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Approximation theorems of a solution of amperometric enzymatic reactions based on Green’s fixed point normalS iteration
Advances in Difference Equations volume 2021, Article number: 128 (2021)
Abstract
In this paper, the authors present a strategy based on fixed point iterative methods to solve a nonlinear dynamical problem in a form of Green’s function with boundary value problems. First, the authors construct the sequence named Green’s normalS iteration to show that the sequence converges strongly to a fixed point, this sequence was constructed based on the kinetics of the amperometric enzyme problem. Finally, the authors show numerical examples to analyze the solution of that problem.
Introduction
The development of a mathematical model based on diffusion has received a great deal of attention in recent years, many scientist and mathematician have tried to apply basic knowledge about the differential equation and the boundary condition to explain and approximate the diffusion and reaction model [1–11].
In 2017, Abukhaled and Khuri [12] solved a solution of amperometric enzymatic reaction based on Green’s function by using the fixed point iteration
subject to
They defined an operator based on the Picard iteration and proved that the operator is a contraction mapping that shows the sequence convergence with regard to Banach’s theorem.
Theorem Banach
([13])
Let \((M, d)\) be a complete metric space and \(P : M \rightarrow M\) be Banach’s contraction map (that is, there exists \(a \in [0, 1)\) such that
for all \(x, y \in M\). Then P has a unique fixed point \(p \in M\). Furthermore, for each \(x_{0} \in M\), the sequence \(\lbrace x_{n}\rbrace \) defined by
for each \(n\geq 0\) converges to the fixed point p.
In 2018, Khuri and Louhichi [14] presented a new numerical approach for the numerical solution of boundary value problems. The algorithm is defined in terms of Green’s function into the Ishikawa fixed point iteration [15]
where \(\{\beta _{n}\}\) and \(\{\alpha _{n}\}\) are sequences in \([0,1]\). Note that the step of \(y_{n}\) is called Mann’s iteration [16].
Further, the converge theorem was proved by using the theorem of Berinde [17].
Theorem Berinde
Let M be an arbitrary Banach space, K is a closed convex subset of M and \(P: K \to K\), which the operator satisfies the Zamfirescu operator. Let \(\{x_{n}\}_{n=0}^{\infty }\) be an Ishikawa iteration and \(x_{0} \in K\), where \(\{\alpha \}\) and \(\{\beta \}\) are sequences of positive numbers in \([0,1]\) with \(\{\beta _{n}\}\) satisfying \(\sum_{n=0}^{\infty }\beta _{n} = \infty \). Then \(\{x_{n}\}_{n=0}^{\infty }\) strongly converges to the fixed point of P.
The above operator is sometimes called Zamfirescu operator [18].
Theorem Zamfirescu
Let \((M, d)\) be a complete metric space and \(P: M \to M\) be a map for there exist the real numbers \(a_{1}\), \(a_{2}\), and \(a_{3}\) satisfying \(0 \leq a_{1} < 1\), \(0 \leq a_{2}\), \(a_{c} < 0.5\) such that, for each pair x, y in M, at least one of the following is true:
 \((z_{1})\):

\(d(Px,Py) \leq a_{1}d(x,y)\);
 \((z_{2})\):

\(d(Px,Py) \leq a_{2}[d(x,Px) + d(y,Py)]\);
 \((z_{3})\):

\(d(Px,Py) \leq a_{3}[d(x,Py) + d(y,Px)]\).
Then P has a unique fixed point p and the sequence \(\{x_{n}\}_{n=0}^{\infty }\) defined by Picard iteration
converges to p for any \(x_{0} \in M\).
In this paper, the authors use the motivation above to construct Green’s normalS iteration based on the sequence of normalS iteration of Sahu [19]. Let K be a convex subset of the normed space M and a nonlinear mapping P, the sequence \(\{x_{n}\}\) in K is call normalS if it is defined by
for each \(n \geq 1\), where \(\lbrace \beta _{n}\rbrace \) is the sequence in \([0,1]\).
The proof of the convergence theorem is based on Berinde’s idea. Finally, the authors use the sequence to approximate problem (1) subject to (2) by showing a numerical example.
Preliminaries
The mathematical model
Diffusion equations were presented by a mathematical model related to Michaelis–Menten kinetics (4) of the enzymatic reaction
where E is an enzyme, S is a substrate, ES is a complex between enzyme and substrate, and P is a product of reaction.
In biochemistry, the enzyme kinetics in ndimension Ω is modeled by the reactiondiffusion equation [20]
where \(D_{S}\) is the diffusion coefficient of a substrate and ν is the initial reaction velocity. By using the Michaelis–Menton hypothesis, the velocity ν for simple reaction processes without competitive inhibition is given by [20, 21]
where \(K = k_{2}E_{0}/K_{M}\) represents a pseudo first order, in which \(k_{2}\) is the unimolecular rate constant, \(E_{0}\) is the total amount of enzymes, and \(K_{M}\) is the Michaelis constant. The onedimensional form of (5) is given by
with the initial condition given by
By introducing the parameters
we obtain the nonlinear reactiondiffusion equation at steady state
where \(S^{\infty }\) is the substrate concentration in bulk solution (mol dm^{−3}), \(\phi ^{2}\) is the Thiele modulus.
