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# Positive solutions of nonlinear Dirichlet BVPs in ODEs with time and space singularities

- Irena Rachůnková
^{1}Email author, - Alexander Spielauer
^{2}, - Svatoslav Staněk
^{1}and - Ewa B Weinmüller
^{2}

**2013**:6

https://doi.org/10.1186/1687-2770-2013-6

© Rachůnková et al.; licensee Springer. 2013

**Received:**17 October 2012**Accepted:**21 December 2012**Published:**16 January 2013

## Abstract

In this paper, we discuss the existence of positive solutions to the singular Dirichlet boundary value problems (BVPs) for ordinary differential equations (ODEs) of the form

where $a\in (-1,0)$. The nonlinearity $f(t,x,y)$ may be singular for the space variables $x=0$ and/or $y=0$. Moreover, since $a\ne 0$, the differential operator on the left-hand side of the differential equation is singular at $t=0$. Sufficient conditions for the existence of positive solutions of the above BVPs are formulated and asymptotic properties of solutions are specified. The theory is illustrated by numerical experiments computed using the open domain MATLAB code bvpsuite, based on polynomial collocation.

**MSC:**34B18, 34B16, 34A12.

## Keywords

- singular ordinary differential equation of the second order
- time singularities
- space singularities
- positive solutions
- existence of solutions
- polynomial collocation

## 1 Introduction

*f*satisfies the local Carathéodory conditions on $[0,T]\times \mathcal{D}$, where $\mathcal{D}:={\mathbb{R}}_{+}\times {\mathbb{R}}_{0}$ and ${\mathbb{R}}_{+}:=(0,\mathrm{\infty})$, ${\mathbb{R}}_{0}:=\mathbb{R}\setminus \{0\}$. Let us recall that a function $h:[0,T]\times \mathcal{A}\to \mathbb{R}$, $\mathcal{A}\subset \mathbb{R}\times \mathbb{R}$, satisfies the

*local Carathéodory conditions*on $[0,T]\times \mathcal{A}$ if

- (i)
$h(\cdot ,x,y):[0,T]\to \mathbb{R}$ is measurable for all $(x,y)\in \mathcal{A}$,

- (ii)
$h(t,\cdot ,\cdot ):\mathcal{A}\to \mathbb{R}$ is continuous for a.e. $t\in [0,T]$,

- (iii)for each compact set $\mathcal{U}\subset \mathcal{A}$, there exists a function ${m}_{\mathcal{U}}\in {L}^{1}[0,T]$ such that$|h(t,x,y)|\le {m}_{\mathcal{U}}(t)\phantom{\rule{1em}{0ex}}\text{for a.e.}t\in [0,T]\text{and all}(x,y)\in \mathcal{U}.$

*x*and/or

*y*vanish, which means that $f(t,x,y)$ may become unbounded for $x=0$ and a.e. $t\in [0,T]$ and all $y\in {\mathbb{R}}_{0}$, and/or it may be unbounded for $y=0$ and a.e. $t\in [0,T]$ and all $x\in {\mathbb{R}}_{+}$. Finally, since $a\ne 0$, Eq. (1a) has a singularity of the first kind at the time variable $t=0$ because

The differential operator on the left-hand side of Eq. (1a) can be equivalently written as ${({t}^{-a}{({t}^{a}u)}^{\prime})}^{\prime}$ and, after the substitution $x(t)={t}^{a}u(t)$, it takes the form ${({t}^{-a}{x}^{\prime})}^{\prime}$, which arises in numerous important applications. Operators of such type were studied in phase transitions of Van der Waals fluids [1–4], in population genetics, especially in models for the spatial distribution of the genetic composition of a population [5, 6], in the homogeneous nucleation theory [7], in relativistic cosmology for description of particles which can be treated as domains in the universe [8], and in the nonlinear field theory [9], in particular, when describing bubbles generated by scalar fields of Higgs type in the Minkowski spaces [10].

The aim of this paper is to study the case $a\in (-1,0)$ which is fundamentally different from the case $a\in (-\mathrm{\infty},-1)$. The latter setting was studied in [11, 12], where the structure and properties of the set of all positive solutions to (1a) and (1b) were investigated (the cardinality of this set is a continuum).

