Abstract
AbstractThe purpose of this paper is to introduce the notions of Θ-type contractions and Θ-type Suzuki contractions and to establish some new fixed point theorems for these two kinds of mappings in the setting of complete metric spaces. The results presented in the paper are an extension of the Banach contraction principle, the Suzuki contraction theorem, and the Jleli and Samet fixed point theorem. As an application, we utilize our results to study the existence problem of solutions of nonlinear Hammerstein integral equations.
Highlights
Introduction and preliminariesLet (X, d) be a complete metric space and T : X → X be a mapping
The purpose of this paper is to prove some existence theorems of fixed points for θ -type contraction and θ -type Suzuki contraction in the setting of complete metric spaces
We prove that z is the unique fixed point of T in X
Summary
The purpose of this paper is to prove some existence theorems of fixed points for θ -type contraction and θ -type Suzuki contraction in the setting of complete metric spaces. Let (X, d) be a complete metric space and T : X → X be a θ -type Suzuki contraction, i.e., there exist θ ∈ ̃ and k ∈ ( , ) such that for all x, y ∈ X with Tx = Ty, M(x, y) := max d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx) , T has a unique fixed point z ∈ X and for each x ∈ X the sequence {Tnx} converges to z. By the completeness of (X, d), without loss of generality, we can assume that {Tnx}∞ n= converges to some point z ∈ X, i.e., lim d Tnx, z =.
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