This work concerns with the solvability of third-order periodic fully problems with a weighted parameter, where the nonlinearity must verify only a local monotone condition and no periodic, coercivity or super or sublinearity restrictions are assumed, as usual in the literature. The arguments are based on a new type of lower and upper solutions method, not necessarily well ordered. A Nagumo growth condition and Leray–Schauder’s topological degree theory are the existence tools. Only the existence of solution is studied here and it will remain open the discussion on the non-existence and the multiplicity of solutions. Last section contains a nonlinear third-order differential model for periodic catatonic phenomena, depending on biological and/or chemical parameters.

<abstract><p>This work presents a discussion of Ambrosetti-Prodi-type second-order periodic problems. In short, the existence, non-existence, and multiplicity of solutions will be discussed on the parameter $ \lambda $. The arguments rely on a Nagumo condition, to guarantee an apriori bound on the first derivative, lower and upper-solutions method, and the Leray-Schauder's topological degree theory. There are two types of new results based on the parameter's variation: An existence and non-existence theorem and a multiplicity theorem, proving the existence of a bifurcation point. An application to a damped and forced pendulum is studied, suggesting a method to estimate the critical values of the parameter.</p></abstract>

This paper presents sufficient conditions for the existence of a bifurcation point for nonlinear periodic third-order fully differential equations. In short, the main discussion on the parameter s about the existence, non-existence, or the multiplicity of solutions, states that there are some critical numbers σ0 and σ1 such that the problem has no solution, at least one or at least two solutions if s<σ0, s=σ0 or σ0>s>σ1, respectively, or with reversed inequalities. The main tool is the different speed of variation between the variables, together with a new type of (strict) lower and upper solutions, not necessarily ordered. The arguments are based in the Leray–Schauder’s topological degree theory. An example suggests a technique to estimate for the critical values σ0 and σ1 of the parameter.

Este artigo aborda a influência da figura de São Bernardo, fervoroso apologista da Ordem do Templo, no contexto da arquitectura militar erguida por estes cavaleiros nos séculos XII, XIII e princípios de XIV. Insere-se na investigação da tese de mestrado intitulada "Castelos da Ordem do Templo em Portugal, 1120-1314", sendo dela parte integrante. Os escritos do doutor cisterciense permitem colocar novas e pertinentes questões, sendo prestada atenção à analogia entre as realidades do mosteiro e do castelo e ao simbolismo inerente à necessidade de defesa de Jerusalém e, em particular, do Monte do Templo. A análise da obra de São Bernardo permite abrir horizontes que em muito ultrapassam a esfera monástica, projectando outros entendimentos na compreensão da arte militar e religiosa erguida pelos Templários no Oriente latino e em Portugal. Synopsis:This article makes an approach to the influence of Saint Bernard, a fervourous defender of the Templar Order, within the context of the military architecture used by these knights in the twelfth, thirteenth and beginning of fourteenth centuries. It was written within the scope of the research for a master's degree dissertation entitled "Castelos da Ordem do Templo em Portugal, 1120-1314" of which it is part. The writings of the Cistercian scholar enable us to raise new, pertinent questions, paying special attention to the analogy between the realities of the monastery and the castle, and to the symbolism inherent to the necessity of defending Jerusalem and particularly the Temple Mount. The analysis of Saint Bernard's work opens new horizons that largely surpass the monastic sphere of action, casting new light on the understanding of military and religious art erected by the Templars in the Latin Orient and in Portugal."Sed et militamus in eis, tanquam in tabernaculis, prorsus violenti ad regnum. Denique militia est vita hominis super terram, et quamdiu militamus in hoc corpore, peregrinamur a Domino, id est a luce." "Aqui estamos, tal como os guerreiros sob a tenda de campanha, tratando de conquistar o céu por meio da violência. A existência humana sobre a terra é idêntica à do soldado e embora prossigamos este combate afundados em nossos corpos actuais, estaremos porém peregrinando ao Senhor, isto é, à Luz". São Bernardo -Sermão XXVI sobre o "Cântico dos Cânticos" Medievalista on line ano 2 • número 2 • 2006 © IEM -Instituto de Estudos Medievais 1 www.fcsh.unl.pt/iem/medievalista