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The main aim of this survey article is to present several known results about vector-valued almost periodic functions and their applications. We separately consider almost periodic functions depending on one real variable and almost periodic functions depending on two or more real variables. We address several open problems and possibilities for further investigations of almost periodic functions, quoting more than two hundred references about the subject under our consideration.

It is shown that every asymptotically regular or λ-firmly nonexpansive mapping T:C→C has a fixed point whenever C is a finite union of nonempty weakly compact convex subsets of a Banach space X which is uniformly convex in every direction. Furthermore, if {T i}i∈I is any compatible family of strongly nonexpansive self-mappings on such a C and the graphs of T i, i ∈I, have a nonempty intersection, then T i, i∈I, have a common fixed point in C.

Some new fixed point theorems are established in the setting of complex valued G-metric spaces. These new results improve and generalize Kang et al.’s results, the Banach contraction principle, and some well-known results in the literature.

This paper proposes a three-dimensional (3D) human posture estimation system that locates 3D significant body points based on 2D body contours extracted from two cameras without using any depth sensors. The 3D significant body points that are located by this system include the head, the center of the body, the tips of the feet, the tips of the hands, the elbows, and the knees. First, a linear support vector machine- (SVM-) based segmentation method is proposed to distinguish the human body from the background in red, green, and blue (RGB) color space. The SVM-based segmentation method uses not only normalized color differences but also included angle between pixels in the current frame and the background in order to reduce shadow influence. After segmentation, 2D significant points in each of the two extracted images are located. A significant point volume matching (SPVM) method is then proposed to reconstruct the 3D significant body point locations by using 2D posture estimation results. Experimental results show that the proposed SVM-based segmentation method shows better performance than other gray level- and RGB-based segmentation approaches. This paper also shows the effectiveness of the 3D posture estimation results in different postures.

We first present some new existence theorems for fixed point problem and minimization problem in compact metric spaces without assuming that mappings possess convexity property. Some applications of our results to new fixed point theorems for nonself mappings in the setting of strictly convex normed linear spaces and usual metric spaces are also given.

We first establish some existence results concerning approximate coincidence point properties and approximate fixed point properties for various types of nonlinear contractive maps in the setting of cone metric spaces and general metric spaces. From these results, we present some new coincidence point and fixed point theorems which generalize Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and some well-known results in the literature.

We introduce some new iterative schemes based on viscosity approximation method for finding a split common element of the solution set of a pair of simultaneous variational inequalities for inverse strongly monotone mappings in real Hilbert spaces with a family of infinitely nonexpansive mappings. Some strong convergence theorems are also given. Our results generalize and improve some well-known results in the literature and references therein.

Some new fixed point theorems for nonlinear maps are established. By using these results, we can obtain some new coincidence point theorems. Our results are quite different in the literature and references therein.

By applying hybrid inclusion and disclusion systems (HIDS), we establish several vectorial variants of system of Ekeland's variational principle on topological vector spaces, some existence theorems of system of parametric vectorial quasi-equilibrium problem, and an existence theorem of system of the Stampacchia-type vectorial equilibrium problem. As an application, a vectorial minimization theorem is also given. Moreover, we discuss some equivalence relations between our vectorial variant of Ekeland's variational principle, common fixed point theorem, and maximal element theorem.