Approximation theorems of a solution of amperometric enzymatic reactions based on Green’s fixed point normal-S iteration

*Correspondence: poom.kum@kmutt.ac.th 1Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand 4Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand Full list of author information is available at the end of the article Abstract


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][2][3][4][5][6][7][8][9][10][11].
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. for all x, y ∈ M. Then P has a unique fixed point p ∈ M. Furthermore, for each x 0 ∈ M, the sequence {x n } defined by x n+1 = Px n for each n ≥ 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] ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x 0 ∈ M, y n = (1β n )x n + β n Px n , where {β n } and {α 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]. In this paper, the authors use the motivation above to construct Green's normal-S iteration based on the sequence of normal-S 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 normal-S if it is defined by 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.

The mathematical model
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 n-dimension is modeled by the reactiondiffusion equation [20] ∂S ∂t 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] ν(t, X) 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 reaction-diffusion equation at steady state where S ∞ is the substrate concentration in bulk solution (mol dm -3 ), φ 2 is the Thiele modulus.

Green's function
Consider the second order differential equation decomposed into a linear term Li[y] and a nonlinear term f (t, y, y ) as follows: subject to the boundary conditions where a ≤ t ≤ 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 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 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 c 1 y 1 (s) + c 2 y 2 (s) = d 1 y 1 (s) + d 2 y 2 (s); C. G has a unit jump discontinuity at t = s, i.e., 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 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 From (19)- (21), we obtain the following Green's function:

Green's normal-S iteration
Applying the Green's function to the normal-S iterative method, we recall the following differential equation: where Li[u] is a linear operator in y, 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 We then apply the normal-S fixed point iterative form where n ≥ 0, (β n ) is a sequence of real numbers in [0, 1]. That is, which is reduced to

Constructing the normal-S Green's iterative scheme
Let Li[s] = ∂ 2 s ∂x 2 and f (α, K, s) = Ks 1+αs , 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 c 2 = 0 and d 1 + d 2 = 0.
The continuity of G implies that and d dx G(x, z) jump discontinuity implies that Hence, From (25), we introduce the following continuous functions on [0, 1] into itself:

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 defined by normal-S (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∈[0,1] |f (s(x))|. Then P G is a contraction, and hence the sequence {s n } is defined by normal-S 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∈[0,1] |f (s(x))|, we consider the last inequality and C = C c < 1. So, we obtain the following: Hence P G is a contraction mapping. Proof By Zamfirescu's theorem, we know that P has a unique fixed point in K that is p. Consider s, m ∈ 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 Similarly, if (z 3 ) holds Denote δ = max{a 1 , a 2 1-a 2 , a 3 1-a 3 }. Then we have 0 ≤ δ < 1 and get The sequence {s n } ∞ n=0 is defined by normal-S iteration (3) and s 0 ∈ K , by (42) we get Consider again s np = (1β n )s n + β n Ps np = (1β n )(s np) + β n (Ps np) = (1β n ) s np + β n Ps np .
So, we have From which implies lim n→∞ s n+1p = 0.
Therefore {s n } ∞ n=0 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 ≤ t ≤ 1 and subject to The exact solution is x(t) = 4 (1+t) 2 . The initial iterate satisfies x = 0 and boundary conditions (47). This x 0 = 4 -3t. By normal-S Green's iteration (29), (48) 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 ( b a |x nx 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, normal-S, and Ishikawa  sequences, respectively. The figure concludes that Mann and normal-S 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, Normal-S, and Ishikawa sequences, respectively. The figure concludes that normal-S sequence is decreasing to 0 faster than the error of Mann and Ishikawa sequences. The initial iterate satisfies s = 0 and the boundary conditions. Then s 0 = 4 -3x. By normal-S Green's iteration (29) and from (36), (37), and (38), the sequence is defined by w n = s n + β n x 0 (z -1) s n (z) -Ks n (z) 1 + αs n (z) dz dz, where K and α are constants of substrate concentration, and set β n = 0.005 + 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 normal-S 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 normal-S sequence S(x) which compared by different values of K with α = 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 normal-S 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 reaction-diffusion 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.