© 1996, Topology Atlas

Prior to 1968, the main fixed point theorem for computing fixed points was that of S. Banach (1933). Let T be a selfmap of a complete metric space. If, for each x, y in X we have d(Tx, Ty) <= kd(x, y), where k satisfies 0 <= k < 1, then T has a unique fixed point, which is obtained by repeated function iteration, beginning with any point in the space. The main drawback of Banach's theorem is that the contractive definition requires the map to be continuous.

In 1968, R. Kannan provided an example of a selfmap of a complete metric space, satisfying a contractive condition, which is not continuous at every point. In a very short time the literature began to be flooded with fixed point theorems involving contractive definitions that do not require continuity. Most of these theorems involve the same proof technique, and the contractive definitions are artificially contrived.

In 1977, in a paper in the Transactions, I partially ordered many of these definitions, and either stated or proved the most general fixed point theorem along each of the strands.

Since then Sehie Park and I have written several papers containing fixed point theorems which subsume large numbers of specific papers as special cases.

If the contractive definition is weak enough, then one must either add some restriction to the space, place an added condition on the map, or use an iteration procedure other than function iteration to obtain a fixed point.

The standard iteration procedures are those of Mann and Ishikawa.

In addition to fixed point theorems for a single map there is a large literature on fixed point theorems for two or more maps, multivalued maps, and fixed point theorems in probabilistic metric spaces.

Someone interested in learning about the subject in relation to nonexpansive maps should consult

K. Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Math. v. 28, Cambridge Univ. Press (1990).

(A nonexpansive map satisfies the condition d(Tx, Ty) <= d(x,y).)

For other survey articles, I will immodestly suggest the following papers of mine:

• A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 257-290.
• Contrative definitions revisited, Topological Methods in Nonlinear Analysis, Contemporary Math. AMS 21 (1983), 189-205.
• Contractive definitions and continuity, Fixed Point Theory and Its Applications (ed. R.F. Brown), Contemporary Math. AMS 72 (1988), 233-245.
• Some fixed point iteration procedures, Int. J. Math.&Math. Analysis 14 (1991), 1- 16.
• (with G. Jungck) Some fixed point theorems for compatible maps. Int. J. Math.&Math. Sci. 16 (1993), 417-428.

There exist a few papers which apply fixed point theory to the solutions of ordinary and partial differential equations, or to integral equations. These areas appear to be promising ones in which to make application of some of the theory that has been developed.

Further related textbooks:

V. I. Istratescu, Fixed Point Theory. An Introduction, Math. and Its Applications, D. Reidel Publishing Co., Dordrecth, 1981.

Schweizer and Sklar, Probabilistic Metric Spaces, North Holland Series in Probability and Applied Matheamtics,1983.