Abstract
In this paper, we are concerned with positive solutions for the Dirichlet boundary value problem for equations and systems of Kirchhoff type. We obtain existence and localization results of positive solutions using Krasnosel’skiĭ’s fixed point theorem in cones and a weak Harnacktype inequality. The localization is given in terms of energy norm, being of interest from a physical point of view. In the case of systems, the results on the localization are established componentwise using the vector version of Krasnosel’skiĭ’s theorem, which allows some of the equations of the system to satisfy the compression condition and others the expansion one.
Authors
Nataliia Kolun
Department of Fundamental Sciences, Military Academy, 65009 Odessa, Ukraine
Faculty of Mathematics and Computer Science, BabeşBolyai University, ClujNapoca, Romania
Radu Precup
Department of Mathematics BabesBolyai University, ClujNapoca, Romania
Keywords
Kirchhoff equation; positive solution; Dirichlet boundary value problem; Krasnosel’skiĭ’s fixed point theorem in a cone; weak Harnack inequality
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15729176
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Energybased localization of positive solutions for stationary Kirchhoff type equations and systems
Abstract
In this paper, we are concerned with positive solutions for the Dirichlet boundary value problem for equations and systems of Kirchhoff type. We obtain existence and localization results of positive solutions using Krasnosel’skiǐ’s fixed point theorem in cones and a weak Harnack type inequality. The localization is given in terms of energy norm, being of interest from a physical point of view. In the case of systems, the results on the localization are established componentwise using the vector version of Krasnosel’skiǐ’s theorem, which allows some of the equations of the system to satisfy the compression condition and others the expansion one.
Key words: Kirchhoff equation, positive solution, Dirichlet boundary value problem, Krasnosel’skiǐ’s fixed point theorem in a cone, weak Harnack inequality.
Mathematics Subject Classification: 34K10, 47J05.
1 Introduction
In this paper we consider the Dirichlet boundary value problem for stationary Kirchhoff type equations
$$\{\begin{array}{cc}\eta ({u}_{{H}_{0}^{1}}){u}^{\prime \prime}=f(x,u),\hfill & \text{a.e.}x\in (0,1)\hfill \\ u(0)=u(1)=0,\hfill & \end{array}$$  (1.1) 
where $\eta \in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ is an increasing function, $f:[0,1]\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is an ${L}^{2}$Carathéodory function and
$$u{}_{{H}_{0}^{1}}={({\int}_{0}^{1}{u}^{\prime}{}_{}{}^{2})}^{\frac{1}{2}}$$ 
is the energy norm.
More general, we consider the Dirichlet problem for $n$dimensional systems
$$\{\begin{array}{cc}{\eta}_{i}({{u}_{i}}_{{H}_{0}^{1}}){u}_{i}^{\prime \prime}={f}_{i}(x,{u}_{1},{u}_{2},\mathrm{\dots},{u}_{n}),\hfill & \text{a.e.}x\in (0,1)\hfill \\ {u}_{i}(0)={u}_{i}(1)=0,\hfill & \end{array}$$  (1.2) 
where ${\eta}_{i}\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ are increasing functions and ${f}_{i}:[0,1]\times {\mathbb{R}}_{+}^{n}\to {\mathbb{R}}_{+}$are ${L}^{2}$Carathéodory functions, $i=1,2,\mathrm{\dots},n.$
The integraldifferential equation in (1.1) is a generalization of the stationary form of the equation introduced by Kirchhoff [1]
$${u}_{tt}\left(a+b{\int}_{0}^{L}{u}_{x}^{2}\mathit{d}x\right){u}_{xx}=0$$ 
as an extension of the classical D’Alembert linear equation ${u}_{tt}a{u}_{xx}=0$ for free vibrations of elastic strings. In these equations, parameter $a$ depends on the initial axial tension and its perturbation given by $b{\int}_{0}^{L}{u}_{x}^{2}\mathit{d}x$ is proportional with the elastic energy. In the case of nonhomogeneous equations, the term $f(x,u)$ stands for a feedbackbased activation of the forces and thus it is of interest to know how to design it in order to guarantee that the elastic energy remains bounded between two given values. Conversely, for a given feedback term $f,$ it is of interest to find an interval in which the elastic energy will be. Thus our aim is to localize solutions with respect to the energy norm, under the form
$$r\le {\leftu\right}_{{H}_{0}^{1}}\le R.$$ 
Kirchhoff type equations also arise as models of diffusion when $\eta $ gives the change of the diffusion coefficient depending on the diffusion energy. In this direction, systems of equations of type (1.2) can model, for example, diffusion of several physical quantities which change their diffusion coefficients as a result of the interractions.
The Kirchhoff equations have been studied by several researchers mainly due to their applications, but also due to the theoretical difficulties caused by their double integraldifferential nature; see, for example, [2], [3], [4], [5], [6], [7], [9], [10] and [8].
The plan of this paper is as follows: In Preliminaries, there are stated the norm version of Krasnosel’skiǐ’s fixed point theorem in a cone and its vector analogue. Also there are given some basic results concerning Sobolev spaces in one dimension. Next, in Section 3.1, there is stated and proved a Harnack type inequality in terms of the energy norm, which is extremely useful for the estimation from below of solutions. Section 3.2 gives the definition and some properties of the Kirchhoff solution operator. Furthemore, in Section 3.3 we present the main result on the localization of positive solutions for problem (1.1) and its specific version for autonomous equations. Finally in Section 3.4, the results are extended to systems by obtaining the location of the solutions on components.
