# Second-order initial value problems with singularities

## Abstract

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

MSC: 34B15, 34B16, 34B18.

## 1 Introduction

In this paper we study the solvability of initial value problems (IVPs) of the form

${x}^{″}=f\left(t,x,{x}^{\prime }\right),$
(1.1)
$x\left(0\right)=A,\phantom{\rule{2em}{0ex}}\underset{t\to {0}^{+}}{lim}{x}^{\prime }\left(t\right)=B,\phantom{\rule{1em}{0ex}}B>0.$
(1.2)

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

IVPs of the form

$\begin{array}{c}{\left(\phi \left(t\right){x}^{\prime }\left(t\right)\right)}^{\prime }=\phi \left(t\right)f\left(x\left(t\right)\right),\hfill \\ x\left(0\right)=A,\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(0\right)=0,\hfill \end{array}$

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

Agarwal and O’Regan  have studied the problem

$\begin{array}{c}{x}^{″}=\phi \left(t\right)f\left(t,x,{x}^{\prime }\right),\phantom{\rule{1em}{0ex}}t\in \left(0,T\right],\hfill \\ x\left(0\right)={x}^{\prime }\left(0\right)=0,\hfill \end{array}$

where $f\left(t,x,p\right)$ may be singular at $x=0$ and/or $p=0$. The obtained results give a positive ${C}^{1}\left[0,T\right]\cap {C}^{2}\left(0,T\right]$-solution under the assumptions that $\phi \in C\left[0,T\right]$, $\phi \left(t\right)>0$ for $t\in \left(0,T\right]$, $f:\left[0,T\right]×{\left(0,\mathrm{\infty }\right)}^{2}\to \left(0,\mathrm{\infty }\right)$ is continuous and

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

IVPs of the form

$\begin{array}{c}{x}^{″}\left(t\right)=f\left(t,x\left(t\right),{x}^{\prime }\left(t\right)\right),\phantom{\rule{1em}{0ex}}0

where $f\left(t,x,p\right)\in C\left(\left(0,1\right)×{\left(0,\mathrm{\infty }\right)}^{2}\right)$, maybe singular at $t=0$, $t=1$, $x=0$ or $p=0$, have been studied by Yang , . The solvability in ${C}^{1}\left[0,1\right]$ and $C\left[0,1\right]\cap {C}^{2}\left(0,1\right)$ is established in these works, respectively, under the assumption that

where k, F, and G are suitable functions.

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

Here, as usual, we use regularization and sequential techniques. Namely, we proceed as follows. First, by means of the topological transversality theorem , we prove an existence result guaranteeing ${C}^{2}\left[a,T\right]$-solutions to the nonsingular IVP for equations of the form (1.1) with boundary conditions

$x\left(a\right)=A,\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(a\right)=B.$

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

$\begin{array}{r}{x}^{″}=f\left(t,x,{x}^{\prime }\right),\\ x\left(0\right)=A+{n}^{-1},\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(0\right)=B-{n}^{-1},\end{array}$
(1.3)

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

We have used variants of the approach described above for various boundary value problems (BVPs); see Grammatikopoulos et al. , Kelevedjiev and Popivanov  and Palamides et al. . For example in , we have established the existence of positive solutions to the BVP

$\begin{array}{c}g\left(t,x,{x}^{\prime },{x}^{″}\right)=0,\phantom{\rule{1em}{0ex}}t\in \left(0,1\right),\hfill \\ x\left(0\right)=0,\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(1\right)=B,\phantom{\rule{1em}{0ex}}B>0,\hfill \end{array}$

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 .

## 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 ${\mathbf{L}}_{\partial u}\left(\overline{U},Y\right)$.

A map F in ${\mathbf{L}}_{\partial u}\left(\overline{U},Y\right)$ is essential if every map G in ${\mathbf{L}}_{\partial u}\left(\overline{U},Y\right)$ 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

(, Chapter I, Theorem 2.2])

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

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

### Theorem 2.2

(, Chapter I, Theorem 2.6])

Let Y be a convex subset of a Banach space E and$U\subset Y$be open. Suppose:

1. (i)

$F,G:\overline{U}\to Y$ are compact maps.

2. (ii)

$G\in {\mathbf{L}}_{\partial U}\left(\overline{U},Y\right)$ is essential.

3. (iii)

$\mathrm{H}\left(x,\lambda \right)$, $\lambda \in \left[0,1\right]$, is a compact homotopy joining F and G, i.e.

