As far as I know the most known Ciric's result is in the paper

Proc. Amer. Math. Soc. 45 (1974), 267-273 (cited 745).

The abstract of this paper is enclosed:

Let $ T:M \to M$ be a mapping of a metric space $(M,d)$ into itself. A mapping $ T$ will be called a quasi-contraction iff

$ d(Tx,Ty) \leqslant q\max \{d(x,y);d(x,Tx);d(y,Ty);d(x,Ty);d(y,Tx)\}$

for some $ q < 1$ and all $ x,y \in M$. In the present paper the mappings of this kind are investigated. The results presented here show that the condition of quasi-contractivity implies all conclusions of Banach's contraction principle. Multi-valued quasi-contractions are also discussed.

In order to start discussion we will mention only a few well known results(Atiyah–Bott, Banach,Browder,Brouwer, Caristi, Kakutani, Knaster–Tarski,Lefschetz, Nielsen,Poincaré–Birkhoff,Schauder,Tychonoff).

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.[3]

the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. More precisely,

it states that any contraction mapping of a non-empty complete metric space into-self has a unique fixed point.

By contrast, 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, but it doesn't describe how to find the fixed point (See also Sperner's lemma).

The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology is notable because it gives, in some sense, a way to count fixed points.

Several converses of the Banach contraction principle exist. The following is due to Czeslaw Bessaga, from 1959:

\begin{thm}

Let $f : X \rightarrow X$ be a map of an abstract set such that each iterate $f^n$ has a unique fixed point. Let $q \in (0, 1)$, then there exists a complete metric on $X$

such that $f$ is contractive, and $q$ is the contraction constant.

\end{thm}

However, in most applications the existence and unicity of a fixed point can be shown directly with the standard Banach fixed point theorem,

by a suitable choice of the metric that makes the map T a contraction. Indeed, the above result by Bessaga strongly suggests to look for such a metric.

At first glance,  it seems that the next example contradicts Bessaga result?

Example.  Let $f(z)= e^{i \alpha} z$ be irrational rotation($\alpha= a\pi$, where $a$ is irrational) and let  $d$  be   a metric on the unit disk. Set   $ M=M(r) =\sup_{|z|=r} d(0,z)$ for 0

There is generalization of finite dimensional Browder fpt to

uniformly convex Banach spaces.

The Browder fixed point theorem is a refinement of the Banach fixed point

theorem for uniformly convex Banach spaces. It asserts that if K

{\displaystyle K} is a nonempty convex closed bounded set in uniformly

convex Banach space and f {\displaystyle f} is a mapping of K

{\displaystyle K} K into itself such that {\displaystyle \|f(x)-f(y)\|\leq

\|x-y\|} (i.e. {\displaystyle f} f is non-expansive), then f

{\displaystyle f} f has a fixed point.

Note that in this version we do not suppose neither that K is compact

nor f and therefore it is an important generalization of finite dimensional case.

Question.  Can we apply this theorem to holomorphic functions and subsets with

Caratheodory metric in corresponding situation.

We will add   a few  further information.

Following the publication in 1995 of two independent versions of the

theorem by Felix Browder and by William Kirk, a new proof of Michael

Edelstein showed that, in a uniformly convex Banach space, every iterative

sequence {\displaystyle f^{n}x_{0}} of a non-expansive map {\displaystyle

f} f has a unique asymptotic center, which is a fixed point of

{\displaystyle f} f. (An asymptotic center of a sequence {\displaystyle

(x_{k})_{k\in \mathbb {N} }}, if it exists, is a limit of the Chebyshev

centers {\displaystyle c_{n}} c_{n} for truncated sequences

{\displaystyle (x_{k})_{k\geq n}} {\displaystyle (x_{k})_{k\geq n}}.) A

stronger property than asymptotic center is Delta-limit of T.C. Lim, which

in the uniformly convex space coincides with the weak limit if the space

has the Opial property.

F. E. Browder, Nonexpansive nonlinear operators in a Banach space.

Proc. Natl. Acad. Sci. U.S.A. 54 (1965) 1041–1044

W. A. Kirk, A fixed point theorem for mappings which do not increase

distances, Amer. Math. Monthly 72 (1965) 1004–1006.

M. Edelstein, The construction of an asymptotic center with a

fixed-point property, Bull. Amer. Math. Soc. 78 (1972), 206-208.

The Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926.

The counting is subject to an imputed multiplicity at a fixed point called the fixed point index. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).

For a formal statement of the theorem, let

f : X → X {\displaystyle f:X\rightarrow X\,}

be a continuous map from a compact triangulable space X to itself. Define the Lefschetz number Λf of f by

Λ f := ∑ k ≥ 0 ( − 1 ) k T r ( f_ ∗ | H _k ( X , Q ) ) , {\displaystyle \Lambda _{f}:=\sum _{k\geq 0}(-1)^{k}\mathrm {Tr} (f_{*}|H_{k}(X,\mathbb {Q} )),}

the alternating (finite) sum of the matrix traces of the linear maps induced by f on the H_k(X,Q), the singular homology of X with rational coefficients.

A simple version of the Lefschetz fixed-point theorem states: if

Λ f ≠ 0 {\displaystyle \Lambda _{f}\neq 0\,}

then f has at least one fixed point, i.e. there exists at least one x in X such that f(x) = x. In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to f has a fixed point as well.

Note however that the converse is not true in general: Λf may be zero even if f has fixed points.

If   X and Y are two topological spaces with the same homotopy type, then

H n ( X ) = H n ( Y ) {\displaystyle H_{n}(X)=H_{n}(Y)\,}

for all n ≥ 0. This means homology groups are topological invariants.

In particular, if X is a connected contractible space, then all its homology groups are 0, except H_0 ( X ) = Z {\displaystyle H_{0}(X)=\mathbb {Z} } .

A proof for the homotopy invariance of singular homology groups can be sketched as follows. A continuous map f: X → Y induces a homomorphism

f ♯ : C_ n ( X ) → C _n ( Y ) . {\displaystyle f_{\sharp }:C_{n}(X)\rightarrow C_{n}(Y).}