Asymptotic regimes of an integro-difference equation with discontinuous kernel (Q6567202)

The authors study the integro-difference equation\N\[\Nu_{n+1}(x)=\int_\Omega k(x,y)F(u_n(y)) \ dy,\N\]\Nwhere \(\delta<k=k(x,y)\leq \Lambda\), with \(\delta>0\) for \(x,y\in \Omega=(-a,a)\), is continuous in \(\bigcup_{i,j}\Omega_i\times \Omega_j\) for\N\[\N\Omega_i=(a_i, a_{i+1}), \ 0\leq i\leq m, \quad a_0=-a, \quad a_{m+1}=a\N\]\NThe nonlinearity \(0\leq F=F(u)\leq M\), \(u\in {\mathbf R}\) is continuous, strictly increasing on \([0,\infty)\) and vanishes elsewhere. It is assumed that \(F\) is differentiable at \(0\), \(r_0=F'(0)>1\), and\N\[\Nu\in (0,\infty) \ \mapsto \ F(u)/u\N\]\Nis strictly decreasing. The linearized operator\N\[\N(T_0)u(x)=r_0\int_\Omega k(x,y)u(y) \ dy\N\]\Nis positive in \(L^2(\Omega)\), and its principal eigenvalue is denoted by \(\lambda_0\). Then the sequence \(u_n\) is constructed in \(X\), which denotes the set of \(L^2(\Omega\) functions continuous except finite many points. First, if \(F(0)=0\) and \(\lambda_0\leq 1\), then \(0\) is the only stationary solution and \((u_n)\) converges \(0\) in \(L^2(\Omega)\). Second, if either \(F(0)=0\) and \(\lambda_0>1\) or \(F(0)>0\), there is a unique positive stationary solution \(w\), and if \(u_0\geq 0\) is not identical to \(0\), \(u_n\) converges to this \(w\) in \(L^2(\Omega)\).