FUZZY-PREŠIC-CIRIC OPERATORS AND APPLICATIONS TO CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS

Satish Shukla,Dhananjay Gopal,Rosana Rodrıguez-Lopez

doi:10.3846/13926292.2016.1247303

Abstract

In this paper, we introduce a new class of operators called fuzzy-Preši´cCiri´c operators. For this type of operators, the existence and uniqueness of fixed point ´ in M-complete fuzzy metric spaces endowed with H-type t-norms are established. The results proved here generalize and extend some comparable results in the existing literature. An example is included which illustrates the main result of this paper. Moreover, some applications of our main theorem to the study of certain types of nonlinear differential equations are provided.

Highlights

Introduction and preliminariesThe contraction mapping principle appeared in the explicit form in Banach’s thesis in 1922 [1], where it was used to establish the existence of solution of integral equations
In 2002, Gregori and Sapena [9] introduced the notion of fuzzy contractive mapping and established Banach contraction theorem in various classes of complete fuzzy metric spaces in the sense of George and Veeramani [4], Kramosil and Michalek [12] and Grabiec [7]
The results obtained by Gregori and Sapena [9] have become recently of interest to many authors. Following this direction of research, we extend and generalize here the fuzzy contractive mappings of Gregori and Sapena [9] by introducing fuzzy-PresicCiric operators and prove some fixed point results for such operators in fuzzy metric spaces under H-type t-norms

Summary

Introduction and preliminaries

The contraction mapping principle appeared in the explicit form in Banach’s thesis in 1922 [1], where it was used to establish the existence of solution of integral equations. In 2002, Gregori and Sapena [9] introduced the notion of fuzzy contractive mapping and established Banach contraction theorem in various classes of complete fuzzy metric spaces in the sense of George and Veeramani [4], Kramosil and Michalek [12] and Grabiec [7]. The results obtained by Gregori and Sapena [9] have become recently of interest to many authors (see [5, 8, 13, 14, 15, 21, 22]) Following this direction of research, we extend and generalize here the fuzzy contractive mappings of Gregori and Sapena [9] by introducing fuzzy-PresicCiric operators and prove some fixed point results for such operators in fuzzy metric spaces under H-type t-norms. In the three sections, we state our main results

Fixed point theorems

Some properties of fuzzy contractive sequences

Applications to differential equations