In the sequel, we introduce the basic notation and state the preliminary results required in the analysis of problem (1a) and (1b). Here, we focus our attention on the case $a\in (-1,0)$ and prove the existence of at least one positive solution of (1a) and (1b). In contrast to [11, 12], we consider the more general situation in which *f* may be also singular at $y=0$. This means that we have to deal with the following additional difficulties.

Let *u* be a positive solution of problem (1a) and (1b), where $f(t,x,y)$ has a singularity at $y=0$. Then there exists ${t}_{0}\in (0,T)$ such that $u({t}_{0})>0$, ${u}^{\prime}({t}_{0})=0$ and hence *f* is unbounded in a neighborhood of the point $({t}_{0},u({t}_{0}),{u}^{\prime}({t}_{0}))$. Unfortunately, we do not know the exact position of ${t}_{0}$ and therefore, it is not possible to construct a universal Lebesgue integrable majorant for all functions $f(t,{u}_{n}(t),{u}_{n}^{\prime}(t))$, where ${u}_{n}$ are positive solutions of a sequence of auxiliary regular problems. Consequently, the Lebesgue dominated convergence theorem is not applicable and we have to use arguments based on the Vitali convergence theorem instead; see Lemma 2. Another tool used in the proofs is a combination of regularization and sequential techniques with the Leray-Schauder nonlinear alternative.

The investigation of singular Dirichlet BVPs has a long history and a lot of methods for their analysis are available. One of the most important ones is the topological degree method providing various fixed point theorems and existence alternative theorems; see, *e.g.*, Lemma 1. For more information on the topological degree method and its application to numerous BVPs, including Dirichlet problems, we refer the reader to the monographs by Mawhin [13–15].

Throughout this paper, we work with the following conditions on the function *f* in (1a):

(H_{1}) $f\in Car([0,T]\times \mathcal{D})$, where $\mathcal{D}={\mathbb{R}}_{+}\times {\mathbb{R}}_{0}$.

_{2}) There exists an $\epsilon >0$ such that

_{3}) For a.e. $t\in [0,T]$ and all $(x,y)\in \mathcal{D}$, the estimate

*h*is nondecreasing in both its arguments,

*g*and

*r*are nonincreasing, and

we denote the norms in $C[0,T]$ and ${L}^{1}[0,T]$, respectively. $A{C}^{1}[0,T]$ denotes the set of functions whose first derivative is absolutely continuous on $[0,T]$, while $A{C}_{\mathrm{loc}}^{1}(0,T]$ is the set of functions having absolutely continuous first derivative on each compact subinterval of $(0,T]$. We use the symbol $meas(\mathcal{M})$ to denote the Lebesgue measure of ℳ.

**Definition 1** We say that a function $u\in A{C}^{1}[0,T]$ is *a positive solution of problem* (1a) and (1b) if $u>0$ on $(0,T)$, *u* satisfies the boundary conditions (1b) and (1a) holds for a.e. $t\in [0,T]$.

**Remark 1**Let a function

*g*have the properties specified in (H

_{3}). Then for each $b,c\in {\mathbb{R}}_{+}$, ${\int}_{0}^{b}g(cs)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s<\mathrm{\infty}$, and it follows from the inequality

_{1}) that ${f}_{n}\in Car([0,T]\times {\mathbb{R}}^{2})$ and (H

_{2}) and (H

_{3}) yield

To show the solvability of problem (7a) and (7b), we use the following alternative of Leray-Schauder type which follows from [[16], Theorem 5.1].

**Lemma 1** *Let* *E* *be a Banach space*, *U* *be an open subset of* *E* *and* $\ell \in U$. *Assume that* $\mathcal{F}:\overline{U}\to E$ *is a compact operator*. *Then either*

A1: ℱ *has a fixed point in* $\overline{U}$, *or*

A2: *there exists an element* $u\in \partial U$ *and* $\lambda \in (0,1)$ *with* $u=\lambda \mathcal{F}(u)+(1-\lambda )\ell $.

In limit processes, we apply the following Vitali convergence theorem; *cf.* [17–19].

