Positive solutions of difference equations with two-point right focal boundary conditions
The author considers the \(n\)th-order difference equation \[ (-1)^{n-p} \Delta^ny= F(k,y, \Delta y,\dots, \Delta^{n-1}y), \quad k\in[0,N], \tag{1} \] together with the two-point right focal boundary conditions \[ \Delta^i y(0)=0, \quad 0\leq i\leq p-1, \qquad \Delta^i y(N+1)=0, \quad p\leq i\leq n-1, \tag{2} \] where \(n,p,N\) are fixed integers with \(n\geq 2\), \(0\leq p\leq n-1\) and \(N\geq p\), and \(\Delta y(k)= y(k+1)- y(k)\) and \(\Delta^ny(k)= \Delta(\Delta^{n-1}y(k))\) for \(n\geq 2\). It is assumed that there exist continuous functions \(f,F\) and \(a,b: [0,N]\to \mathbb{R}\) such that \[ a(k)\leq \frac{F(k,u,u_1,\dots, u_{n-1})} {f(u)}\leq b(k), \quad u\in (0,+\infty), \] \(a(k)\geq 0\), \(k\in [0,N]\), \(a(k)\not\equiv 0\), \(k\in [p,N]\). By a positive solution of (1), (2) we mean a nontrivial \(y: [0,N+n]\to [0,+\infty)\) satisfying (1), (2). By defining an appropriate Banach space and cone, the author uses a fixed point theorem due to Krasnosel'skij, to give existence results for two positive solutions of (1), (2). Further, the author considers a special case of (1) for \(n=2\), \(p=1\), subject to some boundary conditions, and establishes upper and lower bounds for the positive solutions.
two-point right focal boundary conditions
scientific article; zbMATH DE number 1215753
scientific article