2 Preliminaries
2.1 Krasnosel’skiǐ type fixed point theorems in cones
In this paper, the main tool for obtaining the existence and localization of positive solutions to problem (1.1) is given by the normversion of Krasnosel’skiǐ’s theorem in a cone [11].
Theorem 2.1 (Krasnosel’skiǐ’s fixed point theorem)
Let $(X,\cdot {}_{X})$ be a Banach space, $K\subset X$ a cone, $r,R\in {\mathbb{R}}_{+}$, $$ and let $N:K\to K$ be a completely continuous operator. Assume that one of the following conditions is satisfied:
 (a)

${N(u)}_{X}\le {u}_{X}$ for all $u\in K$ if ${u}_{X}=R,$ and ${N(u)}_{X}\ge {u}_{X}$ for all $u\in K$ if ${u}_{X}=r$ (compression);
 (b)

${N(u)}_{X}\ge {u}_{X}$ for all $u\in K$ if ${u}_{X}=R,$ and ${N(u)}_{X}\le {u}_{X}$ for all $u\in K$ if ${u}_{X}=r$ (expansion).
Then $N$ has a fixed point $u$ in $K$ with
$$r\le {u}_{X}\le R.$$ 
In the case of systems, we use the vector version of Theorem 2.1 given in [12], which allows us to localize individually the components of a solution under different behaviors of the nonlinear terms of the system.
Theorem 2.2 (Vector version of Krasnosel’skiǐ’s fixed point theorem)
Let $(X,\cdot {}_{X})$ be a Banach space, ${K}_{1},{K}_{2},\mathrm{\dots},{K}_{n}\subset X$ cones; $K:={K}_{1}\times {K}_{2}\times \mathrm{\dots}\times {K}_{n}$, $r,R\in {\mathbb{R}}_{+}^{n}$, $r=({r}_{1},{r}_{2},\mathrm{\dots},{r}_{n})$, $R=({R}_{1},{R}_{2},\mathrm{\dots},{R}_{n})$ with $$ for all $i$, and let $N:K\to K$, $N=({N}_{1},{N}_{2},\mathrm{\dots},{N}_{n}),$ ${N}_{i}:K\to {K}_{i}$ $(i=1,2,\mathrm{\dots},n)$ be completely continuous. Assume that for each $i=1,2,\mathrm{\dots},n,$ one of the following conditions is satisfied:
 (a)

${{N}_{i}(u)}_{X}\le {{u}_{i}}_{X}$ for all $u\in K$ if ${{u}_{i}}_{X}={R}_{i},$ and ${{N}_{i}(u)}_{X}\ge {{u}_{i}}_{X}$ for all $u\in K$ if ${{u}_{i}}_{X}={r}_{i};$
 (b)

${N({u}_{i})}_{X}\ge {{u}_{i}}_{X}$ for all $u\in K$ if ${{u}_{i}}_{X}={R}_{i},$ and ${{N}_{i}(u)}_{X}\le {{u}_{i}}_{X}$ for all $u\in K$ if ${{u}_{i}}_{X}={r}_{i}.$
Then $N$ has a fixed point $u=({u}_{1},{u}_{2},\mathrm{\dots},{u}_{n})$ in $K$ with
$${r}_{i}\le {{u}_{i}}_{X}\le {R}_{i}$$ 
for $i=1,2,\mathrm{\dots},n.$
2.2 Sobolev spaces in one dimension
Looking for Carathéodory solutions of equations we are led to use Sobolev spaces.
The Sobolev space ${H}_{0}^{1}(0,1)$ given by
$${H}_{0}^{1}(0,1)=\{u:[0,1]\to \mathbb{R}:u\text{is absolutely continuous,}{u}^{\prime 2}(0,1),u(0)=u(1)=0\}$$ 
is endowed with scalar product and energy norm
$${(u,v)}_{{H}_{0}^{1}}={\int}_{0}^{1}uv,u{}_{{H}_{0}^{1}}={({\int}_{0}^{1}{u}^{\prime}{}_{}{}^{2})}^{\frac{1}{2}}.$$ 
Its dual space is denoted by ${H}^{1}(0,1)$ and for any $h\in {H}^{1}(0,1)$ and $\nu \in {H}_{0}^{1}(0,1),$ by $(h,\nu )$ we mean the value of the linear functional $h$ on $\nu $. In case that $h\in {L}^{2}(0,1),$ one has $(h,v)={(h,v)}_{{L}^{2}}={\int}_{0}^{1}hv.$
Also, ${H}^{2}(0,1)$ is the space
$${H}^{2}(0,1)=\{u\in {C}^{1}[0,1]:{u}^{\prime}\text{is absolutely continuous,}{u}^{\prime \prime}\in {L}^{2}(0,1)\}.$$ 
Note that for any $h\in {H}^{1}(0,1),$ there is a unique $u\in {H}_{0}^{1}(0,1)$ (weak solution) such that ${u}^{\prime \prime}=h$ in the sense of distributions, i.e.
