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# Second-order initial value problems with singularities

- Petio Kelevedjiev
^{1}Email author and - Nedyu Popivanov
^{2}

**2014**:161

https://doi.org/10.1186/s13661-014-0161-z

© Kelevedjiev and Popivanov; licensee Springer. 2014

**Received:**30 January 2014**Accepted:**13 June 2014**Published:**26 September 2014

## Abstract

Using barrier strip arguments, we investigate the existence of $C[0,T]\cap {C}^{2}(0,T]$-solutions to the initial value problem ${x}^{\u2033}=f(t,x,{x}^{\prime})$, $x(0)=A$, ${lim}_{t\to {0}^{+}}{x}^{\prime}(t)=B$, which may be singular at $x=A$ and ${x}^{\prime}=B$.

**MSC:** 34B15, 34B16, 34B18.

## Keywords

- initial value problem
- singularities
- existence
- monotone and positive solutions
- barrier strips

## 1 Introduction

Here the scalar function $f(t,x,p)$ is defined on a set of the form $({D}_{t}\times {D}_{x}\times {D}_{p})\mathrm{\setminus}({S}_{A}\cup {S}_{B})$, where ${D}_{t},{D}_{x},{D}_{p}\subseteq \mathbb{R}$, ${S}_{A}={\mathcal{T}}_{1}\times \{A\}\times \mathcal{P}$, ${S}_{B}={\mathcal{T}}_{2}\times \mathcal{X}\times \{B\}$, ${\mathcal{T}}_{i}\subseteq {D}_{t}$, $i=1,2$, $\mathcal{X}\subseteq {D}_{x}$, $\mathcal{P}\subseteq {D}_{p}$, and so it may be singular at $x=A$ and $p=B$.

have been investigated by Rachůnková and Tomeček [1]–[3]. For example in [1], the authors have discussed the set of all solutions to this problem with a singularity at $t=0$. Here $A<0$, $\phi \in C[0,\mathrm{\infty})\cap {C}^{1}(0,\mathrm{\infty})$ with $\phi (0)=0$, ${\phi}^{\prime}(t)>0$ for $t\in (0,\mathrm{\infty})$ and ${lim}_{t\to \mathrm{\infty}}\frac{{\phi}^{\prime}(t)}{\phi (t)}=0$, *f* is locally Lipschitz on $(-\mathrm{\infty},L]$ with the properties $f(L)=0$ and $xf(x)<0$ for $x\in (-\mathrm{\infty},0)\cup (0,L)$, where $L>0$ is a suitable constant.

where *g*, *h*, *r*, and *w* are suitable functions.

where *k*, *F*, and *G* are suitable functions.

The solvability of various IVPs has been studied also by Bobisud and O’Regan [7], Bobisud and Lee [8], Cabada and Heikkilä [9], Cabada *et al*. [10], [11], Cid [12], Maagli and Masmoudi [13], and Zhao [14]. Existence results for problem (1.1), (1.2) with a singularity at the initial value of ${x}^{\prime}$ have been reported in Kelevedjiev-Popivanov [15].

*a priori*bounds by the barrier strips technique. Further, the obtained existence theorem assures ${C}^{2}[0,T]$-solutions for each nonsingular IVP included in the family

where $n\in \mathbb{N}$ is suitable. Finally, we apply the Arzela-Ascoli theorem on the sequence $\{{x}_{n}\}$ of ${C}^{2}[0,T]$-solutions thus constructed to (1.3) to extract a uniformly convergent subsequence and show that its limit is a $C[0,T]\cap {C}^{2}(0,T]$-solution to singular problem (1.1), (1.2). In the case $A\ge 0$, $B\ge 0$ we establish $C[0,T]\cap {C}^{2}(0,T]$-solutions with important properties - monotony and positivity.

*et al*. [17], Kelevedjiev and Popivanov [18] and Palamides

*et al*. [19]. For example in [17], we have established the existence of positive solutions to the BVP

which may be singular at $x=0$. Note that despite the more general equation of this problem, the conditions imposed here as well as the results obtained are not consequences of those in [17].

## 2 Topological transversality theorem

In this short section we state our main tools - the topological transversality theorem and a theorem giving an important property of the constant maps.

So, let *X* be a metric space and *Y* be a convex subset of a Banach space *E*. Let $U\subset Y$ be open in *Y*. The compact map $F:\overline{U}\to Y$ is called *admissible* if it is fixed point free on *∂U*. We denote the set of all such maps by ${\mathbf{L}}_{\partial u}(\overline{U},Y)$.

