Existence and comparison results for first order periodic implicit difference equations with maxima. (English) Zbl 1005.39012

Let \(\varphi: \mathbb{R}^{T+1} \to\mathbb{R}^{T+1}\) denote the operator defined by \[ (\varphi u)_k= \begin{cases} \max_{i\in \{0,\dots,k\}} u_i, & 0\leq k\leq h, \\ \max_{i\in\{k-h, \dots,k\}}u_i, & h<k\leq T. \end{cases} \] Consider the so-called quasi-linear difference equation with maxima \[ \Delta u_k+M u_{k+ 1}+ N(\varphi u)_{k+1}= \sigma_k,\;k\in\{0,1, \dots,T-1\}. \tag{*} \] The authors obtain uniqueness and comparison results for solution of (*) satisfying the boundary condition \(u_\delta =u_T\). Next the similar problem is studied for nonlinear version of (*) including the proof of existence of solution on the basis of concept of lower and upper solutions.
A lot of examples illustrates the obtained results.


39A11 Stability of difference equations (MSC2000)