# Third order problems with nonlocal conditions of integral type

- Abdelkader Boucherif
^{1}Email author, - Sidi Mohamed Bouguima
^{2}, - Zehour Benbouziane
^{2}and - Nawal Al-Malki
^{3}

**2014**:137

https://doi.org/10.1186/s13661-014-0137-z

© Boucherif et al.; licensee Springer 2014

**Received: **23 January 2014

**Accepted: **23 May 2014

**Published: **10 September 2014

## Abstract

We discuss the existence of solutions of nonlinear third order ordinary differential equations with integral boundary conditions. We provide sufficient conditions on the nonlinearity and the functions appearing in the boundary conditions that guarantee the existence of at least one solution to our problem. We rely on the method of lower and upper solutions to generate an iterative technique, which is not necessarily monotone.

## Keywords

## Introduction

## Preliminaries

### Definition 1

A solution of problem (1)-(4) is a function $u\in {D}_{0}$ that satisfies (1) for every $t\in I$ and the conditions (3) and (4).

### Definition 2

Let $\alpha ,\beta \in {D}_{0}$ satisfy ${\alpha}^{\prime}(t)\le {\beta}^{\prime}(t)$ for every $t\in I$. We denote by $[{\alpha}^{\prime},{\beta}^{\prime}]$ the set of all $v\in {D}_{0}$ such that ${\alpha}^{\prime}(t)\le v(t)\le {\beta}^{\prime}(t)$ for every $t\in I$.

It is clear that if ${u}^{\prime}\in [{\alpha}^{\prime},{\beta}^{\prime}]$ and $u\in {D}_{0}$, then $u\in [\alpha ,\beta ]$.

### Definition 3

Let $\alpha ,\beta \in {D}_{0}$ satisfy ${\alpha}^{\prime}(t)\le {\beta}^{\prime}(t)$ for every $t\in I$. Let $S(\alpha ,\beta )$ denote the set of all functions $u\in {D}_{0}$ such that $u\in [\alpha ,\beta ]$ and ${u}^{\prime}\in [{\alpha}^{\prime},{\beta}^{\prime}]$.

### Remark 1

It is clear that $u\in {D}_{0}$ and ${u}^{\prime}\in [{\alpha}^{\prime},{\beta}^{\prime}]$ imply that $u\in S(\alpha ,\beta )$.

### Definition 4

### Remark 2

The operators $p$ and $q$ are continuous and bounded.

## Main results

In this section we state and prove our main results. The first result is of independent interest and plays a key role in the proof of our second result.

### Theorem 1

*Let*$\varphi :I\times \mathbb{R}\to \mathbb{R}$*be continuous*, *bounded and satisfy the following condition*:

(H_{
ϕ
}): $(\varphi (t,{v}_{2})-\varphi (t,{v}_{1}))({v}_{2}-{v}_{1})<0$*for all*${v}_{1},{v}_{2}\in \mathbb{R}$*such that*${v}_{1}\le {v}_{2}$*and*$t\in I$.

*Then for any*$\delta $, $\rho $

*the boundary value problem*

*has a unique solution*$u$.

### Proof

*Uniqueness*. Suppose that problem (5) has two solutions $x$ and $u$ in ${D}_{0}$. Put $z={x}^{\prime}-{u}^{\prime}$. Then $z(1)={x}^{\prime}(1)-{u}^{\prime}(1)=0$. We show that $z(0)=0$. Suppose this is not true. Then either $z(0)>0$ or $z(0)<0$. We consider the case $z(0)>0$. From the condition at $t=0$ it follows that $0<z(0)=a{z}^{\prime}(0)$. Since $a$ is nonnegative we have ${z}^{\prime}(0)>0$, which implies that $z$ is increasing to the right of $t=0$. Since $z(1)=0$ there must exist $\xi \in [0,1)$ such that $z(\xi )={max}_{t\in I}z(t)$. Then

