# Positive solutions to second-order differential equations with dependence on the first-order derivative and nonlocal boundary conditions

- Tadeusz Jankowski
^{1}Email author

**2013**:8

https://doi.org/10.1186/1687-2770-2013-8

© Jankowski; licensee Springer. 2013

**Received: **20 September 2012

**Accepted: **15 January 2013

**Published: **17 January 2013

## Abstract

In this paper, we consider the existence of positive solutions for second-order differential equations with deviating arguments and nonlocal boundary conditions. By the fixed point theorem due to Avery and Peterson, we provide sufficient conditions under which such boundary value problems have at least three positive solutions. We discuss our problem both for delayed and advanced arguments *α* and also in the case when $\alpha (t)=t$, $t\in [0,1]$. In all cases, the argument *β* can change the character on $[0,1]$, see problem (1). It means that *β* can be delayed in some set $\overline{J}\subset [0,1]$ and advanced in $[0,1]\setminus \overline{J}$. An example is added to illustrate the results.

**MSC:**34B10.

## Keywords

## 1 Introduction

involving Stieltjes integrals with suitable functions *A* and *B* of bounded variation on *J*. It is not assumed that ${\lambda}_{1}$, ${\lambda}_{2}$ are positive to all positive *x*. As we see later, the measures *dA*, *dB* can be signed measures.

We introduce the following assumptions:

H_{1}: $f\in C(J\times {\mathbb{R}}_{+}\times \mathbb{R},{\mathbb{R}}_{+})$, $\alpha ,\beta \in C(J,J)$, *A* and *B* are functions of bounded variation;

H_{2}: $h\in C(J,{\mathbb{R}}_{+})$ and *h* does not vanish identically on any subinterval;

H_{3}: $1-\gamma -{\lambda}_{1}[p]>0$ or $1-\xi -{\lambda}_{2}[p]>0$ for $p(t)=1$, $t\in J$, $\gamma ,\xi \ge 0$.

Recently, the existence of multiple positive solutions for differential equations has been studied extensively; for details, see, for example, [1–31]. However, many works about positive solutions have been done under the assumption that the first-order derivative is not involved explicitly in nonlinear terms; see, for example, [3, 6, 8–14, 17, 20, 25–27, 30]. From this list, only papers [9–12, 14, 20, 30] concern positive solutions to problems with deviating arguments. On the other hand, there are some papers considering the multiplicity of positive solutions with dependence on the first-order derivative; see, for example, [2, 4, 5, 7, 15, 16, 18, 19, 21–24, 28, 29, 31]. Note that boundary conditions (BCs) in differential problems have important influence on the existence of the results obtained. In this paper, we consider problem (1) which is a problem with dependence on the first-order derivative with BCs involving Stieltjes integrals with signed measures of *dA*, *dB* appearing in functionals ${\lambda}_{1}$, ${\lambda}_{2}$; moreover, problem (1) depends on deviating arguments.

or $1-\alpha \eta >0$, respectively.

here ${\lambda}_{1}$ has the same form as in problem (1) with signed measure *dA* appearing in functional ${\lambda}_{1}$.

The main results of papers [25, 26] have been obtained by the fixed point index theory for problems without deviating arguments. The study of positive solutions to boundary value problems with Stieltjes integrals in the case of signed measures has also been done in papers [3, 7, 13, 14, 27] for second-order differential equations (also impulsive) or third-order differential equations by using the fixed point index theory, the Avery-Peterson fixed point theorem or fixed point index theory involving eigenvalues.

for some constants ${a}_{i}$, ${b}_{i}$ and some functions ${g}_{1}$, ${g}_{2}$. In our paper, the assumption that the measures *dA*, *dB* in the definitions of ${\lambda}_{1}$, ${\lambda}_{2}$ are positive is not needed. More precisely, one needs to choose the above functions ${g}_{1}$, ${g}_{2}$ in such a way that the assumption H_{4} holds. It means that ${g}_{1}$, ${g}_{2}$ can change sign on *J*.