Green’s function
Consider the second order differential equation decomposed into a linear term \(\operatorname{Li}[y]\) and a nonlinear term \(f(t,y,y')\) as follows:
subject to the boundary conditions
where \(a \leq t \leq b\). Bernfeld and Lakshmikantham [22] presented the existence and uniqueness theorems for solutions of (11).
The Green’s function \(G(t,s)\) corresponding to the linear term \(\operatorname{Li}[y]\) is defined as the solution of the following boundary value problem:
and has the piecewise form
where \(y_{1}\) and \(y_{2}\) form a fundamental set of solutions for \(\operatorname{Li}[y] = 0\). The unknowns could be found using the homogeneous conditions given in (12) and the fact that the Green’s function is continuous and its first derivative has a unit jump discontinuity. More precisely, the constants are determined using the following properties:

A.
G satisfies the corresponding homogeneous boundary conditions
$$ \operatorname{BC}_{a} \bigl[G(t, s) \bigr] = \operatorname{BC}_{b} \bigl[G(t, s) \bigr] = 0; $$(15) 
B.
G is continuous at \(t = s\), i.e.,
$$ c_{1}y_{1}(s) + c_{2}y_{2}(s) = d_{1}y_{1}(s) + d_{2}y_{2}(s); $$(16) 
C.
\(G'\) has a unit jump discontinuity at \(t = s\), i.e.,
$$ d_{1}y'_{1}(s)+d_{2}y'_{2}(s)  c_{1}y'_{1}(s)  c_{2}y'_{2}(s) = 1. $$(17)
A particular solution to \(y'' = f(t, y, y', y'')\) is expressed in terms of G and is given by the following structure:
We construct the Green’s function for the differential operator \(\operatorname{Li}[y] = y'' = 0\), which has two linearly independent solutions \(y_{1}(t) = 1\) and \(y_{2}(t) = t\). From (14), the Green’s function will have the form
where the unknowns are found by the properties A, B, and C listed above. To find the homogeneous boundary conditions, we have
\(G(t, s)\) is continuous and \(G'(t, s)\) discontinues at \(t = s\) then
From (19)–(21), we obtain the following Green’s function:
Green’s normalS iteration
Applying the Green’s function to the normalS iterative method, we recall the following differential equation:
where \(\operatorname{Li}[u]\) is a linear operator in y, \(\operatorname{No}[y]\) is a nonlinear operator in y, and \(f(t, y)\) is a linear or nonlinear function in y. Let \(y_{p}\) be a particular solution of (23). We define the linear integral operator in terms of the Green’s function and the particular solution \(y_{p}\) as follows:
Here, G is the Green’s function corresponding to the linear differential operator \(\operatorname{Li}[y]\). For convenience, we set \(y_{p} = v\). Adding and subtracting \(\operatorname{No}[v]  f(s, v)\) from within the integral in (24) yield
We then apply the normalS fixed point iterative form
where \(n \geq 0\), \((\beta _{n})\) is a sequence of real numbers in \([0,1]\). That is,
which is reduced to
Main results
Constructing the normalS Green’s iterative scheme
Let \(\operatorname{Li}[s] = \frac{\partial ^{2} s}{\partial x^{2}}\) and \(f(\alpha ,K,s) = \frac{Ks}{1+\alpha s}\), consider the enzyme substrate reaction equation, which takes the form of the following nonlinear equation:
with boundary condition (2), then the required Green’s function
subject to the corresponding homogenous boundary conditions
Using boundary condition (32) in Green’s function (19) then \(G(x,z)\), we obtain the equations
The continuity of G implies that
and \(\frac{d}{dx}G(x,z)\) jump discontinuity implies that
Hence,
From (25), we introduce the following continuous functions on \([0, 1]\) into itself:
then equations (27)–(29) become
Convergence theorems
In Theorem 1 we show that the operator \(P_{G}\) is a contraction mapping, and in Theorem 2 we show that if the operator P satisfies condition Z, then the sequence \(\{s_{n}\}_{n=0}^{\infty }\) defined by normalS (29) converges strongly to the fixed point of P.
Theorem 1
Assume that the function f, which appears in the definition of the operator \(P_{G}\), is such that
where \(C_{c} = \max_{x\in [0,1]}f'(s(x))\). Then \(P_{G}\) is a contraction, and hence the sequence \(\{s_{n}\}\) is defined by normalS iteration (29).
