Nonimprovable, in a certain sense, sufficient conditions for the unique solvability of the boundary value problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right),u\left(a\right)+\lambda u\left(b\right)=c$$ are established, where $\ell :C\left(\right[a,b];R)\to L\left(\right[a,b];R)$ is a linear bounded operator, $q\in L\left(\right[a,b];R)$, $\lambda \in R_{+}$, and $c\in R$. The question on the dimension of the solution space of the homogeneous problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right),u\left(a\right)+\lambda u\left(b\right)=0$$ is discussed as well.

In this paper, we are interested in the dynamic evolution of an elastic body, acted by resistance forces depending also on the displacements. We put the mechanical problem into an abstract functional framework, involving a second order nonlinear evolution equation with initial conditions. After specifying convenient hypotheses on the data, we prove an existence and uniqueness result. The proof is based on Faedo-Galerkin method.

We study oscillatory properties of solutions of the Emden-Fowler type differential equation $$u^{\left(n\right)}\left(t\right)+p\left(t\right){\left|u\left(\sigma \left(t\right)\right)\right|}^{\lambda }signu\left(\sigma \left(t\right)\right)=0,$$ where $0<\lambda <1$, $p\in L_{\mathrm{loc}}({ℝ}_{+};ℝ)$, $\sigma \in C({ℝ}_{+};{ℝ}_{+})$ and $\sigma \left(t\right)\ge t$ for $t\in {ℝ}_{+}$. Sufficient (necessary and sufficient) conditions of new type for oscillation of solutions of the above equation are established. Some results given in this paper generalize the results obtained in the paper by Kiguradze and Stavroulakis (1998).

The asymptotic properties of solutions of the equation $u^{\text{'}\text{'}\text{'}}\left(t\right)=p_1\left(t\right)u\left({\tau }_1\left(t\right)\right)+p_2\left(t\right)u^{\text{'}}\left({\tau }_2\left(t\right)\right)$, are investigated where $p_i:[a,+\infty [\to R(i=1,2)$ are locally summable functions, ${\tau }_i:[a,+\infty [\to R(i=1,2)$ measurable ones and ${\tau }_i\left(t\right)\ge t(i=1,2)$. In particular, it is proved that if $p_1\left(t\right)\le 0$, $p_2^2\left(t\right)\le \alpha \left(t\right)\left|p_1\left(t\right)\right|$, $${\int }_a^{+\infty }{[{\tau }_1\left(t\right)-t]}^2{p}_1\left(t\right)dt<+\infty \text{and}{\int }_a^{+\infty }\alpha \left(t\right)dt<+\infty ,$$ then each solution with the first derivative vanishing at infinity is of the Kneser type and a set of all such solutions forms a one-dimensional linear space.

We define a non-smooth guiding function for a functional differential inclusion and apply it to the study the asymptotic behavior of its solutions.

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $$\mathrm{d}x=\mathrm{d}A\left(t\right)·f(t,x),h\left(x\right)=0$$ is established, where $f:[a,b]\times {ℝ}^n\to {ℝ}^n$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A:[a,b]\to {ℝ}^{n\times n}$ with bounded total variation components, and $h:{BV}_s([a,b],{ℝ}^n)\to {ℝ}^n$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x\left(t_1\left(x\right)\right)=ℬ\left(x\right)·x\left(t_2\left(x\right)\right)+c_0,$ where $t_i:{BV}_s([a,b],{ℝ}^n)\to [a,b]$$...$

On the segment $I=[a,b]$ consider the problem $$u^{\text{'}}\left(t\right)=f\left(u\right)\left(t\right),u\left(a\right)=c,$$ where $fC(I,ℝ)\to L(I,ℝ)$ is a continuous, in general nonlinear operator satisfying Carathéodory condition, and $c\in ℝ$. The effective sufficient conditions guaranteeing the solvability and unique solvability of the considered problem are established. Examples verifying the optimality of obtained results are given, as well.