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\markboth{ Existence and number of solutions to semilinear equations }
{ P. S. Milojevi\'c }
\begin{document}
\setcounter{page}{201}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Nonlinear Differential Equations, \newline
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 201--221\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 Existence and number of solutions to semilinear equations with 
 applications to boundary-value problems
% 
\thanks{ {\em 1991 Mathematics Subject Classifications:} 47H15,47H09, 35J40. 
 \hfil\break\indent 
{\em Key words:} Existence, number of solutions, (pseudo) A-proper
maps, elliptic BVP's, \hfil\break\indent
 parabolic and hyperbolic equations.
 \hfil\break\indent
\copyright 2000 Southwest Texas State University. 
\hfil\break\indent Published October 25, 2000.  } } 

\date{}
\author{  P. S. Milojevi\'c } 
\maketitle
\begin{abstract} 
 We present recent and some new existence results on the number of 
 solutions to nonlinear equations and to (non)resonant semilinear 
 equations involving nonlinear perturbations of Fredholm maps of 
 index zero.  We apply our results to semilinear elliptic, 
 and to semilinear parabolic and  hyperbolic periodic  
 boundary-value problems.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]             
\newtheorem{corollary}{Corollary}[section] 
 
\section{Introduction}

Let $X$ and $Y$ be Banach spaces and $T:X\to Y$ be a nonlinear
map of $A$-proper type. Under various conditions on $T$, we
study in Section 2 the surjectivity and the finitness of the
solution set of the equation $Tx=f$. In particular, we
look at nonresonant semilinear equations of the form
$ Ax+Nx=f$
where $A$ is a Fredholm map of index zero and the nonlinear
map $N$ is such that $A+N$ is (pseudo) $A$-proper.
We say that this equation is not at resonance if $A$ and $N$ are
are such that it is solvable for each $f\in Y$.
Applications to semi-abstract nonresonant semilinear equations are
given in Section 3. Section 4 is devoted to applications
of the results of Section 3 to boundary-value problems (BVP) for 
semilinear elliptic equations. 
In Section 5, some comments on periodic BVP's for semilinear parabolic and
hyperbolic equations assuming nonuniform nonresonance conditions
are made. The existence of solutions for such problems has been studied
earlier in \cite{Ma-1, Ma-2, MW, NW, FN, G, INW}.


\section{Number of solutions to operator equations}


In this section, we shall study the number of solutions to the
equation $Tx=f$. The unique (approximate)
solvability of this equation has been studied in detail in \cite{Mi-6},
 using the $A$-proper mapping approach.

\paragraph{Definition} A map $T:D\subset X\to Y$ is
(pseudo) $A$-proper with respect to a scheme
$\Gamma=\{X_n,Y_n,Q_n\}$ with $\dim X_n=\dim Y_n$ on $D$ if
whenever $\{x_{n_k}\in D\cap X_{n_k}\}$ is bounded and
such that $Q_{n_k}Tx_{n_k}-Q_{n_k}f\to 0$ for some $f\in Y$, then
$\{x_n\}$ has a subsequence converging to $x\in D$ (there is
$x\in D$) with $Tx=f$.

Next, we shall define $A$-proper homotopies.

\paragraph{Definition} A homotopy $H:[0,1]\times D\to Y$ is $A$-proper
with respect to $\Gamma$ on $D$ if $Q_nH_t:D\cap X_n\to Y_n$ is continuous for
each $t$ and $n$, and if $\{x_{n_k}\in D\cap X_{n_k}\}$ is bounded
and $t_k\in [0,1]$ with $t_k\to t$ are such that
$Q_{n_k}H(t_k,x_{n_k})-Q_{n_k}f\to 0$ as $k\to \infty$
for some $f\in Y$, then a subsequence of $\{x_{n_k}\}$ converges to
$x\in D$ and $H(t,x)=f$. \smallskip

The classes of $A$-proper and pseudo $A$-proper maps are very general.
For many examples of such maps, we refer the reader to \cite{Mi-1}-\cite{Mi-5}.

\subsection*{Nonlinear equations}

We say that a map $T:X\to Y$ satisfies condition (+) if $\{x_n\}$ is
bounded whenever $Tx_n\to f$ in $Y$.
Let $\Sigma$ be the set of all points $x\in X$ where $T$ is not locally
invertible and card$T^{-1}(\{f\})$ be the cardinal number of the set
$T^{-1}(\{f\})$.

\begin{theorem}[\cite{Mi-7}]  % Theorem 2.1
 Let $T:X\to Y$ be continuous, $A$-proper and satisfy condition (+). Then
\begin{description}

\item{(a)} The set $T^{-1}(\{f\})$ is compact (possibly empty ) for each
$f\in Y$.

\item{(b)} The range $R(T)$ of $T$ is closed and connected.

\item{(c)} $\Sigma$ and $T(\Sigma)$ are closed subsets of $X$ and $Y$, respectively,
and $T(X\setminus \Sigma)$ is open in $Y$.

\item{(d)} card$T^{-1}(\{f\})$ is constant and finite (it may be 0) on each
connected component of the open set $Y\setminus T(\Sigma)$.

\item{(e)} if $\Sigma =\emptyset$, then $T$ is a homeomorphism from $X$ to $Y$.

\item{(f)} if $\Sigma\ne \emptyset$, then the boundary $\partial T(X\setminus \Sigma)$
of $T(X\setminus \Sigma)$ satisfies
$\partial T(X\setminus \Sigma)\subset T(\Sigma)$.
\end{description}\end{theorem}

\paragraph{Proof.} Since $T$ is proper by Proposition 2.1 in \cite{Mi-7},
it is a closed map.
Since $X\setminus \Sigma$ is an open set, $\Sigma$ is a closed set. Hence
(a)-(c) hold, where $T(X\setminus \Sigma)$ is open since $T$ is locally
invertible on $X\setminus \Sigma$.
(d) follows from the Ambrosetti
theorem [A] and (e) follows from the global inversion theorem. Next,
(b) and (c) imply that
$$T(X)=T(\Sigma)\cup T(X\setminus \Sigma)
=T(\Sigma)\cup \overline{F(X\setminus \Sigma)}
=\overline{T(X)}.\eqno (2.1) $$
Moreover, $\partial T(X\setminus  \Sigma)
=\overline{T(X\setminus \Sigma)}\setminus
T(X\setminus \Sigma)$, which together with (2.1) imply (f). 
\hfill$\diamondsuit$ \smallskip


Next, we shall look at another surjectivity result.
Let $J:X\to 2^{X^*}$ be the normalized duality map and $G:X\to Y$ be a
bounded map such that $Gx\ne 0$ for all x with $\|x\|\ge r_0$ for
some $r_0>0$ and
$$
\mbox{For each large $r>0$, $\deg(Q_nG, B(0,r)\cap X_n,0)\ne 0$ for all
large $n$.} \eqno (2.2)
$$ 

\begin{theorem} %Theorem 2.2
Let $T:X\to Y$ satisfy conditions $(+)$ and (2.2), and let  \begin{description}

\item{(i)} For each $f\in Y$ there is an $r_f>0$ such that
$$Tx\ne \lambda Gx\;\hbox{for}\;\;x\in \partial B(0,r_f), \lambda <0.
\eqno (2.3)$$
\item{(ii)} $H(t,x)=tTx+(1-t)Gx$ is an $A$-proper with respect to $\Gamma$ homotopy on
$[0,1]\times X$.
\end{description}
Then $T$ is surjective. Moreover, if $T$ is continuous, then $T^{-1}(\{f\})$
is compact for each $f\in Y$ and the cardinal number
card$T^{-1}(\{f\})$  is constant, finite and positive on each connected 
component of the set $Y\setminus T(\Sigma)$.
\end{theorem}

\paragraph{Proof.} The surjectivity of $T$ has been established 
earlier by the author (see, eg \cite{Mi-3,Mi-5}).
Moreover, $T$ is continuous and proper by Proposition 2.1 in
\cite{Mi-7}. Hence,
the other assertions of the theorem follow from Theorem 2.1. 

\begin{corollary} % Corollary 2.1
 Let $F,K:X\to X$ be continuous ball-condensing maps
and $T=I-F$ and $G=I-K$ satisfy (2.2)-(2.3). Then the conclusions of
Theorem 2.2 hold for $T$.
\end{corollary}

This corollary is also valid for general condensing maps (see \cite{S}).
For a map $M$, define its quasinorm by $|M|=\limsup_{\|x\|\to \infty}
\|Mx\|/\|x\|$.

\begin{theorem}[cf. \cite{Mi-5}] %  Theorem 2.3
Let $A:D(A) \subset X \to Y$ be a linear densely
defined map and $N:X \to Y$ be bounded and of the form $Nx=B(x)x+Mx$ for some
linear maps $B(x):X \to X$. Assume that there is a $c>|M|$ and a positively
homogeneous map $C:X\to Y$ such that
 $$\|Ax-(1-t)Cx-tB(x)x\| \ge c\|x\|, \; x \in D(A) \setminus
B(0,R). \eqno (2.4)$$
\begin{description}
\item{(i)} $H_t=A-(1-t)C-tN$ is $A$-proper with respect to $ \Gamma =
\{ X _{ n},Y _{ n}, Q _{ n} \}$ for $t \in [0,1)$ and $A-N$ is pseudo
$A$-proper