*G*denotes a Green function connected with our problem, so in our case it is given by

In our paper, we eliminate ${\lambda}_{1}$ and ${\lambda}_{2}$ from problem (2) to obtain the equation $x=\overline{\mathcal{W}}x$ with a corresponding operator $\overline{\mathcal{W}}$, and then we seek solutions as fixed points of this operator $\overline{\mathcal{W}}$.

Note that if we put $\gamma =\xi =0$ in the BCs of problem (1), then this new problem is more general than the previous one because in this case someone, for example, can take ${\lambda}_{1}[x]=\gamma x(\eta )$, ${\lambda}_{2}[x]=\xi x(\eta )$. In this paper, we try to explain why for some cases we have to discuss problem (1) with constants $\gamma >0$ or $\xi >0$.

*K*. Usually, we need to find a nonnegative function

*κ*and a constant $\overline{\rho}\in (0,1]$ such that $G(t,s)\le \kappa (s)$ for $t,s\in J$ and $G(t,s)\ge \overline{\rho}\kappa (t)$ for $t\in [\eta ,\overline{\eta}]\subset [0,1]$ and $s\in J$ (see, for example, [25–27]) to work with the inequality

*J*, then we can take $\gamma =\xi =0$ in the boundary conditions of problem (1) to work with the inequality

for *ζ*, *ϱ* such that $\zeta +\varrho <1$, $0<\zeta <\varrho <1$ with $\kappa =min(\zeta ,1-\varrho )$; see Section 5.

*J*. In this case, we can put $\xi =0$ in the boundary conditions of problem (1) to find a constant $\rho \in (0,1)$ to work with the inequality

see Section 3. For the case $\alpha (t)\ge t$ on *J*, we can put $\gamma =0$ to work similarly as in Section 3; see Section 4. Note that in the above three cases for the argument *β*, we need only the assumption $\beta \in C(J,J)$, which means that *β* can change the character in *J*.

Note that in cited papers, positive solutions to differential equations with dependence on the first-order derivative have been investigated only for problems without deviating arguments, see [2, 4, 5, 7, 15, 16, 18, 19, 21–24, 28, 29, 31]. Moreover, BCs in problem (1) cover some nonlocal BCs discussed earlier.

Motivated by [25–27], in this paper, we apply the fixed point theorem due to Avery-Peterson to obtain sufficient conditions for the existence of multiple positive solutions to problems of type (1). In problem (1), an unknown *x* depends on deviating arguments which can be both of advanced or delayed type. To the author’s knowledge, it is the first paper when positive solutions have been investigated for such general boundary value problems with functionals ${\lambda}_{1}$, ${\lambda}_{2}$ and with deviating arguments *α*, *β* in differential equations in which *f* depends also on the first-order derivative. It is important to indicate that problems of type (1) have been discussed with signed measures of *dA*, *dB* appearing in Stieltjes integrals of functionals ${\lambda}_{1}$, ${\lambda}_{2}$.

The organization of this paper is as follows. In Section 2, we present some necessary lemmas connected with our main results. In Section 3, we first present some definitions and a theorem of Avery and Peterson which is useful in our research. Also in Section 3, we discuss the existence of multiple positive solutions to problems with delayed argument *α*, by using the above mentioned Avery-Peterson theorem. At the end of this section, an example is added to verify theoretical results. In Section 4, we formulate sufficient conditions under which problems with advanced argument *α* have positive solutions. In the last section, we discuss problems of type (1) when $\alpha (t)=t$ on *J*.

## 2 Some lemmas

**Lemma 1**

*Let*$x\in {C}^{1}(J,\mathbb{R})$, $p(t)=1$, $t\in J$.

*Assume that*

*A*

*and*

*B*

*are functions of bounded variation and*,

*moreover*,

*with*

- (i)
$1-\gamma -{\lambda}_{1}[p]\ne 0$

*or* - (ii)
$1-\xi -{\lambda}_{2}[p]\ne 0$.

*Then*

*Here*, Var*A* *denotes the variation of a function* *A* *on* *J*.

