The Fixed Point Method for Fuzzy Approximation of a Functional Equation Associated with Inner Product Spaces

Th. M. Rassias (1984) proved that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer ≥2,∑=1‖∑−(1/)=1‖2=∑=1‖‖2∑−‖(1/)=1‖2 holds for all 1,…,∈. The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation ∑=1(∑−(1/)=1∑)==1(∑)−((1/)=1) which is said to be a functional equation associated with inner product spaces

Evaluation of FEM modelling for stress propagation under pressure wheel of corn planter

Seeds need a certain range of pressure in the soil bed to germinate and grow ideally. Usually pressure from machinery wheels applies more pressure and prevents seed ideal germination. A finite element model (FEM) was developed to investigate stress propagation in the soil. The pressure wheel of corn planter with 4 km/h speed was chosen to analyze the stress in a sandy-loamy soil. A real corn planter tire was modeled with its mechanical characteristics and imported into ABAQUS/Explicit environment. Frictional contact (based on Mohr-coulomb theory) was used for the soil-tire interaction. The soil was considered as an elastic-perfectly plastic material. Drucker-Prager model was used for soil behavior in plastic region. To evaluate the stress under pressure wheel, FEM results were compared with the Boussinesq theoretical model. On both models, soil stresses decrease with soil depth increasing from zero depth on soil surface to 0.2 m depth. On FEM, stress distribution varied between 47.8 to 8.1 kPa in depth of 0.01 to 0.2 m. FEM and Boussinesq models showed high correlation with each other (R2=95). Our results indicate that the stress under pressure wheels can be properly predicted by using FEM, allowing the pressure simulation to reduce the negative impacts on seed germination and crop yield