\item{(ii)} For all $r>R$, $\deg (Q_n(A-C),B(0,r)\cap X_n,0)
\ne 0$ for each large $n$.
\end{description}
\noindent
Then the equation $Ax-Nx=f$ is solvable for each $f\in Y$. If, in addition,
$A-N$ is continuous and $A$-proper, then $(A-N)^{-1}(\{f\})$ is compact
for each $f\in Y$ and card$(A-N)^{-1}(\{f\})$ is
constant, finite and positive on each connected component of the
set $Y\setminus (A-N)(\Sigma)$.
\end{theorem}

\paragraph{Proof.} Regarding the surjectivity of $A-N$,
it suffices to solve $Ax-Nx=0$.
Define $H(t,x)=Ax-(1-t)Cx-tNx$ on $[0,1]\times D(A)$. Then there is an $r>0$ 
such that 
$$H(t,x)\ne 0\;\;\hbox{for }\;x\in \partial B(0,r)\cap D(A),\;t\in [0,1].
\eqno (2.5)$$
If not, then there are $x_n\in H$ and $t_n\in [0,1]$ such that
$\|x_n\|\to \infty$ and $H(t_n,x_n)=0$.
Let $\epsilon>0$ be small such that $|M|\le (|M|+\epsilon)\|x\|$
for $\|x\|\ge R_1$ and $|M|+\epsilon <c$. For each $x_n$ with $\|x_n\|\ge R_1$
we have that
$$c\|x_n\|\le\|Ax_n-(1-t)Cx_n-tB(x_n)x_n\|\le (|M|+\epsilon )\|x_n\|.$$
Dividing by $\|x_n\|$, this leads to a contradiction and (2.5) holds.
Hence, $A-N$ is surjective by the homotopy result in \cite{Mi-2,Mi-3}. Next,
it is easy to see
that $\|(A-N)x\|\to \infty$ as $\|x\|\to \infty$ by (2.4). Hence,
the other assertions follow from Theorem 2.1. 

\section{Semi-abstract nonresonance problems}

Let $Q\subset R^n$ be a bounded domain, $V$ be a closed subspace of
$W_2^{2m}(Q)$ containing the test functions
and $L:V\to L_2$ be a linear map
with closed range in $H=L_2(Q)$. Let
$V_1$ be a closed subspace of $V$ and $L_1$ be the restriction of
$L$ to $V_1$. Assume
\begin{description}
\item{(L1)} Each eigenvalue $\lambda _j$ of $L_1$ has a finite multiplicity and the
corresponding  eigenfunctions $\{\dots , w_{-1}, w_0,
w_1,\dots\}$ form a complete set in $V_1$.
\end{description}

\noindent
Let $A=A_1+L$ for some linear map $A_1:V\to H$. For a fixed integer
$j$, define $B:V\to H$ by $Bu=-Au+\lambda_ju$.

\begin{description}
\item{(B1)} There is $\lambda \ne \lambda _j$, $j=1,2,\dots $, such that the 
map $B-\lambda I=-A_1-L-(\lambda-\lambda_j)I:V\to L_2$ is bijective.
\end{description}

\noindent
Let $\lambda \ne \lambda _j$ for each $j=1,2,\dots $ be fixed,
$\Gamma =\{Y_n,Q_n\}$ be a projectionally complete scheme for $L_2$ and 
$X_n=(B-\lambda I)^{-1}(Y_n)\subset V$ for each $n$.
Then $\Gamma _B=\{X_n,Y_n,Q_n\}$ is an admissible or a projectionally 
complete scheme for $(V,L_2)$. Since $B-\lambda I:V\to L_2$ is linear,
one-to-one and $A$-proper with respect to $\Gamma _B$, there is a constant 
$c>0$ (depending)only on $\lambda )$ such that
$$\|(B-\lambda I)u\|\ge c\|u\|_V,\;\; u\in V. \eqno (3.1)$$
Consider the following semilinear equation in $V$
$$Au+g(x,u,Du,\dots ,D^{2m-1}u)u+f(x,u,Du,\dots ,D^{2m}u)=h(x)\eqno(3.2)$$

For $u\in H$, set $u^{\pm}=\max(\pm u,0)$. Let $r=\lambda_{j+1}-\lambda_j$.
We require that $B$ has the following
properties: 

\paragraph{Property I} $B$ is a closed densely defined map in $H$ with
closed range $R(B)$, $(Bu,u)\ge -r^{-1}\|Bu\|^2$
and $R(B)= N(B)^{\perp}$ in $H$, $N(-L_1+\lambda_jI)
\subset N(B)$ and $(Bu,u)=(-Lu+\lambda_ju,u)$ on $V$.

\paragraph{Property II} If $(Bu,u)=-r^{-1}\|Bu\|^2$ for some $u\in V$,
then $u\in N(-L_1+\lambda_jI)\oplus N(-L_1+\lambda_{j+1}I)$. \smallskip
 

Let us note that if $B^{-1}$ is a partial inverse of $B$ and
$B^{-1}+r^{-1}I$ is strongly monotone on $R(B)$, i.e. it is a
bounded linear map on $R(B)$ and $((B^{-1}+r^{-1}I)u,u)=c_0
\|(B^{-1}+r^{-1}I)u\|^2$ on $R(B)$ for some $c_0>0$, then ([BF])
Property II holds in the sense that if $(Bu,u)= -r^{-1}\|Bu\|^2$
for some $u\in V$, then $u\in N(B)\oplus N(B+rI)$. If $B$ is
selfadjoint or angle bounded in the sense of H. Amann, it is
known that $B^{-1}+r^{-1}I$ is strongly monotone. If $B\ne B^*$
and $B$ is a normal map, the strong monotonicity of $B^{-1}+r^{-1}I$
has been discussed in Hetzer \cite{H}.

Some properties of $B$ are given next.

\begin{lemma} %Lemma 3.1
Let $B$ have Properties I and II. Suppose that $p_{\pm}
\in L_{\infty}(Q)$ are such that $0\le p_{\pm}(x)\le r$ for a.e. $x\in Q$
and
$$\int_Q[p_+(v^+)^2+p_-(v^-)^2]>0\;\hbox{for all}\;v\in N(-L_1+\lambda_jI)
\setminus \{0\}$$
and
$$\int_Q[(r-p_+)(w^+)^2+(r-p_-)(w^-)^2]>0\;\hbox{for all}\;w\in
N(-L_1+\lambda_{j+1}I)\setminus \{0\}.$$
Then the equation
$$Bu+p_+u^+-p_-u^-=0 \eqno (3.3)$$
has only the trivial solution.
\end{lemma} 

\paragraph{Proof.} Define $p:Q\times R\to R$ by
$$ \displaylines{
 p(x,u)=p_+(x)\;\hbox{if}\;\;u\ge 0, \cr
 p(x,u)=p_-(x)\;\hbox{if}\;\;u\le 0.
}$$
Then
$$
0\le p(x,u)\le r\;\;\hbox{for}\;\;(x,u)\in Q\times R \eqno (3.4)
$$
and, for $u\in H$ and a.e. $x\in Q$,
\begin{eqnarray*}
p(x,u(x))u(x)&=&p(x,u(x))u^+(x)-p(x,u(x))u^-(x) \\
&=& p_+(x)u^+(x)-p_-(x)u^-(x).
\end{eqnarray*}
Define $P:V\subset H\to H$ by $(Pu)(x)=p(x,u(x))u(x)$ for a.e. $x\in Q.$
Then (3.3) is equivalent to
$$ Bu+Pu=0,\;\;u\in V. \eqno (3.5) $$
By (3.4), we have that $\|Pu\|^2\le r(Pu,u)$ on $V$. Moreover, for each
solution $u\in V$ of (3.5), we get by Property I that
$$ -r^{-1}\|Pu\|^2=-r^{-1}\|Bu\|^2\le (Bu,u)=(-Pu,u) $$
and so $\|Pu\|^2\ge r(Pu,u)$. Hence, $\|Pu\|^2=r(Pu,u)$ and
$(Bu,u)=-r^{-1}\|Bu\|^2$. By Property II, we get that $u\in N(-L_1+\lambda_jI)
\oplus N(-L_1+\lambda_{j+1}I)$.
Hence, $u=v+w$ with $v\in N(-L_1+\lambda_jI)$ and $w\in N(-L_1
+\lambda_{j+1}I)$.
Since $u$ is a solution of (3.3), we get that
\begin{eqnarray*}
(Bu,u)&=&(-Lu+\lambda_ju,u) \\
&=&(-Lv+\lambda_jv,v)+(-Lv+\lambda_jv,w)+(-Lw+\lambda_jw,v+w) \\
&=&(-Lw+\lambda_{j+1}w-rw,v+w) \\
&=& (-rw,w)
\end{eqnarray*}
and so $(-rw,w)+(p(.,u(.))(v+w),v+w)=0$. Then
\begin{eqnarray*}
\lefteqn{ (v-w,-rw+p(.,u(.))(v+w)) }\\
&=&(v+w,-rw+p(.,u(.))(v+w))
-2(w,-rw+p(.,u(.))(v+w)) \\
&=&-2(w,-rw+p(.,u(.))(v+w)) \\
&=&-2(v+w,-rw-B(v+w))+2(v,-rw-B(v+w)) \\
&=&-2(v,rw+B(v+w))=-2(v,B(v+w))=0
\end{eqnarray*}
since $v\in N(-L_1+\lambda_jI)\subset N(B)$ and $R(B)=N(B)^{\perp}$.
Since
\begin{eqnarray*}
(p(.,u(.))(v+w),v-w) 
&=& (p(.,u(.))v,v)+([r-p(.,u(.))]w,w)\\
&&  +([r-p(.,u(.))]w,-v) +(p(.,u(.))v,-w) \\
&=& (p(.,u(.))v,v)+([r-p(.,u(.))]w,w)
\end{eqnarray*}
we get that
$$(p(.,u(.))v,v)+([r-p(.,u(.))]w,w)=(p(.,u(.))(v+w),v-w)
+(rw,-v+w)=0.\eqno (3.6)$$
Since each term in (3.6) is nonnegative by (3.4), we get that each term is
zero, i.e.,
$$\int_Qp(x,v(x)+w(x))v^2(x)dx =0\eqno (3.7)$$
$$\int_Q[(r-p(x,v(x)+w(x))]w^2(x)dx=0.\eqno (3.8)$$
Set $Q_v=\{x\in Q\;|v(x)\ne 0\}$ and $Q_w=\{x\in Q\;|w(x)\ne 0\}$.

