On some applications of theorems on the spectral radius to differential equations (Q1363056)

The paper deals with the Darboux problem of neutral type in an implicit form \[ z_{xy}(x,y)= f(x,y,z(h(x,y)),z_{xy}(H(x,y))),\quad\text{for }(x,y)\in I^2,\tag{1} \] \[ z(x,0)= 0,\quad z(0,y)= 0,\tag{2} \] where \(I= [0,a]\), \(a>0\). The author proves uniqueness results for the problem (1)--(2) using fixed point theorems and theorems for the spectral radius of the sum of operators.