Naoki Shioji

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

In this paper, we study the convergence of the sequence defined by x0 ∈ C, xn+1 = αnx + (1− αn)Txn, n = 0, 1, 2, . . . , where 0 ≤ αn ≤ 1 and T is a nonexpansive mapping from a closed convex subset of a Banach space into itself.

Contractive mappings, Kannan mappings and metric completeness

In this paper, we first study the relationship between weakly contractive mappings and weakly Kannan mappings. Further, we discuss characterizations of metric completeness which are connected with the existence of fixed points for mappings. Especially, we show that a metric space is complete if it has the fixed point property for Kannan mappings.

A further generalization of the knaster-kuratowski-mazurkiewicz theorem

Granas and Dugundji obtained the following generalization of the Knaster-Kuratowski-Mazurkiewicz theorem. Let X be a subset of a topological vector space E and let G be a setvalued map from X into E such that for each finite subset {xl, .., xn} of X, co{xl. x, } C Un=I Gxi and for each x e X, Gx is finitely closed, i.e., for any finite-dimensional subspace L of E, Gx n L is closed in the Euclidean topology of L. Then {Gx: x e X} has the finite intersection property. By relaxing, among others, the condition that X is a subset of E , we obtain a further generalization of the theorem and show some of its applications.

Fixed point theory in weak second-order arithmetic

Abstract We develop a basic part of fixed point theory in the context of weak subsystems of second-order arithmetic. RCA0 is the system of recursive comprehension and Σ01 induction. WKL0 is RCA0 plus the weak Konigs lemma: every infinite tree of sequences of 0s and 1s has an infinite path. A topological space X is said to possess the fixed point property if every continuous function f:X→X has a point x ϵ X such that f(x) = x. Within WKL0 (indeed RCA0), we prove Brouwers theorem asserting that every nonempty compact convex closed set C in R n has the fixed point property, provided that C is expressed as the completion of a countable subset of Q n. We then extend Brouwers theorem to its infinite dimensional analogue (the Tychonoff-Schauder theorem for R N ) still within RCA0. As an application of this theorem, we prove the Cauchy-Peano theorem for ordinary differential equations within WKL0, which was first shown by Simpson without reference to the fixed point theorem. Within RCA0, we also prove the Markov-Kakutani theorem which asserts the existence of a common fixed point for certain families of affine mappings. Adapting Kakutanis ingenious proof for deducing the Hahn–Banach theorem from the Markov-Kakutani theorem, we also establish the Hahn-Banach theorem for seperable Banach spaces within WKL0, which was first shown by Brown and Simpson in a different way.

Invariant sets for nonlinear evolution equations, Cauchy problemsand periodic problems

In the case of K≠D(A)¯, we study Cauchy problems and periodic problems for nonlinear evolution equation u(t)∈K, u′(t)

Existence of periodic solutions for nonlinear evolution equations with pseudo monotone operators

In this paper, we study the existence of T -periodic solutions for the problem u′(t) + A(t)u(t) = 0, t ∈ R, where A(t) is a T -periodic, pseudo monotone mapping from a reflexive Banach space into its dual.

Fan's theorem concerning systems of convex inequalities and its applications

On etend le theoreme de Fan aux fonctions convexes semi-continues inferieurement avec des valeurs dans (−∞,+∞)

Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms

Let N > 4, let 2* = 2N/(N - 2) and let Ω C R N be a bounded domain with a smooth boundary ∂Ω. Our purpose in this paper is to consider the existence of solutions of the problem: u > 0 in Ω, u = 0 on ∂Ω, where 0 < μ< (N-2/2) 2 .

Radial symmetry of

Let

n

f \in C((0,1)\times (0,\infty),\mathbb{R})