*Proof*Note that in case (i), we have

This proves case (i).

we get the result in case (ii). This ends the proof. □

**Remark 1**If we assume that

*A*and

*B*are increasing functions, then there exists $\sigma \in J$ such that

*M*from Lemma 1 has the form

Let us introduce the assumption.

_{0}:

*A*and

*B*are functions of bounded variation and

We require the following result.

**Lemma 2**

*Let the assumption*H

_{0}

*hold and let*$y\in {L}^{1}(J,\mathbb{R})$.

*Then problem*(3)

*has a unique solution given by*

*Proof*Integrating the differential equation in (3) two times, we have

*u*is a solution of (6), then

Solving this system with respect to ${\lambda}_{1}[u]$, ${\lambda}_{2}[u]$ and then substituting to (6), we have the assertion of this lemma. This ends the proof. □

*T*by

Let us introduce the following assumption.

_{4}:

*A*and

*B*are functions of bounded variation and

- (i)
$\delta >0$, $\mathrm{\Delta}>0$, ${A}_{j}\ge 0$, ${B}_{j}\ge 0$, ${\mathcal{G}}_{j}(s)\ge 0$ for $j=1,2$ where ${A}_{j}$, ${B}_{j}$, ${\mathcal{G}}_{j}$,

*δ*, Δ are defined as in the assumption H_{0}, - (ii)
$\gamma ({A}_{1}-{A}_{2})+\xi {A}_{2}\ge 0$, $\gamma ({B}_{1}-{B}_{2})+\xi {B}_{2}\ge 0$, $\eta \gamma {B}_{1}+(1-\gamma ){B}_{2}\ge 0$, $(1-\xi \eta ){A}_{1}-(1-\xi ){A}_{2}\ge 0$, ${B}_{1}-{B}_{2}\ge 0$, $\delta -\eta \gamma {B}_{1}-(1-\gamma ){B}_{2}\ge 0$, $\eta \gamma {A}_{1}+(1-\gamma ){A}_{2}\ge 0$, $1-\xi \eta -{B}_{2}\ge 0$, $\delta -(1-\xi \eta ){A}_{1}+(1-\xi ){A}_{2}\ge 0$, $(1-\xi \eta ){B}_{1}-(1-\xi ){B}_{2}\ge 0$.

**Lemma 3** *Let the assumptions* H_{1}-H_{4} *hold*. *Then* $T:K\to K$.

*Proof*Clearly, $u\in K$ is a positive solution of problem (1) if and only if $u\in K$ solves the operator equation $u=Tu$. Then

Note that ${\lambda}_{1}[Fu]\ge 0$, ${\lambda}_{2}[Fu]\ge 0$ in view of the assumptions H_{1}, H_{2}, H_{4} and the positivity of Green’s function *G*.

Hence, *Tu* is concave and $Tu(t)\ge 0$ on *J*.

To do it, we consider two steps. Let ${\parallel Tu\parallel}_{1}=Tu({t}^{\ast})$.

Step 1. Let $Tu(0)<Tu(\eta )$. Then ${t}^{\ast}\in (0,\eta )$ or ${t}^{\ast}\in (\eta ,1)$ and ${min}_{[0,\eta ]}Tu(t)=Tu(0)$.

It shows $T:K\to K$. This ends the proof. □

**Remark 2**Take $dB(t)=(bt-1)\phantom{\rule{0.2em}{0ex}}dt$, $b>1$. Note that the measure changes the sign and is increasing. It is easy to show that

If we assume that $b\ge 3$, then ${B}_{1}>0$, ${B}_{2}>0$, ${\mathcal{G}}_{2}(s)\ge 0$, $s\in J$.

**Remark 3**Take $dA(t)=(a{t}^{2}-1)\phantom{\rule{0.2em}{0ex}}dt$, $a>1$. Note that the measure changes the sign and is increasing. It is easy to show that

If we assume that $a\ge 6$, then ${A}_{1}>0$, ${A}_{2}>0$, ${\mathcal{G}}_{1}(s)\ge 0$, $s\in J$.

