A fixed point theorem for contractions in modular metric spaces

The notion of a (metric) modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by the author [V. V. Chistyakov, Metric modulars and their application, Dokl. Math. 73(1) (2006) 32-35, and Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72(1) (2010) 1-14]. In this paper we establish a fixed point theorem for contractive maps in modular spaces. It is related to contracting rather ``generalized average velocities'' than metric distances, and the successive approximations of fixed points converge to the fixed points in a weaker sense as compared to the metric convergence.Comment: 31 pages, LaTeX, uses elsarticle.st

A Millennium Learning Goal for education post‐2015: a question of outcomes or processes

As the target year for the current Millennium Development Goal of universal completion of primary education approaches, three World Bank economists have proposed its replacement with a Millennium Learning Goal. This is part of a trend of increased privileging of learning outcomes. The proposal is assessed from the perspective of human rights-based and social justice conceptualisations of education quality. A Millennium Learning Goal may enhance information on inclusion, conceived as equal opportunity to achieve learning outcomes. However, there is a danger that it would be misused to generate high stakes tests that can be detrimental to the achievement of goals that are not readily measurable and hence to the relevance of education. It is argued that a process goal with qualitative targets for the assessment of learning, for the monitoring of educational processes and for the processes by which learning goals are determined would be more appropriate for the international level.As the target year for the current Millennium Development Goal of universal completion of primary education approaches, three World Bank economists have proposed its replacement with a Millennium Learning Goal. This is part of a trend of increased privileging of learning outcomes. The proposal is assessed from the perspective of human rights-based and social justice conceptualisations of education quality. A Millennium Learning Goal may enhance information on inclusion, conceived as equal opportunity to achieve learning outcomes. However, there is a danger that it would be misused to generate high stakes tests that can be detrimental to the achievement of goals that are not readily measurable and hence to the relevance of education. It is argued that a process goal with qualitative targets for the assessment of learning, for the monitoring of educational processes and for the processes by which learning goals are determined would be more appropriate for the international level

Asymptotically Nonexpansive Mappings in Modular Function Spaces

AbstractIn this paper, we prove that if ρ is a convex, σ-finite modular function satisfying a Δ2-type condition, C a convex, ρ-bounded, ρ-a.e. compact subset of Lρ, and T: C→C a ρ-asymptotically nonexpansive mapping, then T has a fixed point. In particular, any asymptotically nonexpansive self-map defined on a convex subset of L1(Ω,μ) which is compact for the topology of local convergence in measure has a fixed point