By (3.7)-(3.8), we get $p(x,v(x)+w(x))=0$ for a.e. $x\in Q_v$ and
$p(x,v(x)+w(x))=r$ for a.e.
$x\in Q_w$ and so $Q_v\cap Q_w=\emptyset$.
If $Q_v=\emptyset$, then $u=w$ and the  equation (3.8) becomes
$$0=\int_Q[(r-p(x,w(x))]w^2(x)dx=\int_Q(r-p_+)(w^+)^2+(r-p_-)(w^-)^2$$
so that by our hypothesis, $w=0$ and therefore $u=0$.

Next, suppose that $Q_v\ne \emptyset$. Then we have that
$p(x,v(x)+w(x))=0$ on $Q_v$ and, by (3.8), $\int_{Q_v}rw^2(x)=0$,
i.e., $w(x)=0$ for a.e. $x\in Q_v$. Then by (3.7)
$$
0=\int_{Q_v}p(x,v(x))v^2(x)=\int_{Q_v}(p_+(v^+)^2+p_-(v^-)^2)=
\int_{Q}(p_+(v^+)^2+p_-(v^-)^2).$$
By our assumption, this implies that $v=0$, in contradiction
to $Q_v\ne \emptyset$. Hence, $Q_v=\emptyset$ and $u=0$.

\begin{lemma} %Lemma 3.2
 Let (L1) and (B1) hold and $B$ have Properties I
and II. Suppose that $a_{\pm}$,
$b_{\pm}\in L_{\infty}(Q)$ are such that $0\le a_{\pm}(x)\le b_{\pm}\le r$
for a.e. $x\in Q$ and
$$\int_Q[a_+(v^+)^2+a_-(v^-)^2]>0\;\hbox{for all}\;
v\in N(-L_1+\lambda_jI)\setminus \{0\}\eqno (3.9)$$
and
$$\int_Q[(r-b_+)(w^+)^2+(r-b_-)(w^-)^2]>0\;\hbox{for all}\;w\in
N(-L_1+\lambda_{j+1}I)\setminus \{0\}.\eqno (3.10)$$
Then there exists $\epsilon=\epsilon(a_{\pm},b_{\pm})>0$ and
$\delta=\delta(a_{\pm},b_{\pm})>0$ such that for all
$p_{\pm}\in L_{\infty}(Q)$
with
$$\displaylines{
\hfill a_+(x)-\epsilon\le p_+(x)\le b_+(x)+\epsilon \hfill\llap{(3.11)}\cr
\hfill a_-(x)-\epsilon\le p_-(x)\le b_-(x)+\epsilon \hfill\llap{(3.12)}
}$$
for a.e. $x\in Q$ and for all $u\in V$, one has
$$\|Bu+p_+u^+-p_-u^-\|\ge \delta \|u\|_V.\eqno (3.13)$$
\end{lemma}

\paragraph{Proof.}
If this is not the case, then we can find the sequences $\{u_k\}\subset V$,
with $\|u_k\|_V=1$ for each $k$ and $\{p_{\pm}^k\}\subset L_{\infty}(Q)$
such that
$$a_{\pm}(x)-k^{-1}\le p_{\pm}(x)\le b_{\pm}(x)
+k^{-1}\;\hbox{a.e. on}\; Q \eqno (3.14)$$
and
$$Bu_k+p_+^ku_k^--p_-^ku_k^-=v_k\to 0\;\;\hbox{as}\;\;k\to\infty.
\eqno (3.15)$$
Then $p_{\pm}^k\to p_{\pm}$ weakly in $H$ with
$a_{\pm}(x)\le p_{\pm}(x)\le b_{\pm}(x)$ a.e. on $Q$. Let $\mu\ne \lambda_j$
and consider the identity
$$\displaylines{
\hfill u_k+(B-\mu I)^{-1}[(p_+^k-p_+)u_k^+-(p_-^k-p_-)u_k^-] 
\hfill\llap{(3.16)} \cr
\quad=(B-\mu I)^{-1}(-p_+u_k^++p_-u_k^--\mu u_k+v_k).
}$$
By the compactness of the embedding of $V$ into $L_2$, we have
that $u_k\to u$ in $L_2$ as well as $u_k^{\pm}\to u^{\pm}$ in $L_2$.
Since $(B-\mu I)^{-1}$ is continuous both as a map from $L_2$ to $V$ and from
$L_2$ to $L_2$, we get that
$$(B-\mu I)^{-1}(-p_+u_k^++p_-u_k^--\mu u_k+v_k)\to
(B-\mu I)^{-1}(-p_+u^++p_-u^--\mu u) \eqno (3.17)$$
in $L_2$ and $V$.
Next, we shall show that $p_{\pm}^k\to p_{\pm}u^{\pm}$ weakly in $H$. For
$\phi \in C_0^{\infty}(Q)$, we have that
\begin{eqnarray*}
(p_+^ku_k^+-p_+u^+,\phi)&=&(p_+^k(u_k^+-u^+),\phi)+((p_+^k-p_+)u^+,\phi) \\
&\le& c\|u_k^+-u^+\|+((p_+^k-p_+)u^+,\phi)
\end{eqnarray*}
which approaches zero as $k$ approaches $\infty$. 
Hence, $p_+^ku_k^+\to p_+u^+$ weakly in $L_2$ by the
density of $C_0^{\infty}(Q)$ in $L_2$, and similarly, $p_-^ku_k^-\to p_-u^-$
weakly in $L_2$. Hence, (3.16)-(3.17) imply that $u=(B-\mu I)^{-1}(-p_+u^++
p_-u^--\mu u)$, i.e., $Bu+p_+u^+-p_-u^-=0$. Moreover, for each
$v\in N(-L_1+\lambda _jI)\setminus \{0\}$, we have that
$$\int_Qp_+(v^+)^2+p_-(v^-)^2\ge \int_Qa_+(v^+)^2+a_-(v^-)^2>0$$
and, for each $w\in N(-L_1+\lambda _{j+1}I)\setminus\{0\}$, we have that
$$\int_Q[(r-p_+)(w^+)^2+(r-p_-)(w^-)^2]\ge \int_Q[(r-b_+)(w^+)^2+(r-b_-)(w^-)^2
]>0.$$
Hence, by Lemma 3.1, $u=0$ a.e.on $Q$. Thus, $u_k\to 0$ in $L_2$, $\|u_k\|_V=1$
and $\|p_+^ku_k^+-p_-^ku_k^--\mu u_k\|\to 0$. By (3.1), we get that
\begin{eqnarray*}
\|Bu_k+p_+^k-p_-^ku_k^-\|&\ge&
 \|Bu_k- \mu u_k\|-\|\mu u_k-p_+^ku_k^++p_-^ku_k^-\| \\
&\ge& c-\|p_+^ku_k^+-p_-^ku_k^--\mu u_k\|.
\end{eqnarray*}
By (3.15), passing to the limit as $k\to \infty$, we get that $0\ge c>0$,
a contradiction. Hence, the lemma is valid.

