### Bounded solutions of Carathéodory differential inclusions: a bound sets approach.

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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...

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.

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