Abstract
We prove some common fixed point theorems for left reversible and near-commutative semigroups in compact and complete metric spaces, respectively. As applications, we get the existence and uniqueness of solutions for a class of nonlinear Volterra integral equations.
Highlights
We show that f has a unique fixed point in X
It is easy to see that every element in F is a local contraction and that F is commutative. It follows from Theorem 2.7 that F has a unique fixed point
Let f be a self-mapping of a complete metric space (X,d) and satisfy the following: (viii) for each x ∈ X, O f (x) is bounded; (ix) there exists φ ∈ Φ such that, for any x, y ∈ X, d( f x, f y) ≤ φ δd O f (x, y)
Summary
Huang et al [17] obtained a few fixed point theorems for left reversible and nearcommutative semigroups of contractive self-mappings in compact and complete metric spaces, respectively. These results subsume some theorems in Boyd and Wong [1], Edelstein [3], and Liu [20]. In this paper, motivated by the results in [14, 15, 16, 17], we establish common fixed point theorems for certain left reversible and near-commutative semigroups of selfmappings in compact and complete metric spaces. The semigroup F is said to have diminishing orbital diameters if, for any x ∈ X with δd(Fx) > 0, there exists g ∈ F such that δd(Fgx) < δd(Fx)
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