$${(u,\nu )}_{{H}_{0}^{1}}=(h,\nu )\phantom{\rule{1em}{0ex}}\text{for all}\nu \in {H}_{0}^{1}(0,1)$$ 
and one has
$${u}_{{H}_{0}^{1}}={h}_{{H}^{1}}.$$  (2.1) 
In particular, if $h\in {L}^{2}(0,1),$ then $u$ is a Carathéodory solution and can be expressed using the Green function as
$$u(x)={\int}_{0}^{1}G(x,s)h(s)\mathit{d}s,$$ 
where $G(x,s)=s(1x)$ for $0\le s\le x\le 1,$ $G(x,s)=x(1s)$ for $0\le x\le s\le 1.$
The following continuous embeddings take place
$${H}_{0}^{1}(0,1)\subset {L}^{2}(0,1)\subset {H}^{1}(0,1)$$ 
and the Poincaré inequalities hold
$${u}_{{L}^{2}}\le \frac{1}{\sqrt{{\lambda}_{1}}}{u}_{{H}_{0}^{1}}\phantom{\rule{1em}{0ex}}(u\in {H}_{0}^{1}(0,1)),$$ 
$${h}_{{H}^{1}}\le \frac{1}{\sqrt{{\lambda}_{1}}}{h}_{{L}^{2}}\phantom{\rule{1em}{0ex}}(h\in {L}^{2}(0,1)),$$  (2.2) 
where ${\lambda}_{1}={\pi}^{2}$ is the first eigenvalue of the Dirichlet problem for the operator $Lu:={u}^{\prime \prime}$ in $(0,1)$ (see Remark 3.3 and Lemma 9.2 [13]).
3 Main results
3.1 A Harnack type inequality
We seek positive solutions of $(\text{1.1})$ which are symmetric with respect to the middle of the interval $[0,1]$, that is $u(1x)=u(x)$ for every $x\in [0,\frac{1}{2}]$. Note that the symmetry on $[0,1]$ of any ${C}^{1}$ function implies that ${u}^{\prime}\left(\frac{1}{2}\right)=0,$ which is useful in the proof of the next theorem. In this respect we consider the Hilbert space
$$X={\widehat{H}}_{0}^{1}(0,1):=\{u\in {H}_{0}^{1}(0,1):u(1x)=u(x)\phantom{\rule{1em}{0ex}}\text{for all}x\in [0,\frac{1}{2}]\},$$ 
endowed with the scalar product and norm ${(.,.)}_{{H}_{0}^{1}},.{}_{{H}_{0}^{1}}.$
In order to apply Theorem 2.1 we need a weak Harnack type inequality for the differential operator $Lu:=\eta ({u}_{{H}_{0}^{1}}){u}^{\prime \prime}$ subject to the boundary conditions $u(0)=u(1)=0$.
Theorem 3.1
Let $u\in {\widehat{H}}_{0}^{1}(0,1)\cap {H}^{2}(0,1)$ be such that $Lu\ge 0$ on $(0,1)$ and $Lu$ is increasing on $(0,\frac{1}{2})$. Then for each ${x}_{0}\in (0,\frac{1}{2})$, there exists $\gamma =\gamma ({x}_{0})>0$ such that
$$u(x)\ge \gamma {u}_{{H}_{0}^{1}}\phantom{\rule{1em}{0ex}}\text{for all}x\in [{x}_{0},1{x}_{0}].$$  (3.1) 
Proof. Let $h:=Lu=\eta ({u}_{{H}_{0}^{1}}){u}^{\prime \prime},$ where $u$ satisfies all the conditions of the theorem. Thus $h$ is nonnegative on $[0,1]$ and increasing on $(0,\frac{1}{2})$. Since $\eta $ is a nonnegative function we have that $u$ is concave on $[0,1]$ and so ${u}^{\prime}$ is decreasing in $[0,1].$ Then
$$u(x)={\int}_{0}^{x}{u}^{\prime}(s)\mathit{d}s\ge x{u}^{\prime}(x).$$  (3.2) 
Furthermore, it is not difficult to prove the inequality
$${u}^{\prime}(x)\ge (12x){u}^{\prime}(0)\phantom{\rule{1em}{0ex}}\text{for all}x\in [0,\frac{1}{2}].$$  (3.3) 
Indeed, if we let $\sigma (x):={u}^{\prime}(x)(12x){u}^{\prime}(0)$ for $x\in [0,\frac{1}{2}],$ then
$${\sigma}^{\prime}(x)={u}^{\prime \prime}(x)+2{u}^{\prime}(0)=\frac{h(x)}{\eta ({u}_{{H}_{0}^{1}})}+2{u}^{\prime}(0).$$ 
Since $h$ is increasing on $[0,\frac{1}{2}]$ and $\eta ({u}_{{H}_{0}^{1}})$ is a fixed positive number, we deduce that ${\sigma}^{\prime}$ is decreasing, so $\sigma $ is concave on $[0,\frac{1}{2}]$. In addition $\sigma (0)=0$ and since $u$ is symmetric with respect to $\frac{1}{2},$ one also has $\sigma (1/2)={u}^{\prime}(1/2)=0.$ Then the concavity of $\sigma $ on $[0,\frac{1}{2}]$ and $\sigma (0)=\sigma (1/2)=0$ guarantee that $\sigma (x)\ge 0$ for all $x\in [0,\frac{1}{2}]$. Thus (3.3) is true. An other remark is that
$${u}_{{H}_{0}^{1}}^{2}={\int}_{0}^{1}u_{}^{\prime}{}_{}{}^{2}=2{\int}_{0}^{\frac{1}{2}}{u}^{\prime 2}\le u_{}^{\prime}{}_{}{}^{2}(0),$$ 
whence
$${u}^{\prime}(0)\ge {u}_{{H}_{0}^{1}}.$$  (3.4) 
Now (3.2), (3.3) and (3.4) give
$$u(x)\ge x(12x){u}_{{H}_{0}^{1}}\phantom{\rule{1em}{0ex}}\text{for all}x\in [0,\frac{1}{2}].$$ 
Next fix any number ${x}_{0}\in (0,\frac{1}{2})$. Then
$$u(x)\ge u({x}_{0})\ge {x}_{0}(12{x}_{0}){u}_{{H}_{0}^{1}}\phantom{\rule{1em}{0ex}}\text{for all}x\in [{x}_{0},1{x}_{0}].$$ 
This shows that (3.1) holds with $\gamma =\gamma \left({x}_{0}\right)={x}_{0}(12{x}_{0})>0.$
Thus Theorem 3.1 is proved.
