In the proposed work, blockchain is implemented on the Base Stations (BSs) and Cluster Heads (CHs) to register the nodes using their credentials and also to tackle various security issues. Moreover, a Machine Learning (ML) classifier, termed as Histogram Gradient Boost (HGB), is employed on the BSs to classify the nodes as malicious or legitimate. In case, the node is found to be malicious, its registration is revoked from the network. Whereas, if a node is found to be legitimate, then its data is stored in an Interplanetary File System (IPFS). IPFS stores the data in the form of chunks and generates hash for the data, which is then stored in blockchain. In addition, Verifiable Byzantine Fault Tolerance (VBFT) is used instead of Proof of Work (PoW) to perform consensus and validate transactions. Also, extensive simulations are performed using the Wireless Sensor Network (WSN) dataset, referred as WSN-DS. The proposed model is evaluated both on the original dataset and the balanced dataset. Furthermore, HGB is compared with other existing classifiers, Adaptive Boost (AdaBoost), Gradient Boost (GB), Linear Discriminant Analysis (LDA), Extreme Gradient Boost (XGB) and ridge, using different performance metrics like accuracy, precision, recall, micro-F1 score and macro-F1 score. The performance evaluation of HGB shows that it outperforms GB, AdaBoost, LDA, XGB and Ridge by 2-4%, 8-10%, 12-14%, 3-5% and 14-16%, respectively. Moreover, the results with balanced dataset are better than those with original dataset. Also, VBFT performs 20-30% better than PoW. Overall, the proposed model performs efficiently in terms of malicious node detection and secure data storage.

In this article the coincidence points of a self mapping and a sequence of multivalued mappings are found using the graphic F-contraction. This generalizes Mizoguchi-Takahashi?s fixed point theorem for multivalued mappings on a metric space endowed with a graph. As applications we obtain a theorem in homotopy theory, an existence theorem for the solution of a system of Urysohn integral equations, and for the solution of a special type of fractional integral equations.

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The analysis in the present paper provides insights into the Liouville-type results for an Eyring-Powell fluid considered as having an incompressible and unsteady flow. The gradients in the spatial distributions of the initial data are assumed to be globally (in the sense of energy) bounded. Under this condition, solutions to the Eyring-Powell fluid equations are regular and bounded under the L2 norm. Additionally, a numerical assessment is provided to show the mentioned regularity of solutions in the travelling wave domain. This exercise serves as a validation of the analytical approach firstly introduced.