**Remark 4**Let $dA(t)=(3t-1)\phantom{\rule{0.2em}{0ex}}dt$, $dB(t)=(\frac{7}{2}t-1)\phantom{\rule{0.2em}{0ex}}dt$, $t\in J$. Then the assumptions H

_{3}, H

_{4}hold if one of the following conditions is satisfied:

- (i)
$\xi =0$, $0<\gamma <\frac{1}{2}$,

- (ii)
$\gamma =0$, $0<\xi <\frac{1}{4}$,

- (iii)
$\gamma =\xi =0$.

*dA*,

*dB*change the sign and are increasing. Indeed, for $p=1$, $t\in J$, we have

It proves that the assumption H_{4} holds.

By a similar way, we prove the assertion in case (ii) or (iii).

## 3 Positive solutions to problem (1) with delayed arguments

Now, we present the necessary definitions from the theory of cones in Banach spaces.

**Definition 1**Let

*E*be a real Banach space. A nonempty convex closed set $P\subset E$ is said to be a cone provided that

- (i)
$ku\in P$ for all $u\in P$ and all $k\ge 0$, and

- (ii)
$u,-u\in P$ implies $u=0$.

Note that every cone $P\subset E$ induces an ordering in *E* given by $x\le y$ if $y-x\in P$.

**Definition 2**A map Φ is said to be a nonnegative continuous concave functional on a cone

*P*of a real Banach space

*E*if $\mathrm{\Phi}:P\to {\mathbb{R}}_{+}$ is continuous and

for all $x,y\in P$ and $t\in [0,1]$.

*φ*is a nonnegative continuous convex functional on a cone

*P*of a real Banach space

*E*if $\phi :P\to {\mathbb{R}}_{+}$ is continuous and

for all $x,y\in P$ and $t\in [0,1]$.

**Definition 3** An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

*φ*and Θ be nonnegative continuous convex functionals on

*P*, let Φ be a nonnegative continuous concave functional on

*P*, and let Ψ be a nonnegative continuous functional on

*P*. Then, for positive numbers

*a*,

*b*,

*c*,

*d*, we define the following sets:

We will use the following fixed point theorem of Avery and Peterson to establish multiple positive solutions to problem (1).

**Theorem 1** (see [1])

*Let*

*P*

*be a cone in a real Banach space*

*E*.

*Let*

*φ*

*and*Θ

*be nonnegative continuous convex functionals on*

*P*,

*let*Φ

*be a nonnegative continuous concave functional on*

*P*,

*and let*Ψ

*be a nonnegative continuous functional on*

*P*

*satisfying*$\mathrm{\Psi}(kx)\le k\mathrm{\Psi}(x)$

*for*$0\le k\le 1$

*such that for some positive numbers*$\overline{M}$

*and*

*d*,

*for all*$x\in \overline{P(\phi ,d)}$.

*Suppose*

*is completely continuous and there exist positive numbers* *a*, *b*, *c* *with* $a<b$ *such that*

(S_{1}): $\{x\in P(\phi ,\mathrm{\Theta},\mathrm{\Phi},b,c,d):\mathrm{\Phi}(x)>b\}\ne 0$ *and* $\mathrm{\Phi}(Tx)>b$ *for* $x\in P(\phi ,\mathrm{\Theta},\mathrm{\Phi},b,c,d)$,

(S_{2}): $\mathrm{\Phi}(Tx)>b$ *for* $x\in P(\phi ,\mathrm{\Phi},b,d)$ *with* $\mathrm{\Theta}(Tx)>c$,

(S_{3}): $0\notin R(\phi ,\mathrm{\Psi},a,d)$ *and* $\mathrm{\Psi}(Tx)<a$ *for* $x\in R(\phi ,\mathrm{\Psi},a,d)$ *with* $\mathrm{\Psi}(x)=a$.