$\mathrm{H}\left(x,1\right)=F\left(x\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\mathrm{H}\left(x,0\right)=G\left(x\right).$
1. (iv)

$\mathrm{H}\left(x,\lambda \right)$, $\lambda \in \left[0,1\right]$, is fixed point free on ∂U.

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

## 3 Nonsingular problem

Consider the IVP

$\left\{\begin{array}{l}{x}^{″}=f\left(t,x,{x}^{\prime }\right),\\ x\left(a\right)=A,\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(a\right)=B,\phantom{\rule{1em}{0ex}}B\ge 0,\end{array}$
(3.1)

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

We include this problem into the following family of regular IVPs constructed for $\lambda \in \left[0,1\right]$

$\left\{\begin{array}{l}{x}^{″}=\lambda f\left(t,x,{x}^{\prime }\right),\\ x\left(a\right)=A,\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(a\right)=B,\end{array}$
(3.2)

and suppose the following.

1. (R)

There exist constants $T>a$, ${m}_{1}$, ${\overline{m}}_{1}$, ${M}_{1}$, ${\overline{M}}_{1}$, and a sufficiently small $\tau >0$ such that

$\begin{array}{c}{m}_{1}\ge 0,\phantom{\rule{2em}{0ex}}{\overline{M}}_{1}-\tau \ge {M}_{1}\ge B\ge {m}_{1}\ge {\overline{m}}_{1}+\tau ,\hfill \\ \left[a,T\right]\subseteq {D}_{t},\phantom{\rule{2em}{0ex}}\left[A-\tau ,{M}_{0}+\tau \right]\subseteq {D}_{x},\phantom{\rule{2em}{0ex}}\left[{\overline{m}}_{1},{\overline{M}}_{1}\right]\subseteq {D}_{p},\hfill \end{array}$

where ${M}_{0}=A+{M}_{1}\left(T-a\right)$,

(3.3)

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

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$x\in {C}^{2}\left[a,T\right]$to the family (3.2) λ , $\lambda \in \left[0,1\right]$, satisfies the bounds

where

$\begin{array}{c}{m}_{2}=min\left\{f\left(t,x,p\right):\left(t,x,p\right)\in \left[a,T\right]×\left[A,{M}_{0}\right]×\left[{m}_{1},{M}_{1}\right]\right\},\hfill \\ {M}_{2}=max\left\{f\left(t,x,p\right):\left(t,x,p\right)\in \left[a,T\right]×\left[A,{M}_{0}\right]×\left[{m}_{1},{M}_{1}\right]\right\}.\hfill \end{array}$

### Proof

Suppose that the set

${S}_{-}=\left\{t\in \left[a,T\right]:{M}_{1}<{x}^{\prime }\left(t\right)\le {\overline{M}}_{1}\right\}$

is not empty. Then

${x}^{\prime }\left(a\right)=B\le {M}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{x}^{\prime }\in C\left[a,T\right]$

imply that there exists an interval $\left[\alpha ,\beta \right]\subset {S}_{-}$ such that

${x}^{\prime }\left(\alpha \right)<{x}^{\prime }\left(\beta \right).$

This inequality and the continuity of ${x}^{\prime }\left(t\right)$ guarantee the existence of some $\gamma \in \left[\alpha ,\beta \right]$ for which

${x}^{″}\left(\gamma \right)>0.$

Since $x\left(t\right)$, $t\in \left[a,T\right]$, is a solution of the differential equation, we have $\left(t,x\left(t\right),{x}^{\prime }\left(t\right)\right)\in \left[a,T\right]×{D}_{x}×{D}_{p}$. In particular for γ we have

$\left(\gamma ,x\left(\gamma \right),{x}^{\prime }\left(\gamma \right)\right)\in {S}_{-}×{D}_{x}×\left({M}_{1},{\overline{M}}_{1}\right].$

Thus, we apply (R) to conclude that

${x}^{″}\left(\gamma \right)=\lambda f\left(\gamma ,x\left(\gamma \right),{x}^{\prime }\left(\gamma \right)\right)\le 0,$

which contradicts the inequality ${x}^{″}\left(\gamma \right)>0$. This has been established above. Thus, ${S}_{-}$ is empty and as a result

Now, by the mean value theorem for each $t\in \left(a,T\right]$ there exists a $\xi \in \left(a,t\right)$ such that