**Lemma 2**

*Let*$\{{\rho}_{n}\}\subset {L}^{1}[0,T]$

*and let*${lim}_{n\to \mathrm{\infty}}{\rho}_{n}(t)=\rho (t)$

*for a*.

*e*. $t\in [0,T]$.

*Then the following statements are equivalent*:

- (i)
$\rho \in {L}^{1}[0,T]$

*and*${lim}_{n\to \mathrm{\infty}}{\parallel {\rho}_{n}-\rho \parallel}_{1}=0$, - (ii)
*the sequence*$\{{\rho}_{n}\}$*is uniformly integrable on*$[0,T]$.

*uniformly integrable*on $[0,T]$ if for any $\epsilon >0$ there exists $\delta >0$ such that if $\mathcal{M}\subset [0,T]$ and $meas(\mathcal{M})<\delta $, then

The paper is organized as follows. In Section 2, we collect auxiliary results used in the subsequent analysis. Section 3 is devoted to the study of limit properties of solutions to Eq. (7a). In Section 4, we investigate auxiliary regular problems associated with the singular problem (1a) and (1b). We show their solvability and describe properties of their solutions. An existence result for the singular problem (1a) and (1b) is given in Section 5. Finally, in Section 6, we illustrate the theoretical findings by means of numerical experiments.

Throughout the paper $a\in (-1,0)$.

## 2 Preliminaries

In this section, auxiliary statements necessary for the subsequent analysis are formulated.

*Then*

- (i)
*r**can be extended on*$[0,T]$*with*$r\in C[0,T]$*and*$r(0)=0$, - (ii)
*H**can be extended on*$[0,T]$*with*$H\in A{C}^{1}[0,T]$,*and the equality*${H}^{\u2033}(t)+\frac{a}{t}{H}^{\prime}(t)-\frac{a}{{t}^{2}}H(t)=-\rho (t)$(8)

*holds for a*.*e*. $t\in [0,T]$.

*Proof*(i) It is clear that $r\in C(0,T]$. Since

- (ii)Let$p(t)={\int}_{t}^{T}\frac{1}{{s}^{a+2}}({\int}_{0}^{s}{\xi}^{a+1}\rho (\xi )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\xi )\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\phantom{\rule{1em}{0ex}}\text{for}t\in (0,T].$

*p*can be extended on $[0,T]$ in such a way that $p\in C[0,T]$. Integrating by parts yields

*H*can be extended on $[0,T]$ with $H\in {C}^{1}[0,T]$. Moreover,

*cf.*(10),

and therefore, ${H}^{\u2033}\in {L}^{1}[0,T]$. Consequently, $H\in A{C}^{1}[0,T]$. Finally, it follows from ${H}^{\prime}(t)=p(t)-r(t)$ and $p(t)=\frac{H(t)}{t}$ that $r(t)=\frac{H(t)}{t}-{H}^{\prime}(t)$. Since, by (11), $\frac{a}{t}r(t)={H}^{\u2033}(t)+\rho (t)$, we see that equality (8) is satisfied for a.e. $t\in [0,T]$ which completes the proof. □

**Lemma 4**

*Let*$\{{\rho}_{n}\}\subset {L}^{1}[0,T]$

*be a uniformly integrable sequence on*$[0,T]$

*and let*${lim}_{n\to \mathrm{\infty}}{\rho}_{n}(t)=\rho (t)$

*for a*.

*e*. $t\in [0,T]$.

*Then the sequence*

*Proof*It follows from Lemma 2 that ${\parallel {\rho}_{n}\parallel}_{1}\le L$ for $n\in \mathbb{N}$, where

*L*is a positive constant. Recall that by Lemma 3(i), $\{\frac{1}{{t}^{a+1}}{\int}_{0}^{t}{s}^{a+1}{\rho}_{n}(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\}\subset C[0,T]$. Let us assume that (12) does not hold. Then there exist $\epsilon >0$, $\{{k}_{n}\}\subset \mathbb{N}$ and $\{{\xi}_{n}\},\{{\eta}_{n}\}\subset [0,T]$ such that ${lim}_{n\to \mathrm{\infty}}{k}_{n}=\mathrm{\infty}$, ${lim}_{n\to \mathrm{\infty}}({\xi}_{n}-{\eta}_{n})=0$ and

*cf.*(9))

The last equality contradicts (13). Consequently, (12) holds and the result follows. □

**Lemma 5**

*Let*$\rho \in {L}^{1}[0,T]$.

