\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 280, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/280\hfil Existence of three solutions]
{Existence of three solutions for higher order BVP with parameters via
Morse theory}

\author[M. Jurkiewicz, B. Przeradzki \hfil EJDE-2016/280\hfilneg]
{Mariusz Jurkiewicz, Bogdan Przeradzki}

\address{Mariusz Jurkiewicz \newline
Institute of Mathematics and Cryptology,
Military University of Technology,
 Gen. Sylwestra Kaliskiego 2, 00-908 Warszawa 49, Poland}
\email{mjurkiewicz@wat.edu.pl}

\address{Bogdan Przeradzki \newline
Institute of Mathematics,
Lodz University of Technology,
W\'{o}lcza\'{n}ska 215, 90-924 \L\'{o}d\'{z}, Poland}
\email{bogdan.przeradzki@p.lodz.pl}

\thanks{Submitted April 26, 2016. Published October 19, 2016.}
\subjclass[2010]{34B15, 37B30}
\keywords{Multidimensional spectrum; Lidstone BVP; critical point theory; 
\hfill\break\indent Palais-Smale condition; critical group; Morse's theory}

\begin{abstract}
 We prove the existence of at least three solutions to a general
 Lidstone problem using the Morse Theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

In 1929, Lidstone introduced a generalization of the Taylor series.
 It approximates a given function in the neighbourhood of two points instead of one. 
Thus, the initial research was devoted to description of a maximal set of 
functions that could be expressed as a Lidstone series. Those considerations 
led to so-called functions of exponential type and next to general Lidstone
boundary value problem (BVP) (comp. \cite{RPA,FM,GG}), 
examined today in various configurations. Motivated by many papers 
(see for example (\cite{JMDJH}, \cite{BZXL}), in which authors studied 
the existence of multiple solutions to Lidstone BVP, and being inspired 
by ideas given by Chang \cite{KCC,KCC2}, we have decided to study the case 
of existence of at least three solutions to the BVP that is being described. 
The crucial results of our research are presented in this paper.

Coming to the point, we shall consider a special case of a general Lidstone BVP 
for a nonlinear ordinary differential equation
of the $2k$-th order studied earlier in \cite{MJ2}:
\begin{equation} \label{general}
\begin{gathered}
x^{(2k)}-\sum_{j=1}^k \lambda_j x^{(2k-2j)}=f(t,x,x'',\ldots, x^{(2k-2)}),\\
x^{(2j)}(0)=0=x^{(2j)}(1),\quad j=0,1,\ldots, k-1.
\end{gathered}
\end{equation}
 It is a natural generalization of the beam equation with fixed ends 
(comp. \cite{MJ1}). The topological methods used in the mentioned papers 
admit arbitrary parameters $\lambda :=(\lambda_1,\ldots ,\lambda_k)$ but need very 
restrictive conditions on an asymptotic behaviour of the nonlinear term $f$.
The problem \eqref{general} is equivalent to the fixed point problem 
for completely continuous operator if the linear homogeneous problem $(f=0)$ 
has only the trivial solution. In \cite{MJ1}, it has been shown that it is 
for $\lambda$ which does not belong to a sequence of $(k-1)$-dimensional hyperplanes; 
the paper contains mainly the case when $\lambda$ sits in this sequence 
-- the resonant case.

Here, we  study a special nonlinear case of the general Lidstone problem,
 that takes the form
\begin{equation}
\begin{gathered}
(-1)^{k}x^{(2k)}+\sum_{j=1}^{k}\lambda
_{j}x^{(2k-2j)}=(-1)^{i-1}f( t,x^{(2i-2)})  \\
x^{(2j)}(0) =x^{(2j)}(1)=0,\quad j=0,\ldots ,k-1,
\end{gathered} \label{prob_simp}
\end{equation}
with fixed parameters $k\geq 2$ and $i\in\{1,\ldots ,k-1\}$. 
Note that continuous $f$ depends on one of the even order derivatives only. 
Then, the problem is equivalent to looking for a critical point of a $C^1$-functional
on a Hilbert space. We have considered it in \cite{JP} and we have found 
assumptions that guarantee the existence of at least one solution. 
Furthermore, in \cite{MJ4} one of the authors has answered a question 
concerning existence of infinitely many solutions to \eqref{prob_simp}. 
After joining those results a natural problem about intermediate case arose. 
Actually, this is the goal of the paper to fill this hiatus, what means 
to formulate the assumptions that lead us to the conclusion of existence no 
fewer than three solutions.

