Research Article | Open Access
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
In this paper, we focus on a class of singular fractional differential equation, which arises from many complex processes such as the phenomenon and diffusion interaction of the ecological-economic-social complex system. By means of the iterative technique, the uniqueness and nonexistence results of positive solutions are established under the condition concerning the spectral radius of the relevant linear operator. In addition, the iterative scheme that converges to the unique solution is constructed without request of any monotonicity, and the convergence analysis and error estimate of unique solution are obtained. The numerical example and simulation are also given to demonstrate the application of the main results and the effectiveness of iterative process.
This paper is inspired by a singular fractional differential equation with Riemann-Stieltjes integral conditionwhich arises from some complex system of economic and engineering science, where is a bounded variation function satisfying , , . For example, usually phenomenologic viscoelastic models are based on springs and dashpots which obey Hooke’s law; however the problem is that the corresponding ordinary differential equations have a relatively restricted class of solutions, which is, in general, too limited to provide an adequate description for the complex systems discussed; see . To overcome this shortcoming, by defining the stress and the strain , Schiessel et al.  introduced a fractional order viscoelasticity Kelvin-Voigt system whose stress decays after a shear jump in an algebraic manner where , is a constant and is Riemann-Liouville derivative with an order of . This implies that the fractional order Kelvin-Voigt model can be generalized by the following mathematical model: with Thus the equation (1) belongs to a particular fractional order viscoelasticity Kelvin-Voigt system with the stress , which is important to study and understand dynamic behaviour for the corresponding viscoelasticity process, also see . In addition, here we also notice that the nonlinearity of (1) exhibits a blow-up behaviour at singular time variable . In particular, the singular problems [3–17] as well as impulsive phenomena [18–35] often lead to some blow-up behaviour [28, 36–40] in various complex process of economic and engineering science such as in; ecological-economic-social complex system (Eco-economic System); since the energy consumption fiercely increases resulting in a rapid decrease in total energy, blow-up phenomena will happen at certain time point . In mechanics process, a blow-up behaviour also occurs near the crack tip in elastic fracture like , where is the distance from the crack tip .
Motivated by the above problems, in this paper, we consider the iterative scheme and the convergence analysis of unique solution for the following singular fractional viscoelasticity complex system with Riemann-Stieltjes integral condition:where are the standard Riemann-Liouville derivatives with and , is denoted by a Riemann-Stieltjes integral, and is a bounded variation function with a sign-changing measure ; the nonlinearity may be singular at .
Mathematical models involving fractional derivatives can describe many advection-dispersion processes [43–45], viscoelasticity characteristics [46–48], thermostat model  and the bioprocesses with long memory . Especially when one wants to describe long-term ecological-economic-social 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 [51–53]. For example, Teng et al.  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.
Normally, only positive solutions are meaningful for most practical problems; thus nonlinear analysis methods, such as iterative methods [37, 55–65], variational methods [66–73], the fixed point theorems [74–86], and upper and lower solution methods [87–89], play an important role for the study of various differential equtions. Recently, by means of monotone iterative technique, Zhang et al.  established the existence and uniqueness of the positive solution for a fractional differential equation with derivativeswhere , , , with , and are the standard Riemann-Liouville derivative. is continuous and increasing on second and third variables. However, in order to apply the iterative technique, most of works require monotonicity conditions, see [37, 55–57, 59–61, 64]. Thus the aim of this paper is to weaken the monotonicity request; that is, we establish the iterative scheme and the convergence analysis of unique solution for singular fractional differential equation (4) without any monotonicity conditions.
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.
2. Preliminaries and Lemmas
For further discussion, here we briefly recall some definitions, notations, and known results, which will be found in the recent monographs.
The work space of this paper is Banach space with the norm . Let be a cone in and construct a subset of as follows:
Definition 1. Let with ; the th Riemann-Liouville fractional integral for a function is defined by provided that the right-hand side is point-wise defined on .
