\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amscd} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 86, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/86\hfil Bernstein approximations] {Bernstein approximations of Dirichlet problems for elliptic operators on the plane} \author[J. Gulgowski\hfil EJDE-2007/86\hfilneg] {Jacek Gulgowski} \address{Jacek Gulgowski \newline Institute of Mathematics \\ University of Gda\'{n}sk \\ ul. Wita Stwosza 57, 80-952 Gda\'{n}sk, Poland} \email{dzak@math.univ.gda.pl} \thanks{Submitted January 2, 2007. Published June 15, 2007.} \subjclass[2000]{35J25, 41A10} \keywords{Dirichlet problems; Bernstein polynomials; global bifurcation} \begin{abstract} We study the finitely dimensional approximations of the elliptic problem \begin{gather*} (Lu)(x,y) + \varphi(\lambda,(x,y),u(x,y) ) = 0 \quad \text{for } (x,y)\in\Omega\\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega, \end{gather*} defined for a smooth bounded domain $\Omega$ on a plane. The approximations are derived from Bernstein polynomials on a triangle or on a rectangle containing $\Omega$. We deal with approximations of global bifurcation branches of nontrivial solutions as well as certain existence facts. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \def\T{ {\mathbb{T}} } \def\S{ {\mathbb{S}} } \def\cj{C^1([0,1])} \def\cz{C([0,1])} \def\cd{C^2([0,1])} \def\nat{{\mathbb{N}}} \def\one{{\bf{1}}} \def\czo{C(\overline{\Omega})} \def\czzo{C_0(\overline{\Omega})} \def\wtt{W^{2,2}(\Omega)} \def\wzt{W^{0,2}(\Omega)} \def\wttz{W_0^{2,2}(\Omega)} \section{Preliminaries} We consider the elliptic problem $$\label{pr:phi} \begin{gathered} (Lu)(x,y) + \varphi(\lambda,(x,y),u(x,y) ) = 0 \quad \text{for } (x,y)\in\Omega \\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega, \end{gathered}$$ where $\Omega\subset\mathbb{R}^2$ is a smooth, bounded domain and the uniformly elliptic operator $L$ is $Lu = a(x,y) u_{xx} + b(x,y)u_{xy}+c(x,y)u_{yy},$ where $a,b,c:\overline{\Omega}\to\mathbb{R}$ are continuous. We assume as well, that the map $\varphi:A\times\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is continuous, and $A\subset\mathbb{R}$ is an open interval. We are going to define two finitely dimensional approximations of the problem (\ref{pr:phi}). These approximations will be based on the Bernstein polynomials on the triangle (see \cite{CD}, \cite{F}, \cite{L}, \cite{S} ) and on the rectangle (see \cite{D}, \cite{L}). So, let us start with some basic information about Bernstein polynomials on a triangle and on a rectangle. Let $\T\subset\mathbb{R}^2$ be a closed triangle with the vertices $P,Q,R\in\mathbb{R}^2$. Let $(p(x,y),q(x,y),r(x,y))\in[0,1]^3$ denote the barycentric coordinates of the point $(x,y)\in\T$ with respect to the triangle $\T$ (when it is not confusing we will write $p,q,r$ instead of $p(x,y),q(x,y),r(x,y)$). So each point $(x,y)\in\T$ is uniquely expressed by coordinates $(p,q,r)$ such that $p,q,r\geq 0$ and $p+q+r=1$. The relation between the coordinates is given by $$(x,y) = pP+qQ+rR.$$ Let now $n\in\nat$ be fixed and $i,j,k$ be nonnegative integers such that $i+j+k=n$. We call the set of functions $T_{i,j,k}^n:\T\to\mathbb{R}$, given by $$T_{i,j,k}^n(x,y) = \frac{n!}{i!j!k!} p^iq^jr^k,$$ the Bernstein basis polynomials on the triangle. For the continuous function $f:\T\to\mathbb{R}$ we call the function $$\label{def:bn1} (B_n^1f)(x,y) = \sum_{i,j,k\geq 0; i+j+k=n} f(\frac{i}{n},\frac{j}{n},\frac{k}{n}) T_{i,j,k}^n(x,y)$$ the Bernstein polynomial of degree $n$ of the function $f$. In the above formula the triple $(\frac{i}{n},\frac{j}{n},\frac{k}{n})$ expresses the barycentric coordinates of the point in the triangle $\T$. On the other hand, let $\S = [\alpha_1,\alpha_2]\times[\beta_1,\beta_2]\subset \mathbb{R}^2$ be the rectangle. Let now $n\in\nat$ be fixed and let $i,j\in\{0,1,\dots ,n\}$. We call the set of functions $S_{i,j}^n:\S\to\mathbb{R}$ given by $$S_{i,j}^n(x,y) = \binom{n}{i} \binom{n}{j} \Bigl(\frac{x-\alpha_1}{\alpha_2-\alpha_1}\Bigr)^i \Bigl(\frac{\alpha_2-x}{\alpha_2-\alpha_1}\Bigr)^{n-i} \Bigl(\frac{y-\beta_1}{\beta_2-\beta_1}\Bigr)^j \Bigl(\frac{\beta_2-y}{\beta_2-\beta_1}\Bigr)^{n-j}$$ the Bernstein basis polynomials on the rectangle. Similarly as above, for the continuous function $f:\S\to\mathbb{R}$ we call the function $$\label{def:bn2} (B_n^2f)(x,y) = \sum_{0\leq i,j\leq n} f(\alpha_1 +\frac{i}{n}(\alpha_2-\alpha_1),\beta_1 +\frac{j}{n}(\beta_2-\beta_1)) S_{i,j}^n(x,y)$$ the Bernstein polynomial of degree $n$ of the function $f$. The most important properties of Bernstein polynomials on a triangle and a rectangle will be given in the lemmas below. Here $\omega_u$ denotes the modulus of continuity of the function $u$, i.e. $$\omega_u(\delta) = \max\{ |u(x_1,y_1)-u(x_2,y_2)| : |x_1-x_2| + |y_1-y_2| \leq \delta \}.$$ The first lemma (see \cite{S}) refers to Bernstein polynomials on a triangle. \begin{lemma}\label{conv:triangle} If $u:\T\to\mathbb{R}$ is continuous, then $B_n^1 u$ converges uniformly to $u$. Moreover, the estimation holds $$\Vert u - B_n^1u\Vert_0 \leq 2\omega_u(\frac{1}{\sqrt{n}}).