The Iterative Scheme and the Convergence Analysis of Unique Solution for a Singular Fractional Differential Equation from the Eco-Economic Complex System ’ s Co-Evolution Process

1School of Transportation and Logistics, Central South University of Forestry and Technology, Changsha 410018, China 2Business School, Hunan Normal University, Changsha 410081, China 3School of Business Administration, Hunan University, Changsha 410082, China 4School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, Shandong, China 5School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China 6Department of Mathematics, Shandong University of Science and Technology, Qingdao 266590, Shandong, China

Mathematical models involving fractional derivatives can describe many advection-dispersion processes [43][44][45], viscoelasticity characteristics [46][47][48], thermostat model [49] and the bioprocesses with long memory [50].Especially when one wants to describe long-term ecological-economicsocial complex system phenomena and diffusive interaction, fractional differential operator possesses a higher accuracy than the traditional integer order differential model in depicting the coevolution process of economic, social, and ecological subsystems and the transport of solute in highly heterogeneous porous media [51][52][53].For example, Teng et al. [54] considered the maximum and minimum solutions for a fractional order differential system, involving a -Laplacian operator and nonlocal boundary conditions, which arises from a complex process of ecological economy phenomena and diffusive interaction.
The present paper has some new features.Firstly, both the nonlinear term and the boundary conditions involve fractional order derivatives of unknown functions.Secondly, the uniqueness and nonexistence results are established under the condition concerning the spectral radius of the relevant linear operator; that is, we do not require any monotonicity conditions for nonlinearity.Thirdly, the iterative scheme is constructed and the convergence analysis of unique solution is carried out.Finally, the nonlocal boundary condition possesses weaker positivity since  can be changing-sign measure.
The rest of the paper is organized as follows.In Section 2, we firstly recall some definitions and basic properties on Riemann-Liouville derivative and integral and then give some properties of the Green function.In Sections 3 and 4, the uniqueness and nonexistence results are established under the condition concerning the spectral radius of the relevant linear operator.In Section 5, numerical example and simulation are given to demonstrate the main results and the effectiveness of iterative process.

Preliminaries and Lemmas
For further discussion, here we briefly recall some definitions, notations, and known results, which will be found in the recent monographs.
Definition 2. The Riemann-Liouville fractional derivative with order  > 0 for a function is given by where  ∈ N is the unique positive integer satisfying  − 1 ≤  <  and  > .
Lemma 3. The Riemann-Liouville fractional derivative and integral enjoy the following properties.
Proof.Firstly, it follows from Definitions 1 and 2 and Lemma 3 that Substituting ( 14) into (4), then (4) reduces to the following boundary value problem: that is the modified boundary value problem (13).
Conversely, using (13) again, the modified boundary value problem (13) is also transformed to the form (4). Thus the problem ( 13) is indeed equivalent to (4) and if  solves the modified boundary value problem (13), from the monotonicity and property of   that thus () =   () also solves (4). Let Then we have the following lemma.
Lemma 8 (Gelfand's formula).For a bounded linear operator  and the operator norm ‖ ⋅ ‖, the spectral radius of   satisfies

Existence Results
To obtain the existence result for (4), we use the following assumptions.
Remark 9. Clearly, if (H2) holds, then we have Now let us define a nonlinear operator  :  →  and a linear operator  :  →  as follows: Clearly, if  solves the operator equation  = , then  is a solution of the boundary value problem (13).
Proof.For any  ∈ , it follows from Lemma 6 that which implies that where Thus we have that  :  → .By the uniform continuity of (, ), (, ) ∈ [0, 1] × [0, 1], we know that the linear operator  is a completely continuous operator.
Theorem 12. Suppose that (H0) − (H3) hold.If the spectral radius of the linear operator () ∈ (0, 1), then (4) has a unique positive solution  * , and there exist two constants  2 >  1 > 0 such that Moreover, for any initial  0 ∈  \ , construct successively a sequence and then the iterative sequence   () converges uniformly to as  → ∞.Furthermore, there exists an error estimation with the rate of convergence where  is the positive eigenfunction of the linear operator .
Proof.Firstly, it follows from Lemma 11 that  :  →  is completely continuous.Since (, 0) ̸ ≡ 0, we know that  does not have zero fixed point.Thus we only need show that  has a unique fixed point in .
Step 1.We shall prove that  has fixed points in .
In fact, for any  ∈  \ {}, it follows from Lemma 10 that there exist two positive numbers   ≥ 1 ≥   > 0 such that On the other hand, by Lemma 10,  has a positive eigenfunction , i.e,  =  () .
It follows from (56) that Now let  0 ∈  \ {} be given; we construct an iterative sequence Without loss of generality, suppose that | 1 −  0 | ̸ =  (otherwise, the proof is completed), and then it follows from (56) Consequently, for any ,  ∈ N, it follows from ( 61)-( 63) Thus That is, there exists a constant  2 >  1 > 0 such that Step 2. Next we shall show that the fixed point of  is unique.
In fact, for any positive fixed point  ̸ =  * ∈  of , similar to ( 56) and ( 60 By induction, we have Thus it follows from (71) and () < 1 that which implies that  =  * , a contradiction.So the positive fixed point of  is unique.
Step 3. In the following, we consider the convergence analysis of solution.Similar to Step 1, for any initial  0 ∈ \, construct successively a sequence and then the iterative sequence   () converges uniformly to the unique fixed point  * of  satisfying  * () =    * (); i.e.,   () converges uniformly to as  → ∞.Furthermore, we have error estimation with the rate of convergence Remark 13.In Theorem 12, we not only establish the condition of existence of unique positive solution for (4), but also construct an iterative sequence which converges uniformly to  order derivative of the unique solution of (4).In particular, the error estimation of between exact solution and approximate solution and the corresponding rate of convergence are also obtained.