\paragraph{Remark 3.1} Modifying suitably the proof of Lemma 3.2, condition
(B1) can be replaced by \begin{description}

\item{(B2)} $\dim N(B)<\infty$ and the partial inverse of $B$ is compact.
\end{description} 
\noindent
Let $R^{s_k}$ be the vector space whose elements are
$\xi=\{\xi_{\alpha}:|\alpha=(\alpha_1,\dots ,\alpha_n)|\le k\}$.
Each $\xi\in {\mathbb R}^k$ may be written as a pair $\xi=(\eta, \zeta)$
with $\eta\in R^{s_{k-1}}$, $\zeta=\{\xi_{\alpha}\;|\;|\alpha|=k\}\in
R^{s_k-s_{k-1}}=R^{s'_k}$ and $|\xi|=(\sum_{|\alpha|\le k}|\xi_{\alpha}|^2
)^{1/2}$. Set $\eta(u)=(Du,\dots ,D^{2m-1}u)$ and $\xi(u)=(u,Du,\dots ,D^{2m}u)$.
Define $Nu=g(x,u,\eta(u))u+Fu$, where
$Fu=f(x,\xi(u))$. Set $k=s_{2m-1}-1$. 

For our next result, suppose that $\dim N(-L_1+\lambda_jI)=1$ and is spanned
by a positive function $w_j$.

\begin{lemma} %  Lemma 3.3
Let $B$ have properties I and II. Suppose that $p_{\pm}
\in L_{\infty}(Q)$ are such that $0\le p_{\pm}(x)\le r$ a.e. $x\in Q$ and
for $r=\lambda_{j+1}-\lambda_j$
$$\int[(r-p_+)(w^+)^2+(r-p_-)(w^-)^2]>0\;\;\hbox{for all}\;\;w\in N(-L_1+
\lambda_{j+1})\setminus \{0\}.$$
Then, if $u$ is a solution of
$Bu+p_+u^+-p_-u^-=0$,
then $u\in N(-L_1+\lambda_jI)$. 
\end{lemma}

\paragraph{Proof.} If not, then arguing as in Lemma 3.1 we get that $u=0$ if
$Q_v=\emptyset$. Next, suppose that $Q_v\ne \emptyset$. Then it follows
from the properties of the eigenfunction $w_j$ and the fact that
$v=aw_j(x)$ for some $a\in R$ and therefore $Q_v=Q$. Hence, we must have
that $p(x,u(x)=0$ for a.e. $x\in Q$. By (3.8), we get that $w=0$ and hence
$u=v$. Thus, in both cases, $u=v\in N(-L_1+\lambda_jI)$. 

\begin{theorem} % Theorem 3.1
Let (L1) and (B1) hold, $B$ have properties I and II
and there are functions $\gamma_{\pm}, \Gamma_{\pm}\in L_{\infty }(Q)$ such
that for some $j\in J\subset Z$, one has
$$\displaylines{
\lambda _j\le \gamma _{\pm}(x)\le \Gamma _{\pm}(x)\le \lambda _{j+1}
\;\;\hbox{for a.e.}\;\;x\in Q\,, \cr
\hfill
\int_Q[(\gamma_+-\lambda _j)(v^+)^2+(\gamma_--\lambda_j)(v^-)^2]>0
\hfill\llap{\rm (3.18)}
}$$
for all $v\in N(L_1-\lambda_jI)\setminus \{0\}$, and
$$\int_Q[(\lambda_{j+1}-\Gamma_+)(w^+)^2+(\lambda_{j+1}-
\Gamma_-)(w^-)^2]>0 \eqno (3.19)$$
for all $w\in N(L_1-\lambda_{j+1}I)\setminus \{0\}$.
Also suppose that for $\epsilon>0$ and $\delta >0$ given in Lemma 3.2,
\begin{description}

\item{(G1)} there is $\rho>0$ such that for a.e. $x\in Q$
$$\gamma_+(x)-\epsilon\le g(x,u,\eta(u))\le \Gamma_+(x)
+\epsilon\;\;\hbox{if}\;\;u>\rho,
\eta(u)\in {\mathbb R}^k$$
$$\gamma_-(x)-\epsilon \le g(x,u,\eta(u))\le
\Gamma_-(x)+\epsilon\;\;\hbox{if}\;\;
u<-\rho , \eta(u)\in {\mathbb R}^k$$

\item{(G2)} There are functions $b(x)\in L_{\infty}(Q)$ and $k_s(x)\in L_2(Q)$
for each $s>0$ such that
$$|g(x,u,\eta(u))|\le sb(x)(\sum _{|\alpha|\le 2m-1}|D^{\alpha}u|^2)^{1/2}+
k_s(x), u\in V.$$

\item{(F)} $\|Fu\|=\|f(x,u,Du,\dots ,D^{2m}u)\|\le \beta \|u\|_V+\gamma$ for
$\beta \in (0,\delta), \gamma>0$.

\item{(H)} $H_t=A-\lambda_jI-tF:V\to H$ is $A$-proper with respect to $\Gamma_B$
for $t\in [0,1)$
and $H_1=B-F$ is pseudo $A$-proper. \end{description}

\noindent
Then (3.2) has at least one solution in $V$ for each $h\in H$.
If $H_1$ is $A$-proper, the set of solutions $S(h)$ of (3.2)
is compact for each
$h\in L_2$ and card $S(h)$ is constant, finite and positive on each
connected component of the set $L_2\setminus (A-N)(\Sigma)$.
\end{theorem}

\paragraph{Proof.} Let $g_1:Q\times R\times {\mathbb R}^k\to R$ be given by
$g_1(x,u,\eta(u))=g(x,u,\eta(u))-
\lambda_j$.
Then define functions
$$\displaylines{
g_+(x,u,\eta(u))=g_1(x,u,\eta(u))\;\hbox{for all }\;\;(x,\eta(u))\in Q\times
  {\mathbb R}^k, u\ge \rho \cr
g_+(x,u,\eta(u))=g_1(x,\rho,\eta(u))\;\;\hbox{for all}\;\;
  (x,\eta(u))\in Q\times R^{s_{2m-1}}, 0\le u\le \rho \,,\cr
g_-(x,u,\eta (u))=g_1(x,u,\eta(u))\;\;\hbox{for all}\;\;(x,\eta(u))\in
  Q\times {\mathbb R}^k, u\le -\rho\,,\cr
g_-(x,u,\eta (u))=-g_1(x,-\rho,\eta(u))\;\;\hbox{for all}
  \;\;(x,\eta(u))\in Q\times R^{s_{2m-1}}, -\rho \le u\le 0 \,,\cr
q(x,0,\eta(u))=g_1(x,0,\eta(u))\;\;\hbox{for all}\;\;(x,\eta(u))\in Q\times
  {\mathbb R}^k \,,\cr
q(x,u,\eta(u))=g_1(x,u,\eta(u))u-g_+(x,u,\eta(u))u\;\;\hbox{for all}\;\;
(x,\eta(u))\in Q\times {\mathbb R}^k
}$$
and $u>0$. Also define
$$q(x,u,\eta(u))=g_1(x,u,\eta(u))u-g_-(x,u,\eta(u))u\;\;\hbox{for all}\;\;
(x,\eta(u))\in Q\times {\mathbb R}^k$$
and $u<0$. Then $q$ satisfies Caratheodory conditions. Set $a_{\pm}(x)=\gamma_{\pm}(x)
-\lambda_j$ and $b_{\pm}(x)=\Gamma_{\pm}(x)-\lambda_j$. Then
$$\displaylines{
a_+(x)-\epsilon\le g_+(x,u,\eta(u))\le b_+(x)+\epsilon\;\;\hbox{on}\;\;
Q\times R_+\times {\mathbb R}^k \cr
a_-(x)-\epsilon\le g_-(x,u,\eta(u))\le b_-(x)+\epsilon\;\;\hbox{on}\;\;
Q\times R_-\times {\mathbb R}^k.
}$$
Then in $V$, problem (3.2) is equivalent  to
$$Bu+g_+(x,u^+,\eta(u))u^+-g_-(x,-u^-,\eta(u))u^-+f(x,\xi(u))
+q(x,u,\eta (u))=-h.$$
Then
for $u\in H$, set $Q_+(u)=\{x\in Q\;|\;u(x)>0\}$, $Q_-(u)=\{x\in Q\;|\;
u(x)<0\}$ and let $\chi_{Q_{\pm}}$ be the corresponding
characteristic functions.
Define the maps $E:H\to L_{\infty}(Q)$, $F,G,H:V\to H$, respectively, by
$$E(u)(x)=g_+(x,u^+(x),\eta(u))\chi_{Q_+(u)}+g_-(x,-u^-(x),\eta(u))
\chi_{Q_-(u)}$$
$G(u)(x)=[E(u)(x)]u(x)=(E(u)u)(x)$ so that
$$G(u)(x)=g_+(x,u^+(x),\eta(u))u^+(x)-g_-(x,-u^-(x),\eta(u))u^-(x),$$
$F(u)(x)=f(x,\xi(u))$ and $H(u)(x)=q(x,u(x),\eta (u))$.
Hence  (3.2) can be written in the operator form
$$Bu+Gu+Fu+Hu=-h,\;\;u\in V.\eqno (3.20)$$
We know that $G, F$ and $H$ are well defined, continuous and bounded in $H$.