$x\left(t\right)-x\left(a\right)={x}^{\prime }\left(\xi \right)\left(t-a\right),$

which yields

This allows us to use (3.3) to show similarly to above that the set

${S}_{+}=\left\{t\in \left[a,T\right]:{\overline{m}}_{1}\le {x}^{\prime }\left(t\right)<{m}_{1}\right\}$

is empty. Hence,

and so

To estimate ${x}^{″}\left(t\right)$, we observe firstly that (R) implies in particular

and

which yield ${m}_{2}\le 0$ and ${M}_{2}\ge 0$. Multiplying both sides of the inequality $\lambda \le 1$ by ${m}_{2}$ and ${M}_{2}$, we get, respectively, ${m}_{2}\le \lambda {m}_{2}$ and $\lambda {M}_{2}\le {M}_{2}$. On the other hand, we have established

Thus,

and each $\lambda \in \left[0,1\right]$ and so

□

Let us mention that some analogous results have been obtained in Kelevedjiev . 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${C}^{2}\left[a,T\right]$.

### Proof

Preparing the application of Theorem 2.2, we define first the set

$U=\left\{x\in {C}_{I}^{2}\left[a,T\right]:A-\tau

where ${C}_{I}^{2}\left[a,T\right]=\left\{x\in {C}^{2}\left[a,T\right]:x\left(a\right)=A,{x}^{\prime }\left(a\right)=B\right\}$. It is important to notice that according to Lemma 3.1 all ${C}^{2}\left[a,T\right]$-solutions to family (3.2) are interior points of U. Further, we introduce the continuous maps

and for $t\in \left[a,T\right]$ and $x\left(t\right)\in j\left(\overline{U}\right)$ the map

Clearly, the map Φ is also continuous since, by assumption, the function $f\left(t,x\left(t\right),{x}^{\prime }\left(t\right)\right)$ is continuous on $\left[a,T\right]$ if

In addition we verify that ${V}^{-1}$ exists and is also continuous. To this aim we introduce the linear map

$W:{C}_{{I}_{0}}^{2}\left[a,T\right]\to C\left[a,T\right],$

defined by $Wx={x}^{″}$, where ${C}_{{I}_{0}}^{2}\left[a,T\right]=\left\{x\in {C}^{2}\left[a,T\right]:x\left(a\right)=0,{x}^{\prime }\left(a\right)=0\right\}$. It is one-to-one because each function $x\in {C}_{{I}_{0}}^{2}\left[a,T\right]$ has a unique image, and each function $y\in C\left[a,T\right]$ has a unique inverse image which is the unique solution to the IVP

${x}^{″}=y,\phantom{\rule{2em}{0ex}}x\left(a\right)=0,\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(a\right)=0.$

It is not hard to see that 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

where $\ell \left(t\right)=B\left(t-a\right)+A$ is the unique solution of the problem

${x}^{″}=0,\phantom{\rule{2em}{0ex}}x\left(a\right)=A,\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(a\right)=B.$

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

We already can introduce a homotopy

$\mathrm{H}:\overline{U}×\left[0,1\right]\to {C}_{I}^{2}\left[a,T\right],$

defined by

$\mathrm{H}\left(x,\lambda \right)\equiv {\mathrm{H}}_{\lambda }\left(x\right)\equiv \lambda {V}^{-1}\mathrm{\Phi }j\left(x\right)+\left(1-\lambda \right)\ell .$

It is well known that j is completely continuous, that is, j maps each bounded subset of ${C}_{I}^{2}\left[a,T\right]$ into a compact subset of ${C}^{1}\left[a,T\right]$. Thus, the image $j\left(\overline{U}\right)$ of the bounded set U is compact. Now, from the continuity of Φ and ${V}^{-1}$ it follows that the sets $\mathrm{\Phi }\left(j\left(\overline{U}\right)\right)$ and ${V}^{-1}\left(\mathrm{\Phi }\left(j\left(\overline{U}\right)\right)\right)$ are also compact. In summary, we have established that the homotopy is compact. On the other hand, for its fixed points we have

$\lambda {V}^{-1}\mathrm{\Phi }j\left(x\right)+\left(1-\lambda \right)\ell =x$

and

$Vx=\lambda \mathrm{\Phi }j\left(x\right),$

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

So, all assumptions of Theorem 2.2 are fulfilled. Hence ${\mathrm{H}}_{1}\left(x\right)$ has a fixed point in U which means that the IVP of (3.2) obtained for $\lambda =1$ (i.e. (3.1)) has at least one solution $x\left(t\right)$ in ${C}^{2}\left[a,T\right]$. 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${C}^{2}\left[a,T\right]$.

### 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${C}^{2}\left[a,T\right]$.

### 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${C}^{2}\left[a,T\right]$with positive values for$t\in \left(a,T\right]$.