A map *F* in ${\mathbf{L}}_{\partial u}(\overline{U},Y)$ is *essential* if every map *G* in ${\mathbf{L}}_{\partial u}(\overline{U},Y)$ such that $G|\partial U=F|\partial U$ has a fixed point in *U*. It is clear, in particular, every essential map has a fixed point in *U*.

### Theorem 2.1

([16], Chapter I, Theorem 2.2])

*Let*$p\in U$*be fixed and*$F\in {\mathbf{L}}_{\partial u}(\overline{U},Y)$*be the constant map*$F(x)=p$*for*$x\in \overline{U}$. *Then* *F* *is essential*.

*We say that the homotopy*$\{{\mathrm{H}}_{\lambda}:X\to Y\}$, $0\le \lambda \le 1$, *is compact if the map*$\mathrm{H}(x,\lambda ):X\times [0,1]\to Y$*given by*$\mathrm{H}(x,\lambda )\equiv {\mathrm{H}}_{\lambda}(x)$*for*$(x,\lambda )\in X\times [0,1]$*is compact*.

### Theorem 2.2

([16], Chapter I, Theorem 2.6])

*Let*

*Y*

*be a convex subset of a Banach space*

*E*

*and*$U\subset Y$

*be open*.

*Suppose*:

- (i)
$F,G:\overline{U}\to Y$

*are compact maps*. - (ii)
$G\in {\mathbf{L}}_{\partial U}(\overline{U},Y)$

*is essential*. - (iii)$\mathrm{H}(x,\lambda )$, $\lambda \in [0,1]$,
*is a compact homotopy joining**F**and**G*,*i*.*e*.$\mathrm{H}(x,1)=F(x)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\mathrm{H}(x,0)=G(x).$

- (iv)
$\mathrm{H}(x,\lambda )$, $\lambda \in [0,1]$,

*is fixed point free on**∂U*.

*Then*$\mathrm{H}(x,\lambda )$, $\lambda \in [0,1]$, *has at least one fixed point in* *U* *and in particular there is a*${x}_{0}\in U$*such that*${x}_{0}=F({x}_{0})$.

## 3 Nonsingular problem

where $f:{D}_{t}\times {D}_{x}\times {D}_{p}\to \mathbb{R}$, ${D}_{t},{D}_{x},{D}_{p}\subseteq \mathbb{R}$.

- (R)There exist constants $T>a$, ${m}_{1}$, ${\overline{m}}_{1}$, ${M}_{1}$, ${\overline{M}}_{1}$, and a sufficiently small $\tau >0$ such that$\begin{array}{c}{m}_{1}\ge 0,\phantom{\rule{2em}{0ex}}{\overline{M}}_{1}-\tau \ge {M}_{1}\ge B\ge {m}_{1}\ge {\overline{m}}_{1}+\tau ,\hfill \\ [a,T]\subseteq {D}_{t},\phantom{\rule{2em}{0ex}}[A-\tau ,{M}_{0}+\tau ]\subseteq {D}_{x},\phantom{\rule{2em}{0ex}}[{\overline{m}}_{1},{\overline{M}}_{1}]\subseteq {D}_{p},\hfill \end{array}$

where ${D}_{{M}_{0}}={D}_{x}\cap (-\mathrm{\infty},{M}_{0}]$.

Our first result ensures bounds for the eventual ${C}^{2}$-solutions to (3.2). We need them to prepare the application of the topological transversality theorem.

### Lemma 3.1

*Let*(R)

*hold*.

*Then each solution*$x\in {C}^{2}[a,T]$

*to the family*(3.2)

_{ λ }, $\lambda \in [0,1]$,

*satisfies the bounds*

*where*

### Proof

*γ*we have

□

Let us mention that some analogous results have been obtained in Kelevedjiev [20]. For completeness of our explanations, we present the full proofs here.

Now we prove an existence result guaranteeing the solvability of IVP (3.1).

### Theorem 3.2

*Let* (R) *hold*. *Then nonsingular problem* (3.1) *has at least one non*-*decreasing solution in*${C}^{2}[a,T]$.

### Proof

*U*. Further, we introduce the continuous maps

*W*is bounded and so, by the bounded inverse theorem, the map ${W}^{-1}$ exists and is linear and bounded. Thus, it is continuous. Now, using ${W}^{-1}$, we define

Clearly, ${V}^{-1}$ is continuous since ${W}^{-1}$ is continuous.