$$ \end{lemma} The second lemma refers to Bernstein polynomials on a rectangle and will be proved below. \begin{lemma}\label{conv:rectangle} If $u:\S\to\mathbb{R}$ is continuous, then $B_n^2 u$ converges uniformly to $u$. Moreover, the estimation holds $$\Vert u - B_n^2u\Vert_0 \leq \frac{5}{2}\omega_u(\frac{\gamma}{\sqrt{n}}),$$ where $\gamma = \max\{ \alpha_2-\alpha_1, \beta_2 - \beta_1 \}$. \end{lemma} \begin{proof} As we can see in \cite{L} the Bernstein approximation of the continuous function $v_0:[0,1]\to\mathbb{R}$ may be estimated by $|(B_n v_0)(t)-v_0(t)|\leq \frac{5}{4}\omega_{v_0}(\frac{1}{\sqrt{n}})$. The similar estimation may be obtained for a continuous function $v:[a,b]\to\mathbb{R}$, for any interval $[a,b]\subset\mathbb{R}$. Let us define $v_0(t) = v(tb + (1-t) a) = v(x)$. \begin{align*} (B_n v)(x) &= \sum_{k=0}^n \binom{n}{k} v(a+\frac{k}{n}(b-a)) \Bigl(\frac{x-a}{b-a}\Bigr)^k \Bigl(\frac{b-x}{b-a}\Bigr)^{n-k} \\ &= \sum_{k=0}^n \binom{n}{k} v_0(\frac{k}{n}) t^k(1-t)^{n-k} = (B_nv_0)(t). \end{align*} So, we have $$\label{appr:real} |(B_nv)(x) - v(x)| = |(B_nv_0)(t) - v_0(t)| \leq \omega_{v_0}(\frac{1}{\sqrt{n}}) = \omega_v(\frac{b-a}{\sqrt{n}}).$$ To prove the estimation for the interval in $\mathbb{R}^2$ we will repeat the reasoning given in \cite{Lo}. Let $v_y(x) = u(x,y)$ for the fixed $y\in[\beta_1,\beta_2]$, and $w_x(y) = u(x,y)$ for fixed $x\in[\alpha_1,\alpha_2]$. Let $\omega_v$ and $\omega_w$ denote the moduli of continuity of $v$ and $w$ respectively. Then $\omega_v(\delta)\leq\omega_u(\delta)$ and $\omega_w(\delta)\leq\omega_u(\delta)$. The functions \begin{gather*} B_n^v(x,y) = \sum_{i=0}^n \binom{n}{i} u(\alpha_1 + \frac{i}{n}(\alpha_2-\alpha_1),y) \Bigl( \frac{x-\alpha_1}{\alpha_2-\alpha_1} \Bigr)^i \Bigl( \frac{\alpha_2-x}{\alpha_2-\alpha_1} \Bigr)^{n-i},\\ B_n^w(x,y) = \sum_{j=0}^n \binom{n}{j} u(x,\beta_1 + \frac{j}{n}(\beta_2-\beta_1)) \Bigl( \frac{y-\beta_1}{\beta_2-\beta_1} \Bigr)^j \Bigl( \frac{\beta_2-y}{\beta_2-\beta_1} \Bigr)^{n-j} \end{gather*} are Bernstein polynomials of $v_y$ and $w_x$ respectively. So from estimation (\ref{appr:real}) we have \begin{gather*} | B_n^v(x,y) - u(x,y)| \leq \frac{5}{4} \omega_v(\frac{\alpha_2-\alpha_1}{\sqrt{n}}) \leq \frac{5}{4} \omega_u(\frac{\gamma}{\sqrt{n}}), \\ | B_n^w(x,y) - u(x,y)| \leq \frac{5}{4} \omega_w(\frac{\beta_2-\beta_1}{\sqrt{n}})\leq \frac{5}{4} \omega_u(\frac{\gamma}{\sqrt{n}}). \end{gather*} We can see that $$(B_nu)(x,y) = \sum_{i=0}{n} B_n^w(\frac{i}{n},y) \Bigl( \frac{x-\alpha_1}{\alpha_2-\alpha_1} \Bigr)^i \Bigl( \frac{\alpha_2-x}{\alpha_2-\alpha_1} \Bigr)^{n-i}$$ So, \begin{align*} &| (B_nu)(x,y) - u(x,y)| \leq | (B_nu)(x,y) - B_n^v(x,y)| + | B_n^v(x,y) - u(x,y)| \\ &\leq \sum_{i=0}{n} |B_n^w(\frac{i}{n},y)-u(\frac{i}{n},y)| \Bigl( \frac{x-\alpha_1}{\alpha_2-\alpha_1} \Bigr)^i \Bigl( \frac{\alpha_2-x}{\alpha_2-\alpha_1} \Bigr)^{n-i} + \frac{5}{4}\omega_u(\frac{\gamma}{\sqrt{n}})\\ &\leq \frac{5}{4}\omega_u(\frac{\gamma}{\sqrt{n}}) + \frac{5}{4}\omega_u(\frac{\gamma}{\sqrt{n}}). \end{align*} This completes the proof. \end{proof} Now we return to problem (\ref{pr:phi}). First let us consider the linear spectral problem $$\label{lin:eig} \begin{gathered} (Lu)(x,y) + \lambda u(x,y) = 0 \quad \text{for } (x,y)\in\Omega \\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega \end{gathered}$$ It is well known (see \cite{GT}) that there exists the minimal eigenvalue $\mu_0$ of the problem (\ref{lin:eig}). This eigenvalue is positive, simple and the associated eigenvector has constant sign. Moreover, $\mu_0$ is the only eigenvalue with the corresponding eigenvector having constant sign. Let $m>0$ be fixed and $A\subset (0,+\infty)$ be an open interval such that $\frac{\mu_0}{m}\in A$. Let $\varphi:A\times\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ be a continuous function such that $$\label{F1} \forall_{\varepsilon > 0} \exists_{\delta > 0} \forall_{(x,y)\in\overline{\Omega}, \lambda\in B, s\in \mathbb{R}} 0\leq s\leq\delta \Rightarrow |\varphi(\lambda,(x,y),s)-\lambda ms| \leq \varepsilon|s|,$$ for any bounded $B\subset A$; $$\label{F2} \forall_{(x,y)\in\overline{\Omega}, \lambda\in A, s < 0} \varphi(\lambda,(x,y),s) > 0.$$ Note that from \eqref{F1} the following conclusion may be drawn $$\label{phi:zero} \varphi(\lambda,(x,y),0)=0, \quad \text{for } (x,y)\in\overline{\Omega}, \lambda\in A.$$ Let us now fix the closed triangle $\T\subset\mathbb{R}^2$, such that $\Omega\subset \T$. Assume, that continuous $u:\overline{\Omega}\to\mathbb{R}$ satisfies boundary conditions, i.e. $u(x,y)=0$ for $(x,y)\in\partial\Omega$. Because of (\ref{phi:zero}) we may continuously extend the superposition $\varphi(\lambda,\cdot,u(\cdot))$ to the triangle $\T$, in such a way that this extension achieves 0 on $\T\setminus\Omega$. Let $\tilde\varphi(\lambda,\cdot,u(\cdot))$ denote this extension. So we may consider the boundary-value problem $$\label{pr:appr1} \begin{gathered} (Lu)(x,y) + (B_n^1\tilde\varphi(\lambda,\cdot,u(\cdot)))(x,y) = 0 \quad \text{for } (x,y)\in\Omega \\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega. \end{gathered}$$ Let us now fix the closed rectangle $\S\subset\mathbb{R}^2$, such that $\Omega\subset \S$. As above, we may continuously extend the function $\varphi(\lambda,\cdot,u(\cdot))$ to the rectangle $\S$, in such a way that this extension achieves 0 on $\S\setminus\Omega$. Let $\hat\varphi(\lambda,\cdot,u(\cdot))$ denote such extension on the rectangle $\S$. So let us consider the problem $$\label{pr:appr2} \begin{gathered} (Lu)(x,y) + (B_n^2\hat\varphi(\lambda,\cdot,u(\cdot)))(x,y) = 0 \quad \text{for } (x,y)\in\Omega \\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega. \end{gathered}$$ For the above boundary-value problems (\ref{pr:phi}), (\ref{pr:appr1}), (\ref{pr:appr2}), we are looking for the weak solutions $(\lambda,u)\in A\times\wtt$. From Sobolev embedding theorem we have $\wtt\subset\czo$ (see \cite{GT}), so $B_n^1\tilde\varphi(\lambda,\cdot,u(\cdot))$ and $B_n^2\hat\varphi(\lambda,\cdot,u(\cdot))$ are well