## 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.

(S1): There are constants $T>0$, ${m}_{1}$, ${\overline{m}}_{1}$ and a sufficiently small $\nu >0$ such that

$\begin{array}{c}{m}_{1}>0,\phantom{\rule{2em}{0ex}}B>{m}_{1}\ge {\overline{m}}_{1}+\nu ,\hfill \\ \left[0,T\right]\subseteq {D}_{t},\phantom{\rule{2em}{0ex}}\left(A,{\stackrel{˜}{M}}_{0}+\nu \right]\subseteq {D}_{x},\phantom{\rule{2em}{0ex}}\left[{\overline{m}}_{1},B\right)\subseteq {D}_{p},\hfill \end{array}$

where ${\stackrel{˜}{M}}_{0}=A+BT+1$,

$f\left(t,x,p\right)\in C\left(\left[0,T\right]×\left(A,{\stackrel{˜}{M}}_{0}+\nu \right]×\left[{m}_{1}-\nu ,B\right)\right),$
(4.1)
(4.2)

and

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

(S2): For some $\alpha \in \left(0,T\right]$ and $\mu \in \left({m}_{1},B\right)$ there exists a constant $k<0$ such that $k\alpha +B>\mu$ and

where T, ${m}_{1}$ and ${\stackrel{˜}{M}}_{0}$ are as in (S1).

Now, for $n\ge {n}_{\alpha ,\mu }$, where ${n}_{\alpha ,\mu }>max\left\{{\alpha }^{-1},{\left(B+k\alpha -\mu \right)}^{-1}\right\}$, and α, μ, and k are as in (S2), we construct the following family of regular IVPs:

$\left\{\begin{array}{l}{x}^{″}=f\left(t,x,{x}^{\prime }\right),\\ x\left(0\right)=A+{n}^{-1},\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(0\right)=B-{n}^{-1}.\end{array}$
(4.3)

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

### Lemma 4.1

Let (S1) and (S2) hold and let${x}_{n}\in {C}^{2}\left[0,T\right]$, $n\ge {n}_{\alpha ,\mu }$, be a solution to (4.3) such that

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

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

### Proof

Since for each $n\ge {n}_{\alpha ,\mu }$ we have

${x}_{n}^{\prime }\left(0\right)=B-{n}^{-1}>\mu -k\alpha >\mu ,$

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

Case 1. From $\mu <{x}_{n}^{\prime }\left(t\right)\le B$, $t\in \left[0,\alpha \right]$, and (S2) we have

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

${x}_{n}^{\prime }\left(t\right)-{x}_{n}^{\prime }\left(0\right)\le kt,\phantom{\rule{1em}{0ex}}t\in \left[0,\alpha \right],$

which yields

Now ${m}_{1}\le {x}_{n}^{\prime }\left(t\right), $t\in \left[0,T\right]$, and (4.2) imply

In particular ${x}_{n}^{″}\left(t\right)\le 0$ for $t\in \left[\alpha ,T\right]$, thus

Case 2. As in the first case, we derive

On the other hand, since ${m}_{1}\le {x}_{n}^{\prime }\left(t\right) for $t\in \left[\beta ,T\right]$, again from (4.2) it follows that

which yields

and

So, as a result of the considered cases we get

from which the assertion follows immediately. □

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

### Theorem 4.2

Let (S1) and (S2) hold. Then singular IVP (1.1), (1.2) has at least one strictly increasing solution in$C\left[0,T\right]\cap {C}^{2}\left(0,T\right]$such that

### Proof

For each fixed $n\ge {n}_{\alpha ,\mu }$ introduce $\tau =min\left\{{\left(2n\right)}^{-1},\nu \right\}$,

${M}_{1}=B-{n}^{-1},\phantom{\rule{2em}{0ex}}{\overline{M}}_{1}=B-{\left(2n\right)}^{-1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{M}_{0}=\left(B-{n}^{-1}\right)T+A+1<{\stackrel{˜}{M}}_{0}$

having the properties

$\begin{array}{c}{\overline{M}}_{1}-\tau >{M}_{1}=B-{n}^{-1}>\mu -k\alpha >\mu >{m}_{1}\ge {\overline{m}}_{1}+\tau ,\hfill \\ \left[0,T\right]\subseteq {D}_{t},\phantom{\rule{2em}{0ex}}\left[A+{n}^{-1}-\tau ,{M}_{0}+\tau \right]\subseteq \left(A,{\stackrel{˜}{M}}_{0}+\tau \right]\subseteq {D}_{x}\hfill \end{array}$

and $\left[{\overline{m}}_{1},{\overline{M}}_{1}\right]\subseteq {D}_{p}$ since ${\overline{M}}_{1}=B-{\left(2n\right)}^{-1}. Besides,

and, in view of (4.1),

$f\left(t,x,p\right)\in C\left(\left[0,T\right]×\left[A+{n}^{-1}-\tau ,{M}_{0}+\nu \right]×\left[{m}_{1}-\tau ,{M}_{1}+\tau \right]\right).$