*Then*

*Proof*Since (

*cf.*(10))

for $t\in [0,T]$, estimate (14) holds. □

## 3 Limit properties of solutions to Eq. (7a)

Here, we investigate asymptotic properties of solutions of (7a). We also provide a related integral equation this solution satisfies.

**Lemma 6**

*Let*(H

_{1})

*hold*.

*Let*$u\in A{C}_{\mathrm{loc}}^{1}(0,T]$

*satisfy Eq*. (7a)

*for a*.

*e*. $t\in [0,T]$

*and*$L:=sup\{|u(t)|+|{u}^{\prime}(t)|:t\in (0,T]\}<\mathrm{\infty}$.

*Then*

*u*

*can be extended on*$[0,T]$

*with*$u\in A{C}^{1}[0,T]$,

*and there exists*$c\in \mathbb{R}$

*such that the integral equation*

*holds for* $t\in [0,T]$.

*Proof*Choose $n\in \mathbb{N}$ and denote by $\rho (t)={f}_{n}(t,u(t),{u}^{\prime}(t))$ for a.e. $t\in [0,T]$. In order to prove that $\rho \in {L}^{1}[0,T]$, define for $m\in \mathbb{N}$, $\frac{1}{m}<T$,

Then ${\rho}_{m}\in {L}^{1}[0,T]$ and ${lim}_{m\to \mathrm{\infty}}{\rho}_{m}(t)=\rho (t)$ for a.e. $t\in [0,T]$. Moreover, $|{\rho}_{m}(t)|\le \mu (t)$ for a.e. $t\in [0,T]$ and all $m\in \mathbb{N}$, where $\mu (t)=sup\{|{f}_{n}(t,x,y)|:|x|\le L,|y|\le L\}\in {L}^{1}[0,T]$. Consequently, by the Lebesgue dominated convergence theorem, $\rho \in {L}^{1}[0,T]$.

Let *H* be the function given in Lemma 3. By Lemma 3(ii), *H* can be extended on $[0,T]$ with $H\in A{C}^{1}[0,T]$ and −*H* satisfies (16) for a.e. $t\in [0,T]$. Therefore, each function $v\in A{C}_{\mathrm{loc}}^{1}(0,T]$ which satisfies Eq. (16) a.e. on $[0,T]$ has the form $v(t)={c}^{\ast}t+{d}^{\ast}{t}^{-a}-H(t)$ for $t\in (0,T]$, with some ${c}^{\ast},{d}^{\ast}\in \mathbb{R}$. By assumption we know that $u\in A{C}_{\mathrm{loc}}^{1}(0,T]$ satisfies (16) a.e. on $[0,T]$, and therefore there exist $c,d\in \mathbb{R}$ such that $u(t)=ct+d{t}^{-a}-H(t)$, $t\in (0,T]$. Since by assumption $|{u}^{\prime}|\le L$ on $(0,T]$, we have $d=0$. Consequently, the function *u* can be extended on the interval $[0,T]$ in the class $A{C}^{1}[0,T]$ and (15) holds on $[0,T]$. □

**Corollary 1** *Let* (H_{1}) *hold*. *Let* $u\in A{C}^{1}[0,T]$ *be a solution of Eq*. (7a). *Then there exists a constant* $c\in \mathbb{R}$ *such that equality* (15) *is satisfied for* $t\in [0,T]$.

*Proof* The result holds by Lemma 6 with $sup\{|u(t)|+|{u}^{\prime}(t)|:t\in [0,T]\}<\mathrm{\infty}$. □

**Remark 2** Corollary 1 says that the set of all solutions $u\in A{C}^{1}[0,T]$ of Eq. (7a) depends on one parameter $c\in \mathbb{R}$ and $u(0)=0$.