defined. Because of (\ref{phi:zero}) each pair $(\lambda,0)\in A\times W^{2,2}(\Omega)$ is the solution of problems (\ref{pr:phi}), (\ref{pr:appr1}) and (\ref{pr:appr2}). We call such pairs {\it trivial solutions}. Let ${\mathcal R}$ denote the closure (in $A \times\czo$) of the set of nontrivial solutions of the problem (\ref{pr:phi}). Let ${\mathcal R}_n^i$ $(i=1,2)$ denote the closure (in $A \times\czo$) of the set of nontrivial solutions of the problems $(\ref{pr:appr1})$ and $(\ref{pr:appr2})$ respectively. The classical global bifurcation theorem given by Rabinowitz (see \cite{R} and also \cite{CH,N}) may be applied to elliptic boundary-value problems. Such applications were considered by many authors (see e.g. \cite{R,R2,Ry,Kie}). Below we prove a similar result \begin{theorem}\label{th1} There exists the noncompact component $C$ of ${\mathcal R}$ such that $(\frac{\mu_0}{m},0)\in C$. \end{theorem} We are also going to show that the similar thesis may be proved for approximating problems (\ref{pr:appr1}) and (\ref{pr:appr2}). We are going to prove that the connected component $C$ of ${\mathcal R}$ is, in a sense explained below, approximated by the branches of sets ${\mathcal R}_n^i$ $(i=1,2)$. Let us further assume that $i\in\{1,2\}$ is fixed. \begin{theorem}\label{th2} Let $\varepsilon>0$, $\frac{\mu_0}{m}\in(a,b)\subset[a,b]\subset A$ and $R>0$. Then, for almost all $n\in\nat$, there exists a component $C_n^i$ of the set ${\mathcal R}_n^i$, such that \begin{itemize} \item[(i)] $C_n^i \cap \Bigl( (\frac{\mu_0}{m}-\varepsilon, \frac{\mu_0}{m}+\varepsilon)\times\{0\} \Bigr) \neq \emptyset$; \item[(ii)] $C_n^i \cap \partial ( [a,b]\times \overline{B(0,R)} ) \neq\emptyset$. \end{itemize} \end{theorem} The relation between components $C_n^i$ and $C$ will be established in Theorem \ref{th3} given below. But first we need to define some notation. For a set $U\subset A\times\czo$ let us denote $$O_\varepsilon( U ) = \{ (\lambda,u)\in A\times\czo : \exists_{(\mu,v)\in U} |\lambda-\mu|+\Vert u-v\Vert_{0} < \varepsilon \}.$$ where $\Vert\cdot\Vert_0$ denotes the norm in $\czo$. \begin{theorem}\label{th3} Let $\varepsilon>0$, $\frac{\mu_0}{m}\in(a,b)\subset[a,b]\subset A$ and $R>0$ be fixed. Assume that \begin{itemize} \item[(i)] $C\subset {\mathcal R}$ is a noncompact component, such that $(\frac{\mu_0}{m},0)\in C$; \item[(ii)] $C_n^i\subset {\mathcal R}_n^i$ is a component, such that $C_i^n\cap \Bigl((\frac{\mu_0}{m}-\varepsilon,\frac{\mu_0}{m} +\varepsilon)\times\{0\}\Bigr) \neq\emptyset$ and $C_n^i \cap \partial ( [a,b]\times \overline{B(0,R)} )\neq\emptyset$; \item[(iii)] $S$ is a component of $C \cap \Bigl([a,b]\times \overline{B(0,R)}\Bigr)$, such that $(\frac{\mu_0}{m},0)\in S$; \item[(iv)] $S_n^i$ is a component of $C_n^i\cap \Bigl([a,b]\times \overline{B(0,R)}\Bigr)$, such that $S_n^i \cap \Bigl((\frac{\mu_0}{m}-\varepsilon,\frac{\mu_0}{m} +\varepsilon)\times\{0\}\Bigr) \neq\emptyset$. \end{itemize} Then, for almost all $n\in\nat$, the relation $S_n^i\subset O_\varepsilon(S)$ holds. \end{theorem} \begin{remark} \label{rmk1} \rm From Theorem \ref{th3} we may conclude that $$\mathop{\rm Li}_{n\to+\infty} S_n^i \subset \mathop{\rm Ls}_{n\to+\infty} S_n^i \subset S,$$ where $\mathop{\rm Li}$ and $\mathop{\rm Ls}$ denote Kuratowski lower and upper limit respectively (see \cite{Ku}). \end{remark} \section{Approximation of global bifurcation branches} We are now going to introduce the necessary notation. First of all, let $\czo$ denote the space of all continuous functions $u:\overline{\Omega}\to\mathbb{R}$, with the norm $\Vert u\Vert_0 = \sup_{(x,y)\in\overline{\Omega}} |u(x,y)|$. Let $\czzo$ denote the subspace of $\czo$ consisting of all functions $u:\overline{\Omega}\to\mathbb{R}$ satisfying boundary conditions $u(x,y)=0$ for $(x,y)\in\partial\Omega$. Let the maps $B_n^i:\czzo\to\czo$ ($i=1,2$) be given by (\ref{def:bn1}) and (\ref{def:bn2}) respectively (here we assume that the appropriate formulas are applied to extensions $\tilde f:\T\to\mathbb{R}$ and $\hat f:\S\to\mathbb{R}$, of the function $f\in\czzo$). It is easy to observe that both maps are bounded linear maps and $\Vert B_n^i\Vert \leq 1$, for $n\in\nat$ and $i=1,2$. It is also well known (see \cite{GT}), that there exists a continuous map $\hat T:\wzt\to \wttz$, such that $$\hat Th = u \Leftrightarrow \begin{cases} (Lu)(x,y) + h(x,y) = 0 & \text{for } (x,y)\in\Omega \\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega. \end{cases}$$ Let $\Phi:A\times\czo\to\czo$ be given by $\Phi(\lambda,u)(x,y) = \varphi(\lambda,(x,y),u(x,y))$. From (\ref{phi:zero}) we may conclude that $\Phi(A\times\czzo)\subset\czzo$. Additionally, let $j:\wttz\to\czzo$ be the inclusion. From the Sobolev embedding theorem (see \cite{GT}) we have the compactness of $j$. Let us denote $T=j\circ \hat T$. Hence the superposition $T\circ \Phi:A\times\czzo\to\czzo$ is completely continuous. The same conclusion may be drawn for maps $T\circ B_n^i\circ\Phi:A\times\czzo\to\czzo$ where $n\in\nat$ and $i=1,2$. The situation is described by the following diagrams $$\begin{CD} A\times\czzo \\ @VV{\Phi }V \\ \czo @>i_0>> \wzt @>\hat T>> \wttz @>j>> \czzo \\ \end{CD}$$ $$\begin{CD} A\times\czzo \\ @VV{\Phi }V \\ \czzo @>B_n^i>> \czo @>i_0>> \wzt @>\hat T>> \wttz @>j>> \czzo \\ \end{CD}$$ Here $i_0$ denotes the natural inclusion $i_0:\czo\to W^{0,2}(\Omega)$. Let $f_n^1:A\times\czzo\to\czzo$ be given by $$f_n^1(\lambda,u) = u - T B_n^1\Phi(\lambda,u).$$ The zeros of the map $f_n^1$ correspond to the solutions of the problem (\ref{pr:appr1}). Similarly let $f_n^2:A\times\czzo\to\czzo$ be given by $$f_n^2(\lambda,u) = u - T B_n^2\Phi(\lambda,u).