*j*is completely continuous, that is,

*j*maps each bounded subset of ${C}_{I}^{2}[a,T]$ into a compact subset of ${C}^{1}[a,T]$. Thus, the image $j(\overline{U})$ of the bounded set

*U*is compact. Now, from the continuity of Φ and ${V}^{-1}$ it follows that the sets $\mathrm{\Phi}(j(\overline{U}))$ and ${V}^{-1}(\mathrm{\Phi}(j(\overline{U})))$ are also compact. In summary, we have established that the homotopy is compact. On the other hand, for its fixed points we have

which is the operator form of family (3.2). So, each fixed point of ${\mathrm{H}}_{\lambda}$ is a solution to (3.2), which, according to Lemma 3.1, lies in *U*. Consequently, the homotopy is fixed point free on *∂U*.

Finally, ${\mathrm{H}}_{0}(x)$ is a constant map mapping each function $x\in \overline{U}$ to $\ell (t)$. Thus, according to Theorem 2.1, ${\mathrm{H}}_{0}(x)=\ell $ is essential.

*U*which means that the IVP of (3.2) obtained for $\lambda =1$ (

*i.e.*(3.1)) has at least one solution $x(t)$ in ${C}^{2}[a,T]$. From Lemma 3.1 we know that

from which its monotony follows. □

The validity of the following results follows similarly.

### Theorem 3.3

*Let*$B>0$*and let* (R) *hold for*${m}_{1}>0$. *Then problem* (3.1) *has at least one strictly increasing solution in*${C}^{2}[a,T]$.

### Theorem 3.4

*Let*$A>0$ ($A=0$) *and let* (R) *hold for*${m}_{1}=0$. *Then problem* (3.1) *has at least one positive* (*nonnegative*) *non*-*decreasing solution in*${C}^{2}[a,T]$.

### Theorem 3.5

*Let*$A\ge 0$, $B>0$*and let* (R) *hold for*${m}_{1}>0$. *Then problem* (3.1) *has at least one strictly increasing solution in*${C}^{2}[a,T]$*with positive values for*$t\in (a,T]$.

## 4 A problem singular at *x* and ${x}^{\prime}$

In this section we study the solvability of singular IVP (1.1), (1.2) under the following assumptions.

_{1}): There are constants $T>0$, ${m}_{1}$, ${\overline{m}}_{1}$ and a sufficiently small $\nu >0$ such that

where ${D}_{{\tilde{M}}_{0}}=(-\mathrm{\infty},{\tilde{M}}_{0}]\cap {D}_{x}$.

_{2}): For some $\alpha \in (0,T]$ and $\mu \in ({m}_{1},B)$ there exists a constant $k<0$ such that $k\alpha +B>\mu $ and

where *T*, ${m}_{1}$ and ${\tilde{M}}_{0}$ are as in (S_{1}).

*α*,

*μ*, and

*k*are as in (S

_{2}), we construct the following family of regular IVPs:

Notice, for $n\ge {n}_{\alpha ,\mu}$, that we have $B-{n}^{-1}>\mu -k\alpha >\mu >{m}_{1}>0$.

### Lemma 4.1

*Let*(S

_{1})

*and*(S

_{2})

*hold and let*${x}_{n}\in {C}^{2}[0,T]$, $n\ge {n}_{\alpha ,\mu}$,

*be a solution to*(4.3)

*such that*

*Then the following bound is satisfied for each*$n\ge {n}_{\alpha ,\mu}$:

*where*
${\varphi}_{\alpha}(t)=\{\begin{array}{ll}kt+B,& t\in [0,\alpha ],\\ k\alpha +B,& t\in (\alpha ,T].\end{array}$

### Proof

we will consider the proof for an arbitrary fixed $n\ge {n}_{\alpha ,\mu}$, considering two cases. Namely, ${x}_{n}^{\prime}(t)>\mu $ for $t\in [0,\alpha ]$ is the first case and the second one is ${x}_{n}^{\prime}(t)>\mu $ for $t\in [0,\beta )$ with ${x}_{n}^{\prime}(\beta )=\mu $ for some $\beta \in (0,\alpha ]$.

*Case*1. From $\mu <{x}_{n}^{\prime}(t)\le B$, $t\in [0,\alpha ]$, and (S

_{2}) we have

*i.e.*${x}_{n}^{\u2033}(t)\le k$ for $t\in [0,\alpha ]$. Integrating the last inequality from 0 to

*t*we get

*Case*2. As in the first case, we derive

from which the assertion follows immediately. □

Having this lemma, we prove the basic result of this section.

### Theorem 4.2

*Let*(S

_{1})

*and*(S

_{2})

*hold*.