## 4 Auxiliary regular problems

depending on the parameter *λ*. Here, $\epsilon >0$ is from (H_{2}) and $n\in \mathbb{N}$.

The following result shows that the solvability of problem (17a) and (17b) is equivalent to the solvability of an integral equation in the set ${C}^{1}[0,T]$.

**Lemma 7**

*Let*(H

_{1})

*hold*.

*Then*

*u*

*is a solution of problem*(17a)

*and*(17b)

*if and only if*

*u*

*is a solution of the integral equation*

*in the set* ${C}^{1}[0,T]$.

*Proof*Let

*u*be a solution of Eq. (17a). Then $u\in A{C}^{1}[0,T]$, and by Corollary 1 (with ${f}_{n}$ replaced by $\lambda {f}_{n}-(1-\lambda )\epsilon $), there exists $c\in \mathbb{R}$ such that the equation

holds for $t\in [0,T]$. Hence, $u(0)=0$ and $u(T)=0$ if and only if $c=0$. Consequently, if *u* is a solution of problem (17a) and (17b), then *u* is a solution of Eq. (18) in ${C}^{1}[0,T]$.

Let *u* be a solution of Eq. (18) in ${C}^{1}[0,T]$. Then ${f}_{n}(t,u(t),{u}^{\prime}(t))\in {L}^{1}[0,T]$. Hence, Lemma 3(ii) (with *ρ* replaced by $-\lambda {f}_{n}+(1-\lambda )\epsilon $) guarantees that $u\in A{C}^{1}[0,T]$ and *u* is a solution of Eq. (17a). Moreover, $u(0)=u(T)=0$. Consequently, *u* is a solution of problem (17a) and (17b) which completes the proof. □

The following results provide bounds for solutions of problem (17a) and (17b).

**Lemma 8**

*Let*(H

_{1})-(H

_{3})

*hold*.

*Then there exists a positive constant*

*S*(

*independent of*$n\in \mathbb{N}$

*and*$\lambda \in [0,1]$)

*such that for all solutions*

*u*

*of problem*(17a)

*and*(17b),

*the estimates*

*hold*.

*Moreover*,

*for any solution*

*u*

*of problem*(17a)

*and*(17b),

*there exists*$\xi \in (0,T)$

*such that*

*Proof*Let

*u*be a solution of problem (17a) and (17b). Then by Lemma 7, equality (18) holds for $t\in [0,T]$. Since by (5), $\lambda {f}_{n}(t,u(t),{u}^{\prime}(t))-(1-\lambda )\epsilon \le -\epsilon $ for a.e. $t\in [0,T]$, the relation

*g*is nonincreasing on ${\mathbb{R}}_{+}$. Due to Remark 1, $L={\parallel g(\frac{\epsilon}{a+2}t(T-t))\parallel}_{1}<\mathrm{\infty}$, which means that

It is clear that *L* is independent of the choice of solution *u* to problem (17a) and (17b) and independent of $n\in \mathbb{N}$, $\lambda \in [0,1]$.

_{3}),

Consequently, there exists $S>0$ such that $Kh(1+Tz,1+z)+M<z$ for $z\ge S$. Now, due to (26), ${\parallel {u}^{\prime}\parallel}_{\mathrm{\infty}}<S$, and therefore, by (25), ${\parallel u\parallel}_{\mathrm{\infty}}<ST$. □

We are now in the position to prove the existence result for problem (7a) and (7b).

**Lemma 9** *Let* (H_{1})-(H_{3}) *hold*. *Then for each* $n\in \mathbb{N}$, *problem* (7a) *and* (7b) *has a solution* *u* *satisfying inequalities* (19)-(21), *where* *S* *is a positive constant independent of* *n*.