$$ The zeros of the map $f_n^2$ correspond to the solutions of the problem (\ref{pr:appr2}). We are going to prove Theorems \ref{th1}, \ref{th2} and \ref{th3} in the sequence of lemmas. \begin{lemma}\label{positive:solutions} If $(\lambda,u)\in{\mathcal R}$, then $u\geq 0$. \end{lemma} \begin{proof} Assume that $U = \{(x,y)\in\Omega | u(x,y)<0\} \neq \emptyset$. Because of \eqref{F2} the relation $(Lu)(x,y) \leq 0$ holds for all $(x,y)\in U$. That is why, by the maximum principle (see \cite{GT}), there exists $(x,y)\in\partial U$ such that $u(x,y) <0$. On the other hand the definition of $U$ implies that if $(x,y)\in\partial U$ then either $u(x,y)=0$ or $(x,y)\in\partial\Omega$. The latter and boundary conditions imply that $u(x,y)=0$ as well. So we have the contradiction with the maximum principle. \end{proof} \begin{corollary}\label{abs:sol} If $u = \lambda mT|u|$ and $u\neq 0$, then $\lambda = \frac{\mu_0}{m}$. \end{corollary} \begin{proof} Because of Lemma \ref{positive:solutions}, we can see that $|u|=u$. Hence $\lambda$ is the eigenvalue of the problem (\ref{lin:eig}) with the corresponding nonnegative eigenvector. So $\lambda m = \mu_0$. \end{proof} \begin{remark} \label{rmk2} \rm Without loss of generality we may assume that $\varphi(\lambda,(x,y),s) = \lambda m|s|$ for $s<0$, $\lambda\in A$ and $(x,y)\in\overline{\Omega}$. Because of Lemma \ref{positive:solutions}, both the original and modified problem have the same set of solutions. Hence, we may assume that the strenghtened version of \eqref{F1} holds $$\label{F1'} \forall_{\varepsilon > 0} \exists_{\delta > 0} \forall_{(x,y) \in\overline{\Omega}, \lambda\in B, s\in\mathbb{R}} |s|\leq\delta \Rightarrow \Bigl|\varphi(\lambda,(x,y),s)-\lambda m|s| \Bigr| \leq \varepsilon|s|,$$ for any bounded $B\subset A$. \end{remark} \begin{lemma} \label{gamma:est} For any compact $B\subset A\setminus\{ \frac{\mu_0}{m}\}$, there exist $\gamma>0$ and $\delta>0$, such that $$\Vert f(\lambda,u)\Vert_0 \geq \gamma\Vert u\Vert_0 \quad \text{for } \lambda\in B, \Vert u\Vert_0\leq \delta.$$ \end{lemma} \begin{proof} Let us first observe, that the inequality holds $$\gamma_0 = \inf_{\lambda\in B, \Vert u\Vert_0 = 1} \Bigl \Vert u - \lambda mT|u|\Bigr\Vert_0 > 0.$$ Assume, contrary to our claim, that there exists the sequence $\{(\lambda_n,u_n)\}\subset B\times\czzo$, such that $\Vert u_n\Vert_0 =1$, $(\lambda_n,u_n)\in B\times\czzo$ and $u_n - \lambda_n m T|u_n| \to 0$. Because $T$ is completely continuous, we can see that $\{T|u_n|\}$ contains convergent subsequence, so we may assume that $u_n\to u_0\in\czzo$. Of course, we may also assume, that $\lambda_n\to \lambda_0\in B$. Hence, without the loss of generality, we have $u_0 = \lambda_0 m T |u_0|$, so by corollary \ref{abs:sol} $\lambda_0 m = \mu_0$, a contradiction. Because of \eqref{F1'} there exists $\delta_0>0$, such that for $\lambda\in B$, $$\Vert u\Vert_0\leq \delta_0 \Rightarrow \Bigl \Vert T\Phi(\lambda,u) - m\lambda T|u|\Bigr \Vert_0 \leq \frac{\gamma_0}{2}\Vert u\Vert_0.$$ So, for $\Vert u\Vert_0\leq \delta_0$ and $\lambda\in B$, the relation holds \begin{align*} \Vert f(\lambda,u)\Vert_0 &\geq \Bigl\Vert u - \lambda mT|u| \Bigr\Vert_0 - \Bigl\Vert T\Phi(\lambda,u) - m\lambda T|u|\Bigr\Vert_0 \\ &\geq \gamma_0\Vert u\Vert_0 - \frac{\gamma_0}{2}\Vert u\Vert_0 \\ &= \frac{\gamma_0}{2}\Vert u\Vert_0 > 0. \end{align*} This completes the proof. \end{proof} \begin{lemma}\label{deg:phi} \begin{itemize} \item[(i)] If $\lambda<\frac{\mu_0}{m}$, then $\deg(f(\lambda,\cdot),B(0,r),0)=1$ for $r>0$ small enough. \item[(ii)] If $\lambda>\frac{\mu_0}{m}$, then $\deg(f(\lambda,\cdot),B(0,r),0)=0$ for $r>0$ small enough. \end{itemize} \end{lemma} \begin{proof} First, let us observe that for any $[a,b]\subset A\setminus\{\frac{\mu_0}{m}\}$ there exists $r>0$, such that for $\lambda\in[a,b]$ the map $f(\lambda,\cdot):\overline{B(0,r)}\to\czzo$ may be joined by homotopy with $f_0(\lambda,\cdot):\overline{B(0,r)}\to\czzo$ given by $f_0(\lambda,u) = u - \lambda m T|u|$. By Lemma \ref{gamma:est} there exist $r_1>0$ and $\gamma>0$ such that $$\Vert f(\lambda,u)\Vert_0 \geq \gamma\Vert u\Vert_0$$ for $\lambda\in [a,b]$ and $\Vert u\Vert_0\leq r_1$. From \eqref{F1'} we may conclude that for $\Vert u\Vert_0\leq r_2$ the inequality holds $$\Vert \Phi(\lambda,u) - m\lambda|u| \Vert_0 \leq \frac{\gamma}{2\Vert T\Vert}\Vert u\Vert_0.$$ Let us take $r=\min\{r_1,r_2\}$ and define the homotopy $h:[0,1]\times\overline{B(0,r)}\to\czzo$, by $$h(\tau,u) = f_0(\lambda,u) - \tau [f(\lambda,u)-f_0(\lambda,u)].$$ Then for $\Vert u\Vert_0 = r$ we have $$\Vert h(\tau,u)\Vert_0 \geq \gamma\Vert u\Vert_0 - \frac{\gamma}{2}\Vert u\Vert_0 > 0,$$ so the homotopy is well defined. Moreover, if $\lambda<\frac{\mu_0}{m}$ the homotopy $h_1 :[0,1]\times \overline{B(0,r)}\to\czzo$, given by $h_1(\tau,u) = u - \tau\lambda mT|u|$, joins $f_0(\lambda,\cdot)$ with identity map. As we can see from corollary \ref{abs:sol} the homotopy may have nontrivial zero only when $\tau \lambda = \frac{\mu_0}{m}$, which is not possible. So (i) is proved. On the other hand if $\lambda > \frac{\mu_0}{m}$, then the map $f_0(\lambda,\cdot)$ may be joined by homotopy with the map $f_1:\overline{B(0,r)}\to\czzo$ given by $f_1(u) = f_0(\lambda,u) - u_0$, where $u_0$ is positive eigenvector of (\ref{lin:eig}), associated with the eigenvalue $\mu_0$. The homotopy may be given by $$h_1(\tau,u) = u - \lambda m T|u| - \tau u_0.