It is shown that for a given system of linearly independent linear continuous functionals $l_i{C}^{n-1}\to ℝ$, $i=1,\cdots ,n$, the set of all $n$-th order linear differential equations such that the Green function for the corresponding generalized boundary value problem (BVP for short) exists is open and dense in the space of all $n$-th order linear differential equations. Then the generic properties of the set of all solutions to nonlinear BVP-s are investigated in the case when the nonlinearity in the differential equation has...

We consider the equation $$-{\left(r\left(x\right)y^{\text{'}}\left(x\right)\right)}^{\text{'}}+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ,$$ where $f\in L_p\left(ℝ\right)$, $p\in (1,\infty )$ and $$r>0,\frac{1}{r}\in L_1^{\mathrm{loc}}\left(ℝ\right),q\in L_1^{\mathrm{loc}}\left(ℝ\right).$$ For particular equations of this form, we suggest some methods for the study of the question on requirements to the functions $r$ and $q$ under which the above equation is correctly solvable in the space $L_p\left(ℝ\right),$$p\in (1...$