*Then singular IVP*(1.1), (1.2)

*has at least one strictly increasing solution in*$C[0,T]\cap {C}^{2}(0,T]$

*such that*

### Proof

*t*, $t\in (0,T]$, we get

and $n\ge {n}_{\alpha ,\mu}$.

*n*, such that

*a priori*bounds for ${x}_{n}(t)$, ${x}_{n}^{\prime}(t)$ and ${x}_{n}^{\u2033}(t)$ on the interval $[\alpha ,T]$, we apply the Arzela-Ascoli theorem to conclude that there exists a subsequence $\{{x}_{{n}_{k}}\}$, $k\in \mathbb{N}$, ${n}_{k}\ge {n}_{\alpha ,\mu}$, of $\{{x}_{n}\}$ and a function ${x}_{\alpha}\in {C}^{1}[\alpha ,T]$ such that

*i.e.*, the sequences $\{{x}_{{n}_{k}}\}$ and $\{{x}_{{n}_{k}}^{\prime}\}$ converge uniformly on the interval $[\alpha ,T]$ to ${x}_{\alpha}$ and ${x}_{\alpha}^{\prime}$, respectively. Obviously, (4.7) and (4.8) are valid in particular for the elements of $\{{x}_{{n}_{k}}\}$ and $\{{x}_{{n}_{k}}^{\prime}\}$, respectively, from which, letting $k\to \mathrm{\infty}$, one finds

_{2}) holds for some $\alpha >0$, then it is true also for an arbitrary ${\alpha}_{0}\in (0,\alpha )$. We will use this fact considering a sequence $\{{\alpha}_{i}\}\subset (0,\alpha )$, $i\in \mathbb{N}$, with the properties

which contradicts to the fact that ${x}_{i,{n}_{k}}^{\prime}(0)=B-{n}_{k}^{-1}$ and ${x}_{i,{n}_{k}}^{\prime}\in C[0,T]$. This contradiction proves that (4.11) is true.

is a $C[0,T]\cap {C}^{2}(0,T]$-solution to (1.1), (1.2). This function is strictly increasing because ${x}^{\prime}(t)={x}_{0}^{\prime}(t)\ge {m}_{1}>0$ for $t\in (0,T]$, and the bounds for $x(t)$ and ${x}^{\prime}(t)$ follows immediately from the corresponding bounds for ${x}_{0}(t)$ and ${x}_{0}^{\prime}(t)$. □

The following results provide information about the presence of other useful properties of the assured solutions. Their correctness follows directly from Theorem 4.2.

### Theorem 4.3

*Let*$A\ge 0$*and let* (S_{1}) *and* (S_{2}) *hold*. *Then the singular IVP* (1.1), (1.2) *has at least one strictly increasing solution in*$C[0,T]\cap {C}^{2}(0,T]$*with positive values for*$t\in (0,T]$.

## 5 Examples

### Example 5.1

Let us note that here ${D}_{t}=(-c,c)$, ${D}_{x}=[-b,b]$ and ${D}_{p}=\mathbb{R}$.

and ${P}_{k}(p)\ne 0$ for $p\in [{p}_{1}-\theta ,{p}_{1})\cup ({p}_{1},{p}_{1}+\theta ]\cup [{p}_{2}-\theta ,{p}_{2})\cup ({p}_{2},{p}_{2}+\theta ]$.

*i.e.*$[-\theta /2,{p}_{2}{T}_{0}+\theta /2]\subseteq [-b,b]$, we get the following conditions for

*θ*and

*T*:

we really can apply Theorem 3.2 to conclude that the considered problem has a strictly increasing solution $x\in {C}^{2}[0,T]$ with $x(t)>0$ on $t\in (0,T]$ for each $T<min\{c,\frac{2b-\theta}{2{p}_{2}}\}$.

### Example 5.2

_{1}) holds, for example, for ${\overline{m}}_{1}=4$, ${m}_{1}=5$, $\nu =0.1$, and an arbitrary fixed $T>0$, moreover, ${\tilde{M}}_{0}=10T+3$. Besides, for $k=-24/{(10T+1)}^{2}$, $\alpha =T/100$ and $\mu =9$, for example, we have

and $f(t,x,p)\le -24/{(10T+1)}^{2}$ on $[0,T/100]\times (2,10T+3]\times [9,10)$, which means that (S_{2}) also holds. By Theorem 4.3, the considered IVP has at least one positive strictly increasing solution in $C[0,T]\cap {C}^{2}(0,T]$.

## Declarations

### Acknowledgements

The work is partially supported by the Sofia University Grant 158/2013 and by the Bulgarian NSF under Grant DCVP - 02/1/2009.

## Authors’ Affiliations

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