Without restriction of generality, we can suppose that δ ≤ ϵ.
Introducing dimensionless quantities (x/ c → x, y/ c → y, k1/ k2 c) → k1, k3/ k2 → k3, k4/ k2 c) → k4, tk2 c → t), we set k2 = 1, without restriction of generality (k1, k3, k4 > 0).
Without restriction of the generality, we may assume that (xi_{n}neq0) for every (ngeq N).
Without restriction of the generality, we may assume that (t_{1}geq t_{2}).
Without restriction of the generality, we may assume that (t_{n}geq t) for n large enough.
In the first case, without restriction of the generality, we can suppose that (T^{n ( k ) }x_{1}neq x_{ast}) for all k.
Without restriction of the generality, we can suppose that T n x ≠ z for all n (or for n large enough).
In this case, we get (d(x_{p},Tx_{p})=d(A,B)), that is, (x_{p}) is a best proximity point of T. So, without restriction of the generality, we may suppose that x_{n}neq x_{n+1},quad n=0,1,2,ldots.
Without any restriction of the generality, we can assume that ( T ( x, y ), T ( y, x ) ) ≤ ( T ( u, v ), T ( v, u ) ) and ( T ( z, t ), T ( t, z ) ) ≤ ( T ( u, v ), T ( v, u ) ).
If for some p ∈ N we have T p x = T p + 1 x, then T p x will be a fixed point of T. So, without restriction of the generality, we can suppose that d ( T n x, T n + 1 x ) > 0 for all n ∈ N. Now, from (1), for all n ∈ N, we have θ ( d ( T n x, T n + 1 x ) ) ≤ [ θ ( M ( T n − 1 x, T n x ) ) ] k, (3).
If for some p ∈ N, we have T p x = T p + 1 x, then T p x will be a fixed point of T. So, without restriction of the generality, we can suppose that d ( T n x, T n + 1 x ) > 0 for all n ∈ N. Now, from (1), for all n ∈ N, we have θ ( d ( T n x, T n + 1 x ) ) ≤ [ θ ( d ( T n − 1 x, T n x ) ) ] k ≤ [ θ ( d ( T n − 2 x, T n − 1 x ) ) ] k 2 ≤ ⋯ ≤ [ θ ( d ( x, T x ) ) ] k n.
