Existence and multiplicity for fractional differential equations with \(m(\xi )\)-Kirchhoff type-equation

In this paper, we first investigate the Palais-Smale compactness condition of the energy functional associated to a \(m(\xi )\) $m(\xi )$ -Kirchhoff-type operator in the appropriate fractional space setting. In this sense, using the Mountain Pass Theorem and the Fountain Theorem, we investigate the existence and multiplicity of weak solutions for a new class of fractional differential equations with \(m(\xi )\) $m(\xi )$ -Kirchhoff-type equation.