_{ ϕ }) imply

*Existence*. For $\lambda \in [0,1]$ consider the family of problems

- (i)$u$ is a solution of (6) if and only if it satisfies, for all $t\in I$,Indeed, it is clear that the differential equation in (6) implies$\begin{array}{rcl}u(t)& =& \frac{\lambda t}{a+1}(\delta +a\rho +a{\int}_{0}^{1}(1-s)\varphi (s,{u}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds)\\ +\frac{\lambda {t}^{2}}{2(a+1)}(\rho -\delta +{\int}_{0}^{1}(1-s)\varphi (s,{u}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds)\\ -\lambda {\int}_{0}^{t}\frac{{(t-s)}^{2}}{2}\varphi (s,{u}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds.\end{array}$(7)Then$u(t)={u}^{\prime}(0)t+{u}^{\u2033}(0)\frac{{t}^{2}}{2}-\lambda {\int}_{0}^{t}\frac{{(t-s)}^{2}}{2}\varphi (s,{u}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds.$(8)It follows that${u}^{\prime}(t)={u}^{\prime}(0)+{u}^{\u2033}(0)t-\lambda {\int}_{0}^{t}(t-s)\varphi (s,{u}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds.$(9)But ${u}^{\prime}(0)=a{u}^{\u2033}(0)+\lambda \delta $, so that$\lambda \rho ={u}^{\prime}(1)={u}^{\prime}(0)+{u}^{\u2033}(0)-\lambda {\int}_{0}^{1}(1-s)\varphi (s,{u}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds.$and consequently${u}^{\u2033}(0)=\frac{\lambda}{a+1}[\rho -\delta +{\int}_{0}^{1}(1-s)\varphi (s,{u}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds],$Now, substitute the expressions of ${u}^{\prime}(0)$ and ${u}^{\u2033}(0)$ into (8) to get (7).${u}^{\prime}(0)=\frac{\lambda}{a+1}[\delta +a\rho +a{\int}_{0}^{1}(1-s)\varphi (s,{u}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds].$
- (ii)We show that there exists a positive constant ${L}_{0}$, independent of $\lambda $, such that any possible solution $u$ of (6) satisfiesThe boundedness of $\varphi $ implies that there exists ${M}_{\varphi}>0$ such that $|\varphi (t,{u}^{\prime}(t))|\le {M}_{\varphi}$ for all $t\in I$, so that ${\parallel {u}^{\u2034}\parallel}_{0}\le {M}_{\varphi}$. Then${\parallel u\parallel}_{D}\le {L}_{0}.$(10)and$|{u}^{\prime}(0)|\le \frac{1}{a+1}[|\delta |+a|\rho |+a\frac{{M}_{\varphi}}{2}]$(11)Combining relations (9), (11), and (12) we see that$|{u}^{\u2033}(0)|\le \frac{1}{a+1}[|\delta |+|\rho |+\frac{{M}_{\varphi}}{2}].$(12)Since $u(t)={\int}_{0}^{t}{u}^{\prime}(s)\phantom{\rule{0.2em}{0ex}}ds$ it follows that${\parallel {u}^{\prime}\parallel}_{0}\le {M}_{2}:=\frac{1}{a+1}[2|\delta |+(a+1)|\rho |+(a+1){M}_{\varphi}].$(13)Also, ${\parallel {u}^{\u2034}\parallel}_{0}\le {M}_{\varphi}$ and (12) imply${\parallel u\parallel}_{0}\le {M}_{2}.$Let ${L}_{0}={M}_{1}+2{M}_{2}+{M}_{\varphi}$. Then any possible solution $u$ of (6) satisfies (10).${\parallel {u}^{\u2033}\parallel}_{0}\le {M}_{1}:=\frac{1}{a+1}[|\delta |+|\rho |+\frac{(2a+3){M}_{\varphi}}{2}].$
- (iii)
Define an operator $\mathrm{\Psi}:{D}_{0}\to {D}_{0}$ by $(\mathrm{\Psi}u)(t)$ = the right-hand side of (7). Let $\mathrm{\Omega}:=\{u\in {D}_{0};{\parallel u\parallel}_{D}\le {L}_{0}\}$. Then it is easily seen that $(\mathrm{\Psi}(\mathrm{\Omega}))$ is uniformly bounded and equicontinuous. Ascoli-Arzela theorem implies that the operator $\mathrm{\Psi}$ is compact. Moreover, the set of all solutions $u$ of the equation $u=\lambda \mathrm{\Psi}u$ is bounded (see (10)). It follows from Schaefer theorem (see [10]) that $u=\mathrm{\Psi}u$ has at least one solution. Thus, (6) has at least one solution for $\lambda =1$, which is, in fact, unique from the previous step. Thus, $u$ is a solution of (5). This completes the proof of the theorem. □

