Hermite continuous and discrete boundary value problems (Q2763891)

The authors study the existence of one or more solutions to Hermite continuous and discrete boundary value problems. In the first part of the paper, the authors discuss the Hermite continuous problem NEWLINE\[NEWLINE y^{(n)}(t) = \varphi (t)f(t, y(t)), \;0<t<1, \quad y^{(j)}(a_{i}) =0, \;j = 0, \ldots , n_{i} -1, \;i = 1, \ldots , k. \tag{1}NEWLINE\]NEWLINE Here, \(n\geq 2\), \(2\leq k \leq n\) are integers, \(0=a_{1}<a_{2}< \cdots < a_{k} =1\), and \(n_{1}, \ldots ,n_{k}\) are positive integers with \(\sum ^{k}_{i=1} n_{i}=n\). In the second part, the authors discuss the Hermite discrete problem NEWLINE\[NEWLINE \Delta^{n}y(m) = f(m, y(m)),\;m\in I_{N}=Z[0, N],\quad \Delta^{j}y(a_{i}) = 0, \;j = 0, \ldots , n_{i}-1,\;i = 1, \ldots,k. \tag{2}NEWLINE\]NEWLINE Here, \(N\geq 2\), \(n\geq 2\), \(2\leq k \leq n\) are integers, \(n_{i}\geq 1\) for \(i=1, \ldots, k\) with \( \sum ^{k}_{i=1} n_{i}=n\), and \(0=a_{i} < a_{1}+n_{1} < a_{2}+n_{2} < \cdots < a_{k}\leq a_{k}+n_{k}-1 = N+n\). NEWLINENEWLINENEWLINEUsing either Krasnoselskij's fixed-point theorem in a cone or a nonlinear alternative of Leray-Schauder type together with properties of Green functions, existence criteria (single and multiple) are presented for nonsingular continuous and discrete Hermite boundary value problems.