Generalized contraction mapping principle and differential equations in probabilistic metric spaces

Abstract

Abstract. A new generalized contraction mapping principle in probabilistic metric spaces is obtained. As an application, we utilize this principle to prove the existence theorems of solutions to differential equations in probabilistic metric spaces. All the results presented in this paper are new.

Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces

“…Since F has the mixed monotone property, it follows from (3.5) and (2.1) that 6) and also it follows from (3.5) and (2.2) that…”

“…Since this principle is a powerful tool in nonlinear analysis, many mathematicians have much contributed to the improvement and generalization of this principle in many ways (see [2][3][4][5][6][7][8][9][10] and others).…”

On common fixed points in generalized Menger spaces

“…The fundamental importance of PM-theory in probabilistic functional analysis due to its extensive applications in random differential as well as random integral equations, for example, the work due to chang and et al [16]. In the field of fixed point, Sehgal [17] presented an active study about the contraction mapping in PM-spaces. Segal and Bharucha-Reid [18] studied Banach's contraction theorem in complete Menger space.…”

$\bar \lambda $-statistically convergent double sequences in probabilistic normed spaces

“…It has a wide range of applications in functional analysis [4]. An important family of probabilistic metric spaces are probabilistic normed spaces (briefly, PN-spaces).…”