Let $C:H\to H$ be defined by $C(u)(x)=b_+(x)u^+(x)-
b_-(x)u^-(x)$. Clearly, $C$ is a positively homogeneous map
and $C,G,H:V\to H$ are completely continuous maps, i.e. they map
weakly convergent sequences in $V$ into strongly convergent sequences in $H$.
Indeed, let us show this, for example, for $G$.
Since $V$ is compactly
embedded in $H$, it follows from the construction of $G$ and
(G2) that if $\{u_k\}\subset V$ converges weakly to $u_0$ in $V$,
then ([K])
$$\|g_+(x,u_k^+,\eta (u_k))-g_-(x,-u_k^-,\eta (u_k))-g_+(x,u_0^+,
\eta (u_0))+
g_-(x,-u_0^-,\eta (u_0))\|$$
approaches $0$. 
Hence, the map $G:V\to L_p$ is completely continuous since
\begin{eqnarray*}
\|Gu_k-Gu_0\|&=&\|E(u_k)u_k-E(u_0)u_0\| \\
&\le& \|E(u_k)(u_k-u_0)\|+
\|E(u_k)-E(u_0)\| \|u_0\| \\
&\le& \max\{\|a_+\|_{\infty}+\|a_-\|_{\infty}+2\epsilon, \|b_+\|_{\infty}+
\|b_-\|_{\infty}\\
&& +2\epsilon\}\|u_k-u_0\|+\|E(u_k)-E(u_0)\|\;\|u_0\|\to 0\,.
\end{eqnarray*}
Thus, we have that
$H_t=B+(1-t)C+t(F+G+H)$ is $A$-proper for each $t\in [0,1)$ from
$V\to H$ and $H_1:V\to H$ is pseudo $A$-proper.

Next, by  construction
\begin{eqnarray*}
(1-t)Cu+tGu&=&[(1-t)b_+(x)+tg_+(x,u^+,\eta(u))]u^+(x) \\
&&-[(1-t)b_-(x)+tg_-(x,-u^-,\eta(u))]u^-(x)
\end{eqnarray*}
and, for a.e. $x\in Q$, $\eta(u)\in {\mathbb R}^k$
$$\displaylines{
a_+(x)-\epsilon\le (1-t)b_+(x)+tg_+(x,u^+(x),\eta(u))\le b_+(x)+\epsilon \cr
a_-(x)-\epsilon\le (1-t)b_-(x)+tg_-(x,-u^-(x),\eta(u))\le b_(x)+\epsilon
}$$
and $|q(x,u,\eta(u))|\le d_{\rho}(x)$ for a.e. $x\in Q$ and all
$(u,\eta(u))\in R\times {\mathbb R}^k$, where $d_{\rho}\in L_2(Q)$ is
independent
of $u$ since $g$ (and hence $g_1$) grows at most linearly. Hence, by
Lemma 3.2 with $p_+(x)=(1-t)b_+(x)+tg_+(x,u^+(x),\eta(u))$ and
$p_-(x)=(1-t)b_-(x)+tg_-(x,-u^-(x),\eta(u))$, we get for some $c>0$
$$\|Bu+(1-t)Cu+tGu\|\ge \delta \|u\|^2\;\hbox{for all}\; u\in V.
$$
It is left to show that $\deg (Q_n(B+C),B_R\cap V_n,0)\ne 0$ for all $n$.
Let $\eta\in (0,r)$ be fixed. Then, for each $t\in [0,1]$, and a.e.
$x\in Q$, we have that $0\le (1-t)\eta +tb_{\pm}\le r$. It
is easy to show that $p_+=(1-t)\eta +tb_+$ and $p_-=(1-t)\eta +tb_-$
satisfy $0\le p_{\pm}\le r$ for a.e. $x\in Q$, and
$$\int_Q[p_+(v^+)^2+p_-(v^-)^2]>0
\;\hbox{for all}\;v\in N(-L_1+\lambda_jI)\setminus \{0\}$$
and
$$\int_Q[r-p_+)(w^+)^2+(r-p_-)(w^-)^2]>0\;\hbox{for all}\;
w\in N(-L_1+\lambda_{j+1}I)\setminus \{0\}.
$$
Hence, one gets that the equation
$$Bu+[(1-t)\eta +tb_+]u^++[(1-t)\eta +tb_-]u^-=0 \eqno (3.21)$$
has only the trivial solution for each $t\in [0,1]$. Since the homotopy
given by (3.21) is $A$-proper, there is an $n\ge n_0$ such that for each $R>0$
and all $n\ge n_0$,
\begin{eqnarray*}
\lefteqn{\deg (P_n(B+b_+(.)^+-b_-(.)^-,B(0,R)\cap H_n,0) }\\
&=&\deg (P_n(B+\eta I),B(0,R)\cap H_n,0)={\pm}1\,.
\end{eqnarray*}
Hence, (3.2) is solvable in $V$ by Theorem 2.3. The
other assertions also follow from this theorem.


\paragraph{Remark 3.2} Conditions (3.18)-(3.19) hold for a wide class of
nonlinearities g. For example, they are implied by $\lambda_j<
\lambda_j+\epsilon\le \gamma_+(x)\le \Gamma_+(x)\le \lambda_{j+1}$
and $\lambda_j\le \gamma_-(x)\le \Gamma_-(x)\le \lambda_{j+1}-
\epsilon<\lambda_{j+1}$, or $\lambda_j\le \gamma_+(x)\le \Gamma_+(x)
\le \lambda_{j+1}-\epsilon<\lambda_{j+1}$
and $\lambda_j<\lambda_j+\epsilon\le \gamma_-(x)\le \Gamma_-(x)
\le \lambda_{j+1}$, in the case when the
eigenfunctions associated to $\lambda_j$ and $\lambda_{j+1}$
change sign in $Q$. \medskip

Next, we shall give some concrete assumptions on $f$ and $g$ that
imply (F)-(H) in Theorem 3.1.

\begin{theorem} Assume that (L1) and (B1) hold, $B$ have properties I 
and II and there be functions $\gamma_{\pm}, \Gamma_{\pm}\in L_{\infty }(Q)$ such
that for some $j\in J\subset Z$, one has
$$ \displaylines{
\lambda _j\le \gamma _{\pm}(x)\le \Gamma _{\pm}(x)\le \lambda _{j+1}
\;\;\hbox{for a.e.}\;\;x\in Q \,\cr
\int_Q[(\gamma_+-\lambda _j)(v^+)^2+(\gamma_--\lambda_j)(v^-)^2]>0
\;\hbox{for all}\;v\in N(L_1-\lambda_jI)\setminus \{0\}\,,\cr
\int_Q[(\lambda_{j+1}-\Gamma_+)(w^+)^2+(\lambda_{j+1}-
\Gamma_-)(w^-)^2]>0\;\hbox{for all}\;w\in N(L_1-\lambda_{j+1}I)\setminus
\{0\}\,.
}$$
Suppose that for the $\epsilon>0$ and $\delta >0$ given in Lemma 2.2,
\begin{description}

\item{(G1)} there is $\rho>0$ such that for a.e. $x\in Q$, $\eta(u)\in {\mathbb R}^k$
$$\displaylines{
\gamma_+(x)-\epsilon\le g(x,u,\eta(u))\le \Gamma_+(x)
+\epsilon\;\;\hbox{if}\;\;u>\rho \cr
\gamma_-(x)-\epsilon \le g(x,u,\eta(u))\le \Gamma_-(x)
+\epsilon\;\;\hbox{if}\;\;u<-\rho
}$$

\item{(G2)} There are functions $b(x)\in L_{\infty}(Q)$ and $k_s(x)\in L_2(Q)$
for each $s>0$ such that
$$|g(x,u,\eta(u))|\le sb(x)(\sum _{|\alpha|\le 2m-1}|D^{\alpha}u|^2)^{1/2}+
k_s(x), u\in V.
$$

\item{(F1)} There are functions $a(x)\in L_{\infty}(Q)$ and $d_r(x)\in L_2(Q)$
for each $r>0$ such that
$$|f(x,\xi(u))|\le ra(x)(\sum_{|\alpha |\le 2m}|D^{\alpha}u|^2)^{1/2}+
d_r(x),\;\;\hbox{for all}\;\;u\in V.
$$

\item{(F2)} There is a constant $k>0$ such that $k\le c$ and
$$|f(x,\eta ,\zeta)-f(x,\eta ,\zeta')|\le k\sum_{|\alpha |=2m}|\zeta _{\alpha}-
\zeta_{\alpha}'|$$
for a.e. $x\in Q$, all $\eta\in {\mathbb R}^k$ and
$\zeta,\zeta'\in R^{s_{2m}'}=R^{s_{2m}}-R^{s_{2m-1}}$, where
$c$ is a constant in (3.1).
\end{description}

\noindent
Then there is a $u\in V$ that satisfies (3.2) for a.e. $x\in Q$.
If $k<c$, then all other assertions of Theorem 3.1 also hold.
\end{theorem}

