Dynamic boundary value problems of the second-order: Bernstein-Nagumo conditions and solvability (Q885309)

The existence of solutions to the dynamic boundary value problem \[ y^{\triangle\triangle}=f(t,y^\sigma,y^\triangle),\quad t\in[a,b]_T,\quad y(a)=A,\quad y(\sigma^2(b))=B, \] is studied. Here, \(T\) is the so-called ``time scale'' (in this paper \(T\equiv \mathbb{R}\) or all points in \(T\) are isolated), \([a,b]_T=\{t\in T:\;a\leq t\leq b\},\) \(f:[a,b]_T\times \mathbb{R}^d\times \mathbb{R}^d\to \mathbb{R}^d,\;A,B\in \mathbb{R}^d,\;d\geq1,\) the operator \(\sigma(t)\) is defined for \(t<\sup T\) by \[ \sigma(t):=\inf\{\tau>t:\;\tau\in T\},\quad \sigma^2(t)=\sigma(\sigma(t))\text{ and }y^\sigma(t)=y(\sigma(t)). \] Besides, the (delta) derivative \(y^\triangle(t)\) of \(y:T\to \mathbb{R}\) is the vector with the property that given \(\varepsilon>0\) there is a neighbourhood \(U\) of \(t\) such that, for all \(s\in U\) and each \(i=1,\dots,d\), \[ | [y_i(\sigma(t))-y_(s)]-{y_i^{\triangle}}(t)[\sigma(t)-s]| \leq\varepsilon| \sigma(t)-s| . \] First, the authors prove the following general existence result: If all potential solutions to the family \[ y^{\triangle\triangle}=\lambda f(t,y^\sigma,y^\triangle),\quad t\in[a,b]_T,\quad y(a)=\lambda A,\quad y(\sigma^2(b))=\lambda B,\quad \lambda\in[0,1], \] that satisfy \(| | y(t)| | \leq M\) for \(t\in[a,\sigma^2(b)]_T\) and \(| | y^\triangle(t)| | \leq N\) for \(t\in[a,\sigma(b)]_T,\) where the constants \(M\) and \(N\) are independent of \(\lambda\), also satisfy \(| | y(t)| | < M\) for \(t\in[a,\sigma^2(b)]_T\) and \(| | y^\triangle(t)| | < N\) for \(t\in[a,\sigma(b)]_T,\) then the original problem has at least one solution. Next, they apply it to prove existence results for the considered problem, assuming that \(f\) satisfies one of the following Bernstein-Nagumo type conditions: \[ | | f(t,p,q)| | \leq\alpha| | q| | +K\text{ for }(t,p,q)\in C_L=\{(t,p,q)\in[a,b]_T\times \mathbb{R}^d\times \mathbb{R}^d:\;| | p| | \leq L\} \] with \(\alpha(\sigma(b)-a)<1\) and \[ | | f(t,p,q)| | \leq\alpha[2<p,f(t,p,q)>+| | q| | ^2]+K \text{ for }(t,p,q)\in C_L \text{ with } 2\alpha L<\sigma^2(b)-a, \] where \(\alpha,\;\;K\) and \(L\) are nonnegative constants. These growth conditions guarantee the a priori bounds which are required for the general existence result.