Two point boundary value problem for the linear system of ordinary differential equations with deviating arguments $$x^{\text{'}}\left(t\right)=A\left(t\right)x\left({\tau }_{11}\left(t\right)\right)+B\left(t\right)u\left({\tau }_{12}\left(t\right)\right)+q_1\left(t\right),u^{\text{'}}\left(t\right)=C\left(t\right)x\left({\tau }_{21}\left(t\right)\right)+D\left(t\right)u\left({\tau }_{22}\left(t\right)\right)+q_2\left(t\right),{\alpha }_{11}x\left(0\right)+{\alpha }_{12}u\left(0\right)=c_0,{\alpha }_{21}x\left(T\right)+{\alpha }_{22}u\left(T\right)=c_T$$ is considered. For this problem the sufficient condition for existence and uniqueness of solution is obtained. The same approach as in [2], [3] is applied.

The problem on the existence of a positive in the interval $]a,b[$ solution of the boundary value problem $$u^{\text{'}\text{'}}=f(t,u)+g(t,u)u^{\text{'}};u(a+)=0,u(b-)=0$$ is considered, where the functions $f$ and $g]a,b[\times ]0,+\infty [\to ℝ$ satisfy the local Carathéodory conditions. The possibility for the functions $f$ and $g$ to have singularities in the first argument (for $t=a$ and $t=b$) and in the phase variable (for $u=0$) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.

We prove the existence of solutions of four-point boundary value problems under the assumption that $f$ fulfils various combinations of sign conditions and no growth restrictions are imposed on $f$. In contrast to earlier works all our results are proved for the Carathéodory case.

We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri...

New sufficient conditions of the existence and uniqueness of the solution of a boundary problem for an ordinary differential equation of $n$-th order with certain functional boundary conditions are constructed by the method of a priori estimates.