Remark 14.
Here we also briefly state how the design parameters affect the control performance and how to choose these parameters.
(1) To design parameters effect of the condition (H1) for system, we can choose nonlinearity  as linear functions or sine (cosine) function which shall very easily satisfy the control condition (H1).
(2) For control condition (H2), we can choose a function  ∈  1 (0, 1) such that ()() is a power function satisfying the power exponent larger than −1, and then the control condition (H2) will naturally hold.
(3) For the selection of the control condition () ∈ (0, 1), according to the Gelfand's formula, the spectral radius of   satisfies In particular, taking  = 1, we have Thus (H2) can be replaced: That is, we can choose the coefficient of function  such that () ∈ (0, 1).Consequently in Theorem 12, the restrictions () ∈ (0, 1) can be omitted.
Remark 15.Noticing that  −−1 ∈ \{}, the iterative process can be started from the initial value  0 =  −−1 , which will simplify the whole iterative process.

Nonexistence Results
In this section, we focus on the nonexistence results of positive solution of (4).
Then ( 4) has no positive solution provided that the spectral radius () > 1.
Proof.The proof is similar to Theorem 16, so here we omit the proof.

Numerical Result and Simulation
In this section, we recall the example (1) in introduction and consider the existence of positive solutions for (1).
On the other hand, we have Therefore, the condition (H0) is satisfied.Let and then  ∈ ((0, 1) × [0, +∞), [0, +∞)) with (, 0) ̸ ≡ 0, and      (, Thus the condition (H1) is also satisfied.Now we check the condition (H2).In fact, since we have Thus according to Theorem 12, (1) has a unique positive solution  * , and there exist two constants  2 >  1 > 0 such that In the following, we are going to show the effectiveness of the proposed iterative method.Since let  0 = − 3/2 , and then the simulations for the iterative process (see Figure 1 and Table 1) show that iterative process is very effective and the iterative convergence speed (see Table 2) is robust.
Remark 18.The maximum errors between iterative values and exact solution are exposed in Table 2, which validates the convergence of the maximum errors to zero at a fast speed.
Thus by using the suggested iterative process, the simulation results are robust against nonlinearities and singularity at time variable and also exhibit a reasonable performance for exploring the uniqueness of solution for the singular fractional order viscoelasticity complex system.

Conclusion
Mathematical models involving fractional derivatives can describe many chaotic systems, advection-dispersion process, viscoelasticity characteristics, and the bioprocesses with long memory.In this paper, we offer some control conditions to govern a complex system arising from ecological economy phenomena and diffusive interaction.These control conditions make us easily judge the existence of unique solution of system and further obtain the approximate solution of system according to the required precision of physical problem.Numerical results show that the iterative convergence speed is very fast, which also implies that the suggested method is more accurate to depict the coevolution process of economic, social, and ecological subsystems and the transport of solute in highly heterogeneous porous media compared to what is mentioned before.The extension of the infinite time controller design based on noncompactness measure for complex viscoelasticity systems with multiple stresses will be presented in future work.

Figure 1 :
Figure 1: The approximation of the solution for (1).

Table 1 :
The numerical approximation of the solution for the Eq.(1).

Table 2 :
Maximum errors between iterative values and exact solution.