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The paper deals with the following second order Dirichlet boundary value problem with p ∈ ℕ state-dependent impulses: z″(t) = f (t,z(t)) for a.e. t ∈ [0, T], z(0) = z(T) = 0, z′(τ i+) − z′(τ i−) = I i(τ i, z(τ i)), τ i = γ i(z(τ i)), i = 1,..., p. Solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses.

The paper deals with the singular nonlinear problem $$u^{\text{'}\text{'}}\left(t\right)+f(t,u\left(t\right),u^{\text{'}}\left(t\right))=0,u\left(0\right)=0,u^{\text{'}}\left(T\right)=\psi \left(u\left(T\right)\right),$$ where $f\in \mathrm{𝐶𝑎𝑟}\left(\right(0,T)\times D)$, $D=(0,\infty )\times$. We prove the existence of a solution to this problem which is positive on $(0,T]$ under the assumption that the function $f(t,x,y)$ is nonnegative and can have time singularities at $t=0$, $t=T$ and space singularity at $x=0$. The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.

The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half-line of the form $${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=p\left(t\right)f\left(u\left(t\right)\right),$$ $$u^{\text{'}}\left(0\right)=0,u\left(\infty \right)=L.$$ The existence of a strictly increasing solution (a homoclinic solution) of this problem is proved by the dynamical systems approach and the lower and upper functions method.