*Proof*Let

*S*be a positive constant in Lemma 8 and let us define

- (i)
$\mathcal{K}(1,\cdot ):\overline{\mathrm{\Omega}}\to {C}^{1}[0,T]$ is a compact operator, and

- (ii)
$\mathcal{K}(\lambda ,x)\ne x$ for each $\lambda \in (0,1]$ and $x\in \partial \mathrm{\Omega}$.

holds for a.e. $t\in [0,T]$ and all $x\in \overline{\mathrm{\Omega}}$ (*cf.* (9)). Consequently, the set $\{\mathcal{K}{(1,x)}^{\prime}:x\in \overline{\mathrm{\Omega}}\}$ is equicontinuous on $[0,T]$. Hence, the set $\mathcal{K}(1,\overline{\mathrm{\Omega}})$ is relatively compact in ${C}^{1}[0,T]$ by the Arzelà-Ascoli theorem. As a result, $\mathcal{K}(1,\cdot )$ is a compact operator and the condition (i) follows.

Due to the fact that by Lemma 7 any fixed point *u* of the operator $\mathcal{K}(\lambda ,\cdot )$ is a solution of problem (17a) and (17b), Lemma 8 guarantees that *u* satisfies inequality (20). Therefore, $\mathcal{K}$ has property (ii). Consequently, by Lemmas 1 and 8, for each $n\in \mathbb{N}$, problem (7a) and (7b) has a solution *u* satisfying estimates (19)-(21). □

Let ${u}_{n}$ be a solution of problem (7a) and (7b) for $n\in \mathbb{N}$. The following property of the sequence $\{|{f}_{n}(t,{u}_{n}(t),{u}_{n}^{\prime}(t))|\}$ is an important prerequisite for solving problem (1a) and (1b).

**Lemma 10** *Let* (H_{1})-(H_{3}) *hold*. *Let* ${u}_{n}$ *be a solution of problem* (7a) *and* (7b) *for* $n\in \mathbb{N}$. *Then the sequence* $\{|{f}_{n}(t,{u}_{n}(t),{u}_{n}^{\prime}(t))|\}$ *is uniformly integrable on* $[0,T]$.

*Proof*

*S*is a positive constant and ${\xi}_{n}\in (0,T)$. Hence, by (3) and (4),

for a.e. $t\in [0,T]$, where ${g}_{\ast}(t)=g(\frac{\epsilon}{a+2}t(T-t))\in {L}^{1}[0,T]$, see Remark 1. Since the sequence $\{r(\frac{2\epsilon}{a+2}|t-{\xi}_{n}|)\}$ is uniformly integrable on $[0,T]$ (*cf.* [[20], criterion A.4], [21, 22]), it follows from (32) that $\{|{f}_{n}(t,{u}_{n}(t),{u}_{n}^{\prime}(t))|\}$ is uniformly integrable on $[0,T]$ and the result follows. □

## 5 The existence result for BVP (1a) and (1b)

This section is devoted to the main result on the existence of positive solutions to the original BVP (1a) and (1b).

**Theorem 1** *Let* (H_{1})-(H_{3}) *hold*. *Then problem* (1a) *and* (1b) *has at least one positive solution*.

*Proof*By Lemma 9, for each $n\in \mathbb{N}$, problem (7a) and (7b) has a solution ${u}_{n}$ satisfying inequalities (29)-(31), where

*S*is a positive constant and ${\xi}_{n}\in (0,T)$. Moreover, by Lemma 10, the sequence $\{|{f}_{n}(t,{u}_{n}(t),{u}_{n}^{\prime}(t))|\}$ is uniformly integrable on $[0,T]$. We now prove that $\{{u}_{n}^{\prime}\}$ is equicontinuous on $[0,T]$. Since ${u}_{n}$ is a fixed point of the operator $\mathcal{K}(1,\cdot )$ given in (27), the equality

and $u\in A{C}^{1}[0,T]$ by Lemma 3(ii). This means that *u* is a positive solution of problem (1a) and (1b) and the result follows. □

## 6 Numerical simulations

where $c>0$ is a parameter. We can use the above formulation because problem (1a) and (1b) is solvable for *f* satisfying the assumptions of Theorem 1 and, therefore, solutions of problem (1a) and (1b) can be computed as solutions of problem (36a) and (36b) using the proper value $c\in (0,{c}^{\ast})$ depending on *f*. The values ${c}^{\ast}$ are provided for given *f* in Examples 1 and 2, below.

