℏ  (0), with degree Then,  has at least one fixed point.
In view of Definition 12, we give the following definition.
Definition 13.The system of Hammerstein type integral equations is called Hyers-Ulam stable such that, for   > 0 ( = 1, 2, 3, 4) and for all  1 ,  2 > 0 and for every solution there exists a unique solution ( 1 ,  2 ) of ( 21) satisfying the following system of inequalities:

Some Data Dependence Assumptions
To proceed further, let the following hypothesis hold: (A 1 ) The nonlocal functions  1 and  2 where , , ],  ∈ R satisfy the following: where (A 2 ) To satisfy the following growth conditions by the constants (A 3 ) The functions F 1 and F 2 satisfy the following growth conditions under the constants , ,

Main Results
Theorem 14.Let F 1 () ∈ [0, 1] be integrable function for FDEs and with integral boundary conditions; then the solution of is provided by where Ω  1 (, ) is Green's function, given by Proof.Applying the operator   1 on (28) and using Lemma 3, we get from problem (28) the following equivalent integral form as By using conditions D  1 ()| =0 = 0, we get  0 = 0. From (31), we have Applying the operator   1 on (32) and using Lemma 3 again, we get from problem (32) the following equivalent integral form given by By substituting the values of  1 ,  2 , and  3 in (33), we get the following integral equation: According to Theorem 14, the equivalent system of Hammerstein type integral equations corresponding to coupled system ( 6) is given by where Ω  2 (, ) is defined as From Ω  1 (, ) and Further, we define the operators  1 : L 1 → L 1 and  2 : and Hence we have (, V) = ( 1 ,  2 )(, V), Υ(, V) = (Υ 1 , Υ 2 ) (, V), and T(, V) = (, V) + Υ(, V).Therefore the operator Complexity equation of Hammerstein type integral equations ( 36) is given by Thus the solution of Hammerstein type equation ( 36) is the fixed points of operator equation (41).
Theorem 15.In view of hypotheses (A 1 ) and (A 4 ), the operator  is -Lipschitz and satisfies the growth condition given by Proof.From condition (A 1 ) and using  ≤ 1, we get ).To obtain the growth condition we have Upon simplification, we get from (44) Similarly, we get where Theorem 16.In view of hypothesis (A 3 ), the operator Υ is continuous and satisfies the growth condition given by where ϝ = ℘(+), ℘ = max{((−1) Proof.Consider bounded set B  = {(, ) ∈ L : ‖(, )‖ ≤ } with sequence (  ,   ) converging to (, ) in B  .We have to show that ‖Υ(  ,   ) − Υ(, )‖ → 0 as  → ∞.Therefore, we have to consider Due to continuity of F 1 , one has as  → ∞.Thus in view of Lebesgue dominated convergent theorem, we have Hence Υ 1 is continuous.Now for growth condition (49), we have ] ( ‖‖ Thus the operator T is a contraction.Hence the uniqueness of solution to system (6) follows due to the Banach fixed point theorem.

Hyers-Ulam Stability of Coupled System
In the present section, we derive the Hyers-Ulam type stability for the solution of the considered problem.Hence in view of (62) and (64), the system of integral equation ( 36) is Hyers-Ulam stable, and consequently, the solution of system ( 6) is Hyers-Ulam stable.

Illustrative Examples
Therefor from Theorem 20, we concluded that (65) has unique solution.Similarly, the conditions of Theorem 21 can be verified easily.Thus the solution of system (65) is Hyers-Ulam stable.

Conclusion
We have investigated sufficient conditions for existence and uniqueness of solutions to a coupled system of nonlinear FDEs with fractional integral boundary conditions with nonlinear -Laplacian operator by using topological degree method.Further, we have established some adequate conditions for the Hyers-Ulam stability to the proposed problem.