Definition 2. The Riemann-Liouville fractional derivative with order for a function is given by where is the unique positive integer satisfying and
Lemma 3. The Riemann-Liouville fractional derivative and integral enjoy the following properties.
(1) If with order , then (2) If , then (3) If , then (4) Let , then where .
Now let and consider the following modified equation of (4): and we have the following lemma.
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 5. Given , then the following boundary value problemhas the unique solution where
Proof. It follows from Lemma 3 that (18) reduces to the following equivalent integral equation: Noticing that and (21), we have , which implies that By (22) and Lemma 3 (3), one has and then On the other hand, we have It follows from (18) that Consequently, the unique solution of problem (18) is The proof is completed.
Lemma 6. The function and have the following properties provided that and for any .
(1) and are nonnegative and continuous for
(2) , where (3) , where (4) where
Proof. It follows from (17) and (19) that the properties (1) and (2) hold. Obviously, (4) also holds if (3) is satisfied. So we only need to prove (3). We divide the proof into two cases,
Case i. If , we have Since we have Case ii. If , we have On the other hand, since , we have Thus This completes the proof of (3).
Lemma 7 (Krein-Rutmann). Let be a continuous linear operator, be a total cone, and If there exist and a positive constant such that , then the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue .
Lemma 8 (Gelfand’s formula). For a bounded linear operator and the operator norm , the spectral radius of satisfies
3. Existence Results
To obtain the existence result for (4), we use the following assumptions.(H0) is a function of bounded variation such that for and .(H1) and there exists a function such that (H2) where (H3)
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).
Lemma 10. Assume that and hold. Then the linear operator is a completely continuous operator, and the spectral radius ; moreover has a positive eigenfunction corresponding to its first eigenvalue .
Proof. For any , it follows from Lemma 6 that which implies that where Thus we have that . By the uniform continuity of , we know that the linear operator is a completely continuous operator.
Next we show that has the first eigenvalue . In fact, it follows from Lemma 6 and that there exists a constant such that , which implies that there exists such that and for all Take such that and for all . So for any , Thus there exists such that for By using Lemma 7, the spectral radius of linear operator is not zero; moreover has a positive eigenfunction corresponding to its first eigenvalue such that The proof is completed.
Lemma 11. If hold, then is completely continuous.
Proof. Similar to Lemma 10, we have . By means of the Arzela-Ascoli Theorem, is completely continuous.
Theorem 12. Suppose that hold. If the spectral radius of the linear operator , then (4) has a unique positive solution , and there exist two constants such that Moreover, for any initial , construct successively a sequence and then the iterative sequence converges uniformly to on as , i.e., as . Furthermore, there exists an error estimation with the rate of convergencewhere is the positive eigenfunction of the linear operator .
Proof. Firstly, it follows from Lemma 11 that is completely continuous. Since , 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 such that On the other hand, by Lemma 10, has a positive eigenfunction , i.e, It follows from (56) thatNow let be given; we construct an iterative sequence Without loss of generality, suppose that (otherwise, the proof is completed), and then it follows from (56) and (58) that So we haveConsequently, for any , it follows from (61)–(63) thatThus we have Noticing that , for any , one gets which implies that is a Cauchy sequence. Consequently, there exists such that converges to . Thus is a fixed point of . It follows from Lemma 4 that is a positive solution of (4). Moreover, since , there exist two positive constants such that Thus That is, there exists a constant 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), there exists a constant such that Thus we haveBy induction, we have Thus it follows from (71) and thatwhich 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 , construct successively a sequence and then the iterative sequence converges uniformly to the unique fixed point of satisfying ; i.e., converges uniformly to on as ; that is, 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.
To design parameters effect of the condition for system, we can choose nonlinearity as linear functions or sine (cosine) function which shall very easily satisfy the control condition .
For control condition , we can choose a function such that is a power function satisfying the power exponent larger than , and then the control condition will naturally hold.
For the selection of the control condition , according to the Gelfand’s formula, the spectral radius of satisfies In particular, taking , we have Thus (H2) can be replaced: <