Search results for " Fractional differential equations"
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Self-similarity and response of fractional differential equations under white noise input
2022
Self-similarity, fractal behaviour and long-range dependence are observed in various branches of physical, biological, geological, socioeconomics and mechanical systems. Self-similarity, also termed self-affinity, is a concept that links the properties of a phenomenon at a certain scale with the same properties at different time scales as it happens in fractal geometry. The fractional Brownian motion (fBm), i.e. the Riemann-Liouville fractional integral of the Gaussian white noise, is self-similar; in fact by changing the temporal scale t -> at (a > 0), the statistics in the new time axis (at) remain proportional to those calculated in the previous axis (t). The proportionality coeffi…
Fixed points of α-type F-contractive mappings with an application to nonlinear fractional differential equation
2016
Abstract In this paper, we introduce new concepts of α-type F-contractive mappings which are essentially weaker than the class of F-contractive mappings given in [21, 22] and different from α-GF-contractions given in [8]. Then, sufficient conditions for the existence and uniqueness of fixed point are established for these new types of contractive mappings, in the setting of complete metric space. Consequently, the obtained results encompass various generalizations of the Banach contraction principle. Moreover, some examples and an application to nonlinear fractional differential equation are given to illustrate the usability of the new theory.