Notice that such kind of estimations from below in terms of the energy norm have been given for the first time in [15] (see also [ppv]).
3.2 The Kirchhoff solution operator
To give the operator form of the Dirichlet problem (1.1) we need to associate the solution operator.
Theorem 3.2
Let $\eta \in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ be increasing and Lipschitz continuous on ${\mathbb{R}}_{+}$ with$\eta (0)\ne 0.$ For each $h\in {H}^{1}(0,1)$, the Dirichlet problem
$$\{\begin{array}{cc}\eta ({u}_{{H}_{0}^{1}}){u}^{\prime \prime}=h,\hfill & \text{a.e.}x\in (0,1)\hfill \\ u(0)=u(1)=0,\hfill & \end{array}$$  (3.5) 
has a unique weak solution ${u}_{h}\in {H}_{0}^{1}(0,1),$ i.e.,
$$\eta ({{u}_{h}}_{{H}_{0}^{1}}){({u}_{h},\nu )}_{{H}_{0}^{1}}=(h,\nu )\phantom{\rule{1em}{0ex}}\text{for all}\nu \in {H}_{0}^{1}(0,1),$$  (3.6) 
and the solution operator $S:{H}^{1}(0,1)\to {H}_{0}^{1}(0,1),$ $S\left(h\right):={u}_{h}$ is continuous and satisfies
$$\eta ({S\left(h\right)}_{{H}_{0}^{1}}){S\left(h\right)}_{{H}_{0}^{1}}\le {h}_{{H}^{1}}\phantom{\rule{1.5em}{0ex}}\text{for all}h\in {H}^{1}(0,1).$$  (3.7) 
Proof. (a) Existence: Let $h\in {H}^{1}(0,1)$ be fixed and consider the operator ${S}_{h}:{H}_{0}^{1}(0,1)\to {H}_{0}^{1}(0,1)$ defined by
$${S}_{h}(\nu )=\frac{1}{\eta ({\nu }_{{H}_{0}^{1}})}w,$$ 
where $w$ is the unique weak solution in ${H}_{0}^{1}(0,1)$ of the equation ${w}^{\prime \prime}=h.$ Clearly, ${S}_{h}$ is completely continuous. In addition, according to (2.1), one has
$${{S}_{h}(\nu )}_{{H}_{0}^{1}}\le \frac{1}{\eta (0)}{h}_{{H}^{1}},\nu \in {H}_{0}^{1}(0,1).$$  (3.8) 
Hence, if we consider the ball
$$B=\{\nu \in {H}_{0}^{1}(0,1):{\nu }_{{H}_{0}^{1}}\le \frac{1}{\eta (0)}{h}_{{H}^{1}}\},$$ 
then ${S}_{h}(B)\subset B$ and according to Schauder’s fixed point theorem, there exists at least one $u$ such that ${S}_{h}(u)=u.$ Clearly $u$ is a weak solution of the Dirichlet problem (3.5).
