Dear Muhammad,

everything important has been said before.

Here is the book you are interested about.

Greetings, Tadeusz

Both

 I think it is more pure than applied

I  am agree with followers

Panchatcharam Mariappan,  Miguel Maldonado    and  George Stoica .

It is both  (perhaps more pure, but   important for the applications).

For example, there is  journal  Fixed Point Theory and Applications. See also

Brown, R. F. (Ed.) (1988). Fixed Point Theory and Its Applications. American Mathematical Society. ISBN 0-8218-5080-

Over the last 50 years the theory of fixed points has been used as important tool in the study of nonlinear phenomena. In particular, fixed point techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory, and physics.

The Banach fixed-point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.

 The Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point. It doesn't describe how to find the fixed point (See also Sperner's lemma).

Fixed point theorems give the conditions under which maps (single or multivalued) have fp or the corresponding equations   have   solutions.

For example, consider a mapping  f  and equation  (i)  f(x)=a with respect to x  for given  a.  Set  g(x)=f(x) +x -a. Then (i) is equivalent to (ii)    g(x)=x

There are a number of generalisations to Banach fixed-point theorem and further; these are applied in PDE theory. See fixed-point theorems in infinite-dimensional spaces.

Dear Muhammad,

everything important has been said before.

Here is the book you are interested about.

Greetings, Tadeusz

It is more pure than applied math. Applications in natural sciences are about existence of solution (for example, for integral equations). This can help before applying numerical methods. In social sciences - mainly for the existence of equilibrium; often used in economic theory.

Thank you all for the contributions, it really adds to my knowledge. Further I am grateful for the book, because I found it interesting and helpful @Tadeusz.

Dear Muhammad,

First recall that  I  am agree with followers

Panchatcharam Mariappan,  Miguel Maldonado    and  George Stoica .

It is both  (perhaps more pure, but   important for the applications).

In particulat it has  important application in economy.

Fixed point theory is one of the most powerful and fruitful tools of modern

mathematics and may be considered a core subject of nonlinear analysis.

The Brouwer fixed point theorem states that if

C is a nonempty compact convex

subset of a Euclidean space that it is fp  set.  Around 1950, most notably through the work of Nash (1950, 1951) on noncooperative games, and the work of Arrow and Debreu (1954) on general equilibrium

theory, it emerged that in economists' most basic and general models, equilibria

are fixed points. The most obvious consequence of this is that fixed

point theorems provide proofs that these models are not vacuous. But fixed

point theory also informs our understanding of many other issues such as comparative statics, robustness under perturbations, stability of equilibria with respect to dynamic adjustment  processes, etc.

The A-D model is one of the most general models of competitive economy and is a crucial part of general equilibrium theory, as it can be used to prove the existence of general equilibrium (or Walrasian equilibrium) of an economy. In general, there may be many equilibria; however, with extra assumptions on consumer preferences, namely that their utility functions be strongly concave and twice continuously differentiable, a unique equilibrium exists.

See Andrew McLennan, Advanced Fixed Point Theory for

Economics, April 8, 2014,

http://cupid.economics.uq.edu.au/mclennan/Advanced/advanced_fp.pdf

Dear all,

Fixed point theory is developed in abstract spaces such as Banach spaces or manifolds which are theoretical but it gains traction and lands in the markets of applied sciences as other parts of mathematics which dominate applied sciences today, in a reversible manner, where pure mathematics also gains nutrition from applied sciences to expand its territories.   

Mohammed I see you have read some of my academic contributions. How is Sokoto Uni? All the best - let me know if I can be of any help.