For a vector of real coefficients $\lambda = (\lambda_{1},\ldots ,\lambda_{n})$, 
let us define
$$
\Lambda (n):=\sum_{j=1}^k (-1)^{k-j} \lambda_j (n^2\pi^2)^{k-j},
$$
$H_n:=\{ \lambda\in\mathbb{R}^{k} : \Lambda (n)+(n^2\pi ^2)^k=0\}$ for $n=1,2,\ldots $. 
Then the linear homogeneous problem has nontrivial solutions if and only if
$$
\lambda\in \sigma^k:=\bigcup_{n=1}^{\infty} H_n.
$$ 
We shall assume that $\lambda\in \Delta_+$, 
where
$$
\Delta_+:=\cap_{n=1}^{\infty} \{\lambda: \Lambda (n)\ge 0\}.
$$ 
This set is nonempty and even large since it contains all $\lambda$ such that
$(-1)^j\lambda_j\ge 0$ for any $j$ if $k$ is even and $(-1)^j\lambda_j\le 0$ 
if $k$ is odd. Obviously, $\Delta_+\cap\sigma^k =\emptyset$.
It can be proved that \eqref{prob_simp} is equivalent (comp. \cite{JP}) to
$$
x^{(2i-2)}(t)=\int_0^1 \mathcal{H}_i(t,s) f(s,x^{(2i-2)}(s))\, ds,
$$
where
$$
\mathcal{H}_i(t,s):=2\sum_{n=1}^{\infty}
\frac{(n^2\pi^2)^{i-1}}{\Lambda (n)+(n^2\pi^2)^k}\sin(n\pi s)\sin(n\pi t).
$$
The last formula is obtained by using the spectral theory of compact 
self-adjoint operators in Hilbert space. Hence, one should find
fixed points of an operator which is a superposition of the Nemytski\v{i} 
operator $\mathbf{f}$ defined by $f$ and the linear integral operator 
$\mathbf H$ with
the kernel $\mathcal{H}_i$. Denote this superposition as $T: C[0,1]\to C[0,1]$. 
The spectrum of $H$ is composed of eigenvalues
$$
\mu_n^2:=\frac{(n^2\pi^2)^{i-1}}{\Lambda (n)+(n^2\pi^2)^k},\quad n=1,2,\ldots
$$
and the limit of this sequence -- $0$ which belongs to the continuous spectrum.
 We can define $H$ by the same formula on the Hilbert space
$L^2(0,1)$. The assumption $\lambda\in \Delta_+$ is necessary to get 
$H\ge 0$ that enables us to define a square root
$S: L^2(0,1)\to L^2(0,1)$ --
a nonnegative linear operator such that $S^2=H$.
From the spectral theory (comp. \cite{MET}), we know that $S$ is an integral 
operator with the kernel
$$
\mathcal{S}(t,s):=\sum_{n=1}^{\infty}{\frac{2(n\pi)^{i-1}}
{\sqrt{\Lambda (n)+(n^2\pi^2)^{k}}}\sin(n\pi s)\sin(n\pi t)}.
$$
In \cite{MJ4}, it is proved that fixed points of $T$ are exactly critical 
points of the  functional $\varphi :L^2(0,1)\to\mathbb{R}$,
$$
\varphi(y):=\frac{1}{2}\| y\|^2-\int_0^1 F(t, Sy(t))\, dt,
$$
since
$$
\varphi'(y)\cdot u=\langle  y,u\rangle -\langle {\mathbf{f}}(Sy), Su\rangle,
$$
where $F(t,w)=\int_{0}^{w}f(t,u) \, du$, $\langle\cdot,\cdot\rangle$ stands 
for the scalar product in $L^2(0,1)$, and $S$ is self-adjoint.

In \cite{JP}, we proved the following result.

\begin{theorem}\label{main-thm}
Assume that $\lambda \in \Delta _{+}$ and let $f:[0,1] \times \mathbb{R}\to \mathbb{R}$ 
be a continuous function that
satisfies the following conditions
\begin{itemize}
\item[(i)] $\limsup_{u\to 0} \frac{f(t,u)}{u}<\mu _1^{-2}$,

\item[(ii)] $\liminf_{u\to +\infty } \frac{f(t,u)}{u}>\mu_1^{-2}$

\item[(iii)] there exist $k\in [0,1/2) $ and $N>0$, such that
for $\| w\| \geq N$, we have
$$
F(t,w)\leq kwf(t,w),
$$
uniformly with respect to $t\in [0,1] $. 
\end{itemize}
Then problem \eqref{prob_simp} possesses at least one nonzero solution.
\end{theorem}

If $f(t,\cdot)$ is odd for any $t$ and the limit in condition $(ii)$ equals
 $+\infty$, then in \cite{MJ4}
it has been proved that $\varphi$ has infinitely many critical points which 
are solutions of \eqref{prob_simp}.
We used slightly different notations in both papers but one can see that 
$$
\mu_1^{2} =\Big(\frac{\Lambda (1)+\pi^{2k}}{\pi^{2i-2}}\Big)^{-1}=\| H\| .
$$
Observe that condition (i) implies $f(t,0)\equiv 0$ and we get the null solution.