(b) Uniqueness: Assume that ${u}_{1},$ ${u}_{2}$ are two solutions of (3.5). Then
$$\eta ({{u}_{1}}_{{H}_{0}^{1}}){{u}_{1}}_{{H}_{0}^{1}}^{2}=(h,{u}_{1}),$$ 
$$\eta ({{u}_{2}}_{{H}_{0}^{1}}){({u}_{1},{u}_{2})}_{{H}_{0}^{1}}=(h,{u}_{1}).$$ 
It follows that
$$\eta ({{u}_{1}}_{{H}_{0}^{1}}){{u}_{1}}_{{H}_{0}^{1}}^{2}=\eta ({{u}_{2}}_{{H}_{0}^{1}}){({u}_{1},{u}_{2})}_{{H}_{0}^{1}}\le \eta ({{u}_{2}}_{{H}_{0}^{1}}){{u}_{1}}_{{H}_{0}^{1}}{{u}_{2}}_{{H}_{0}^{1}}.$$ 
Simplifying gives
$$\eta ({{u}_{1}}_{{H}_{0}^{1}}){{u}_{1}}_{{H}_{0}^{1}}\le \eta ({{u}_{2}}_{{H}_{0}^{1}}){{u}_{2}}_{{H}_{0}^{1}}.$$ 
The function $\eta (x)x$ being strictly increasing on ${\mathbb{R}}_{+},$ the last inequality gives
$${{u}_{1}}_{{H}_{0}^{1}}\le {{u}_{2}}_{{H}_{0}^{1}}.$$ 
By symmetry, the converse inequality also holds. Thus
$${{u}_{1}}_{{H}_{0}^{1}}={{u}_{2}}_{{H}_{0}^{1}}$$ 
and correspondingly
$$\eta ({{u}_{1}}_{{H}_{0}^{1}})=\eta ({{u}_{2}}_{{H}_{0}^{1}}).$$ 
Now the uniqueness of solution of the Dirichlet problem for the operator ${u}^{\prime \prime}$ yields ${u}_{1}={u}_{2}.$
(c) Continuity: Let ${h}_{k}\to h$ in ${H}^{1}(0,1)$ and let ${u}_{k}:=S({h}_{k}).$ Using (3.7) we have that the sequence $({u}_{k})$ is bounded. Hence, passing if necessary to a subsequence, we may assume that the sequence of real numbers $({{u}_{k}}_{{H}_{0}^{1}})$ is convergent. We now prove that the sequence $({u}_{k})$ is Cauchy. From
$${u}_{k}^{\prime \prime}=\frac{1}{\eta ({{u}_{k}}_{{H}_{0}^{1}})}{h}_{k},$$ 
we have
$${({u}_{k}{u}_{p})}^{\prime \prime}=\frac{1}{\eta ({{u}_{k}}_{{H}_{0}^{1}})}{h}_{k}\frac{1}{\eta ({{u}_{p}}_{{H}_{0}^{1}})}{h}_{p}$$ 
in the weak sense. Consequently
$${\left{u}_{k}{u}_{p}\right}_{{H}_{0}^{1}}^{2}=(\frac{1}{\eta ({{u}_{k}}_{{H}_{0}^{1}})}{h}_{k}\frac{1}{\eta ({{u}_{p}}_{{H}_{0}^{1}})}{h}_{p},{u}_{k}{u}_{p})=$$ 
$$=\frac{1}{\eta ({{u}_{k}}_{{H}_{0}^{1}})}({h}_{k}{h}_{p},{u}_{k}{u}_{p})+\left(\frac{1}{\eta ({{u}_{k}}_{{H}_{0}^{1}})}\frac{1}{\eta ({{u}_{p}}_{{H}_{0}^{1}})}\right)({h}_{p},{u}_{k}{u}_{p}).$$ 
Since $\eta $ is Lipschitz continuous on ${\mathbb{R}}_{+},$ there is an $L>0$ such that
$$\left\eta ({{u}_{p}}_{{H}_{0}^{1}})\eta ({{u}_{k}}_{{H}_{0}^{1}})\right\le L\left{{u}_{p}}_{{H}_{0}^{1}}{{u}_{k}}_{{H}_{0}^{1}}\right.$$ 
Furthermore
$${\left{u}_{k}{u}_{p}\right}_{{H}_{0}^{1}}^{2}\le \frac{1}{\eta (0)}{{h}_{k}{h}_{p}}_{{H}^{1}}{{u}_{k}{u}_{p}}_{{H}_{0}^{1}}+\frac{L\left{{u}_{p}}_{{H}_{0}^{1}}{{u}_{k}}_{{H}_{0}^{1}}\right}{\eta {(0)}^{2}}{{h}_{p}}_{{H}^{1}}{\left{u}_{k}{u}_{p}\right}_{{H}_{0}^{1}},$$ 
whence the simplification gives
$${{u}_{k}{u}_{p}}_{{H}_{0}^{1}}\le \frac{1}{\eta (0)}{{h}_{k}{h}_{p}}_{{H}^{1}}+\frac{L}{{\eta}^{2}(0)}{{h}_{p}}_{{H}^{1}}\left{{u}_{p}}_{{H}_{0}^{1}}{{u}_{k}}_{{H}_{0}^{1}}\right.$$ 
Since ${{h}_{k}}_{{H}^{1}}$ is bounded and $({h}_{k})$ and $({{u}_{k}}_{{H}_{0}^{1}})$ are convergent, one immediately obtain that the sequence $({u}_{k})$ is Cauchy. Hence there is $u$ with ${u}_{k}\to u$ and passing to the limit in
$$\eta ({{u}_{k}}_{{H}_{0}^{1}}){({u}_{k},\nu )}_{{H}_{0}^{1}}=({h}_{k},\nu ),v\in {H}_{0}^{1}(0,1)$$ 
we obtain that $u=S(h).$ Finally the uniqueness of the solution implies that the whole sequence $({u}_{k})$ converges to $S(h),$ that is $S({h}_{k})\to S(h).$
(d) Inequality (3.7) immediately follows from (3.6) by taking $v=u$ and using the obvious inequality $(h,u)\le {h}_{{H}^{1}}{u}_{{H}_{0}^{1}}.$
The next theorem expresses a monotonicity property of the solution operator.
