existence and convergence of fixed points for noncyclic φ-contractions

Et a and b be two nonempty subsets of a metric space (x, d). if self mapping t : a∪b → a∪b be a noncyclic map , i.e., t (a) ⊆ a and t (b) ⊆ b; then x∗ ∈ a∪b is a fixed point of t provided that t x∗ = x∗. if t : a∪b → a∪b be a cyclic map , i.e., t (a) ⊆ b and t (b) ⊆ a and d(a, b) = inf{d(a, b) : a ∈ a, b ∈ b}; then x∗ ∈ a ∪ b is called a best proximity point of t provided that d(x∗, t x∗) = d(a, b). if d(a, b) > 0, then a best proximity point serves as a optima for the operator equation t x = x.in 2005, anthony eldred, kirk and veeremani [5] introduced noncyclic nonexpansive mappings and studied the existence of a fixed point of such mappings. in 2006, cyclic contraction mappings on uniformly convex banach spaces were introduced and studied by anthony eldred and veeremani [6]. since then, the problems of the existence of a best proximity point (fixed point) of cyclic (noncyclic) mappings, have been extensively studied by many authors; see for instance [1, 2, 5, 6, 7, 9, 11, 12, 13, 14] and references therein. in 2009, al-thagafi and shahzad [4] generalized cyclic contraction condition and proved existence of best proximity points.