All this implies that for each $n\ge {n}_{\alpha ,\mu }$ the corresponding IVP of family (4.3) satisfies (R). Thus, we apply Theorem 3.2 to conclude that (4.3) has a solution ${x}_{n}\in {C}^{2}\left[0,T\right]$ for each $n\ge {n}_{\alpha ,\mu }$. We can use also Lemma 3.1 to conclude that for each $n\ge {n}_{\alpha ,\mu }$ and $t\in \left[0,T\right]$ we have

$A
(4.4)

and

${m}_{1}\le {x}_{n}^{\prime }\left(t\right)\le B-{n}^{-1}

Now, these bounds allow the application of Lemma 4.1 from which one infers that for each $n\ge {n}_{\alpha ,\mu }$ and $t\in \left[0,T\right]$ the bounds

${m}_{1}\le {x}_{n}^{\prime }\left(t\right)<{\varphi }_{\alpha }\left(t\right)\le B$
(4.5)

hold. For later use, integrating the least inequality from 0 to t, $t\in \left(0,T\right]$, we get

(4.6)

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

We consider firstly the sequence $\left\{{x}_{n}\right\}$ of ${C}^{2}\left[0,T\right]$-solutions of (4.3) only for each $n\ge {n}_{\alpha ,\mu }$. Clearly, for each $n\ge {n}_{\alpha ,\mu }$ we have in particular

which together with (4.6) gives

where ${A}_{1}={m}_{1}\alpha +A$. On combining the last inequality and (4.4) we obtain

(4.7)

From (4.5) we have in addition

(4.8)

Now, using the fact that (4.1) implies continuity of $f\left(t,x,p\right)$ on the compact set $\left[\alpha ,T\right]×\left[{A}_{1},{\stackrel{˜}{M}}_{0}\right]×\left[{m}_{1},{\varphi }_{\alpha }\left(\alpha \right)\right]$ and keeping in mind that for each $n\ge {n}_{\alpha ,\mu }$

we conclude that there is a constant ${M}_{2}$, independent of n, such that

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

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

Clearly, the functions ${x}_{{n}_{k}}\left(t\right)$, $k\in \mathbb{N}$, ${n}_{k}\ge {n}_{\alpha ,\mu }$, satisfy integral equations of the form

${x}_{{n}_{k}}^{\prime }\left(t\right)={x}_{{n}_{k}}^{\prime }\left(\alpha \right)+{\int }_{\alpha }^{t}f\left(s,{x}_{{n}_{k}}\left(s\right),{x}_{{n}_{k}}^{\prime }\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in \left(\alpha ,T\right].$

Now, since $f\left(t,x,p\right)$ is uniformly continuous on the compact set $\left[\alpha ,T\right]×\left[{A}_{1},{\stackrel{˜}{M}}_{0}\right]×\left[{m}_{1},{\varphi }_{\alpha }\left(\alpha \right)\right]$, from the uniform convergence of $\left\{{x}_{{n}_{k}}\right\}$ it follows that the sequence $\left\{f\left(s,{x}_{{n}_{k}}\left(s\right),{x}_{{n}_{k}}^{\prime }\left(s\right)\right)\right\}$, ${n}_{k}\ge {n}_{\alpha ,\mu }$ is uniformly convergent on $\left[\alpha ,T\right]$ to the function $f\left(s,{x}_{\alpha }\left(s\right),{x}_{\alpha }^{\prime }\left(s\right)\right)$, which means

$\underset{k\to \mathrm{\infty }}{lim}{\int }_{\alpha }^{t}f\left(s,{x}_{{n}_{k}}\left(s\right),{x}_{{n}_{k}}^{\prime }\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds={\int }_{\alpha }^{t}f\left(s,{x}_{\alpha }\left(s\right),{x}_{\alpha }^{\prime }\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds$

for each $t\in \left(\alpha ,T\right]$. Returning to the integral equation and letting $k\to \mathrm{\infty }$ yield

