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The aim of this paper is to study the existence of Q , T -affine-periodic solutions for affine-periodic systems on time scales of the type x Δ t = A t x t + f t and x Δ t = A t x t + g t , x t , t ∈ T , assuming that corresponding homogeneous equation of this system admits exponential dichotomy. The result is also extended to the case of pseudo Q , T -affine-periodic solutions. The main approaches are based on the Banach contraction mapping principle, but certain technical aspects on time scales are more complicated.

With the emergence of light field imaging in recent years, the compression of its elementary image array (EIA) has become a significant problem. Our coding framework includes modeling and reconstruction. For the modeling, the covariancematrix form of the 4-D Epanechnikov kernel (4-D EK) and its correlated statistics were deduced to obtain the 4-D Epanechnikov mixture models (4-D EMMs). A 4-D Epanechnikov mixture regression (4-D EMR) was proposed based on this 4-D EK, and a 4-D adaptive model selection (4-D AMLS) algorithm was designed to realize the optimal modeling for a pseudo video sequence (PVS) of the extracted key-EIA. A linear function based reconstruction (LFBR) was proposed based on the correlation between adjacent elementary images (EIs). The decoded images realized a clear outline reconstruction and superior coding efficiency compared to high-efficiency video coding (HEVC) and JPEG 2000 below approximately 0.05 bpp. This work realized an unprecedented theoretical application by (1) proposing the 4-D Epanechnikov kernel theory, (2) exploiting the 4-D Epanechnikov mixture regression and its application in the modeling of the pseudo video sequence of light field images, (3) using 4-D adaptive model selection for the optimal number of models, and (4) employing a linear function-based reconstruction according to the content similarity.