$$ So, let us now assume that $h_1(\tau,u)=0$ for $\Vert u\Vert_0\leq r$ and $\tau\in[0,1]$. Then we have $u = \lambda mT|u|+\tau u_0\geq 0$ and $$\int_\Omega u u_0 = \lambda m \int_\Omega (Tu)u_0 + \tau \int_\Omega u_0^2.$$ As we can see from the definition of a weak solution (see \cite{GT}) of the Dirichlet problem, for each $u,v\in W_0^{2,2}(\Omega)$ the relation $\int_\Omega (Tu)v = \int_\Omega (Tv)u$ holds, so $$\int_\Omega (Tu)u_0 = \int_\Omega u(Tu_0) = \frac{1}{\mu_0} \int_\Omega uu_0.$$ Hence $$(1-\frac{m\lambda}{\mu_0}) \int_\Omega u u_0 = \tau \int_\Omega u_0^2 > 0,$$ what implies $\lambda<\frac{\mu_0}{m}$, and contradicts our assumption. That is why (ii) holds true. \end{proof} \begin{proof}[Proof of Theorem \ref{th1}] We are going to refer to the generalization of Rabinowitz global bifurcation theorem given in \cite{DG}. This theorem refers to the more general case of convex-valued maps, but may be applied to the single valued case, as in Theorem \ref{th1}. What we need is the interval $[a,b]$, such that the set of all bifurcation points of $f$ is contained in that interval, and the change of local topological degree of the maps $f(\lambda,\cdot):\overline{B(0,r)}\to\czzo$ on the small balls around zero. As we can see, because of Lemma \ref{positive:solutions}, the only bifurcation point is $(\frac{\mu_0}{m},0)$, so we may take $[a,b]=[\frac{\mu_0}{m},\frac{\mu_0}{m}]$. Additionally, by Lemma \ref{deg:phi}, there is the degree change in the neighborhood of $\frac{\mu_0}{m}$. \end{proof} \begin{lemma}\label{lem1} Let $\delta>0$ and ${\mathcal K} = (a,\frac{\mu_0}{m}-\delta) \cup (\frac{\mu_0}{m}+\delta,b)\subset[a,b]\subset A$. Then there exists $r>0$, such that $({\mathcal K}\times \overline{B(0,r)}) \cap {\mathcal R}^i_n = \emptyset$ for almost all $n\in\nat$. \end{lemma} \begin{proof} Let us have $i\in\{1,2\}$ fixed. Assume, contrary to our claim, that there exists the increasing sequence $\{\gamma(n)\}\subset\nat$ and points $(\lambda_n,u_n)\in{\mathcal K}\times\czzo$, such that $f_{\gamma(n)}^i(\lambda_n,u_n)=0$, $u_n\neq 0$, $u_n\to 0$ and $\lambda_n\to\lambda_0\in\overline{\mathcal K}$. Then \begin{gather*} u_n = T B_{\gamma(n)}^i \Phi(\lambda_n,u_n), \\ u_n = \lambda_n m T B_{\gamma(n)}^i |u_n| + T B_{\gamma(n)}^i [\Phi(\lambda_n,u_n) - m\lambda_n |u_n|]. \end{gather*} Let us now denote $v_n = u_n / \Vert u_n\Vert_0$. Then $$v_n = \lambda_n mT B_{\gamma(n)}^i |v_n| + T B_{\gamma(n)}^i \frac{\Phi(\lambda_n,u_n) - m\lambda_n |u_n|}{\Vert u_n\Vert_0}.$$ Because of \eqref{F1'} there is $\Bigl\Vert \frac{\Phi(\lambda_n,u_n) - m\lambda_n |u_n|}{\Vert u_n\Vert_0}\Bigr\Vert_0 \to 0$. Then, because of $\Vert T\circ B_{\gamma(n)}^i\Vert \leq \Vert T\Vert$, letting $n\to+\infty$ gives $$T B_{\gamma(n)}^i \frac{\Phi(\lambda_n,u_n) - m\lambda_n |u_n|}{\Vert u_n\Vert_0} \to 0.$$ Additionally, the sequence $\{B_{\gamma(n)}^i v_n\}$ is bounded, so taking appropriate subsequence of $\{ v_n\}$ we may assume that $v_n\to v_0\in\czzo$. Moreover, we can see that $$\Vert B_{\gamma(n)}^i v_n - v_0\Vert_0 \leq \Vert B_{\gamma(n)}^i( v_n - v_0 ) \Vert_0 + \Vert B_{\gamma(n)}^i v_0 - v_0 \Vert_0.$$ Because $\Vert B_{\gamma(n)}^i\Vert \leq 1$ and $B_{\gamma(n)}^i v_0 \to v_0$, we can see that $B_{\gamma(n)}^i v_n \to v_0$. So, letting $n\to+\infty$ we have $v_0 = \lambda_0 m T|v_0|$. This, for $\Vert v_0\Vert_0=1$, implies $\lambda_0=\frac{\mu_0}{m}\not\in\overline{\mathcal K}$, a contradiction. \end{proof} \begin{lemma}\label{homotopy} Let $\delta>0$ and ${\mathcal K} = (a,\frac{\mu_0}{m}-\delta) \cup (\frac{\mu_0}{m}+\delta,b)\subset[a,b]\subset A$. Then there exists $r_0>0$, such that $$\deg(f(\lambda,\cdot),B(0,r),0)=\deg(f_n(\lambda,\cdot),B(0,r),0),$$ for all $r\in(0,r_0)$, almost all $n\in\nat$ and for all $\lambda\in{\mathcal K}$. \end{lemma} \begin{proof} Let us now take $\gamma>0$ and $\delta>0$ as in Lemma \ref{gamma:est}, for $B=\overline{\mathcal K}$. Let us also take $r\in(0,\delta)$, such that for all $\lambda\in{\mathcal K}$, the implication holds $$\Vert u\Vert_0\leq r \Rightarrow \Vert \Phi(\lambda,u) - m\lambda |u|\Vert_0 \leq \frac{\gamma}{6\Vert T\Vert}\Vert u\Vert_0.$$ Let us, for the fixed $n\in\nat$, $\lambda\in{\mathcal K}$ and $i\in\{1,2\}$ take, the homotopy $h_{n,\lambda}^i:[0,1]\times\overline{B(0,r)}\to\czzo$, given by $$h_{n,\lambda}^i(\tau,u) = f(\lambda,u) - \tau[ f_n^i(\lambda,u) - f(\lambda,u)].$$ Assume now that $h_{n,\lambda}^i(\tau,u) = 0$ for $\Vert u\Vert_0 = r$ and $\tau\in[0,1]$. Then $$u = T[ \Phi(\lambda,u) + \tau (B_n^i\Phi(\lambda,u) - \Phi(\lambda,u)) ].$$ Because the set $${\mathcal A} = \{ \Phi(\lambda,u) + \tau (B_n^i\Phi(\lambda,u) - \Phi(\lambda,u)) : u\in\czzo, \Vert u\Vert_0 = r, \tau\in[0,1] \}$$ is bounded, the set $T({\mathcal A})$ is relatively compact, so all functions $u\in\czzo$, such that $h_{n,\lambda}^i(\tau,u) = 0$ and $\Vert u\Vert_0 = r$, are uniformly continuous. Hence, by Lemma \ref{conv:triangle} and \ref{conv:rectangle}, we may assume, that for $n\in\nat$ large enough, $\lambda\in{\mathcal K}$ and $u\in{\mathcal A}$ the inequality holds $\Vert m\lambda T(|u| - B_n^i |u|)\Vert_0 \leq \frac{\gamma}{6}r = \frac{\gamma}{6}\Vert u\Vert_0$. That is why, for all functions $u\in\czzo$, such that $h_{n,\lambda}^i(\tau,u) = 0$ and $\Vert u\Vert_0 = r$, and $n\in\nat$ large enough \begin{align*} &\Vert f(\lambda,u) - f^i_n(\lambda,u) \Vert_0 \\ &\leq \Vert T(\Phi(\lambda,u) - m\lambda |u|) \Vert_0 + \Vert T B_n^i(m\lambda|u| - \Phi(\lambda,u))\Vert_0 + \Vert m\lambda T(|u| - B_n^i |u|)\Vert_0 \\ &\leq \frac{\gamma}{6\Vert T\Vert}\Vert T\Vert \Vert u\Vert_0 +\frac{\gamma}{6\Vert T\Vert}\Vert T\Vert \Vert u\Vert_0 + \frac{\gamma}{6}\Vert