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94. Khamsi MA, Misane D, Compactness of convexity structures in metric spaces. Math. Japon. 1995;41:321–326. 95. Khamsi MA, Kirk WA, An Introduction to Metric Spaces and Fixed Point Theory. New York:Wiley; 2001. 96. Khamsi MA, Wojciechowski PJ, On the additivity of the Minkowski functionals. Numer. Funct. Anal. Optim. 2013;34:635–647. 97. Kim, TH, Kim ES, Shin SS, Minimization theorems relating to fixed point theorems on complete metric spaces. Math. Japon. 1997;45:97–102. 98. Kirk WA, Caristi’s fixed point theorem and metric convexity.

A) From the condition d(x, f (x)) ≤ ϕ (x) − ϕ ( f (x)), for all x ∈ X, we have that n ∑d f k (x), f k+1 (x) ≤ ϕ (x) − ϕ f n+1 (x) ≤ ϕ (x), for all x ∈ X. k=0 Since (X, →, d) is a Kasahara space it follows that f n (x) → x∗ (x) as n → ∞. But f : (X, →) → (X, →) has a closed graphic. This implies that x∗ (x) ∈ Ff . (b) We remark that d (x, f ∞ (x)) ≤ n ∑d f k (x), f k+1 (x) + d f n+1 (x), f ∞ (x) . k=0 This implies d (x, f ∞ (x)) ≤ ∞ ∑d f k (x), f k+1 (x) ≤ c d(x, f (x)). k=0 This completes the proof.

8. , M(x, y) = max d(x, y), d(x, T x), d(y, Ty), d(x, Ty), d(y, T x) . 35) with α ∈ [0, 1) and L ≥ 0, which is too weak to ensure the existence of a fixed point (see the next example, taken from Ref. [63]; see also Ref. [62]). 4. Let X = N = {0, 1, 2, . } with the usual norm and let T be defined by T (n) = n + 1. 35) with α = 12 and L = 2 but T is fixed point free. Indeed, if we take x = n, y = m, m > n, then d(T x, Ty) = m − n, M(x, y) = m − n + 1, d(y, T x) = m − n − 1. 35) reduces to 3 5 m − n ≤ α (m − n + 1) + 2(m − n − 1) = (m − n) − , 2 2 which is true, since m − n ≥ 1.

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