Systems of coupled clamped beams equations with full nonlinear terms: Existence and location results

This work gives sufficient conditions for the solvability of the fourth order coupled system┊ u⁽⁴⁾(t)=f(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) v⁽⁴⁾(t)=h(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) with f,h: [0,1]×ℝ⁸→ℝ some L¹- Carathéodory functions, and the boundary conditions {┊ u(0)=u′(0)=u′′(0)=u′′(1)=0 v(0)=v′(0)=v′′(0)=v′′(1)=0. To the best of our knowledge, it is the first time in the literature where two beam equations are considered with full nonlinearities, that is, with dependence on all derivatives of u and v. An application to the study of the bending of two elastic coupled campled beams is considered.

Preschoolers’ free play - connections with emotional and social functioning

Play has an important role in various aspects of children’s development. However, time for free play has declined substantially over the last decades. To date, few studies have focused on the relationship between opportunities for free play and children’s social functioning. The aims of this study are to examine whether children ́s free play is related to their social functioning and whether this relationship is mediated by children ́s emotional functioning. Seventy-eight children (age, 55- 77 months) were tested on their theory of mind and emotion understanding. Parents reported on their children’s time for free play, empathic abilities, social competence and externalizing behaviors. The main findings showed that free play and children’s theory of mind are negatively related to externalizing behaviors. Empathy was strongly related to children’s social competence, but free play and social competence were not associated. Less time for free play is related to more disruptive behaviors in preschool children, however certain emotional functioning skills influence these behaviors independently of the time children have for free play. These outcomes suggest that free play might help to prevent the development of disruptive behaviors, but future studies should further examine the causality of this relationship.

Unbounded solutions for functional problems on the half-line

This paper presents an existence and localization result of unbounded solutions for a second-order differential equation on the half-line with functional boundary conditions. By applying unbounded upper and lower solutions, Green's functions and Schauder fixed point theorem, the existence of at least one solution is shown for the above problem. One example and one application to an Emden-Fowler equation are shown to illustrate our results.