## Remarks on contractive conditions of integral type.(English)Zbl 1215.54021

In this excellent paper, the author shows that a number of contractive conditions of integral type are equivalent to corresponding contractive conditions of non-integral type. I shall illustrate this by stating Theorem 1 of the paper. Let $$\mathfrak{F} := \{ F : \mathbb{R_+} \to \mathbb{R_+} : F$$ is increasing, continuous and satisfies $$F^{-1}(0) = \{0\}\}$$ and $$\Phi := \{\varphi : \mathbb{R_+} \to \mathbb{R_+} : \varphi$$ is nondecreasing, right upper continuous, and satisfies $$\varphi(t) < t \text{ for each } t > 0\}$$.
Let $$A, B, S,$$ and $$T$$ be selfmaps of a metric space $$(X, d)$$. For $$x,y \in X$$, set $$M(x,y) :=\max\{d(Ax,by), d(Ax,Sx), d(By,Ty), [d(Ax,By) + d(By,Sx)]/2\}.$$ Then the following are equivalent:
(i)
there exist $$\varphi \in \Phi$$ and $$f \in \mathfrak{F}$$ such that $\int_0^{d(Sx,Ty)}f(s)\,ds \leq \varphi\Big(\int_0^{M(x,y)}f(s)\,ds\Big)\text{ for all }x,y \in X;$
(ii)
there exist $$F \in \mathfrak{F}$$ and $$\varphi \in \Phi$$ such that $$F(d(Sx,Ty)) \leq \varphi(F(M(x,y))$$ for all $$x,y \in X$$;
(iii)
for every $$\alpha \in (0,1)$$, there exists $$F \in\mathfrak{F}$$ such that $$F(d(Sx,Ty)) \leq \alpha(F(M(x,y))$$ for all $$x,y \in X$$;
(iv)
there exists $$\varphi \in\Phi$$ such that $$d(Sx, Ty) \leq \varphi(M(x,y))$$ for all $$x,y \in X$$.
The author also proves the following fixed point theorem. Theorem 8. Let $$T$$ be a selfmap of a complete metric space $$(X,d)$$ such that, for some $$\varphi \in \Phi$$ and $$F \in \mathfrak{F}$$, $$F(d(Tx,Ty)) \leq \varphi(F(d(x,y)))$$ for all $$x, y \in X$$. Then $$T$$ has a unique fixed point $$x_{\epsilon}$$ and, for any $$x \in X, \lim_nT^nx = x_{\epsilon}$$.

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
Full Text:

### References:

 [1] Aliouche, A., Common fixed point theorems of Gregus type for weakly compatible mappings satisfying generalized contractive conditions, J. math. anal. appl., 341, 707-719, (2008) · Zbl 1138.54031 [2] Altun, I.; Türkoğlu, D.; Rhoades, B.E., Fixed points of weakly compatible maps satisfying a general contractive condition of integral type, Fixed point theory appl. 2007, (2007), article ID 17301 · Zbl 1153.54022 [3] Branciari, A., A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. math. math. sci., 29, 531-536, (2002) · Zbl 0993.54040 [4] Browder, F.E., On the convergence of successive approximations for nonlinear functional equations, Nederl. akad. wetensch. proc. ser. A 71 =indag. math., 30, 27-35, (1968) · Zbl 0155.19401 [5] Djoudi, A.; Aliouche, A., Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type, J. math. anal. appl., 329, 1, 31-45, (2007) · Zbl 1116.47047 [6] Djoudi, A.; Merghadi, F., Common fixed point theorems for maps under a contractive condition of integral type, J. math. anal. appl., 341, 2, 953-960, (2008) · Zbl 1151.54032 [7] Fremlin, D.H., Measure theory vol II, (2001), Broad Foundations University of Essex · Zbl 1166.28001 [8] Hegedűs, M.; Szilágyi, T., Equivalent conditions and a new fixed point theorem in the theory of contractive type mappings, Math. japon., 25, 147-157, (1980) · Zbl 0438.54037 [9] Jachymski, J., Common fixed point theorems for some families of maps, Indian J. pure appl. math., 25, 925-937, (1994) · Zbl 0811.54034 [10] Jachymski, J., Equivalent conditions and the meir – keeler type theorems, J. math. anal. appl., 194, 293-303, (1995) · Zbl 0834.54025 [11] Jachymski, J., Equivalence of some contractivity properties over metrical structures, Proc. amer. math. soc., 125, 2327-2335, (1997) · Zbl 0887.47039 [12] Jachymski, J.; Jóźwik, I., Nonlinear contractive conditions: A comparison and related problems. fixed point theory and its applications, (), 123-146 · Zbl 1149.47044 [13] Jachymski, J.; Matkowski, J.; Świa̧tkowski, T., Nonlinear contractions on semimetric spaces, J. appl. anal., 1, 125-134, (1995) · Zbl 1295.54055 [14] Kuczma, M.; Choczewski, B.; Ger, R., Iterative functional equations, () · Zbl 1141.39023 [15] Leader, S., Fixed points for general contractions in metric spaces, Math. japon., 24, 17-24, (1979/80) · Zbl 0409.54052 [16] McAuley, L.F., A relation between perfect separability completeness, and normality in semi-metric spaces, Pacific J. math., 6, 315-326, (1956) · Zbl 0072.17802 [17] Meir, A.; Keeler, E., A theorem on contraction mappings, J. math. anal. appl., 28, 326-329, (1969) · Zbl 0194.44904 [18] Nadler, S.B., Multi-valued contraction mappings, Pacific J. math., 30, 475-488, (1969) · Zbl 0187.45002 [19] Reich, S., Fixed points of contractive functions, Boll. un. mat. ital., 5, 4, 26-42, (1972) · Zbl 0249.54026 [20] Reich, S., Some problems and results in fixed point theory, (), 179-187 · Zbl 0531.47048 [21] Rhoades, B.E., Two fixed-point theorems for mappings satisfying a general contractive condition of integral type, Int. J. math. math. sci., 63, 4007-4013, (2003) · Zbl 1052.47052 [22] Sessa, S., On a weak commutativity condition of mappings in fixed point considerations, Publ. inst. math. (beograd) (N.S.), 32, 46, 149-153, (1982) · Zbl 0523.54030 [23] Singh, S.L.; Mishra, S.N., Remarks on jachymski’s fixed point theorems for compatible maps, Indian J. pure appl. math., 28, 5, 611-615, (1997) · Zbl 0876.54033 [24] Suzuki, T., Meir – keeler contractions of integral type are still meir – keeler contractions, Int. J. math. math. sci. 2007, (2007), article ID 39281 · Zbl 1142.54019 [25] Vijayaraju, P.; Rhoades, B.E.; Mohanraj, R., A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, Int. J. math. math. sci., 15, 2359-2364, (2005) · Zbl 1113.54027 [26] Zhang, X., Common fixed point theorems for some new generalized contractive type mappings, J. math. anal. appl., 333, 2, 780-786, (2007) · Zbl 1133.54028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.