${x}_{\alpha }^{\prime }\left(t\right)={x}_{\alpha }^{\prime }\left(\alpha \right)+{\int }_{\alpha }^{t}f\left(s,{x}_{\alpha }\left(s\right),{x}_{\alpha }^{\prime }\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in \left(\alpha ,T\right],$

which implies that ${x}_{\alpha }\left(t\right)$ is a ${C}^{2}\left(\alpha ,T\right]$-solution to the differential equation ${x}^{″}=f\left(t,x,{x}^{\prime }\right)$ on $\left(\alpha ,T\right]$. Besides, (4.6) implies

Further, we observe that if the condition (S2) holds for some $\alpha >0$, then it is true also for an arbitrary ${\alpha }_{0}\in \left(0,\alpha \right)$. We will use this fact considering a sequence $\left\{{\alpha }_{i}\right\}\subset \left(0,\alpha \right)$, $i\in \mathbb{N}$, with the properties

For each $i\in \mathbb{N}$ we consider sequences

$\left\{{x}_{i,{n}_{k}}\right\},\phantom{\rule{1em}{0ex}}{n}_{k}\ge {n}_{i+1,\mu },k\in \mathbb{N},{n}_{i+1,\mu }>max\left\{{\alpha }_{i+1}^{-1},{\left(B+k{\alpha }_{i+1}-\mu \right)}^{-1}\right\},$

on the interval $\left[{\alpha }_{i+1},T\right]$. Thus, we establish that each sequence $\left\{{x}_{i,{n}_{k}}\right\}$ has a subsequence $\left\{{x}_{i+1,{n}_{k}}\right\}$, $k\in \mathbb{N}$, ${n}_{k}\ge {n}_{i+1,\mu }$, converging uniformly on the interval $\left[{\alpha }_{i+1},T\right]$ to any function ${x}_{{\alpha }_{i+1}}\left(t\right)$, $t\in \left[{\alpha }_{i+1},T\right]$, that is,

(4.9)

which is a ${C}^{2}\left({\alpha }_{i+1},T\right]$-solution to the differential equation ${x}^{″}\left(t\right)=f\left(t,x\left(t\right),{x}^{\prime }\left(t\right)\right)$ on $\left({\alpha }_{i+1},T\right]$ and

The properties of the functions from $\left\{{x}_{{\alpha }_{i}}\right\}$, $i\in \mathbb{N}$, imply that there exists a function ${x}_{0}\left(t\right)$ which is a ${C}^{2}\left(0,T\right]$-solution to the equation ${x}^{″}=f\left(t,x,{x}^{\prime }\right)$ on the interval $\left(0,T\right]$ and is such that

hence ${lim}_{t\to {0}^{+}}{x}_{0}\left(t\right)=A$,

(4.10)

We have to show also that

$\underset{t\to {0}^{+}}{lim}{x}_{0}^{\prime }\left(t\right)=B.$
(4.11)

Reasoning by contradiction, assume that there exists a sufficiently small $\epsilon >0$ such that for every $\delta >0$ there is a $t\in \left(0,\delta \right)$ such that

${x}_{0}^{\prime }\left(t\right)

In other words, assume that for every sequence $\left\{{\delta }_{j}\right\}\subset \left(0,T\right]$, $j\in \mathbb{N}$, with ${lim}_{j\to \mathrm{\infty }}{\delta }_{j}=0$, there exists a sequence $\left\{{t}_{j}\right\}$ having the properties ${t}_{j}\in \left(0,{\delta }_{j}\right)$, ${lim}_{j\to \mathrm{\infty }}{t}_{j}=0$ and

${x}_{0}^{\prime }\left({t}_{j}\right)
(4.12)

It is clear that every interval $\left(0,{\delta }_{j}\right)$, $j\in \mathbb{N}$, contains a subsequence of $\left\{{t}_{j}\right\}$ converging to 0. Besides, from (4.9) and (4.10) it follows that for every $j\in \mathbb{N}$ there are ${i}_{j},{n}_{j}\in \mathbb{N}$ such that ${\alpha }_{{i}_{j}}<{\delta }_{j}$ and

(4.13)

for all $i>{i}_{j}$ and all ${n}_{k}\ge max\left\{{n}_{i,\mu },{n}_{j}\right\}$. Moreover, since the accumulation point of $\left\{{t}_{j}\right\}$ is 0, for each sufficiently large $j\in \mathbb{N}$ there is a ${t}_{j}\in \left[{\alpha }_{i},{\delta }_{j}\right)$ where $i>{i}_{j}$. In summary, for every sufficiently large $j\in \mathbb{N}$, that is, for every sufficiently small ${\delta }_{j}>0$, there are ${i}_{j},{n}_{j}\in \mathbb{N}$ such that for all $i>{i}_{j}$ and ${n}_{k}\ge max\left\{{n}_{i,\mu },{n}_{j}\right\}$ from (4.12) and (4.13) we have