The reason for changing the boundary conditions from (1b) to (36b) is that the differential equation (36a) subject to (1b) is not well posed; see [23]. However, to enable successful numerical treatment, well-posedness of the model is crucial. This property means that Eq. (36a) subject to proper boundary conditions has at least a *locally unique* solution,^{a} and this solution depends continuously on the problem data. The well-posedness of the problem is important for two reasons. First of all, it allows to express errors in the solution of the analytical problem in terms of modeling errors and data errors (all measured via appropriate norms). Therefore, when the errors in the data become smaller due to more precise modeling or smaller measurement inaccuracies, the errors in the solution will decrease. The second reason is that the well-posedness decides if the numerical simulation will be at all successful. If the analytical problem is ill-posed, then the inevitable round-off errors can become extremely magnified and fully spoil the accuracy of the approximation.

whose roots ${\lambda}_{1}=1$ and ${\lambda}_{2}=-a$ are positive. Therefore, conditions for *u* and ${u}^{\prime}$ can be prescribed at $t=1$ as it is done in (36b).

### 6.1 MATLAB Code bvpsuite

To illustrate the analytical results discussed in the previous section, we solved numerically examples of the form (36a) and (36b) using a MATLAB™ software package bvpsuite designed to solve BVPs in ODEs and differential algebraic equations. The solver routine is based on a class of collocation methods whose orders may vary from two to eight. Collocation has been investigated in the context of singular differential equations of first and second order in [24, 25], respectively. This method could be shown to be robust with respect to *singularities in time* and retains its high convergence order in the case that the analytical solution is appropriately smooth. The code also provides an asymptotically correct estimate for the global error of the numerical approximation. To enhance the efficiency of the method, a mesh adaptation strategy is implemented, which attempts to choose grids related to the solution behavior in such a way that the tolerance is satisfied with the least possible effort. Error estimate procedure and the mesh adaptation work dependably provided that the solution of the problem and its global error are appropriately smooth.^{b} The code and the manual can be downloaded from http://www.math.tuwien.ac.at/~ewa. For further information, see [26]. This software proved useful for the approximation of numerous singular BVPs important for applications; see, *e.g.*, [3, 9, 27, 28].

### 6.2 Preliminaries

^{c}model of the form

*a*was chosen as $a=-0.1,-0.5,-0.9$. Since in this case the exact solution is given, $u(t)=\frac{2t(1-\sqrt{t})}{a+1.5}$, the value ${u}^{\prime}(1)$ is available, ${u}^{\prime}(1)=-0.72,-1,-1.67$, respectively. In Figure 1, the numerical solutions of BVPs (39a) and (39b) are shown. They will be used as starting values for the numerical solution of Examples 1 and 2; see Sections 6.3 and 6.4, respectively. All numerical results have been obtained using collocation with five Gaussian collocation points on an equidistant grid (justified by a very simple solution structure) with the step size 0.01.

### 6.3 Example 1

*f*in (40a) has the form

_{1})-(H

_{3}) with $\epsilon =1$, $\phi (t)=\frac{1}{\sqrt{t}}$ for $t\in (0,1]$ and

It follows from Theorem 1 that there exists at least one value of $c>0$ such that the related solution *u* of problem (40a) and (40b) with ${u}^{\prime}(1)=c$ is positive on $(0,1)$ with $u(0)=0$. Using formula (35), we now determine an interval $(0,{c}^{\ast})\subset (0,\mathrm{\infty})$ containing all admissible values of *c*.

*u*be a solution of problem (1a) and (1b) with

*f*from (41). Then by (35), we obtain

We begin the calculations with $\delta =0$ and increase its value gradually until we arrive at $\delta =1$; *cf.* (40a). In each step we use the solution of the previous problem to solve the next one. The aim is to find a good starting value for both the solution *u* and the value ${u}^{\prime}(1)$ before solving the BVP (40a) and (40b), *i.e.*, find the final value of $c\in (0,{c}^{\ast})$ such that $u(0)=0$.

- 1.
Numerical approximation of BVP (39a) and (39b) is used as an initial guess for ODE (45) with $\delta =0$ subject to terminal conditions $u(1)=0$, ${u}^{\prime}(1)=-0.5$.