u\Vert_0 = \frac{\gamma}{2}\Vert u\Vert_0. \end{align*} Consequently $$\Vert h_{n,\lambda}^i(\tau,u)\Vert_0 \geq \gamma\Vert u\Vert_0 - \frac{\gamma}{2}\Vert u\Vert_0 > 0,$$ which is a contradiction. \end{proof} \begin{proof}[Proof of Theorem \ref{th2}] Let us take any $\varepsilon>0$ and the interval $[a,b]\subset A$, such that $\frac{\mu_0}{m}\in(a,b)$. From Lemma \ref{lem1} we can see that for almost all $n\in\nat$ the relation holds ${\mathcal R}_n^i\cap (([a,\frac{\mu_0}{m}-\varepsilon]\cup [\frac{\mu_0}{m}+\varepsilon,b])\times\{0\})=\emptyset$. So the set of bifurcation points of $f_n^i |_{(a,b)\times\czzo}$ is contained in the interval $[\frac{\mu_0}{m}-\varepsilon,\frac{\mu_0}{m}+\varepsilon]$. Moreover, by Lemmas \ref{deg:phi} and \ref{homotopy}, there is the change of topological degree for $\lambda<\frac{\mu_0}{m}-\varepsilon$ and $\lambda>\frac{\mu_0}{m}+\varepsilon$. So, as in the proof of Theorem \ref{th1}, we may apply the global bifurcation theorem given in \cite{DG}. \end{proof} For the rest of this article, let us have an interval $[a,b]\subset A$, such that $\frac{\mu_0}{m}\in(a,b)$, and a constant $R>0$ fixed. Moreover, let us assume, according to Theorem \ref{th3}, that \begin{itemize} \item[(i)] $C\subset {\mathcal R}$ is noncompact component, such that $(\frac{\mu_0}{m},0)\in C$; \item[(ii)] $C_n^i\subset {\mathcal R}_n^i$ is a component, such that $C_i^n\cap \Bigl((\frac{\mu_0}{m}-\varepsilon,\frac{\mu_0}{m}+\varepsilon)\times\{0\}\Bigr) \neq\emptyset$ and $C_n^i \cap \partial ( [a,b]\times \overline{B(0,R)} )\neq\emptyset$; \item[(iii)] $S$ is a component of $C \cap \Bigl([a,b]\times \overline{B(0,R)}\Bigr)$, such that $(\frac{\mu_0}{m},0)\in S$; \item[(iv)] $S_n^i$ is a component of $C_n^i\cap \Bigl([a,b]\times \overline{B(0,R)}\Bigr)$, such that $S_n^i \cap \Bigl((\frac{\mu_0}{m}-\varepsilon,\frac{\mu_0}{m} +\varepsilon)\times\{0\}\Bigr) \neq\emptyset$. \end{itemize} \begin{lemma}\label{lem:oeps} For almost all $n\in\nat$ the inclusion $S_n^i \subset O_\varepsilon( S )$ holds. \end{lemma} \begin{proof} Let us fix $i\in\{1,2\}$. Let us observe that $$\label{temptheseta} \forall_{\eta>0} \exists_{n_0\in\nat} \forall_{n>n_0} (f_n^i)^{-1}(0) \cap ([a,b]\times\overline{B(0,R)}) \subset O_\eta\Bigl( f^{-1}(0)\cap ([a,b]\times\overline{B(0,R)})\Bigr).$$ Assume, contrary to our claim, that there exists $\eta_0>0$ and the sequence $(\lambda_n,u_n)\in (f_{\gamma(n)}^i)^{-1}(0)\cap ([a,b] \times\overline{B(0,R)})$, where $\{\gamma(n)\}\subset \nat$, such that $(\lambda_n,u_n)\not\in O_{\eta_0}( f^{-1}(0)\cap ([a,b]\times\overline{B(0,R)}))$. We may assume that $\lambda_n\to\lambda_0\in[a,b]$. Moreover, the sequence $\{ B_{\gamma(n)}^i\Phi(\lambda_n,u_n)\}$ is bounded, so $u_n = TB_{\gamma(n)}^i\Phi(\lambda_n,u_n)$, contains convergent subsequence. So we may also assume that $u_n\to u_0\in \overline{B(0,R)}$. We can see that \begin{align*} & \Vert TB_{\gamma(n)}^i\Phi(\lambda_n,u_n) - T\Phi(\lambda_n,u_n)\Vert_0 \\ & \leq \Vert TB_{\gamma(n)}^i\Phi(\lambda_n,u_n) - TB_{\gamma(n)}^i\Phi(\lambda_0,u_0)\Vert_0 + \Vert TB_{\gamma(n)}^i\Phi(\lambda_0,u_0) - T\Phi(\lambda_0,u_0)\Vert_0. \end{align*} As we can see the above sum converges to zero, what gives $u_0 = T\Phi(\lambda_0,u_0)$, a contradiction. Let us now show that $$\label{appr:stepone} \forall_{\eta >0} \exists_{n_0\in\nat} \forall_{n > n_0} \; S^i_n \subset O_\eta ({\mathcal R}_0),$$ where ${\mathcal R}_0 = {\mathcal R}\cap([a,b]\times \overline{B(0,R)})$. Assume, contrary to our claim, that there exists the sequence $S_{\gamma(n)}^i$ and positive number $\eta_0>0$, satisfying $S_{\gamma(n)}^i\not\subset O_{\eta_0}({\mathcal R}_0)$. Let $(\lambda_n,u_n)\in S_{\gamma(n)}$ satisfy $(\lambda_n,u_n)\not\in O_{\eta_0}({\mathcal R}_0)$. By Lemma $\ref{lem1}$ (applied for an open open neighbourhood of ${\mathcal K}$) there exists $r_0>0$, such that for almost all $n\in\nat$, $$S_{\gamma(n)}\cap \bigl( {\mathcal K}\times \overline{B(0,r_0)} \bigr) = \emptyset,$$ where ${\mathcal K} = [a,\frac{\mu_0}{m}-\frac{\eta_0}{2}] \cup [\frac{\mu_0}{m}+\frac{\eta_0}{2},b]$. We may assume that $r_0 < {\frac{\eta_0}{2}}$. If $\Vert u_n\Vert_0 r_0$, for any $\lambda\in[a,b]$. So $(\lambda_n,u_n)\not\in O_{r_0}([a,b]\times\{0\})$ and $(\lambda_n,u_n)\not\in O_{r_0}({\mathcal R}_0)$, what contradicts (\ref{temptheseta}), because $f^{-1}(0)\cap ([a,b]\times \overline{B(0,R)}) = ([a,b]\times\{0\}) \cup {\mathcal R}_0$. The contradiction proves \eqref{appr:stepone}. Now we are going to prove $$\label{appr:steptwo} \forall_{\varepsilon >0} \exists_{n_0\in\nat} \forall_{n > n_0}\; S_n^i \subset O_\varepsilon (S).$$ Assume that there exists $\eta_0 > 0$ and the subsequence $S_{\gamma(n)}^i$ of $S_n^i$ such that $$S_{\gamma(n)}^i \not\subset O_{\eta_0} (S).