*and*

*K*instead of

*P*and let ${\overline{P}}_{r}=\{x\in K:\parallel x\parallel \le r\}$. Now, we define the nonnegative continuous concave functional Φ on

*K*by

Note that $\mathrm{\Phi}(x)\le {\parallel x\parallel}_{1}$. Put $\mathrm{\Psi}(x)=\mathrm{\Theta}(x)={\parallel x\parallel}_{1},\phi (x)={\parallel {x}^{\prime}\parallel}_{1}$.

Now, we can formulate the main result of this section.

**Theorem 2**

*Let the assumptions*H

_{1}-H

_{4}

*hold with*$\xi =0$, $\gamma >0$.

*Let*$\alpha (t)\le t$, $t\in J$.

*In addition*,

*we assume that there exist positive constants*

*a*,

*b*,

*c*,

*d*,

*M*, $a<b$

*and such that*

*and*

(A_{1}): $f(t,u,v)\le \frac{d}{\mu}$ *for* $(t,u,v)\in J\times [0,Md]\times [-d,d]$,

(A_{2}): $f(t,u,v)\ge \frac{b}{L}$ *for* $(t,u,v)\in [0,\eta ]\times [b,\frac{b}{\rho}]\times [-d,d]$,

(A_{3}): $f(t,u,v)\le \frac{a}{\mu}$ *for* $(t,u,v)\in J\times [0,a]\times [-d,d]$.

*Then problem*(1)

*has at least three nonnegative solutions*${x}_{1}$, ${x}_{2}$, ${x}_{3}$

*satisfying*${\parallel {x}_{i}^{\prime}\parallel}_{1}\le d$, $i=1,2,3$,

*and* ${\parallel {x}_{3}\parallel}_{1}<a$.

*Proof* Basing on the definitions of *T*, we see that $T\overline{P}$ is equicontinuous on *J*, so *T* is completely continuous.

Let $x\in \overline{P(\phi ,d)}$, so $\phi (x)={\parallel {x}^{\prime}\parallel}_{1}\le d$. By Lemma 1, ${\parallel x\parallel}_{1}\le Md$, so $0\le x(t)\le Md$, $t\in J$. Assumption (A_{1}) implies $f(t,x(\alpha (t)),{x}^{\prime}(\beta (t)))\le \frac{d}{\mu}$.

This proves that $T:\overline{P(\phi ,d)}\to \overline{P(\phi ,d)}$.

_{1}) is satisfied. Take

_{2}) implies $f(t,x(\alpha (t)),{x}^{\prime}(\beta (t)))\ge \frac{b}{L}$. Hence,

This proves that condition (S_{1}) holds.

_{2}) is satisfied. Take $x\in P(\phi ,\mathrm{\Phi},b,d)$ and ${\parallel Tx\parallel}_{1}>\frac{b}{\rho}=c$. Then

so condition (S_{2}) holds.

This shows that condition (S_{3}) is satisfied.

This ends the proof. □

**Example**Consider the following problem:

*J*with fixed $\overline{\rho}\in (0,1)$. Indeed, $f\in C([0,1]\times {\mathbb{R}}_{+}\times [-d,d],{\mathbb{R}}_{+})$, $\gamma =\frac{1}{4}$, $\eta =\frac{1}{2}$, $h(t)=h>0$, $\xi =0$ and

*J*. Moreover,

for $(t,u,v)\in [0,1]\times [0,2d]\times [-d,d]$.

All the assumptions of Theorem 2 hold, so problem (8) has at least three positive solutions.

**Remark 5** We can also construct an example in which, for example, ${\lambda}_{1}[x]={\int}_{0}^{1}x(t)(3t-1)\phantom{\rule{0.2em}{0ex}}dt$ to use the results of Remark 4. Note that also this measure changes the sign.

