Conditions for the well-posedness of nonlocal problems for second-order linear differential equations
The authors investigate the differential equation \[ (r(t) u')' = p(t) u + q(t),\quad t\in [a,b], \] with nonlocal conditions of one of the following three types \[ \begin{aligned} u(a) = l_{1}(u) + c_{1}, & \qquad u(b) = l_{2}(u) + c_{2},\\ u(a) = l_{1}(u) + c_{1}, & \qquad r(b)u'(b) = l_{2}(r u') + c_{2},\\ r(a)u'(a) = l_{1}(r u') + c_{1}, & \qquad r(b)u'(b) = l_{2}(r u') + c_{2},\\ \end{aligned} \] where \(-\infty < a < t_{0} < b < +\infty\), \(r:[a,b]\to(0,+\infty)\) is a continuous function, \(p,q:[a,b]\to\mathbb{R}\) are Lebesgue integrable functions, \(l_{1}: C([a,t_{0}])\to\mathbb{R}\) and \(l_{2}: C([t_{0},b])\to\mathbb{R}\) are linear bounded functionals and \(c_{i}\in\mathbb{R}\) (\(i=1,2\)). The authors provide sharp conditions guaranteeing the well-posedness of the above problems. Let us note that the problem is said to be well-posed if it is uniquely solvable for arbitrarily fixed \(c_{i}\in\mathbb{R}\) (\(i=1,2\)) and Lebesgue integrable function \(q:[a,b]\to\mathbb{R}\) and if there exists a positive constant \(\varrho\) independent of the \(c_{i}\) (\(i=1,2\)) and \(q\) such that the solution admits the estimate \[ \|u\|_{C^{1}} \leq \varrho(|c_{1}| + |c_{2}| + \|\tilde{q}\|_{C}), \] where \(\tilde{q}(t) = \int_{a}^{t} q(s)\, ds\). Finally, let us point out that the above general nonlocal conditions are reduced to multipoint boundary conditions if we take \[ l_{1}(u) = \sum\limits_{k=1}^{m} l_{1 k} u(a_{k}) \text{ and } l_{2}(u) = \sum\limits_{k=1}^{m} l_{2 k} u(b_{k}), \] where \(m\geq 1\), \(l_{ik}\in\mathbb{R}\), and \[ a < a_{m} < \dots < a_{1} \leq b_{1} < \dots < b_{m} < b. \]
scientific article; zbMATH DE number 6008895
scientific article