$$ But for almost all $n\in\nat$ we have $$\emptyset \neq S_{\gamma(n)}^i\cap([a,b]\times\{0\})\subset (\frac{\mu_0}{m}-\eta_0,\frac{\mu_0}{m}+\eta_0)\times\{0\} \subset O_{\eta_0}(S)$$ and consequently $S_{\gamma(n)}^i \cap O_{\eta_0}(S)\neq\emptyset$. Assume that $(\lambda_n,u_n)\in S_{\gamma(n)}^i$ are points such that $(\lambda_n,u_n)\not\in O_{\eta_0}(S)$. Because, by (\ref{appr:stepone}) $$\forall_{\varepsilon >0} \exists_{n_0\in\nat} \forall_{n > n_0} (\lambda_n,u_n) \in O_\varepsilon ({\mathcal R}_0),$$ and ${\mathcal R}_0$ is compact, there exists the subsequence of $\{(\lambda_n,u_n)\}$ converging to $(\lambda_0,u_0)$ in ${\mathcal R}_0$. Because $(\lambda_n,u_n)\not\in O_{\eta_0}(S)$ the relation $(\lambda_0,u_0) \not\in O_{\eta_0} (S)$ holds as well. The set ${\mathcal R}_0$ is a compact metric space, $X = S\cap {\mathcal R}_0$ and $Y = \{ (\lambda_0,u_0) \}$ are its closed subsets, not belonging to the same component of ${\mathcal R}_0$. By separation lemma (see \cite{WD}) there exists the separation ${\mathcal R}_0 = {\mathcal R}_x\cup {\mathcal R}_y$ of ${\mathcal R}_0$ , where ${\mathcal R}_x$ and ${\mathcal R}_y$ are closed and disjoint, and such that $S\cap {\mathcal R}_0 \subset {\mathcal R}_x$ and $(\lambda_0,u_0)\in {\mathcal R}_y$. This implies, that there exist open and disjoint subsets $U_x, U_y \subset [a,b]\times \overline{B(0,R)}$, such that $(\lambda_0,u_0)\in U_y$ and $S\cap {\mathcal R}_0 \subset U_x$ and ${\mathcal R}_0\subset U_x\cup U_y$. Because ${\mathcal R}_0$ is compact, there exists $\eta>0$, such that $O_\eta({\mathcal R}_0)\subset U_x\cup U_y$. Let us observe that, by (\ref{appr:stepone}), for almost all $n\in \nat$ the relation holds $S_{\gamma(n)}^i \subset O_\eta( {\mathcal R}_0 ) \subset U_x\cup U_y$. Moreover, $S_{\gamma(n)}^i\cap U_x\neq\emptyset$ and $S_{\gamma(n)}^i\cap U_y\neq\emptyset$, what (because of $U_x\cap U_y=\emptyset$) contradicts the connectedness of $S_{\gamma(n)}^i$. The contradiction proves $(\ref{appr:steptwo})$ and finishes the proof of the lemma. \end{proof} Now Theorem \ref{th3} follows as a corollary of the above result. \begin{remark} \label{rmk3} \rm Both approximations are, in fact, the finitely dimensional ones. Let us take, as an example, the approximation on the triangle. Let us denote $N=\frac{n(n+1)}{2}$ and associate the coordinates of the point $\xi\in\mathbb{R}^N$ with $\xi_{i,j,k} = u(\frac{i}{n},\frac{j}{n},\frac{k}{n})$, where $i,j,k\geq 0$ and $i+j+k=n$. Then we have $$u(x,y) = T[B_n^1\Phi(\lambda,u)],$$ and substituting $(x,y) = \frac{i_0}{n}P+\frac{j_0}{n}Q+\frac{k_0}{n}R$ we have $$\label{findim:eqn} \xi_{i_0,j_0,k_0} = \sum_{i,j,k\geq 0; i+j+k=n} \varphi(\lambda,\xi_{i,j,k}) [T (T_{i,j,k}^n)](\frac{i_0}{n},\frac{j_0}{n},\frac{k_0}{n}).$$ where $[T (T_{i,j,k}^n)](\frac{i_0}{n},\frac{j_0}{n},\frac{k_0}{n})$ are known coefficients depending on $\T$ and $\Omega$ only. In the above formula we should treat the function $T(T_{i,j,k}^n)\in\czo$ as extended by zero to the whole triangle $\T$. That is why the dimension of the problem (\ref{findim:eqn}) is generally smaller then $N$. \end{remark} \section{Existence theorem} In this section we are going to show how the approximation theorem for global bifurcation branches given above, may be applied in the proof of the existence theorem for the boundary-value problem $$\label{ex:pr} \begin{gathered} (Lu)(x,y) + \varphi(x,y,u(x,y)) = 0 \quad \text{for } (x,y)\in\Omega \\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega. \end{gathered}$$ We will show, that the solution of the above problem exists and is approximated by solutions of $$\label{ex:appr} \begin{gathered} (Lu)(x,y) + (B_n^1\tilde\varphi(\cdot,u(\cdot)))(x,y) = 0 \quad \text{for } (x,y)\in\Omega \\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega. \end{gathered}$$ and $$\label{ex:appr2} \begin{gathered} (Lu)(x,y) + (B_n^2\hat\varphi(\cdot,u(\cdot)))(x,y) = 0 \quad \text{for } (x,y)\in\Omega \\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega. \end{gathered}$$ \begin{theorem} \label{thm4} Let $\varphi:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ be the continuous function, such that there exist the positive numbers $A<\mu_00} \exists_{\delta>0} \forall_{(x,y)\in\overline{\Omega}, s\in\mathbb{R}} 0\leq s\leq \delta\Rightarrow |\varphi((x,y),s)-Bs| \leq \varepsilon|s|;\\ \label{exst:infty} \forall_{\varepsilon>0} \exists_{R >0} \forall_{(x,y)\in\overline{\Omega}, s\in\mathbb{R}} s\geq R \Rightarrow |\varphi((x,y),s)-As|\leq \varepsilon|s|. \end{gather} Then \begin{itemize} \item[(a)] there exists the nonnegative solution$u$of \eqref{ex:pr} and for almost all$n\in\nat$there exists solution$u_n$of \eqref{ex:appr}, such that, there exists the subsequence$\{u_{\gamma(n)}\}$satisfying$\lim_{n\to+\infty} \Vert u_{\gamma(n)}-u\Vert_0 = 0$, and \item[(b)] there exists the nonnegative solution$u$of \eqref{ex:pr} and for almost all$n\in\nat$there exists solution$v_n$of$(\ref{ex:appr2})$, such that, there exists the subsequence$\{v_{\gamma(n)}\}$satisfying$\lim_{n\to+\infty} \Vert v_{\gamma(n)}-u\Vert_0 = 0$. \end{itemize} \end{theorem} \begin{proof} Let us assume that$\varphi((x,y),s) = B|s|$, for$s<0$and$(x,y)\in\overline{\Omega}$. Let us now define the continuous function$\psi:(0,+\infty)\times\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$, by $$\psi(\lambda,(x,y),s) = \lambda \varphi((x,y),s).$$ We may now consider the nonlinear spectral problem $$\label{pr:exst} \begin{gathered} (Lu)(x,y) + \psi(\lambda,(x,y),u(x,y) ) = 0 \quad \text{for } (x,y)\in\Omega \\ u(x,y) = 0 \quad \text{for } (x,y)\in\partial\Omega. \end{gathered}$$ Function$\psi$satisfies \eqref{F1} and \eqref{F2}, so Theorem \ref{th1} may be applied to problem \eqref{pr:exst}. So, there exists the noncompact component$C$of${\mathcal R}$for the problem \eqref{pr:exst}, such that$(\frac{\mu_0}{B},0) \in C$and for$(\lambda,u)\in C$, the inequality holds$u\geq 0$. Let us now observe that, because$C$is not compact, one of the three situations takes place \begin{itemize} \item[(i)] there exists sequence$(\lambda_n,u_n)\in C$, such that$\lambda_n\to 0$; \item[(ii)] there exists sequence$(\lambda_n,u_n)\in C$, such that$\Vert u_n\Vert_0\to+\infty$. \item[(iii)] there exists sequence$(\lambda_n,u_n)\in C$, such that$\lambda_n\to +\infty$; \end{itemize} For all those situations, we may assume that$u_n\neq 0$. In the reasoning