## 4 Positive solutions to problem (1) with advanced arguments

*J*, so the interval $[0,\eta ]$ is now replaced by $[\eta ,1]$. It means that we can put $\gamma =0$ with $\xi >0$ in the boundary conditions of problem (1) because someone can take ${\lambda}_{1}[x]=\overline{\gamma}x(\eta )$ as an example. Let us introduce the cone ${K}_{2}$ by

Now $\mathrm{\Phi}(x)={min}_{[\eta ,1]}|x(t)|$. Functionals Ψ, Θ, *φ* are defined as in Section 3. We formulate only the main result using the cone ${K}_{2}$ instead of *K* (see Theorem 2); the proof is similar to the previous one.

**Theorem 3**

*Let the assumptions*H

_{1}-H

_{4}

*hold with*$\gamma =0$, $\xi >0$.

*Let*$\alpha (t)\ge t$, $t\in J$.

*In addition*,

*we assume that there exist positive constants*

*a*,

*b*,

*c*,

*d*,

*M*, $a<b$

*and such that*

*and*

(B_{1}): $f(t,u,v)\le \frac{d}{\mu}$ *for* $(t,u,v)\in J\times [0,Md]\times [-d,d]$,

(B_{2}): $f(t,u,v)\ge \frac{b}{L}$ *for* $(t,u,v)\in [\eta ,1]\times [b,\frac{b}{\mathrm{\Gamma}}]\times [-d,d]$,

(B_{3}): $f(t,u,v)\le \frac{a}{\mu}$ *for* $(t,u,v)\in J\times [0,a]\times [-d,d]$.

*Then problem*(1)

*has at least three nonnegative solutions*${x}_{1}$, ${x}_{2}$, ${x}_{3}$

*satisfying*${\parallel {x}_{i}^{\prime}\parallel}_{1}\le d$, $i=1,2,3$,

*and* ${\parallel {x}_{3}\parallel}_{1}<a$.

## 5 Positive solutions to problem (1) for the case when $\alpha (t)=t$on *J*

*J*and $\gamma =\xi =0$. It means that now $\mathrm{\Phi}(x)={min}_{[\zeta ,\varrho ]}|x(t)|$ for some fixed constants

*ζ*,

*ϱ*such that $0<\zeta <\varrho <1$. For $0<\zeta +\varrho <1$ we can show that $G(t,s)\ge \kappa G(s,s)$, $t\in [\zeta ,\varrho ]$, $s\in J$. Now, for $\kappa =min(\zeta ,1-\varrho )$, we introduce the cone ${K}_{3}$ by

Functionals Ψ, Θ, *φ* are defined as in Section 3; the cone *K* is now replaced by ${K}_{3}$.

**Theorem 4**

*Let the assumptions*H

_{1}-H

_{4}

*hold with*$\gamma =\xi =0$.

*Let*$0<\zeta +\varrho <1$, $\alpha (t)=t$, $t\in J$.

*In addition*,

*we assume that there exist positive constants*

*a*,

*b*,

*c*,

*d*,

*M*, $a<b$

*and such that*

*and*

(C_{1}): $f(t,u,v)\le \frac{d}{\mu}$ *for* $(t,u,v)\in J\times [0,Md]\times [-d,d]$,

(C_{2}): $f(t,u,v)\ge \frac{b}{L}$ *for* $(t,u,v)\in [\zeta ,\varrho ]\times [b,\frac{b}{\kappa}]\times [-d,d]$,

(C_{3}): $f(t,u,v)\le \frac{a}{\mu}$ *for* $(t,u,v)\in J\times [0,a]\times [-d,d]$.

*Then problem*(1)

*has at least three nonnegative solutions*${x}_{1}$, ${x}_{2}$, ${x}_{3}$

*satisfying*${\parallel {x}_{i}^{\prime}\parallel}_{1}\le d$, $i=1,2,3$,

*and* ${\parallel {x}_{3}\parallel}_{1}<a$.

## 6 Conclusions

In this paper, we have discussed boundary value problems for second-order differential equations with deviating arguments and with dependence on the first-order derivative. In our research, the deviating arguments can be both delayed and advanced. By using the fixed point theorem of Avery and Peterson, new sufficient conditions for the existence of positive solutions to such boundary problems have been derived. An example is provided for illustration.

## Declarations

## Authors’ Affiliations

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