- 2.
Use the above approximation as an initial guess for ODE (45) with $\delta =0.01$ subject to terminal conditions $u(1)=0$, ${u}^{\prime}(1)=-1.0$.

- 3.
Use the above approximation as an initial guess for ODE (45) with $\delta =0.1$ subject to terminal conditions $u(1)=0$, ${u}^{\prime}(1)=-1.0$.

- 4.
Use the above approximation as an initial guess for ODE (45) with $\delta =1.0$ subject to terminal conditions $u(1)=0$, ${u}^{\prime}(1)=-1.0$.

**Problem (**
**40a**
**) and (**
**40b**
**): Complete data of the numerical simulation for different values of**
a

a | K | ${\mathit{c}}^{\mathbf{\ast}}$ | c | $\mathit{u}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|---|---|

−0.1 | 1.5819 | 1.327538328 | 1.00569659944 | 1.40264071382347 E-14 |

−0.5 | 2.2173 | 1.869327784 | 1.41953539630 | 1.18953445347016 E-13 |

−0.9 | 3.6844 | 3.070760848 | 2.33615892300 | 4.13495149060736 E-14 |

### 6.4 Example 2

*f*satisfies conditions (H

_{1})-(H

_{3}) with $\epsilon =1$, $\phi (t)=\frac{1}{\sqrt{t}}$ for $t\in (0,1]$ and

*u*of problem (46a) and (46b) is positive on $(0,1)$ and $u(0)=0$ holds. We now again determine an interval $(0,{c}^{\ast})\subset (0,\mathrm{\infty})$ containing all such values of

*c*. Let

*u*be a solution of problem (1a) and (1b) with

*f*given in (47). Inequality (19) yields

*a*, we choose ${u}^{\prime}(1)\in (-{c}^{\ast},0)$ and analogously carry out the path-following in

*δ*first. The related chain for $a=-0.1$ is as follows.

- 1.
Numerical approximation of BVP (39a) and (39b) is used as an initial guess for ODE (49) with $\delta =0$ subject to terminal conditions $u(1)=0$, ${u}^{\prime}(1)=-0.8$.

- 2.
Use the above approximation as an initial guess for ODE (49) with $\delta =0.01$ subject to terminal conditions $u(1)=0$, ${u}^{\prime}(1)=-0.8$.

- 3.
Use the above approximation as an initial guess for ODE (49) with $\delta =0.1$ subject to terminal conditions $u(1)=0$, ${u}^{\prime}(1)=-1.2$.

- 4.
Use the above approximation as an initial guess for ODE (49) with $\delta =0.5$ subject to terminal conditions $u(1)=0$, ${u}^{\prime}(1)=-1.3$.

- 5.
Use the above approximation as an initial guess for ODE (49) with $\delta =1.0$ subject to terminal conditions $u(1)=0$, ${u}^{\prime}(1)=-1.7$.

**Problem (**
**46a**
**) and (**
**46b**
**): Complete data of the numerical simulation for different values of**
a

a | ${\mathit{c}}^{\mathbf{\ast}}$ | c | $\mathit{u}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|---|

−0.1 | 2.044582190 | 1.63000971355 | 4.57015034596816 e-12 |

−0.5 | 2.528760387 | 2.04278888650 | 1.82943058378521 e-13 |

−0.9 | 3.566085670 | 2.91010561000 | 1.16498324337590 e-12 |

## 7 Conclusions

Here, it is only known that $c\in (0,{c}^{\ast})$, where ${c}^{\ast}$ can be specified depending on functions *f* arising in Examples 1 and 2. Now, a simple shooting method combined with the bisection idea is used to find *c* in such a way that $u(0)=0$.

## Endnotes

^{a} This BVP can have more than one solution, but they may not lay close together.

^{b} The required smoothness of higher derivatives is related to the order of the used collocation method.

^{c} The nonlinear term in *f* has been omitted; see (40a) and (46a).

## Declarations

### Acknowledgements

Dedicated to Jean Mawhin on the occasion of his 70th birthday.

This research was supported by the grant Matematické modely a struktury, PrF-2012-017. The authors thank the referees for suggestions which improved the paper.

## Authors’ Affiliations

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