Theorem 3.3
If $0\le {h}_{1}\le {h}_{2},$ then ${S({h}_{1})}_{{H}_{0}^{1}}\le {S({h}_{2})}_{{H}_{0}^{1}}.$
Proof. Denote $u:=S({h}_{1})$ and $\nu :=S({h}_{2}).$ Since ${h}_{1},{h}_{2}\ge 0,$ one has $u,\nu \ge 0.$ Then
$$\eta ({u}_{{H}_{0}^{1}}){u}_{{H}_{0}^{1}}^{2}=({h}_{1},u)\le ({h}_{2},u)=\eta ({\nu }_{{H}_{0}^{1}})(u,\nu )\le \eta ({\nu }_{{H}_{0}^{1}}){u}_{{H}_{0}^{1}}{\nu }_{{H}_{0}^{1}}$$ 
which gives
$$\eta ({u}_{{H}_{0}^{1}}){u}_{{H}_{0}^{1}}\le \eta ({\nu }_{{H}_{0}^{1}}){\nu }_{{H}_{0}^{1}},$$ 
whence the conclusion
$${u}_{{H}_{0}^{1}}\le {\nu }_{{H}_{0}^{1}}.$$ 
3.3 Existence and localization results
Theorem 3.4
Let $\eta \in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ be increasing and Lipschitz continuous on ${\mathbb{R}}_{+}$ with$\eta (0)\ne 0$ and let $f:[0,1]\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be an ${L}^{2}$Carathéodory function, increasing in the second variable and symmetric in the first variable, i.e.,
$$f(x,s)=f(1x,s)\phantom{\rule{1em}{0ex}}\text{for all}x\in [0,\frac{1}{2}],s\in {\mathbb{R}}_{+}.$$  (3.9) 
Assume that for some ${x}_{0}\in (0,\frac{1}{2})$ and some numbers $\alpha ,\beta >0$ with $\alpha \ne \beta ,$ the following inequalities hold:
$$\frac{{f(\cdot ,\alpha )}_{{L}^{2}}}{\alpha \eta (\alpha )}\le \pi ,$$  (3.10) 
$$\frac{{\int}_{{x}_{0}}^{1{x}_{0}}f(x,\gamma \beta )\mathit{d}x}{\beta \eta (\beta )}\ge \frac{1}{\gamma},$$  (3.11) 
where $\gamma ={x}_{0}\left(12{x}_{0}\right).$ Then there exists at least one solution $u\in {\widehat{H}}_{0}^{1}(0,1)$ of (1.1), which is positive and concave in $(0,1),$ satisfies the Harnack inequality
$$u(x)\ge \gamma {u}_{{H}_{0}^{1}}\phantom{\rule{1em}{0ex}}\text{for all}x\in [{x}_{0},1{x}_{0}],$$  (3.12) 
and
$$r\le {u}_{{H}_{0}^{1}}\le R,$$  (3.13) 
where $r=\mathrm{min}\{\alpha ,\beta \},$ $R=\mathrm{max}\{\alpha ,\beta \}.$
Proof. We shall apply Theorem 2.1 with $X={\widehat{H}}_{0}^{1}(0,1)$ and the cone
$$K=\{u\in {\widehat{H}}_{0}^{1}(0,1):u\ge 0\phantom{\rule{1em}{0ex}}\text{on}(0,1),u(x)\ge \gamma {u}_{{H}_{0}^{1}}\text{for all}x\in [{x}_{0},1{x}_{0}]\}.$$ 
Note that $K$ does not reduce to the origin. For example, the function $S\left(1\right)\in K\setminus \left\{0\right\}.$ As operator $N$ we consider the composed mapping
$$N(u)=Sf(\cdot ,u).$$ 
We first note that $N$ is welldefined from $K$ to $K.$ Indeed, if $u\in K,$ then $f(.,u)\in {L}^{2}(0,1)$ and so $N(u)$ has the expression
$$N(u)(x)=\frac{1}{\eta ({u}_{{H}_{0}^{1}})}{\int}_{0}^{1}G(x,s)f(s,u(s))\mathit{d}s.$$ 
Since the Green’s function $G$ is symmetric and so is $f$ accordingly with (3.9), the function $N\left(u\right)$ is also symmetric. Moreover, since $f(.,u)$ is nonnegative on $(0,1)$ and increasing on $(0,\frac{1}{2}),$ the function $N\left(u\right)$ satisfies the Harnack inequality
$$N(u)\ge \gamma {N(u)}_{{H}_{0}^{1}}.$$ 
Therefore $N(u)\in K$ and thus $N\left(K\right)\subset K.$
In addition, the operator $N$ is completely continuous. Indeed, if $\mathcal{M}\subset K$ is a bounded set, then it is easy to see that $N(\mathcal{M})$ is bounded in ${H}^{2}(0,1)$ and thus it is compact in ${H}^{1}(0,1)$ as claimed.
