Study of differential equation theory has come a long way with applications in the various fields. In 1961, Zygmund and Calderón introduced the notion of derivatives on metric Lr which proved to be better in applications than approximate derivatives. But most of the studies involved are on fuzzy set theory, so it seems likely that intuitionistic fuzzy Lr - norm - based derivatives deserve study. In the study, the fuzzy derivative is extended to intuitionistic fuzzy derivatives with respect.to Lr - norm - based derivatives using intuitionistic fuzzy number valued functions. The efficacy of the proposed method is established by proving the associated theorems along with numerical examples. Further, the Cauchy problem is also studied with respect to intuitionistic fuzzy setting.

Kinematic control of redundant serial manipulators has been carried out for the past half century based primarily on a generalized inverse velocity formulation that is known to have mathematical deficiencies. A recently developed inverse kinematic position mapping is extended to an operational configuration space differentiable manifold formulation for kinematic control with obstacle avoidance is employed to resolve deficiencies in the generalized inverse velocity formulation. Output trajectory tracking with obstacle avoidance for four and twenty degree of redundancy manipulators is carried out to demonstrate the effectiveness of the differentiable manifold approach and that it resolves deficiencies of the conventional generalized inverse velocity formulation.

In this paper, we have established some new bounds of Fejér type Hermite-Hadamard inequality for k−fractional integrals involving r−times differentiable preinvex functions. It is noteworthy that in the past there was no weighted version of left and right sides of the Hermite-Hadamard inequality for k-fractional integrals for generalized convex functions available in literature.

The calculation of echo intensity involved in the SAR image simulation is usually based on the electromagnetic formula or approximate formula derived under certain assumptions. However, the parameters used in these formulas are often difficult to obtain, and the formulas have errors with the actual situation. In this paper, a method of embedding deep neural network (DNN) into the simulation process based on ray tracing is proposed, so that the DNN model can be directly used to fit the calculation formula of echo intensity from real SAR images. Simulation results show that this method can obtain SAR images with high similarity.

In this article, we show the existence of fixed point for the derivative of interval-valued functions. Meanwhile, the fixed point inquiry will utilize the common fixed point methods under the condition of compatibility of the hybrid composite mappings in the sense of the Hausdorff metric. Some examples are given to support the usability of the result of this research .