The periodic boundary value problem u''(t) = f(t,u(t),u'(t)) with u(0) = u(2π) and u'(0) = u'(2π) is studied using the generalized method of upper and lower solutions, where f is a Carathéodory function satisfying a Nagumo condition. The existence of solutions is obtained under suitable conditions on f. The results improve and generalize the work of M.-X. Wang et al. [5].

This paper investigates a class of fractional functional integrodifferential inclusions with nonlocal conditions in Banach spaces. The existence of mild solutions of these inclusions is determined under mixed continuity and Carathéodory conditions by using strongly continuous operator semigroups and Bohnenblust-Karlin's fixed point theorem.

We establish not only sufficient but also necessary conditions for existence of solutions to a singular multi-point third-order boundary value problem posed on the half-line. Our existence results are based on the Krasnosel’skii fixed point theorem on cone compression and expansion. Nonexistence results are proved under suitable a priori estimates. The nonlinearity $f=f(t,x,y)$ which satisfies upper and lower-homogeneity conditions in the space variables $x,y$ may be also singular at time $t=0$. Two examples of applications...

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.

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

An inverse problem for a nonlocal problem describing the temperature of a conducting device is studied.