Next we prove that
$${u}_{{H}_{0}^{1}}\ge {N(u)}_{{H}_{0}^{1}}\phantom{\rule{1em}{0ex}}\text{for all}u\in K\phantom{\rule{1em}{0ex}}\text{with}{u}_{{H}_{0}^{1}}=\alpha .$$  (3.14) 
We suppose the contrary, i.e.,
$$  (3.15) 
Denote $\nu :=N(u)=Sf(\cdot ,u)$ and use (3.7) and (2.2), where $h=f(\cdot ,u),$ to deduce
$$\eta ({\nu }_{{H}_{0}^{1}}){\nu }_{{H}_{0}^{1}}\le {f(\cdot ,u)}_{{H}^{1}}\le \frac{1}{\pi}{f(\cdot ,u)}_{{L}^{2}}.$$  (3.16) 
Furthermore, since for every $x$ one has
$$u(x)={\int}_{0}^{x}1\cdot {u}^{\prime}(\tau )\mathit{d}\tau \le {\left({\int}_{0}^{x}{1}^{2}\mathit{d}\tau \right)}^{\frac{1}{2}}\cdot {\left({\int}_{0}^{x}u_{}^{\prime}{}_{}{}^{2}(\tau )\mathit{d}\tau \right)}^{\frac{1}{2}}\le {u}_{{H}_{0}^{1}}=\alpha $$ 
and $f$ is increasing in the second variable, we have
$${f(\cdot ,u)}_{{L}^{2}}\le {f(\cdot ,\alpha )}_{{L}^{2}}.$$  (3.17) 
Now, from (3.16) and (3.17), we have
$$\eta ({\nu }_{{H}_{0}^{1}}){\nu }_{{H}_{0}^{1}}\le \frac{1}{\pi}{f(\cdot ,\alpha )}_{{L}^{2}}.$$  (3.18) 
On the other hand, since from (3.15) ${\nu }_{{H}_{0}^{1}}>\alpha $ and $s\eta (s)$ is strictly increasing in ${\mathbb{R}}_{+},$ we have
$$\eta ({\nu }_{{H}_{0}^{1}}){\nu }_{{H}_{0}^{1}}>\alpha \eta (\alpha ).$$  (3.19) 
Then (3.18) and (3.19) lead to
$$ 
which contradicts our assumption (3.10). Thus (3.14) is proved.
Next we prove that
$${u}_{{H}_{0}^{1}}\le {N(u)}_{{H}_{0}^{1}}\phantom{\rule{1em}{0ex}}\text{for all}u\in K\phantom{\rule{1em}{0ex}}\text{with}{u}_{{H}_{0}^{1}}=\beta .$$  (3.20) 
We suppose the contrary, i.e.,
$$\text{there exists}\phantom{\rule{1em}{0ex}}u\in K:{u}_{{H}_{0}^{1}}=\beta \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{u}_{{H}_{0}^{1}}>{N(u)}_{{H}_{0}^{1}}.$$  (3.21) 
Letting $v:=N\left(u\right)$ gives
$$\eta ({\nu }_{{H}_{0}^{1}}){\nu}^{\prime \prime}=f(\cdot ,u)$$ 
in the weak sense. Then also using (3.12) we obtain
$$\eta ({\nu }_{{H}_{0}^{1}})({\nu}^{\prime \prime},\nu )=\eta ({\nu }_{{H}_{0}^{1}}){\nu }_{{H}_{0}^{1}}^{2}={(f(\cdot ,u),\nu )}_{{L}^{2}}={\int}_{0}^{1}\nu (x)f(x,u(x))\mathit{d}x$$ 
$$\ge {\int}_{{x}_{0}}^{1{x}_{0}}\nu (x)f(x,u(x))\mathit{d}x\ge {\int}_{{x}_{0}}^{1{x}_{0}}\nu (x)f(x,\gamma \beta )\mathit{d}x\ge \gamma {\nu }_{{H}_{0}^{1}}{\int}_{{x}_{0}}^{1{x}_{0}}f(x,\gamma \beta )\mathit{d}x.$$ 
Consequently
$$\eta ({\nu }_{{H}_{0}^{1}}){\nu }_{{H}_{0}^{1}}\ge \gamma {\int}_{{x}_{0}}^{1{x}_{0}}f(x,\gamma \beta )\mathit{d}x.$$  (3.22) 
On the other hand, $$ whence
$$\beta \eta (\beta )>\gamma {\int}_{{x}_{0}}^{1{x}_{0}}f(x,\gamma \beta )\mathit{d}x,$$ 
which contradicts (3.11). Thus (3.20) is proved. Now Krasnosel’skiǐ’s theorem applies and yields the result.
3.4 The case of systems
In this section we extend the results from the equations to the case of systems. We look for solutions $u=({u}_{1},{u}_{2},\mathrm{\dots},{u}_{n})$ with ${u}_{i}\in K$ for all $i=1,2,\mathrm{\dots},n.$ Under the details given before, to each ${\eta}_{i}$ we associate a solution operator ${S}_{i}:{H}^{1}(0,1)\to {H}_{0}^{1}(0,1)$. Then system $(\text{1.2})$ reads equivalently as
$${u}_{i}={S}_{i}{f}_{i}(\cdot ,u),i=1,2,\mathrm{\dots},n.$$ 
Let $N:{K}^{n}\to {K}^{n},$ $N=({N}_{1},{N}_{2},\mathrm{\dots},{N}_{n})$ be defined by
$${N}_{i}(u)={S}_{i}{f}_{i}(\cdot ,u(\cdot ))\phantom{\rule{1em}{0ex}}(i=1,2,\mathrm{\dots},n).$$ 
If ${u}_{i}\in K$ for each $i,$ then ${N}_{i}(u)\in K.$ Thus the cone ${K}^{n}$ is invariant by $N.$ Moreover, the operator $N$ is completely continuous since, by standard arguments, the components ${N}_{i}$ are completely continuous.
