Common Fixed Point and Invariant Approximation Theorems for Mappings Satisfying Generalized Contraction Principle

We prove common fixed point theorems for weakly compatible mappings satisfying a generalized contraction principle by using a control function. As an application, we have established invariant approximation result. Our theorems generalize recent results existing in the literature.

On the other hand, (Park, S., (1980) and (Khan, M.S., Swaleh, M. & Sessa, S., 1984) proved fixed point theorem for a self mapping by altering distances between the points and using a control function, whereas (Sastry, K.P.R. & Babu, G.V.R., 1999) extended the concept for weakly commuting pairs of self mappings and proved common fixed point theorem in a complete metric space by using the control function.

Definitions and Preliminaries
Definition 2.1 Two self mappings T and f of a metric space (X, d) are said to be weakly compatible, if f T x = T f x whenever f x = T x for all x ∈ X.

Definition 2.2 Let T and f be self mappings of a nonempty subset M of a metric space X. The mapping T is called f -contraction mapping, if there exists a real number
for all x, y ∈ M.
Definition 2.4 (Beg, I. & Abbas, M., 2006) A self mapping T of a metric space (X, d) is said to be weakly contractive with respect to a self mapping f : X → X, if for each x, y ∈ X, If f = I, the identity mapping, then the Definition( 2.4) reduces to the definition of weakly contractive mapping given by (Alber, Ya.I. & Guerre-Delabriere, S.( 1997) and (Rhoades, B.E., 2001).
Definition 2.5 Let M be a nonempty subset of a metric space (X, d).The set of best M-approximants to u ∈ X, denoted as P M (u) is defined by where dist(u, M) = inf{d(x, u) : x ∈ M}.

Main Results
Theorem 3.1 Let T and f be self mappings of a metric space (X, d) satisfying where and ψ, φ : [0, ∞) → [0, ∞) are both continuous and monotone decreasing functions with ψ(t) = 0 = φ(t) if and only if t = 0.If T X is a complete metric space and T X ⊂ f X, then T and f have a coincidence point in X.Further, if T and f are weakly compatible, then they have a unique common fixed point in X.
Proof: Let x 0 ∈ X be arbitrary point.Construct the sequence {x n } such that f x n = T x n−1 for each n = 1, 2, 3, ...∞ which is possible since T X ⊂ f X.Now, where By monotone property of ψ function, we have d(T x n , T x n+1 ) ≤ d(T x n−1 , T x n ).Therefore, the sequence {d(T x n , T x n+1 )} is monotone decreasing and continuous.Hence there exists a real number r ≥ 0 such that, As n → ∞ in ( 7), we have ψ(r) ≤ ψ(r) − φ(r) which is possible only when r = 0. Thus Next, we claim that {T x n } is a Cauchy sequence.Assume the contrary.Then there exists an > 0 and subsequences {n i } and {m i } such that m i < n i < m i+1 along with Then, it follows that By equation ( 9), Now, Using inequalities ( 9) and (11), we have as i → ∞ Now using inequality (4) and ( 10), we have where Using inequalities (9), ( 10) and ( 12), we have As i → ∞ and using (11), inequality (13) becomes, a contradiction, as > 0. Thus {T x n } is a Cauchy sequence in T X which in turn implies that { f x n } is also a Cauchy sequence in X.Since T X is complete, {T x n } converges to some v ∈ T X.Since T X ⊂ f X and v = f u for some u ∈ X.Thus { f x n } converges to f u.Now, where By monotone increasing property of ψ and φ, we have and u is the coincidence point of T and f .Since T and f are weakly compatible, they commute at their coincidence point.Hence and φ : [0, ∞) → [0, ∞) be defined by

.
Hence v is the common fixed point of T and f .Uniqueness: Let v and w be two common fixed points of T and f .(i.e) v = T v = f v and w = T w = f w.Using inequality (4), we haveψ(d(T v, T w)) ≤ ψ(M(v, w)) − φ(M(v, w))where,M(v, w) = max {d( f v, f w), d( f v, T v), d( f w, T w), 1 2 [d( f v, T w) + d( f w, T v)]}.= d(v, w) Therefore, ψ(d(v, w)) = ψ(d(T v, T w)) ≤ ψ(d(v, w)) − φ(d(v, w))which is possible only when v = w.Hence v is the unique common fixed point of T and f .Example 3.1 Let X = [0, 1] with the usual metric.Define two self mappings T and f of X by T x = x 2 and f x = x for all x ∈ X.Let ψ : [0, ∞) → [0, ∞) be defined by