Analytical solution to the nonlinear singular boundary value problem arising in biology
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Abstract
This paper proposes an analytical procedure for the nonlinear singular boundary value problem that arises in biology and in the study of some diseases. As a first step, we present a constructive proof of the existence and uniqueness of solution. Then, we apply the Picard iterative sequence by constructing an integral equation whose Green’s function is not negative. The convergence of this iterative sequence is then controlled by an embedded parameter so that it tends to the unique solution.
Keywords
singular boundary value problem Picard iterative sequence biology1 Introduction
On the other hand, boundary value problem of the form (1.1)(1.2) involving the governing ordinary differential equation (1.1) or slight generalizations of it has been investigated by Gatica et al. [12], Fink et al. [13], Baxley [14], Baxley and Gersdorff [15], Wang and Li [16], Tinio [17], Wang [18], Ebaid [19], Agarwal and O’Regan [20] and also by the authors of [21, 22]. In these works, the fixed point theory or approximation theory was prevalently used, and their studies included assumptions that restrict \(f(x,y)\) to be of one sign and usually continuous for \(y\geq0\). Later, in Refs [23, 24, 25], authors started to allow signchanging nonlinearities but still they have required \(f(x,y)\) to be continuous for \(y\geq0\). We notice that, in all these works, it has been assumed \(n=0\) and they have considered some particular types of boundary conditions (1.2) or some specific forms of governing differential equation (1.1).
The aim of the present work is to provide some constructive existence and uniqueness theorems for problem (1.1)(1.2) of the same type as those provided by Ford and Pennline in [26] by the same assumptions considered by them, i.e., \(f (x,y(x) )\) is not only allowed to be signchanging but it also can be singular with respect to y. Also, we apply the Picard iterative sequence by constructing an integral equation whose Green’s function is not negative. The convergence of this iterative sequence is then controlled by an embedded parameter so that it converges to the unique solution.
2 Constructing the integral equation
3 Some properties of the Green’s function
In this section, we present some properties of the Green’s function as theorems which are extremely important in the analysis of (1.1)(1.2).
Lemma 3.1

The function \(v(x)\) is a positive decreasing function of \(x\in [0,1]\) and unbounded at the origin.

Assuming that m, n and k are such that \(W(1)\) is positive, then \(u(x)\) is a positive increasing function of \(x\in[0,1]\).

Assuming that m, n and k are such that \(W(1)\) is negative, then \(u(x)\) is a negative decreasing function of \(x\in[0,1]\).
Theorem 3.2
Proof
First, suppose that m, n and k are such that \(W(1)\) is positive, then from Lemma 3.1 it follows that \(u(x)\) is a positive function of \(x\in[0,1]\). On the other hand, \(v(x)\) is a positive decreasing function of \(x\in[0,1]\) from Lemma 3.1. Therefore, we conclude that \(G(x,t)\geq0\).
Second, suppose that m, n and k are such that \(W(1)\) is negative, then from Lemma 3.1 it follows that \(u(x)\) is a negative function of \(x\in[0,1]\). On the other hands, \(v(x)\) is a positive decreasing function of \(x\in[0,1]\) from Lemma 3.1. Therefore, we conclude again that \(G(x,t)\geq0\). □
Theorem 3.3
Proof
4 The region of existence and uniqueness
Theorem 4.1
 (a)
\(\frac{\partial f}{\partial y}\) is continuous in \(\mathbf {D}: [0,1]\times[y_{L}(x),y_{U}(x)]\) and satisfies \(0\leq\frac{\partial f}{\partial y}\leq N_{D}\) within it.
 (b)
\(y_{0}(x)=\frac{1}{2}[y_{L}(x)+y_{U}(x)]\) and \(y_{L}(x)\leq y_{n}(x)\leq y_{U}(x), n=1,2,3,\ldots \) .
 (c)
The value of \(k^{2}\) satisfies \(k^{2}\geq\frac{N_{D}}{2}\).
Proof
5 Solution procedure and illustrative physiology models
Example 5.1
Example 5.2
Notes
Acknowledgements
The authors are grateful to anonymous reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.
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