Let X be a real Banach space and G ⊂ X open and bounded. Assume that one of the following conditions is satisfied: (i) X* is uniformly convex and T:Ḡ→ X is demicontinuous and accretive; (ii) T:Ḡ→ X is continuous and accretive; (iii) T:X ⊃ D(T)→ X is m-accretive and Ḡ ⊂ D(T). Assume, further, that M ⊂ X is pathwise connected and such that M ∩ TG ≠ ∅ and $M\cap \overline{T\left(\partial G\right)}=\varnothing$. Then $M\subset \overline{TG}$. If, moreover, Case (i) or (ii) holds and T is of type $\left(S_1\right)$, or Case (iii) holds and T is of type $\left(S_2\right)$, then M ⊂ TG. Various results of Morales,...

Les solutions d’équations d’évolution $\frac{du}{dt}+Au\ni f$ où $A$ est un opérateur maximal monotone d’un espace de Hilbert $H$, et $f\in L^1(0,T,H)$ sont étudiées dans le cas général en introduisant une notion de solution faible. Des résultats particuliers sont donnés lorsque $H$ est de dimension finie ou plus généralement lorsque l’intérieur de $D\left(A\right)$ est non vide.

Let $g$: $𝐑\to 𝐑$ be a continuous function, $e$: $[0,1]\to 𝐑$ a function in $L^2[0,1]$ and let $c\in 𝐑$, $c\ne 0$ be given. It is proved that Duffing’s equation $u^{\text{'}\text{'}}+c{u}^{\text{'}}+g\left(u\right)=e\left(x\right)$, $0<x<1$, $u\left(0\right)=u\left(1\right)$, $u^{\text{'}}\left(0\right)=u^{\text{'}}\left(1\right)$ in the presence of the damping term has at least one solution provided there exists an $𝐑>0$ such that $g\left(u\right)u\ge 0$ for $\left|u\right|\ge 𝐑$ and ${\int }_0^{1}e\left(x\right)dx=0$. It is further proved that if $g$ is strictly increasing on $𝐑$ with ${lim}_{u\to -\infty }g\left(u\right)=-\infty$, ${lim}_{u\to \infty }g\left(u\right)=\infty$ and it Lipschitz continuous with Lipschitz constant $\alpha <4{\pi }^2+c^2$, then Duffing’s equation given above has exactly one solution for every $e\in L^2[0,1]$.

This paper deals with the generalized nonlinear third-order left focal problem at resonance $$\left\{\begin{array}{c}{\left(p\left(t\right)u^{\text{'}\text{'}}\left(t\right)\right)}^{\text{'}}-q\left(t\right)u\left(t\right)=f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right)),t\in ]t_0,T[,m(u\left(t_0\right),u^{\text{'}\text{'}}\left(t_0\right))=0,n(u\left(T\right),u^{\text{'}}\left(T\right))=0,l(u\left(\xi \right),u^{\text{'}}\left(\xi \right),u^{\text{'}\text{'}}\left(\xi \right))=0,\hfill \end{array}\right.$$ where the nonlinear term is a Carathéodory function and contains explicitly the first and second-order derivatives of the unknown function. The boundary conditions that we study are quite general, involve a linearity and include, as particular cases, Sturm-Liouville boundary conditions. Under certain growth conditions on the nonlinearity, we establish the existence of the nontrivial solutions by using the...