\paragraph{Proof.} It is easy to see that (F) of Theorem 3.1 holds.
Hence, it remains to verify (H) of that theorem, i.e. that
$H_t=B-tF$ is $A$-proper with respect to $\Gamma_B$ for each $t\in [0,1)$
and $H_1$ is pseudo $A$-proper. Since the embedding of $V$ into $H$ is
compact, it suffices to show these facts for
$F_t= L-tF$. Set $B_{\mu}=B-\mu I$ for
some $\mu \ne \lambda _j$ for each $j$. Then, for each $t\in [0,1]$,
it follows from (F2), the Holder inequality, and an easy calculation
that
$$(F_tu-F_tv,B_{\mu}(u-v))\ge (1-k/c)\|B_{\mu}(u-v)\|^2+\phi (u-v)\;
\eqno (3.22)$$
where the functional $\phi : V\to R$ is given by
$$\phi (u-v)=t(M(u,v)-M(v,v),B_{\mu}(u-v))+\mu (u-v,B_{\mu}(u-v)),
$$
with $M:V\times V\to H$ being the continuous form 
$M(u,v)=f(x,\eta(u),\zeta(v))$. The functional $\phi$ is weakly
continuous. Indeed, let $u_k\to u$ weakly in $V$. Then $u_k\to u$
in the $W_2^{2m-1}$-norm by the Sobolev imbedding theorem  and by the
results from \cite{K}, it is not hard to show that $\phi (u_k-u)\to 0$
as $k\to \infty$. If $k<c$, then (F2) implies that $F_t$ is
$A$-proper with respect to $\Gamma_B$ (see, e.g., in \cite{Mi-2}-\cite{Mi-4}).
If $k=c$, then $F_t$ is again $A$-proper for each $t\in [0,1)$
and it is easy to see that $F_1$ is pseudo $L_{\mu}$-monotone.
Hence, $F_1$ is pseudo $A$-proper with respect to $\Gamma_B$ (\cite{Mi-4})
and (H) of Theorem 3.1 holds.

\begin{corollary} % Corollary 3.1
Let the conditions of Theorem 3.2 hold with
(G1) replaced by 
$$\displaylines{
\rlap{\rm (G1')}\hfill 
\gamma _{\pm}(x)\le \liminf_{u\to \pm \infty}g(x,u,\eta (u)) 
\le \limsup_{u\to \pm \infty}g(x,u,\eta (u)) \hfill\cr
\le \Gamma _{\pm}(t,x)
}$$
uniformly for a.e. $(x,\eta (u))\in Q\times {\mathbb R}^k$.
Then there is a $u\in V$ that satisfies (3.2) for a.e. $x\in Q$.
\end{corollary}

\paragraph{Proof.} It is easy to see that (G1') implies (G1). 

\section{Strong solvability of elliptic BVP's}

A. We shall apply the results of Section 3 to strong solvability of elliptic
boundary-value problems in $V$ of the form
$$\sum _{|\alpha |\le 2m}A_\alpha (x)D^\alpha u(x)+
g(x,u,Du,\dots ,D^{2m-1}u)u+f(x,u,Du,\dots ,D^{2m})=h,
\eqno (4.1)$$
under non-uniform non-resonance conditions. Here
$Q\subset R^n$ is a bounded smooth domain, $V$ is a closed subspace of
$W_2^{2m}(Q)$ containing the test functions, the linear part is
elliptic and $h\in L_2(Q)$.
Assume the linear map $L:V\to L_2(Q)$, induced by the linear elliptic
operator in (4.1), has
closed range in $H=L_2(Q)$ and satisfies
conditions (L1), (B1) in Section 3 with $B=-L+\lambda_j I$. Here,
$L_1=L$ and $A_1=0$.

Let $\lambda \ne \lambda _j$ for each $j=1,2,\dots $ be fixed,
$\Gamma =\{Y_n,Q_n\}$ be a projectionally complete scheme for $L_2$
and $X_n=(B-\lambda I)^{-1}(Y_n)\subset V$ for each $n$.
Then $\Gamma _L=
\{X_n,Y_n,Q_n\}$ is an admissible or a projectionally complete scheme for
$(V,L_2)$. Since $B-\lambda I:V\to L_2$ is linear, one-to-one and
$A$-proper with respect to $\Gamma _L$, there is a constant $c>0$ (depending)
only on $\lambda )$ such that
$$\|(B-\lambda I)u\|\ge c\|u\|_V, \;\;u\in V. \eqno (4.2)$$

\begin{theorem} % Theorem 4.1
 Let $B=-L+\lambda_j I$ be a closed densely defined map
in $H$ such that $R(B)=N(B)^{\perp}$, $(Bu,u)\ge -r^{-1}\|Bu\|^2$ on $V$
and if $(Bu,u)=-r^{-1}\|Bu\|^2$ for some $u\in V$, then $u\in N(-L+
\lambda_j I)\oplus N(-L+\lambda_{j+1}I)$. Suppose that
there are functions $\gamma_{\pm}, \Gamma_{\pm}\in L_{\infty }(Q)$ such
that for some $j\in J\subset Z$, one has
$$\lambda _j\le \gamma _{\pm}(x)\le \Gamma _{\pm}(x)\le \lambda _{j+1}
\;\;\hbox{for a.e.}\;\;x\in Q$$
and
$$\int_Q[(\gamma_+-\lambda _j)(v^+)^2+(\gamma_--\lambda_j)(v^-)^2]>0
\;\hbox{for all}\;v\in N(L-\lambda_jI)\setminus \{0\}$$
and
$$\int_Q[(\lambda_{j+1}-\Gamma_+)(w^+)^2+(\lambda_{j+1}-
\Gamma_-)(w^-)^2]>0\;\hbox{for all}\;w\in N(L-\lambda_{j+1}I)\setminus
\{0\}.
$$
Suppose that for $\epsilon>0$ and $\delta >0$ given in Lemma 3.2,
\begin{description}
\item{(G1)} there is $\rho>0$ such that for a.e. $x\in Q$
$$\gamma_+(x)-\epsilon\le g(x,u,\eta(u))\le \Gamma_+(x)
+\epsilon\;\;\hbox{if}\;\;u>\rho,
\eta(u)\in {\mathbb R}^k$$
$$\gamma_-(x)-\epsilon \le g(x,u,\eta(u))\le \Gamma_-(x)
+\epsilon\;\;\hbox{if}\;\;u<-\rho,\eta(u)\in {\mathbb R}^k$$

\item{(G2)} There are functions $b(x)\in L_{\infty}(Q)$ and $k_s(x)\in L_2(Q)$
for each $s>0$ such that
$$|g(x,u,\eta(u))|\le sb(x)(\sum _{|\alpha|\le 2m-1}|D^{\alpha}u|^2)^{1/2}+
k_s(x), u\in V.
$$

\item{(F)} $\|Fu\|=\|f(x,u,\dots ,D^{2m}u)\|\le \beta \|u\|_V+\gamma $ for
some $\beta \in (0,\delta )$, $\gamma >0$.

\item{(H)} $H_t=L-tF:V\to H$ is $A$-proper with respect to $\Gamma _L$
for $t\in [0,1)$ and $L-F$ is pseudo $A$-proper.
\end{description}

\noindent
Then (4.1) has a solution $u\in V$ for each $h\in L_2$. If $L-F$ is
$A$-proper, $S(h)=(L-F)^{-1}(\{h\})$ is compact for each $h\in L_2$ and
card $S(h)$ is constant, finite and positive on each
connected component of the set $L_2\setminus (L-F)(\Sigma)$.
\end{theorem}

\paragraph{Proof.} It follows from Theorem 3.1 with $L_1=L$ and $A_1=0$.
\hfill$\diamondsuit$ \smallskip

As before, we give now some concrete conditions on $f, g$ so that
(H) holds.

\begin{theorem} % Theorem 4.2
Let $B=-L+\lambda_j I$ be a closed densely defined map
in $H$ such that $R(B)=N(B)^{\perp}$, $(Bu,u)\ge -r^{-1}\|Bu\|^2$ on $V$
and if $(Bu,u)=-r^{-1}\|Bu\|^2$ for some $u\in V$, then $u\in N(-L+
\lambda_j I)\oplus N(-L+\lambda_{j+1}I)$. Suppose that
there are functions $\gamma_{\pm}, \Gamma_{\pm}\in L_{\infty }(Q)$ such
that for some $j\in J\subset Z$, one has
$$\displaylines{
\lambda _j\le \gamma _{\pm}(x)\le \Gamma _{\pm}(x)\le \lambda _{j+1}
\;\;\hbox{for a.e.}\;\;x\in Q\,, \cr
\int_Q[(\gamma_+-\lambda _j)(v^+)^2+(\gamma_--\lambda_j)(v^-)^2]>0
\;\hbox{for all}\;v\in N(L-\lambda_jI)\setminus \{0\}\,,\cr
\int_Q[(\lambda_{j+1}-\Gamma_+)(w^+)^2+(\lambda_{j+1}-
\Gamma_-)(w^-)^2]>0\;\hbox{for all}\;w\in N(L-\lambda_{j+1}I)\setminus
\{0\}.
}$$
Furthermore, suppose that for the $\epsilon>0$ and $\delta >0$ given in Lemma 3.2,
\begin{description}