\section{Main results}

Now, we can formulate the main result.

\begin{theorem} \label{thm2}
Let $f$ be continuous, and assume the conditions:
\begin{itemize}
\item[(C1)] there exist an integer $m\ge 1$ and the limits, for any $t$, such that
$$
\frac{\partial f}{\partial x}(t,0):=\lim_{u\to 0}\frac{f(t,u)}{u}\in
 (\mu_m^{-2}, \mu_{m+1}^{-2});
$$

\item[(C2)] the infinite system of linear equations
$$
c_j=2\mu_j \sum_{n=1}^{\infty} \mu_n \int_0^1 
\frac{\partial f}{\partial x}(t,0)\sin(n\pi t)\sin(j\pi t)\, dt\;
 c_n,\quad j=1,2,\ldots,
$$
has only trivial solutions: $c_j=0$, $j=1,2,\ldots $.

\item[(C3)] there exist $0<\alpha<\mu_1/2$, $\beta\in\mathbb{R}$ such that 
$F(t,u)\le\alpha u^2+\beta$ for any $t,u$.
\end{itemize}
Then \eqref{prob_simp} has at least three solutions.
\end{theorem}

\begin{proof}
 We apply \cite[Thm. 5.4]{KCC2} 
(first slightly weaker result appears in \cite{KCC}) 
which gives at least three critical points for $C^1$-functional $\psi$ 
defined on the Hilbert space, if $\psi$ is bounded below, 
it satisfies  Palais-Smale condition and has a nondegenerate critical point 
different from the argument of its minimum with finite
Morse index. Here, $L^2(0,1)$ is the Hilbert space, $\varphi$ is the 
functional and the null solution is a critical point with finite Morse index.

Condition (C3) leads to the estimate
$$
\varphi(y)\ge \frac{1}{2}\| y\|^2-\int_0^1 (\alpha (Sy)(t)^2+\beta)\, dt
\ge \big(\frac{1}{2}- \alpha\mu_1^2\big) \|y\|^2-\beta
$$
that implies values of $\varphi$ are bounded from below by a quadratic 
function with finite minimum. It gives that $\varphi$ is bounded from below
and coercive. Let $(y_n)$ be a Palais-Smale sequence, i.e.
$|\varphi(y_n)|\le M$ and $\varphi'(y_n)=y_n-S({\mathbf{f}}(S(y_n)))\to 0$. 
Since $S$ is compact, $S(\mathbf{f}(S(y_n)))$ has a convergent subsequence
and the Palais-Smale condition is satisfied.

Let us observe that $\varphi'(0)=0$ and there exists the second derivative 
of this functional at $0$:
$$
\varphi''(0)(u,v)=\langle u,v\rangle 
-\int_0^1 \frac{\partial f}{\partial x}(t,0)\cdot Su(t)\cdot Sv(t)\, dt 
=\langle u,v\rangle -\langle \frac{\partial f}{\partial x}(\cdot , 0)
\cdot Su,Sv\rangle .
$$
This bilinear functional on $L^2(0,1)$ is symmetric and continuous which means 
that it defines a self-adjoint operator
$L$ on $L^2(0,1)$ given by $\varphi''(0)(u,v)=\langle Lu,v\rangle$. 
It follows that
$$
Lu=u-S\Big(\frac{\partial f}{\partial x}(\cdot,0)\cdot Su\Big) .
$$
If we denote by $e_n(t):=\sqrt{2}\cdot\sin(n\pi t)$ for $n=1,2,\ldots$, 
then it is a complete orthonormal basis of eigenfunctions of $S$ corresponding 
eigenvalues $\mu_n$. Let $H_-$ denotes a subspace spanned by $e_n$ with 
$n\le m$ and $H_+$ its orthogonal complement. We shall show that $L$ is one-to-one.
If it is not the case, there exists $u=\sum_n c_n e_n\neq 0$, $Lu=0$ and then
\begin{equation} \label{coeff}
u=\sum_{j=1}^{\infty}\sum_{n=1}^{\infty} \mu_j c_n\mu_n \int_0^1  \frac{\partial f}{\partial x}(s,0)e_j(s)e_n(s)\, ds\cdot e_j.
\end{equation}
that implies the system from (C2).