Proof
Performing integration by parts in equations (29), (36)–(38), the product is
where \(s = w_{n}\) of (38). Thus
By applying the mean value theorem for \(f(s)\) and using the condition that \(C_{c} = \max_{x\in [0,1]}f'(s(x))\), we consider the last inequality
where \(\sv\ = \max_{x\in [0,1]}s(x)v(x)\) and \(C =C_{c} < 1\). So, we obtain the following:
such that \(0 \leq C < 1\). Hence \(P_{G}\) is a contraction mapping. □
Theorem 2
Let M be an arbitrary Banach space, K be a closed convex subset of M, and \(P : K \rightarrow K\) be an operator satisfying the condition of Zamfirescu. Let \(\{s_{n}\}_{n=0}^{\infty }\) be defined by normalS (3) and \(s_{0} \in K\), where \(\{\beta _{n}\}\) is a sequence in \([0,1]\). Then \(\{s_{n}\}_{n=0}^{\infty }\) converges strongly to the fixed point of P.
Proof
By Zamfirescu’s theorem, we know that P has a unique fixed point in K that is p. Consider \(s, m \in K\). Since P is a Zamfirescu operator, at least one of conditions \((z_{1})\), \((z_{2})\), and \((z_{3})\) is satisfied. If \((z_{2})\) holds, then
so
from \(0 \leq a_{2} <1\)
Similarly, if \((z_{3})\) holds
Denote \(\delta = \max \{a_{1}, \frac{a_{2}}{1a_{2}}, \frac{a_{3}}{1a_{3}} \}\). Then we have \(0 \leq \delta < 1\) and get
The sequence \(\{s_{n}\}_{n=0}^{\infty }\) is defined by normalS iteration (3) and \(s_{0} \in K\), by (42) we get
Consider again
By (42) again,
So, we have
by induction
From \(\delta (1\beta _{k} + \beta _{k} \delta ) < 1\),
which implies
Therefore \(\{s_{n}\}_{n=0}^{\infty }\) converges strongly to the fixed point of P. This is completes the proof. □
Numerical examples
In the first example, we show a simple example to compare the solution with three iterative methods to explain the convergence of the sequences. In the last example, we present the main example to analyze the main problem (10).
Example 1
Consider the following differential equation \(x(t)\):
where \(0 \leq t \leq 1\) and subject to
The exact solution is \(x(t) = \frac{4}{(1+t)^{2}}\). The initial iterate satisfies \(x'' =0\) and boundary conditions (47). This \(x_{0} = 43t\). By normalS Green’s iteration (29),
Table 1 shows the convergence step, Fig. 1 shows the convergence step and the error step of sequence \(\{x_{n}\}\), which the error is calculated from \((\int _{a}^{b}x_{n}  x_{\mathrm{exact}}^{2})^{1/2}\).
Figure 1 shows a sequence of functions to compare three iterative methods. From the boundary condition, the value of problem starts at 4 and stops at 1. The back line is the solution of function, while red, blue, and green lines are Mann, normalS, and Ishikawa sequences, respectively. The figure concludes that Mann and normalS are converging faster than Ishikawa and converging nearly to the solution of the function.
Figure 2 shows the error of three iterative sequences to compare the error value. Red, blue, and green lines mark Mann, NormalS, and Ishikawa sequences, respectively. The figure concludes that normalS sequence is decreasing to 0 faster than the error of Mann and Ishikawa sequences.
Example 2
Consider the differential equation (10)
where \(0 \leq x \leq 1\) and subject to
The initial iterate satisfies \(s'' =0\) and the boundary conditions. Then \(s_{0} = 43x\). By normalS Green’s iteration (29) and from (36), (37), and (38), the sequence is defined by
where K and α are constants of substrate concentration, and set \(\beta _{n} = 0.005+\frac{0.0000001}{n^{2}}\).
Table 2 and Fig. 3 show approximation of substrate concentration sequence \(S(x)\) for different values of α and K.
Explanation of Fig. 3: Firstly, the error of normalS sequence \(S(x)\) compares with different values of α with \(K = 0.00001\), the error sequence of large α converges faster than that of small α. Secondly, the error of normalS sequence \(S(x)\) which compared by different values of K with \(\alpha = 1000\), the error sequence of small K converges faster than that of large K.
Conclusion
This paper presents a strategy based on fixed point iterative methods with normalS iteration (38) to solve a nonlinear dynamical problem in a form of Green’s function with boundary conditions used in Theorem 1 and Theorem 2 to guarantee the solution. Example 2 explains two constants K and α in the nonlinear reactiondiffusion equation at steady state (1). Therefore, the values of K must be small, while the values of α should be large, so the error value of sequence will converge to 0 faster than the other cases.
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Acknowledgements
The authors would like to thank the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Cluster (CLASSIC), Faculty of Science, KMUTT and Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok (KMUTNB), Bangkok, Thailand, contract no. KMUTNBBasicR6422, for financial support.
Funding
This project was supported by the Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) and the Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok (KMUTNB), Bangkok, Thailand. Contract no. KMUTNBBasicR6422.
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Muangchooin, K., Sitthithakerngkiet, K., SaNgiamsunthorn, P. et al. Approximation theorems of a solution of amperometric enzymatic reactions based on Green’s fixed point normalS iteration. Adv Differ Equ 2021, 128 (2021). https://doi.org/10.1186/s1366202103289w
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DOI: https://doi.org/10.1186/s1366202103289w
Keywords
 Fixed point iteration
 Green’s function
 Enzymatic reaction
 Boundary value problems