\item(G1) there is $\rho>0$ such that for a.e. $x\in Q$
$$\gamma_+(x)-\epsilon\le g(x,u,\eta(u))\le \Gamma_+(x)
+\epsilon\;\;\hbox{if}\;\;u>\rho,
\eta(u)\in {\mathbb R}^k$$
$$\gamma_-(x)-\epsilon \le g(x,u,\eta(u))\le \Gamma_-(x)
+\epsilon\;\;\hbox{if}\;\;u<-\rho,\eta(u)\in {\mathbb R}^k$$

\item{(G2)} There are functions $b(x)\in L_{\infty}(Q)$ and $k_s(x)\in L_2(Q)$
for each $s>0$ such that
$$|g(x,u,\eta(u))|\le sb(x)(\sum _{|\alpha|\le 2m-1}|D^{\alpha}u|^2)^{1/2}+
k_s(x), u\in V.$$

\item{(F1)} There are functions $a(x)\in L_{\infty}(Q)$ and $d_r(x)\in L_2(Q)$
for each $r>0$ such that
$$|f(x,\xi(u))|\le ra(x)(\sum_{|\alpha |\le 2m}|D^{\alpha}u|^2)^{1/2}+
d_r(x),\;\;\hbox{for all}\;\;u\in V.$$

\item{(F2)} There is a constant $k>0$ such that $k\le c$ and
$$|f(x,\eta ,\zeta)-f(x,\eta ,\zeta')|\le k\sum_{|\alpha |=2m}|\zeta _{\alpha}-
\zeta_{\alpha}'|$$
for a.e. $x\in Q$, all $\eta\in {\mathbb R}^k$ and
$\zeta,\zeta'\in R^{s_{2m}'}=R^{s_{2m}}-R^{s_{2m-1}}$, where
$c$ is a constant in (4.2).
\end{description} 

\noindent
Then there is a $u\in V$ that satisfies (4.1) for a.e. $x\in Q$
and all other assertions of Theorem 4.1 are valid if $k<c$.
\end{theorem} 

\paragraph{Proof.} It follows from Theorem 4.1 with $L_1=L$ and $A_1=0$.
\hfill$\diamondsuit$ \smallskip 

For our next result, we assume also
\begin{description} 
\item{(L2)} There is an integer $j\ge 1$ such that $\lambda _j<\lambda _{j+1}$ and
$Lw=\lambda_kw$ for $k=j$ and $k=j+1$, has the continuation property,
that is if $w(x)=0$ on a set of
positive measure, then $w(x)=0$ a.e. on $Q$.
\end{description}

\begin{theorem} %  Theorem 4.3
 Let $L$ satisfy (L1)-(L2) and (B1) with $B=-L+\lambda_j I$
and let $\gamma(x), \Gamma(x)\in L_{\infty}(Q)$ be such that
\begin{description}

\item{(H1)} $\lambda _j\le \gamma (x) \le \Gamma (x)\le \lambda _{j+1}$
with $\mathop{\rm meas}\{x\in Q| \lambda _j\ne \gamma (x)\}>0$ and
$\mathop{\rm meas}\{x\in Q | \lambda _{j+1}\ne \Gamma (x)\}>0$.
\end{description}
Suppose that (G1) of Theorem 3.4 holds and for $\epsilon >0$ and
$\delta >0$ given by Lemma 3.2
\begin{description}
\item{(H2)} $\gamma (x)-\epsilon \le g(x,\xi)\le \Gamma (x)+\epsilon$
for all $(x,\xi)\in Q\times R^{s_{2m-1}}$

\item{(H3)} $\|Fu\|=\|f(x,u,\dots ,D^{2m}u)\|\le \beta \|u\|_V+\gamma $ for
some $\beta \in (0,\delta )$, $\gamma >0$.

\item{(H4)} $H_t=L-tF$ is $A$-proper with respect to $\Gamma _L$
for $t\in [0,1)$ and $L-F$ is pseudo $A$-proper.
\end{description}
\noindent
Then (4.1) has a solution $u\in V$ and all other assertions of
Theorem 4.1 are valid.
\end{theorem}

\paragraph{Proof.}
Clearly, (L2) and (H1) imply the integral inequalities in Theorem 4.2.
Hence, the conclusion follows from this theorem.

\begin{theorem} % Theorem 4.4
 Let $L$ and $\gamma (x)$, $\Gamma (x)$ be as in Theorem 4.3.
Let $f:Q\times R^{s_{2m}}\to R$ and $g:Q\times R^{s_{2m-1}}\to R$ be
Caratheodory functions such that
\begin{description} 
\item{(F1)} There are functions $a(x)\in L_{\infty}(Q)$ and $d_r(x)\in L_p(Q)$
for each $r>0$ such that
$$|f(x,\xi(u))|\le ra(x)(\sum_{|\alpha |\le 2m}|D^{\alpha}u|^2)^{1/2}+
d_r(x),\;\;\hbox{for all}\;\;u\in V.$$

\item{(F2)} There is a constant $k>0$ such that $k\le c$ and
$$|f(x,\eta ,\zeta)-f(x,\eta ,\zeta')|\le k\sum_{|\alpha |=2m}|\zeta _{\alpha}-
\zeta_{\alpha}'|$$
for a.e. $x\in Q$, all $\eta\in {\mathbb R}^k$ and
$\zeta,\zeta'\in R^{s_{2m}'}=
R^{s_{2m}}-R^{s_{2m-1}}$.

\item{(G1)} $\lambda _j\le \gamma (x)\le liminf_{|u|\to \infty }g(x,u,\eta (u))
\le limsup_{|u|\to \infty }g(x,u,\eta (u))\le \Gamma(x)\le \lambda _{j+1}$
uniformly for $x\in Q$ and the non-u components $\eta (u)$.

\item{(G2)} There are functions $b(x)\in L_{\infty}(Q)$ and $k_s(x)\in L_p(Q)$
for each $s>0$ such that
$$|g(x,u,\eta(u))|\le sb(x)(\sum _{|\alpha|\le 2m-1}|D^{\alpha}u|^2)^{1/2}+
k_s(x), u\in V.$$
\end{description}
Then there is a $u\in V$ that satisfies Eq. (4.1) for a.e. $x\in Q$
and all other assertions of Theorem 4.1 are valid if $k<c$.
\end{theorem}

\paragraph{Proof.} It follows from Theorem 4.2, as in the case of 
Corollary 3.1. \hfill$\diamondsuit$\smallskip \smallskip

Theorem 4.2 extends the existence result of Beresticki-de Figueiredo \cite{BF} who
assumed $f=0$ and $g$ to depend only on $u$. A simplified proof of
their results has been given by Mawhin \cite{Ma-2}. If $f$ does not depend
on derivatives of order $2m$, the existence part of
Theorem 4.4 reduces to a result of
Mawhin-Ward \cite{MW}. Their proofs are based on the Leray-Schauder and
the coincidence degree theories respectively.

B. In this subsection we shall look at boundary value problems
$$Lu=\lambda _1u+g(x,u)=h,\;\;\hbox{in}\;\;Q,\;\;u|_{\partial Q}=0\eqno (4.3)$$
where $L$ is either selfadjoint or non-selfadjoint second order
elliptic partial differential operator, and
$\lambda_1$ is the first (resp. principal) eigenvalue of the
selfadjoint (resp. nonselfadjoint) operator $-L$, $h\in L_p(Q)$
with $p>n$ and $g:Q\times R\to R$ is a Caratheodory function which
grows at most linearly, i.e. there are a constant $c_1>0$ and a
function $c_2\in L_p(Q)$, $p>n$, such that
$$|g(x,u)|\le c_1|u|+c_2(x)$$
for a.e. $x\in Q$ and all $u\in R$. We assume that $L$ is such that
the Bony's maximum principal (see eg. \cite{B, AC}) and the abstract
Krein-Rutman theorem \cite{KR} imply the existence of a real simple
eigenvalue $\lambda_1>0$ of
$$-Lu=\lambda_1u,\;\;u|_{\partial Q}=0$$
of minimal modulus such that there is a corresponding smooth
eigenfunction $w_1>0$ in $Q$ and $\partial w_1/\partial \eta<0$
on $\partial Q$, where $\partial /\partial \eta$ stands for the
outward normal derivative. Moreover, if $L$ is nonseladjoint then
$\lambda_1$ is also an eigenvalue for the
adjoint problem
$$-L^*u=\lambda_1u,\;\;u|_{\partial Q}=0,$$
such that there is a corresponding smooth eigenfunction $w_1^*>0$
in $Q$ and $\partial w_1^*/\partial \eta<0$ on $\partial Q$.
\vskip 2mm
Now, using Lemma 3.3, we shall prove the following existence result for
(4.3) when the nonlinearity $f(x,u)= \lambda_1u+g(x,u)$ "lies"
between the first two eigenvalues $\lambda_1$ and $\lambda_2$.
We assume,
without loss of generality, that the following upper bounds are
nonnegative
$$\displaylines{
\hfill g_+(x)=\limsup_{u\to \infty}g(x,u)/u\le \Gamma_+(x),\;\;\hbox{a.e. on}\;\;
Q \hfill\llap{(4.4)}\cr 
\hfill g_-(x)=\limsup_{u\to -\infty}g(x,u)/u\le \Gamma_-(x),\;\;\hbox{a.e. on}\;\;
Q.\hfill\llap{(4.5)}
}$$
Since $g$ grows linearly, we can suppose, without loss of generality,
that $\Gamma_{\pm}\in L_p(Q)$, $p>n$.

