Common Fixed Point Theorems for Left Reversible and Near-commutative Semigroups and Applications

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