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Using a suitable version of Mawhin’s continuation principle, we obtain an existence result for the Floquet boundary value problem for second order Carathéodory differential equations by means of strictly localized $C^2$ bounding functions.

The paper deals with the quasi-linear ordinary differential equation ${\left(r\left(t\right)\varphi \left(u^{\text{'}}\right)\right)}^{\text{'}}+g(t,u)=0$ with $t\in [0,\infty )$. We treat the case when $g$ is not necessarily monotone in its second argument and assume usual conditions on $r\left(t\right)$ and $\varphi \left(u\right)$. We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano, [Acta...

We study the existence of a mild solution to the nonlocal initial value problem for semilinear second-order differential inclusions in abstract spaces. The result is obtained by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables getting the result without any requirements for compactness of the right-hand side or of the cosine family generated by the linear operator.

The paper deals with the multivalued boundary value problem $x^{\text{'}}\in A(t,x)x+F(t,x)$ for a.a. $t\in [a,b]$, $Mx\left(a\right)+Nx\left(b\right)=0$, in a separable, reflexive Banach space $E$. The nonlinearity $F$ is weakly upper semicontinuous in $x$. We prove the existence of global solutions in the Sobolev space $W^{1,p}([a,b],E)$ with $1<p<\infty$ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.