Application of a noncompactness technique to an investigation of the existence of solutions to a nonlocal multivalued Darboux problem
For a given multifunction \(F:\Delta\times E\to P_{fc}(E)\) \((F:\Delta\times E\to P_f(E)\) respectively) and, given a \((\phi,\psi) \in C\), satisfying some assumptions, the authors study the existence of a solution to the following Darboux problem \[ \frac{\partial^2u(x,y)}{\partial x\partial y} \in F(x,y,u(x,y))\quad\text{a.e. in \(\Delta\)}, \] \[ u(x,0)+\sum_{j=1}^r h_j(x)u(x,b_j) = \phi(x),\quad x\in I, \] \[ u(0,y)+\sum_{i=1}^p k_i(y) u(a_i,y) = \psi(y),\quad y\in I, \] where \(a_i\) \((i=1,\dots,p)\) and \(b_j\) \((j=1,\dots,r)\) are given numbers such that \[ 0<a_1<\dots<a_p\leq c,\qquad 0<b_1<\dots<b_r\leq c. \] A compactness type condition involving the ball measures of noncompactness is applied.
scientific article; zbMATH DE number 1359783