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A fundamental question in Celestial Mechanics is to analyze the possiblefinal motions of the Restricted $3$-body Problem, that is, to provide thequalitative description of its complete (i.e. defined for all time) orbits astime goes to infinity. According to the classification given by Chazy back in1922, a remarkable possible behaviour is that of oscillatory motions, where themotion $q$ of the massless body is unbounded but returns infinitely ofteninside some bounded region: \[ \limsup_{t\to\pm\infty}|q(t)|=\infty\qquad\qquad\text{and}\qquad\qquad \liminf_{t\to\pm\infty}|q(t)|<\infty. \] In contrast with the other possible final motions in Chazy'sclassification, oscillatory motions do not occur in the $2$-body Problem, whilethey do for larger numbers of bodies. A further point of interest is theirappearance in connection with the existence of chaotic dynamics.  In this paper we introduce new tools to study the existence of oscillatorymotions and prove that oscillatory motions exist in a particular configurationknown as the Restricted Isosceles $3$-body Problem (RI3BP) for almost allvalues of the angular momentum. Our method, which is global and not limited tonearly integrable settings, extends the previous results\cite{guardia2021symbolic} by blending variational and geometric techniqueswith tools from nonlinear analysis such as topological degree theory. To thebest of our knowledge, the present work constitutes the first complete analyticproof of existence of oscillatory motions in a non perturbative regime.

Oscillatory Motions in the Restricted 3-body Problem: A functional analytic approach