Common fixed point theorems for left reversible and near-commutative semigroups and applications
- PDF / 621,655 Bytes
- 14 Pages / 468 x 680 pts Page_size
- 19 Downloads / 168 Views
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. 1. Introduction Recently, Y.-Y. Huang and C.-C. Hong [15, 16], T.-J. Huang and Y.-Y. Huang [14], and Y.-Y. 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. As applications, we use our main results to show the existence and uniqueness of solutions of nonlinear Volterra integral equations. Our results generalize, improve, and unify the corresponding results of Fisher [4, 5, 6, 7, 8, 9, 10, 11, 12], Hegedus and Szilagyi [13], Y.-Y. Huang and C.-C. Hong [16], T.-J. Huang and Y.-Y. Huang [14], Y.-Y. Huang et al. [17], Liu [18, 19, 20], Ohta and Nikaido [21], Rosenholtz [22], Taskovic [23], and others. Recall that a semigroup F is said to be left reversible if, for any s,t ∈ F, there exist u,v ∈ F such that su = tv. It is easy to see that the notion of left reversibility is equivalent to the statement that any two right ideals of F have nonempty intersection. A semigroup F is called near commutative if, for any s,t ∈ F, there exists u ∈ F such that st = tu. Clearly, every commutative semigroup is near commutative, and every near-commutative semigroup is left reversible, but the converses are not true. Throughout this paper, (X,d) denotes a metric space, N, R+ , and R denote the sets of positive integers, nonnegative real numbers, and real numbers, respectively. Let F be a semigroup of self-mappings on X and let f be a self-mapping on X. For A,B ⊆ X, x, y ∈ X, define Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:2 (2005) 175–188 DOI: 10.1155/JIA.2005.175
176
Common fixed point theorems
δd (A,B) = sup d(a,b) : a ∈ A, b ∈ B , δd (A) = δd (A,A),
Fx = {x} ∪ {gx : g ∈ F },
δd (x,A) = δd {x},A ,
O f (x) = f n x : n ∈ {0} ∪ N ,
O f (x, y) = O f (x) ∪ O f (y),
C f = {h : h : X −→ X, f h = h f },
n
n
(1.1)
H f = h : h : X −→ X, h ∩n∈N f x ⊆ ∩n∈N f X ,
HF = h : h : X −→ X, h ∩g ∈F gX ⊆ ∩g ∈F gX ,
Φ = φ : φ : R+ −→ R+ is upper semicontinuous from the right,
φ(0) = 0, φ(t) < t for t > 0 . A denotes the closure of A. Clearly, H f ⊇ C f ⊇ { f n : n ∈ N} ∪ {iX }, where iX is the identity mapping on X. The mapping f is called a closed mapping if, y = f x whenever {xn }n∈N ⊆ X such that limn→∞ xn = x and limn→∞ f xn = y for some x, y ∈ X. It is simple to check that the composition of two closed self-mappings in compact metric spaces is clos
Data Loading...