${x}_{i,{n}_{k}}^{\prime }\left({t}_{j}\right)

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

Now, it is easy to verify that the function

$x\left(t\right)=\left\{\begin{array}{ll}A,& t=0,\\ {x}_{0}\left(t\right),& t\in \left(0,T\right],\end{array}$

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

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 (S1) and (S2) hold. Then the singular IVP (1.1), (1.2) has at least one strictly increasing solution in$C\left[0,T\right]\cap {C}^{2}\left(0,T\right]$with positive values for$t\in \left(0,T\right]$.

## 5 Examples

### Example 5.1

Consider the IVP

$\begin{array}{r}{x}^{″}=\frac{\sqrt{{b}^{2}-{x}^{2}}}{\sqrt{{c}^{2}-{t}^{2}}}{P}_{k}\left({x}^{\prime }\right),\\ x\left(0\right)=0,\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(0\right)=B,\phantom{\rule{1em}{0ex}}B>0,\end{array}$

where $b,c\in \left(0,\mathrm{\infty }\right)$, and the polynomial ${P}_{k}\left(p\right)$, $k\ge 2$, has simple zeroes ${p}_{1}$ and ${p}_{2}$ such that

$0<{p}_{1}

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

Clearly, there is a sufficiently small $\theta >0$ such that

$0<{p}_{1}-\theta ,\phantom{\rule{2em}{0ex}}{p}_{1}+\theta \le B\le {p}_{2}-\theta$

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

We will show that all assumptions of Theorem 3.2 are fulfilled in the case

the other cases as regards the sign of ${P}_{k}\left(p\right)$ around ${p}_{1}$ and ${p}_{2}$ may be treated similarly. For this case choose $\tau =\theta /2$, ${m}_{1}={p}_{1}>0$ and ${M}_{1}={p}_{2}$. Next, using the requirement $\left[A-\tau ,{M}_{0}+\tau \right]\subseteq \left[-b,b\right]$, i.e.$\left[-\theta /2,{p}_{2}{T}_{0}+\theta /2\right]\subseteq \left[-b,b\right]$, we get the following conditions for θ and T:

$-\theta /2\ge -b\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{p}_{2}{T}_{0}+\theta /2\le b,$

which yield $\theta \in \left(0,2b\right]$ and $T\le \frac{2b-\theta }{2{p}_{2}}$. Besides, $\left[0,T\right]\subseteq \left(-c,c\right)$ yields $T. Thus, $0. Now, choosing

${\overline{m}}_{1}={p}_{1}-\theta \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\overline{M}}_{1}={p}_{2}+\theta ,$

we really can apply Theorem 3.2 to conclude that the considered problem has a strictly increasing solution $x\in {C}^{2}\left[0,T\right]$ with $x\left(t\right)>0$ on $t\in \left(0,T\right]$ for each $T.

### Example 5.2

Consider the IVP

$\begin{array}{c}{x}^{″}=\frac{\left({x}^{\prime }-5\right)\left(15-{x}^{\prime }\right)}{{\left(x-2\right)}^{2}\left({x}^{\prime }-10\right)},\hfill \\ x\left(0\right)=2,\phantom{\rule{2em}{0ex}}\underset{t\to {0}^{+}}{lim}{x}^{\prime }\left(t\right)=10.\hfill \end{array}$

Notice that here

${S}_{A}=\mathbb{R}×\left\{2\right\}×\left(\left(-\mathrm{\infty },10\right)\cup \left(10,\mathrm{\infty }\right)\right),\phantom{\rule{2em}{0ex}}{S}_{B}=\mathbb{R}×\left(\left(-\mathrm{\infty },2\right)\cup \left(2,\mathrm{\infty }\right)\right)×\left\{10\right\}.$

It is easy to check that (S1) holds, for example, for ${\overline{m}}_{1}=4$, ${m}_{1}=5$, $\nu =0.1$, and an arbitrary fixed $T>0$, moreover, ${\stackrel{˜}{M}}_{0}=10T+3$. Besides, for $k=-24/{\left(10T+1\right)}^{2}$, $\alpha =T/100$ and $\mu =9$, for example, we have

$k\alpha +B=-24T/100{\left(10T+1\right)}^{2}+10>9=\mu$

and $f\left(t,x,p\right)\le -24/{\left(10T+1\right)}^{2}$ on $\left[0,T/100\right]×\left(2,10T+3\right]×\left[9,10\right)$, which means that (S2) also holds. By Theorem 4.3, the considered IVP has at least one positive strictly increasing solution in $C\left[0,T\right]\cap {C}^{2}\left(0,T\right]$.

