In this paper we discuss some fundamental results in real and functional analysis including the Riesz representation theorem, the Hahn-Banach theorem, and the Baire category theorem. We also discuss applications of these theorems to other topics in analysis.

We work in the set theory without the axiom of choice: ZF. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gâteauxdifferentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f : F → R is a linear functional such that f ≤ p|F , then there exists a...

Solvabilities of generalized vector variational-type inequalities (GVVTI) are discussed in this work. First, the solvability of GVVTI without monotone assumption for mappings is considered in (reflexive) Banach spaces by using Brouwer fixed point theorem (Browder fixed point theorem). Second, the solvability of GVVTI with monotone assumption for mappings is considered in reflexive Banach spaces...

We initiate a study of cohomological aspects of weakly almost periodic group representations on Banach spaces, in particular, isometric representations on reflexive Banach spaces. Using the Ryll-Nardzewski fixed point Theorem, we prove a vanishing result for the restriction map (with respect to a subgroup) in the reduced cohomology of weakly periodic representations. Combined with the Alaoglu-B...

This paper deals with a nonlinear fractional differential equation with integral boundary condition of the following form: { D t x(t) = f(t, x(t), D β t x(t)), t ∈ (0, 1), x(0) = 0, x(1) = ∫ 1 0 g(s)x(s)ds, where 1 < α ≤ 2, 0 < β < 1. Our results are based on the Schauder fixed point theorem and the Banach contraction principle. Keywords—Fractional differential equation; Integral boundary condi...

In 1994, S.G. Matthews introduced the notion of a partial metric space and obtained, among other results, a Banach contraction mapping for these spaces. Later on, S.J. O’Neill generalized Matthews’ notion of partial metric, in order to establish connections between these structures and the topological aspects of domain theory. Here, we obtain a Banach fixed point theorem for complete partial me...

In this paper, we prove a minimization theorem for a proper lower semicontinuous convex function in a real Banach space, applying Takahashi’s nonconvex minimization theorem. Then we give another proof of Bishop-Phelps’ theorem.