Since $L=I-S_1$ where $S_1u=S(\frac{\partial f}{\partial x}(\cdot ,0)\cdot Su)$ 
is compact, then the range of $L$ is closed with codimension $0$. 
It follows that $L$ is an isomorphism of $L^2(0,1)$ onto itself 
and $0$ is a nondegenerate critical point of $\varphi$. We shall show that
\begin{gather*}
\varphi''(0)(x,x)\le -C\| x\|^2,\quad x\in H_-,\\
\varphi''(0)(x,x)\ge C\| x\|^2,\quad x\in H_+,
\end{gather*}
for some positive constant $C$.

From (C1) there exists $\xi > 0$ such that 
$$
\mu_m^{-2}+\xi\le \frac{\partial f}{\partial x}(t,0)\le \mu_{m+1}^{-2}-\xi,
\quad t\in [0,1].
$$
Then 
$$
\varphi''(0)(x,x)=\| x\|^2-\int_0^1 \frac{\partial f}{\partial x}(t,0)
|Sx(t)|^2\, dt
$$ 
and, for $x\in H_-$,
$$
\varphi''(0)(x,x)\le \| x\|^2 -(\mu_m^{-2}+\xi)\| Sx\|^2\le -\xi\mu_m^2 \| x\|^2
$$
and similarly, for $x\in H_+$,
$$
\varphi''(0)(x,x)\ge \| x\|^2 -(\mu_{m+1}^{-2}-\xi)\| Sx\|^2
\ge \xi\mu_{m+1}^2 \| x\|^2.
$$
This means that  $L|H_-$ is negative and $L|H_+$ is positive and the
 Morse index of the critical point $0$ equals $m$ so it is finite.

It remains to show that there exists $x$ such that $\varphi(x)<0 =\varphi (0)$. 
But from $(C1)$ one can take $\xi>0$ and $\varepsilon>0$ such that
$f(t,u)\ge (\mu_m^{-2} +\xi) u$ for any $t$ and $|u|\le \varepsilon$.
It follows that $F(t,u)\ge \frac{1}{2}(\mu_m^{-2}+\xi) u^2$ for 
$|u|\le\varepsilon$.
 Then take $0\neq y\in H_-$ such that $\sup_t |Sy(t)|\le\varepsilon$ and get
$$
\varphi (y) = \frac{1}{2}\| y\|^2-\int_0^1 F(t,Sy(t))\, dt
\le -\frac{\xi\mu_{m}^{2}}{2}\| y\|^2<0.
$$
The proof is complete.
\end{proof}


\subsection*{Remarks} 
If $\frac{\partial f}{\partial x}(t,0)$ does not depend on $t\in [0,1]$, 
then condition (C2) is satisfied since
$$
\int_0^1 \frac{\partial f}{\partial x}(t,0) \sin(n\pi s)\sin(j\pi s)\, ds=0
$$
for $n\neq j $. In other cases, condition (C2) can be verified by a 
finite algorithm. The condition is equivalent to:
$$
1\notin \sigma (S_1)
$$
and this operator is compact selfadjoint, hence its eigenvalues can be 
obtained by the Courant-Hilbert method (see \cite{MET}), and the sequence of
eigenvalues of $S_1$ tends to 0,therefore only finite number of them can 
be greater then $1$. The main result can be obtained (essentially with 
the same proof) by using \cite[Corollary 3]{Am} where only 
the Leray-Schauder degree is applied.

\begin{corollary} \label{cor1}
The equation describing a shape of a beam freely supported on both ends:
$$
u^{(4)}-\lambda_1 u''+\lambda_2 u=f(t,u),\quad u(0)=u(1)=u''(0)=u''(1)=0
$$
with $\lambda_{1,2}\ge 0$ has at least three solutions if there exists 
$m\in\mathbb{N}$ such that
$$
\lambda_2+\lambda_1 m^{2}\pi^{2}+m^4\pi^4
< \frac{\partial f}{\partial x}(t,0)
< \lambda_2+\lambda_1 (m+1)^{2}\pi^{2}+(m+1)^4\pi^4
$$
for all $t$ and {\rm (C2), (C3)} hold.
\end{corollary}

A similar problem with $\lambda_1=0$, $\lambda_2$ depending on $t$ and $f(t,u)=h(t)$ 
was recently studied in \cite{DrH}.

\subsection*{Acknowledgments}
The authors would like to thank to an anonymous referees for their careful 
reading of the original manuscript.


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\end{document}