## References

1. Rachůnková I, Tomeček J: Bubble-type solutions of non-linear singular problem. Math. Comput. Model. 2010, 51: 658-669. 10.1016/j.mcm.2009.10.042

2. Rachůnková I, Tomeček J: Homoclinic solutions of singular nonautonomous second-order differential equations. Bound. Value Probl. 2009., 2009:

3. Rachůnková I, Tomeček J: Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics. Nonlinear Anal. 2010, 72: 2114-2118. 10.1016/j.na.2009.10.011

4. Agarwal RP, O’Regan D: Second-order initial value problems of singular type. J. Math. Anal. Appl. 1999, 229: 441-451. 10.1006/jmaa.1998.6169

5. Yang G: Minimal positive solutions to some singular second-order differential equations. J. Math. Anal. Appl. 2002, 266: 479-491. 10.1006/jmaa.2001.7748

6. Yang G: Positive solutions of some second-order nonlinear singular differential equations. Comput. Math. Appl. 2003, 45: 605-614. 10.1016/S0898-1221(03)00020-8

7. Bobisud LE, O’Regan D: Existence of solutions to some singular initial value problems. J. Math. Anal. Appl. 1988, 133: 215-230. 10.1016/0022-247X(88)90376-9

8. Bobisud LE, Lee YS: Existence of monotone or positive solutions of singular second-order sublinear differential equations. J. Math. Anal. Appl. 1991, 159: 449-468. 10.1016/0022-247X(91)90207-G

9. Cabada A, Heikkilä S: Extremality results for discontinuous explicit and implicit diffusion problems. J. Comput. Appl. Math. 2002, 143: 69-80. 10.1016/S0377-0427(01)00501-5

10. Cabada A, Cid JA, Pouso RL: Positive solutions for a class of singular differential equations arising in diffusion processes. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2005, 12: 329-342.

11. Cabada A, Nieto JJ, Pouso RL: Approximate solutions to a new class of nonlinear diffusion problems. J. Comput. Appl. Math. 1999, 108: 219-231. 10.1016/S0377-0427(99)00112-0

12. Cid JA: Extremal positive solutions for a class of singular and discontinuous second order problems. Nonlinear Anal. 2002, 51: 1055-1072. 10.1016/S0362-546X(01)00879-3

13. Maagli H, Masmoudi S: Existence theorem of nonlinear singular boundary value problem. Nonlinear Anal. 2001, 46: 465-473. 10.1016/S0362-546X(99)00455-1

14. Zhao Z: Positive solutions of nonlinear second order ordinary differential equations. Proc. Am. Math. Soc. 1994, 121: 465-469. 10.1090/S0002-9939-1994-1185276-5

15. Kelevedjiev, P, Popivanov, N: On the solvability of a second-order initial value problem. Paper presented at the 40th international conference on the applications of mathematics in engineering and economics, Technical University of Sofia, Sozopol, 8-13 June 2014

16. Granas A, Guenther RB, Lee JW: Nonlinear boundary value problems for ordinary differential equations. Diss. Math. 1985, 244: 1-128.

17. Grammatikopoulos MK, Kelevedjiev PS, Popivanov N:On the solvability of a singular boundary-value problem for the equation $f\left(t,x,{x}^{\prime },{x}^{″}\right)=0$. J. Math. Sci. 2008, 149: 1504-1516. 10.1007/s10958-008-0079-z

18. Kelevedjiev P, Popivanov N: Second order boundary value problems with nonlinear two-point boundary conditions. Georgian Math. J. 2000, 7: 677-688.

19. Palamides P, Kelevedjiev P, Popivanov N:On the solvability of a Neumann boundary value problem for the differential equation $f\left(t,x,{x}^{\prime },{x}^{″}\right)=0$. Bound. Value Probl. 2012., 2012: 10.1186/1687-2770-2012-77

20. Kelevedjiev P: Positive solutions of nonsingular and singular second order initial value problems. Int. Electron. J. Pure Appl. Math. 2010, 2: 117-127.

## Acknowledgements

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

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Correspondence to Petio Kelevedjiev.

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