New fixed point theorems for theta-phi contraction in complete metric spaces

Abstract

In this paper, we introduce the notions of θ-φ contraction and θ-φ Suzuki contraction and establish some new fixed point theorems for these mappings in the setting of complete metric spaces. The results presented in the paper improve and extend the corresponding results due to Banach, Browder [F. E. Browder, Nederl. Akad. Wetensch.

Some fixed point theorems for θ- ϕ C-contractions

“…As pointed out in [25], Theorem 1.7 improved and extended the corresponding results of Banach, Samet [21], Jleli and Samet [12], Kannan [13], Dugundji-Granas [7], Boyd-Wong [2], Matkowski [14], Browder [3] and so on.…”

“…Just recently, Zheng et al [25] introduced the notion of θ − φ contraction which generalized θ−contraction and other contractions (see [12,25] and the references therein).…”

“…According to [12,25], we denote by Θ the set of functions θ : (0, ∞) → (1, ∞) satisfying the following conditions:…”

Weak theta-phi-contraction and discontinuity

“…Compared to other definitions, our definition is very weak. As a matter of fact, many of the existing results can be derived from our results [15]. An important feature of our definition is that continuity or discontinuity at the fixed point is independent of the definition.…”

“…And we denote by Φ ( [15]) the set of functions φ : [1, ∞) → [1, ∞) satisfying the following conditions:…”