Informal Complete Metric Space and Fixed Point Theorems

The concept of informal vector space is introduced in this paper. In informal vector space, the additive inverse element does not necessarily exist. The reason is that an element in informal vector space which subtracts itself cannot be a zero element. An informal vector space can also be endowed with a metric to define a so-called informal metric space. The completeness of informal metric space can be defined according to the similar concept of a Cauchy sequence. A new concept of fixed point and the related results are studied in informal complete metric space.

A New Concept of Fixed Point in Metric and Normed Interval Spaces

The main aim of this paper is to propose the concept of so-called near fixed point and establish many types of near fixed point theorems in the set of all bounded and closed intervals in R . The concept of null set will be proposed in order to interpret the additive inverse element in the set of all bounded closed intervals. Based on the null set, the concepts of metric interval space and normed interval space are proposed, which are not the conventional metric and normed spaces. The concept of near fixed point is also defined based on the null set. In this case, we shall establish many types of near fixed point theorems in the metric and normed interval spaces.

Shape functions associated with inverse element formulations

Super-convergent element formulations in local co-ordinates are obtained using inverse strategies. In the inverse approach discretization errors of the element formulation are minimized leading to super-convergent solutions. In the development of the inverse element model, no shape functions are introduced and therefore the task of element transformation from local to global co-ordinates system remains a challenge. In this paper, a procedure is proposed to produce shape functions associated with the inverse element formulations via hierarchical polynomials. A membrane element formulation is developed using inverse strategy as an example and its associated shape functions are determined using hierarchical polynomials. Numerical results indicate higher accuracy of the developed model in global co-ordinates compared to the reported models in the literature for the same element.