\begin{theorem} % Theorem 4.5
Let $g:Q\times R\to R$ be a Caratheodory function
that grows linearly, $g_+(x)$ and $g_-(x)$ are different from
zero on a set of nonzero measure, and
$$g(x,u)u\ge 0\eqno (4.6)$$
for a.e. $x\in Q$ and all $u\in R$. Suppose that (4.4)-(4.5) hold and
$$0\le \Gamma_{\pm}(x)\le r (=\lambda_2-\lambda_1),\;\;\hbox{ for a.e.}
x\in Q,\eqno (4.7)$$
$$\int_{w>0}[r-\Gamma_+]w^2dx+\int_{w<0}[r-\Gamma_-]w^2dx>0,\;\;
\hbox{ for all},\;\;w\in N(L+\lambda_2I)\setminus \{0\}.\eqno (4.8)$$
Then Eq. (4.3) has at least one solution $u\in W_p^2(Q)\cap H_0^1(Q)$,
$p>n$, for each $h\in L_p(Q)$. Moreover, $u\in C^{1,\mu}(\bar Q)$.
\end{theorem}

\paragraph{Proof.} Let $\gamma$ be a fixed constant with $0<\gamma<r$ and
define the operator \\
$E:W_p^2(Q)\cap H_0^1(Q)\subset C^1(\bar Q)\to L_p(Q)$
by
$$Eu=Lu+\lambda_1u+ru\,.$$
We shall show that there exists a constant $C>0$ independent of t
such that $\|u\|_{C^1}\le C$
for all possible solutions $u\in W_p^2(Q)\cap H_0^1(Q)$ of
the homotopy
$$H(t,u)=Lu+\lambda_1u+(1-t)\gamma u+tg(x,u)=th, t\in [0,1).\eqno (4.9)$$
Clearly, (4.9) has only the trivial solution for $t=0$. If such a $C$
does not exist, then there exist $t_k\in (0,1)$ and $u_k\in W_p^2(Q)$
such $\|u_k\|\to \infty$ and
$$Eu_k=t_k[\gamma u_k-g(t_k,u_k)+h(x)],\;\;u|_{\partial Q}=0.\eqno (4.10)$$
Set $v_k=u_k/\|u_k\|_{C^1}$. Then, (4.10) becomes
$$Ev_k=t_k[\gamma v_k-g(x,u_k)/\|u_k\|_{C^1}+h/\|u_k\|_{C^1}],\;\;
v_k|_{\partial Q}=0.\eqno (4.11)$$
We may assume that $t_k\to t$ and $g(x,u_k)/\|u_k\|_{C^1}
\rightharpoonup K(x)$ in $L_p(Q)$ since $g$  has a linear growth.
Since $g(x,u_k)/\|u_k\|_{C^1}=g(x,u_k)/u_k(x).v_k(x)\rightharpoonup G(x)v(x)$
with $G(x)\ne 0$ on a set of positive measure, we get that 
$K(x)=G(x)v(x)\ne 0$ on a set of positive measure.
Using $L_p$-estimate and the compact embedding of $W_p^2(Q)$
into $C^1(\bar Q)$, we can deduce from (4.11) that $v_k\to v$ in
$C^1(\bar Q)$, $\|v\|_{C^1}=1$ and $v|_{\partial Q}=0$.
Moreover, $\{Lv_k\}$ is also bounded in $L_p(Q)$ by (4.11). Hence,
by the reflexivity of $L_p$ and the weak closedness of $L$, we
may assume that $Lv_k\rightharpoonup Lv$ in $L_p$ with
$v\in W_p^2(Q)\cap H_0^1(Q)$ and $v$ solves the equation
$$Ev=t[\gamma v-K(x)],\;\;v|_{\partial Q}=0.\eqno (4.12)$$
As in \cite{INW}, Eq. (4.12) is equivalent to
$$Lv+\lambda_1 v+p_+(x)v^+-p_-(x)v^-=0, \;v|_{\partial Q}=0
\eqno (4.13)$$
where $p_+(x)=(1-t)\gamma +tk_v^+(x)$ and $p_-(x)=(1-t)\gamma +tk_v^-(x)$
and $k_v(x)=K(x)/v(x)$ if $v(x)\ne 0$ and $k_v=0$ if $v(x)=0$
since $0\le k_v(x)\le \Gamma_+(x)$ if $v(x)>0$ and $0\le k_v(x)\le
\Gamma _-(x)$ if $v(x)<0$. Hence, by Lemma 3.3 (or Lemma 1 in
\cite{INW}), we get that
$v\in N(L+\lambda_1I)\setminus \{0\}$.

Next, passing to the limit in
$$(Lv_k+\lambda _1v_k+(1-t)\gamma v_k+t_k g(x,u_k)/\|u_k\|_{C^1},v_k)
=(t_kh/\|u_k\|_{C^1},v_k)$$
we get
$$((1-t)\gamma v+tK,v)=0.$$
Note that $t\ne 1$, for otherwise (4.12)-(4.13) imply that
$p_+(x)v^+-p_-(x)v^-=0$ which leads to $K(x)=0$ a.e. on Q,
a contradiction.
Hence, $(K,v)<0$ since $(1-t)\gamma \|v\|^2=-t(K,v)$. This
contradicts the fact that $0\le t_k(g(x,u_k)/\|u_k\|_{C^1},v_k)
\to t(K,v)$. Hence, we have shown that all solutions of (4.9) are bounded
and, by the Leray-Schauder homotopy theorem, Eq (4.3) has a solution
in $W_p^2(Q)\cap H_0^1(Q)$ for each $p\in 
L_p(Q)$.\hfill$\diamondsuit$\smallskip

Theorem 4.5 extends Theorem 1 in Iannacci-Nkashama-Ward \cite{INW} who
showed the solvability of Eq (4.3) only for $h\in L_p(Q)$ that are
orthogonal to $w_1$ but without assuming that $g_+(x)$ and $g_-(x)$
are not zero on a set of positive measure. 
On the other hand, their result extends some
earlier ones of de Figueiredo and Ni \cite{FN}, Gupta \cite{G} and others.
As in Theorems 4.4, it will be shown elsewhere that Theorem 4.5
can be extended to include nonlinearities depending on
derivatives up to the second order.

\section{Time periodic solutions of BVP's for nonlinear parabolic
and hyperbolic equations}

The semi-abstract results in Section 3 have been used in
\cite{Mi-7} to prove the existence and the number of solutions
of generalized periodic solutions (GPS),
under nonuniform nonresonance conditions, for the nonlinear parabolic
equation
$$ u_t+A_0u+g(t,x,u,u_t,D_xu,\dots ,D_x^{2m-1}u)u+f(t,x,u,u_t,D_xu,\dots ,
D_x^{2m}u)=h $$
in $H=L_2(\Omega)$,
where $\Omega =[0,2\pi ]\times Q$ with $Q\subset R^n$, $A_0$ is
a uniformly strongly elliptic operator of order $2m$
in $x\in Q$ for each $t\in [0,2\pi]$,
and the nonlinear hyperbolic equations with damping
\begin{eqnarray*}
 \sigma u_t+u_{tt}+A_0u&=&g(t,x,u,u_t,u_{tt},D_xu,\dots ,D_x^{2m-1}u)u\\
&&+f(t,x,u,u_t,u_{tt},D_xu,\dots ,D_x^{2m}u)+h
\end{eqnarray*} 
with $h$ in $H$, $\sigma\ne 0$, boundary conditions
$$u(t,.)\in H_0^m(Q)\;\;\hbox{for all}\;\;t\in (0,2\pi)\,, $$
and periodicity conditions
$$u(0,x)=u(2\pi,x)\;\;\hbox{for all}\;\;x\in Q\,.$$
These results extend the corresponding
existence results in
Nkashama-Willem \cite{NW}, who assumed only the $u$ dependence in $g$ and
$f=0$ and used the coincidence degree theory.


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\noindent{\sc Petronije S. Milojevi\'c}\\
Department of Mathematical Sciences and CAMS\\
New Jersey Institute of Technology\\
Newark, NJ 07102,  USA \\
e-mail: pemilo@m.njit.edu

\end{document}