below let$\Phi:\czzo\to\czzo$denote the Niemytzki operator associated with the function$\varphi$. The first conclusion is that (i) implies (ii). Assume, contrary to the claim, that$\lambda_n\to 0$and$\Vert u_n\Vert_0\leq M$, for a constant$M>0$. So$u_n = \lambda_n T \Phi(u_n)$implies$u_n\to 0$. Then, $$v_n = \lambda_n B T v_n + \lambda_n T\frac{\Phi(u_n) - B u_n}{\Vert u_n\Vert_0},$$ where$v_n = \frac{u_n}{\Vert u_n\Vert_0}$. We can see that$\{ T v_n\}$contains convergent subsequence. Because of our assumption the sequence$\{ \frac{\Phi(u_n) - B u_n}{\Vert u_n\Vert_0}\}$converges to zero. Letting$n\to+\infty$we have$v_n\to 0$, so we came to the contradiction, with$\Vert v_n\Vert_0 = 1$. So at least one of (ii) and (iii) holds true. That is why in the component$C$, there exist the point$(\lambda,u)$, such that the sum$\Vert u\Vert_0+|\lambda|$is arbitrary large. We are now going to show that there exists$\lambda_1>1$and$R_1>0$, such that $$\forall_{(\lambda,u)\in C} \Vert u\Vert_0 \geq R_1 \Rightarrow \lambda\in(\lambda_1,+\infty).$$ Assume contrary to our claim that, there exists the sequence$(\lambda_n,u_n)\in C$, such that$\Vert u_n\Vert_0\to +\infty$and$\lambda_n\to\lambda_0 \in[0,1]$. Then, for$v_n = \frac{u_n}{\Vert u_n\Vert_0}$, the relation holds $$v_n = \lambda_n T \frac{\Phi(u_n) - Au_n}{\Vert u_n\Vert_0} + \lambda_n A T v_n.$$ Let us now observe that$\frac{\Phi(u_n) - Au_n}{\Vert u_n\Vert_0}\to 0$. We will show, that for any positive$\eta>0$, the inequality$|\frac{\Phi(u_n)(x,y) - Au_n(x,y)}{\Vert u_n\Vert_0}|<\eta$holds for almost all$n\in\nat$. First, let us choose$R>0$, such that for$s>R$the relation holds $$\Bigl| \frac{\varphi((x,y),s) - As}{s} \Bigr| < \eta.$$ Because the function$|\varphi((x,y),s) - As|$is bounded on$\overline{\Omega}\times[0,R]$we can see that for$n\in\nat$large enough the implication holds $$u_n(x,y) \in [0,R] \Rightarrow \Bigl|\frac{\Phi(u_n)(x,y) - Au_n(x,y)}{\Vert u_n\Vert_0}\Bigr|<\eta.$$ Moreover, for$\Vert u_n\Vert_0 > R$the implication holds $$u_n(x,y) > R \Rightarrow \frac{|\Phi(u_n)(x,y) - Au_n(x,y)|}{|u_n(x,y)|} \leq \Bigl|\frac{\Phi(u_n)(x,y) - Au_n(x,y)}{\Vert u_n\Vert_0}\Bigr|<\eta$$ what proves$|\frac{\Phi(u_n)(x,y) - Au_n(x,y)}{\Vert u_n\Vert_0}|<\eta$, for any$(x,y)\in\overline{\Omega}$. Hence$\frac{1}{\Vert u_n\Vert_0}T (\Phi(u_n) - Au_n)\to 0$. Moreover, there exists the convergent subsequence of$\{T v_n\}$. So we may assume that$v_n\to v_0$and then we have $$v_0 = \lambda_0 A T v_0$$ for$v_0\neq 0$and$v_0\geq 0$. This implies$\lambda_0 = \frac{\mu_0}{A}>1$, what contradicts our assumption. By Theorem \ref{th2} there exists the sequence of connected sets$C_n^i\subset(0,+\infty)\times\czzo$, such that for$n\in\nat$large enough$C_n^i \cap ((0,1)\times\{0\})\neq \emptyset$and for any$R>0$the relation holds$C_n^i \cap \partial( [\frac{\mu_0}{2B},2]\times \overline{B(0,R)} \neq \emptyset$. Let us observe that for$R>0$large enough, there exists$(\lambda_n,u_n)\in C_n^i$such that$\lambda_n\in(1,2]$. So, from the connectedness of the sets$C$and$C_n^i$, we may conclude that there exist pairs$(1,u)\in C$,$(1,u_n)\in C_n^1$and$(1,v_n)\in C_n^2$, and because of Theorem \ref{th3} the point$(1,u)$may be selected in such way that there exists infinitely many points$(1,u_n)\in C_n^1$or$(1,v_n)\in C_n^2$being arbitrarily close to$(1,u)$. \end{proof} \begin{thebibliography}{00} \bibitem{CD} G. Chang, P. J. Davis, {\it The Convexity of Bernstein Polynomials over Triangles}, Journal of Approximation Theory, 40 (1984), 11-28. \bibitem{CH} N.-S. Chow, J.K. Hale, {\it Methods of bifurcation theory}, Springer Verlag, 1982. \bibitem{D} P. J. Davis, {\it Interpolation and Approximation}, Blaisdell Publishing Company, 1963. \bibitem{DG} S. Domachowski, J. Gulgowski, {\it A Global Bifurcation Theorem for Convex-Valued Differential Inclusions}, Zeitschrift f\"ur Analysis und ihre Anwedungen, 23 (2004), No. 2, 275-292. \bibitem{F} G. Farin, {\it Triangular Bernstein-B\'ezier patches}, Computer Aided Geometric Design, vol 3, no. 2, August 1986, 83-127. \bibitem{GT} D. Gilbarg, N. S. Trudinger, {\it Elliptic partial differential equations of second order}, Springer-Verlag, New York / Berlin, 1983. \bibitem{Kie} H. Kielh\"ofer, {\it Smoothness and asymptotics of global positive branches of$\Delta u + \lambda f(u) = 0\$}, Z. Angew Math. Phys., 1992, {\bf 43}, 139-153. \bibitem{Ku} K. Kuratowski, {\it Topology}, PWN Warszawa, 1966. \bibitem{L} G. G. Lorentz, \textit{Bernstein Polynomials}, Springer-Verlag, Berlin-Heidelberg-New York, 1993. \bibitem{Lo} S. Lojasiewicz, {\it Wst\c{e}p do teorii funkcji rzeczywistych (Introduction to real functions theory)} (in Polish), PWN Warszawa, 1976. \bibitem{N} L. Nirenberg, \textit{Topics in nonlinear functional analysis}, Courant Institute of Mathematical Sciences New York University, New York, 1974. \bibitem{R} P. Rabinowitz, \textit{Some global results for nonlinear eigenvalue problems}, Journal of Functional Analysis, 7, 487-513, (1971). \bibitem{R2} P. Rabinowitz, {\it On bifurcation from infinity}, Journal of Differential Equations, 14, 462-475, (1973). \bibitem{Ry} B. P. Rynne, {\it The structure of Rabinowitz' Global Bifurcating Continua for Generic Quasilinear Elliptic Equations}, Nonlinear Analysis, Theory, Methods \& Applications, 1998, Vol. 32, No. 2, 167-181 \bibitem{S} D. Stancu, {\it Some Bernstein Polynomials in Two Variables and their Applications}, Soviet Mathematics 1, pp. 1025-1028 (1960). \bibitem{WD} G. Whyburn, E. Duda, \textit{Dynamic topology}, Springer-Verlag, New York (1979). \end{thebibliography} \end{document}