Controllability for the Nonlinear Fuzzy Neutral Integrodifferential Equations with Nonlocal Conditions

S Nayayanamoorthy¹, M. Nagarajan²

¹Assistant Professor, Bharathiar University, Coimbatore, Tamil Nadu, India.

²Research Scholar, Bharathiar University, Coimbatore, Tamil Nadu, India.

*Corresponding Author: snm_phd@yahoo.co.in, mnagarajanphd@gmail.com

ABSTRACT:

In this paper, we devoted study the controllability for the nonlinear fuzzy neutral integrodifferential equations control system in EN. Moreover we study the fuzzy solution for the normal, convex, upper semicontinuous, and compactly supported interval fuzzy number. The results are obtained by using the Banach Fixed point theorem.

KEY WORDS: Fuzzy set, fuzzy number, neutral integrodifferential system, fuzzy solution, fixed point theorem.

1. INTRODUCTION:

The term ‘‘fuzzy differential equation’’ was coined in 1978 by Kandel and Byatt (1978). There are many suggestions to define a fuzzy derivative. One of the earliest was to generalize the Hukuhara derivative of a set-valued function. This generalization was made by Puri and Ralescu (1983) and studied by Kaleva (1987). It soon appeared that the solution of fuzzy differential equation interpreted by Hukuhara derivative has a drawback: it became fuzzier as time goes. Hence, the fuzzy solution behaves quite differently from the crisp solution. To alleviate the situation, Hüllerme-ier (1997) interpreted fuzzy differential equation as a family of differential inclusions. The main shortcoming of using differential inclusions is that we do not have a derivative of a fuzzy number-valued function. There is another approach to solve fuzzy differen-tial equations which is known as Zadeh’s extension principle (Misukoshi, ChalcoCano, Román-Flores, and Bassanezi, 2007; Oberguggenberger and Pittschmann, 1999), the basic idea of the extension principle is: consider fuzzy differential equation as a deterministic differential equation then solve the deterministic differential equation. After getting deterministic solution, the fuzzy solution can be obtained by applying extension principle to deterministic solution. But in Zadeh’s extension principle we do not have a derivative of a fuzzy number-valued function either. In Bede and Gal (2005) and Bede, Rudas, and Bencsik (2007), strongly generalized derivative concept was introduced. This concept allows us to solve the mentioned shortcomings and in Khastan, Bahrami, and Ivaz (2009) authors studied higher order fuzzy differential equations with strongly generalized derivative concept.

Recently, Gasilov, Amrahov, and Fatullayev (2011) proposed a new method to solve a fuzzy initial value problem for the fuzzy linear system of differential equations based on properties of linear transformations. But they used fuzzy bunch of functions instead of fuzzy number valued functions. In recent paper, Y. C Kuwun, J. S Hwang, J.S Park and J. H Park, Controllability for the Impulsive Semilinear Fuzzy Integrodifferential equations with nonlocal conditions can be extended to the fuzzy neutral integrodifferential equations. We establish a synthesis of crisp solution of fuzzy initial value problem and the method proposed in Kaleva (1987) to solve fuzzy initial value problem. To do this firstly we remained the following basic concepts from fuzzy arithmetic and fuzzy calculus.

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