Aseev launched a new branch of functional analysis by introducing the theory of quasilinear spaces in the framework of the topics of norm, bounded quasilinear operators and functionals (Aseev (1986)). Furthermore, some quasilinear counterparts of classical nonlinear analysis that lead to such result as Frechet derivative and its applications were examined deal with. This pioneering work causes a lot of results in such applications such as (Rojas-Medar et al. (2005), Talo and Başar (2010), and Nikol'skiȋ (1993)). His work has motivated us to introduce the concept of quasilinear inner product spaces. Thanks to this new notion, we obtain some new theorems and definitions which are quasilinear counterparts of fundamental definitions and theorems in linear functional analysis. We claim that some new results related to this concept provide an important contribution to the improvement of quasilinear functional analysis.

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It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we first developed the algebraic structure of the class of fuzzy sets F ℝ n and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, u q = ∫ 0 1 sup x ∈ u α x q d α 1 / q where 1 ≤ q ≤ ∞ , we stated that F ℝ n , u q is a complete normed space. Furthermore, we introduced an inner product in this space for the case q = 2 . The inner product must be in the form u , v = ∫ 0 1 u α , v α   K ℝ n d α = ∫ 0 1 a , b ℝ n d α    : a ∈ u α , b ∈ v α . For u , v ∈ F ℝ n . We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F ℝ n . Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.

In this paper, we investigate some set-valued contraction mappings in partial Hausdorff metric spaces and prove the existence of fixed point of this set-valued mappings in partial Hausdorff metric spaces. We also give an example as support of our results.

In this paper, we will present the notion of the biquasilinear functional which is a new concept of quasilinear functional analysis. Just like bilinear functional, the notions of a biquasilinear functional and a quadratic form will not need to have the constitution of an inner product quasilinear space. We were able to define these functionals in any quasilinear space. After giving this new notion, we discuss some examples and prove some theorems for considerable exercises to the theory of biquasilinear functionals in Hilbert quasilinear spaces.