The following result is a generalization of Theorem 3.1 and guarantees the existence of positive solutions to the problem $(\text{1.2})$ and their componentwise localization.
Theorem 3.5
Let for any index $i\in \{1,2,\mathrm{\dots},n\}$ ${\eta}_{i}$ is continuous, increasing and nonnegative function, ${\eta}_{i}(0)\ne 0,$ ${\eta}_{i}$ has a limited derivative and let the functions ${f}_{i}\in C([0,1]\times {\mathbb{R}}_{+}^{n};{\mathbb{R}}_{+})$ is increasing on ${\mathbb{R}}_{+}$ with respect to any variable ${u}_{j},$ $j=1,2,\mathrm{\dots},n$, and
$${f}_{i}(x,s)={f}_{i}(1x,s)\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{1em}{0ex}}x\in [0,\frac{1}{2}].$$ 
Moreover, let the function ${g}_{i}(t)=t{\eta}_{i}(t)$, where $t\ge 0,$ is strictly increasing in ${\mathbb{R}}_{+},$ the point ${x}_{0}$ be fixed in $(0,\frac{1}{2})$ and let exist ${\alpha}_{i},{\beta}_{i},{\gamma}_{i}>0$ with ${\alpha}_{i}\ne {\beta}_{i}$ such that
$$\frac{{{f}_{i}(\cdot ,\alpha )}_{{L}^{2}}}{{\alpha}_{i}{\eta}_{i}({\alpha}_{i})}\le \pi ,\frac{{\int}_{{x}_{0}}^{1{x}_{0}}{f}_{i}(x,\gamma \beta )\mathit{d}x}{{\beta}_{i}{\eta}_{i}({\beta}_{i})}\ge \frac{1}{\gamma}$$ 
for $i=1,2,\mathrm{\dots},n,$ where $\alpha =({\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{n})$ and $\beta =({\beta}_{1},{\beta}_{2},\mathrm{\dots},{\beta}_{n}).$ Then for $(\text{1.2})$ at least one solution $u=({u}_{1},{u}_{2},\mathrm{\dots},{u}_{n}),$ which is positive on $(0,1)$, concave, satisfied the Harnack inequality
$${u}_{i}(x)\ge \gamma {{u}_{i}}_{{\widehat{H}}_{0}^{1}}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}i=1,2,\mathrm{\dots},n\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{1em}{0ex}}x\in [{x}_{0},1{x}_{0}],$$ 
where $\gamma ={x}_{0}(12{x}_{0}),$ and
$${r}_{i}\le {{u}_{i}}_{{\widehat{H}}_{0}^{1}}\le {R}_{i},$$ 
where ${r}_{i}=min\{{\alpha}_{i},{\beta}_{i}\},$ ${R}_{i}=max\{{\alpha}_{i},{\beta}_{i}\}$, $i=1,2,\mathrm{\dots},n.$
Proof. The result is a consequence of the vectorial version of Krasnosel’ski$\stackrel{\u02c7}{\u0131}$’s fixed point theorem in cones.
3.5 Some particular cases
In this section, we shall take into consideration some particular cases of the problem $(\text{1.1})$. We consider the case when $f=f(u),$ i.e. the problem
$$  (3.23) 
Corollary 3.6
Let $\eta $ is continuous, increasing and nonnegative function, $\eta (0)\ne 0,$ $\eta $ has a limited derivative and let the function $f\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ is increasing. Moreover, let the function $g(t)=t\eta (t)$, where $t\ge 0,$ is strictly increasing in ${\mathbb{R}}_{+},$ the point ${x}_{0}$ be fixed in $(0,\frac{1}{2})$ and let exists $\alpha ,\beta >0$ with $\alpha \ne \beta ,$ such that
$$\frac{f(\alpha )}{\alpha \eta (\alpha )}\le \pi ,\frac{f(\gamma \beta )}{\beta \eta (\beta )}\ge \frac{1}{2{x}_{0}\gamma}.$$ 
Then for $(\text{3.23})$ at least one solution $u\in {\widehat{H}}_{0}^{1}(0,1),$ which is positive on $(0,1),$ concave, satisfied the Harnack inequality
$$u(x)\ge \gamma {u}_{{\widehat{H}}_{0}^{1}}\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{1em}{0ex}}x\in [{x}_{0},1{x}_{0}],\text{where}\phantom{\rule{1em}{0ex}}\gamma ={x}_{0}(12{x}_{0}),$$ 
and
$$r\le {u}_{{\widehat{H}}_{0}^{1}}\le R,$$ 
where $r=min\{\alpha ,\beta \},$ $R=max\{\alpha ,\beta \}.$
Proof. The proof of this corollary is similar to the proof of Theorem 3.4, with the operator $N$ replaced by the $\stackrel{~}{N}(u)=Sf(u(\cdot ))$.
4 Conclusions
In this paper, we have studied the positive solutions for the Dirichlet boundary value problem for Kirchhoff equations and systems. We have obtained existence and localization results of positive solutions using Krasnosel’ski$\stackrel{\u02c7}{\u0131}$’s fixed point theorem in cones and a weak Harnack type inequality. In the case of systems, results on the localization of solutions are established using the vector version of Krasnosel’ski$\stackrel{\u02c7}{\u0131}$’s theorem, where the contractionexpansion conditions are expressed on the components.
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