### Remark 3

We should emphasize that, unlike Theorem 6 in [9], our Theorem 1 gives the uniqueness of the solution and this is essentially utilized in the proof of our Theorem 2 below.

For our second main result we introduce the notion of lower and upper solutions of problem (1), (2), (3), (4).

### Definition 5

- (a)We say that $\alpha \in {D}_{0}$ is a lower solution of problem (1), (3), (4) if$\{\begin{array}{l}-{\alpha}^{\u2034}(t)\le f(t,\alpha (t),{\alpha}^{\prime}(t),{\alpha}^{\u2033}(t))\phantom{\rule{1em}{0ex}}\text{for all}t\in I,\\ {\alpha}^{\prime}(0)-a{\alpha}^{\u2033}(0)\le {\int}_{0}^{1}{h}_{1}(\alpha (s),{\alpha}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds,\\ {\alpha}^{\prime}(1)\le {\int}_{0}^{1}{h}_{2}(\alpha (s),{\alpha}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds.\end{array}$
- (b)We say that $\beta \in {D}_{0}$ is an upper solution of problem (1), (3), (4) if$\{\begin{array}{l}-{\beta}^{\u2034}(t)\ge f(t,\beta (t),{\beta}^{\prime}(t),{\beta}^{\u2033}(t))\phantom{\rule{1em}{0ex}}\text{for all}t\in I,\\ {\beta}^{\prime}(0)-a{\beta}^{\u2033}(0)\ge {\int}_{0}^{1}{h}_{1}(\beta (s),{\beta}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds,\\ {\beta}^{\prime}(1)\ge {\int}_{0}^{1}{h}_{2}(\beta (s),{\beta}^{\prime}(s))\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

To state and prove our second main result we introduce the following assumptions.

_{ f }): $f:I\times {\mathbb{R}}^{3}\to \mathbb{R}$ is continuous and satisfies

- (1)
there exists ${C}_{0}>0$ such that any solution $u$ of (1), with $u\in S(\alpha ,\beta )$, satisfies $|{u}^{\u2033}(t)|\le {C}_{0}$, for all $t\in I$;

- (2)
$(f(t,u(t),{v}_{2},w)-f(t,u(t),{v}_{1},w))({v}_{2}-{v}_{1})<0$ for ${v}_{1},{v}_{2}\in \mathbb{R}$ such that ${v}_{1}\le {v}_{2}$, $u\in D\cap [\alpha ,\beta ]$, $w\in \mathbb{R}$ and $t\in I$;

- (3)
$f(t,\alpha (t),v(t),{\alpha}^{\u2033}(t))\le f(t,u(t),v(t),w)\le f(t,\beta (t),v(t),{\beta}^{\u2033}(t))$ for all $u\in D\cap [\alpha ,\beta ]$, $v\in {C}^{2}\cap [{\alpha}^{\prime},{\beta}^{\prime}]$, $w\in \mathbb{R}$, and $t\in I$.

(A_{
h
}): ${h}_{1},{h}_{2}:{\mathbb{R}}^{2}\to \mathbb{R}$ are continuous and nondecreasing with respect to both arguments.

### Remark 4

There are several sufficient conditions that imply (A_{
f
})(1). See for instance [6], Lemma 1], [3], Lemma 1].

