Existence and uniqueness of solutions for fuzzy initial value problems under granular differentiability

Volume 31, Issue 2, pp 197--213 http://dx.doi.org/10.22436/jmcs.031.02.06

Publication Date: May 04, 2023 Submission Date: November 13, 2022 Revision Date: February 13, 2023 Accteptance Date: March 22, 2023

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Authors

N. Soma - Department of of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram-522302, Andhra Pradesh, India. S. K. Grande - Department of of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram-522302, Andhra Pradesh, India. R. P. Agarwal - Department of Mathematics, Texas A \(\&\) M University, Kingsville, TX 78363-8202, Texas, USA. M. S. N. Murty - Sainivas, Bank Street, D.No. 21-47, Opp. State Bank of India, Nuzvid-521201, Andhra Pradesh, India.

Abstract

In this paper, we introduce the notion of second and higher-order granular differentiability for fuzzy number-valued functions. A weighted granular metric is defined for continuously granular differentiable mappings and proves that it is a complete metric space. Fuzzy initial value problems are investigated for second and higher-order fuzzy differential equations under granular differentiability. Sufficient conditions are established for the existence and uniqueness of solutions for the fuzzy initial value problems. An algorithm is developed to determine the solution to the fuzzy initial value problem under granular differentiability. Moreover, examples are presented to verify our theoretical results and algorithm.

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Keywords

• Fuzzy number
• granular differentiability
• contraction mapping
• fuzzy differential equations
• horizontal membership function

MSC

•  34A07
•  34A12

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