### Theorem 2

*Let*$\alpha ,\beta \in {D}_{0}$*be*, *respectively*, *a lower and an upper solution of problem* (1), (3), (4) *such that*${\alpha}^{\prime}\le {\beta}^{\prime}$*on*$I$. *Assume that the conditions* (A_{
f
}) *and* (A_{
h
}) *are satisfied for the pair*$(\alpha ,\beta )$, *where*$\alpha $*is a given lower solution and*$\beta $*is a given upper solution*. *Then problem* (1), (3), (4) *has at least one solution*$u\in S(\alpha ,\beta )$.

### Proof

_{ f })(1). Consider the modified problem

- 1.The sequence ${({u}_{j})}_{j\in \mathbb{N}}$ is well defined. Indeed, for any $t\in I$ and any $z\in \mathbb{R}$ we have $q(z)\in [{\alpha}^{\prime},{\beta}^{\prime}]$ and $p({u}_{j-1}(t))\in [\alpha ,\beta ]$. It follows that the function $\varphi :I\times \mathbb{R}\to \mathbb{R}$, defined byis continuous and bounded for all $t\in I$ and $z\in \mathbb{R}$. Moreover, condition (A$\varphi (t,z)=F(t,{u}_{j-1}(t),z,{u}_{j-1}^{\prime \prime}(t))=f(t,p({u}_{j-1})(t),q(z),{u}_{j-1}^{\prime \prime}(t)),$
_{ f })(2) shows that $\varphi $ satisfies condition (H_{ ϕ }) in Theorem 1. It follows from this theorem that (17) has a unique solution ${u}_{j}$, for each $j=1,2,\dots $ . - 2.
For each $j=0,1,\dots $ the functions ${u}_{j}$ satisfy ${u}_{j}\in S(\alpha ,\beta )$ and the sequence ${({u}_{j}^{\prime \prime})}_{j\in \mathbb{N}}$ is uniformly bounded.

*Claim* 1. There exists $K$ depending only on ${K}_{1}$, ${M}_{f}$, $\overline{h}$, ${\parallel {\alpha}^{\u2033}\parallel}_{0}$, and ${\parallel {\beta}^{\u2033}\parallel}_{0}$ such that ${\parallel {u}_{\ell}^{\prime \prime}\parallel}_{0}\le K$ and ${u}_{\ell}\in S(\alpha ,\beta )$.

*Claim*2. ${u}_{\ell}\in S(\alpha ,\beta )$. Since ${u}_{\ell}(0)=0$ it suffices to show that ${u}_{\ell}^{\prime}\in [{\alpha}^{\prime},{\beta}^{\prime}]$,

*i.e.*${\alpha}^{\prime}\le {u}_{\ell}^{\prime}\le {\beta}^{\prime}$. We, first, prove that ${\alpha}^{\prime}\le {u}_{\ell}^{\prime}$. For this purpose, set $W(t)={u}_{\ell}^{\prime}(t)-{\alpha}^{\prime}(t)$, for $t\in I$. We show that $W(t)\ge 0$ for all $t\in I$. Suppose by contradiction, that there exists ${\xi}_{1}\in I$ such that $W({\xi}_{1})<0$. Since $W$ is continuous, it follows that there exists $\eta \in I$ such that $W(\eta )=min\{W(t);t\in I\}<0$. Hence we have $W(\eta )<0$, ${W}^{\prime}(\eta )=0$ and ${W}^{\u2033}(\eta )>0$. Thus,

_{ f })(3). Now, if $\eta =0$, then $W(0)<0$, ${W}^{\prime}(0)\ge 0$, and ${W}^{\u2033}(0)\ge 0$. It follows that

## Declarations

### Acknowledgements

A Boucherif is grateful to King Fahd University of Petroleum and Minerals for its constant support. The authors are grateful to the anonymous referees and the handling editor for comments that led to the improvement of the presentation of the manuscript.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd.**Open Access** This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.