---
title: Berinde Weak Contraction Principles
url: https://www.emergentmind.com/topics/berinde-weak-contraction-principles
type: topic
---

# Berinde Weak Contraction Principles

Searching arXiv for the cited papers and closely related Berinde weak contraction work.
arxiv_search.query({"search_query":"id:2604.14157 OR ti:\"Derivative Type Mapping Theorem for the Interpolative Berinde Weak Contraction in Metric Spaces with Application\"","max_results":5,"sort_by":"submittedDate","sort_order":"descending"})
Berinde weak contraction principles are fixed-point conditions for self-maps on metric or normed spaces that weaken the strict Banach inequality by introducing comparison functions, auxiliary distance terms, implication-type hypotheses, or transformed distance gauges, while retaining Picard-iteration conclusions under appropriate completeness assumptions. In the supplied literature, the name covers Berinde’s original weak \((a,A)\)-contraction, weak \(\varphi\)-contractions, interpolative Berinde weak contractions, derivative-type variants based on \(\varphi'(d(\cdot,\cdot))\), and Suzuki–Berinde extensions that also characterize completeness of Banach spaces; a related altering-metric framework is described as “almost” covering Berinde’s metric-type theorem [1302.4013].

## 1. Classical metric-space formulations

The literature uses several closely related formulations under the Berinde weak-contraction label. One version is the weak \(\varphi\)-contraction: if \((X,d)\) is a metric space and \(\varphi:[0,\infty)\to[0,\infty)\) is continuous and strictly increasing, with \(\varphi(0)=0\) and \(\varphi(t)>0\) for all \(t>0\), then \(T:X\to X\) is a weak \(\varphi\)-contraction when
\[
d\bigl(Tx,Ty\bigr)\le d(x,y)-\varphi\bigl(d(x,y)\bigr).
\]
Berinde’s 2004 paper also uses the more specialized weak \((a,A)\)-contraction form
\[
d(Tx,Ty)\le a\,d(x,y)+A\,d(Tx,y),\qquad \forall x,y\in X,
\]
with \(a\in[0,1)\) and \(A\ge 0\). When \(A=0\), this recovers the Banach contraction with constant \(a\) [1302.4013].

A second formulation, presented as the “classical Berinde weak contraction” in a normed-space setting, requires nonnegative constants \(\alpha,\beta,\gamma\) satisfying
\[
\alpha+\beta+\gamma<1
\]
and
\[
d(Tx,Ty)\le \alpha\,d(x,Tx)+\beta\,d(y,Ty)+\gamma\,d(x,y),\qquad \forall x,y\in X.
\]
When \(\alpha=\beta=0\), one recovers the Banach contraction \(d(Tx,Ty)\le \gamma\,d(x,y)\) [2209.12554].

| Formulation | Contractive condition | Limiting case |
|---|---|---|
| Weak \(\varphi\)-contraction | \(d(Tx,Ty)\le d(x,y)-\varphi(d(x,y))\) | Strict distance drop |
| Weak \((a,A)\)-contraction | \(d(Tx,Ty)\le a\,d(x,y)+A\,d(Tx,y)\) | \(A=0\) gives Banach |
| Classical Berinde weak contraction | \(d(Tx,Ty)\le \alpha d(x,Tx)+\beta d(y,Ty)+\gamma d(x,y)\) | \(\alpha=\beta=0\) gives Banach |

These formulations share the same structural aim: they relax pure Lipschitz contraction while still controlling successive Picard iterates. The data also states that the original Berinde weak contraction of the form
\[
d(Tx,Ty)\le \lambda\,d(x,y)^\alpha d(x,Tx)^{1-\alpha}
\]
is itself an interpolation between Banach’s and Kannan’s conditions [2604.14157].

## 2. Fixed-point conclusions and Picard behavior

The basic fixed-point conclusion attached to Berinde’s original weak \((a,A)\)-contraction is that, on a complete metric space \((X,d)\), \(T\) is a strong Picard operator: for every \(x_0\in X\), the Picard iteration \(x_{n+1}=Tx_n\) converges to a fixed point of \(T\) [1302.4013]. In this usage, the emphasis is on convergence of the iterates rather than uniqueness.

Stronger conclusions appear once the contractive hypothesis is sharpened. In Turinici’s altering-metric theorem, if \((X,d)\) is complete, \(e(x,y)=\psi(d(x,y))\) is induced by an altering function \(\psi\), \(\phi:[0,\infty)\to[0,1)\) is strictly subunitary and right-upper-semicontinuous in the Boyd–Wong sense, and \(T\) satisfies
\[
e(Tx,Ty)\le \phi(d(x,y))\,M(x,y),\qquad x\neq y,
\]
then \(T\) has exactly one fixed point and every Picard orbit converges to it; the map is a globally strong Picard operator [1302.4013].

An analogous global conclusion holds in the Suzuki–Berinde setting on Banach spaces. If \((X,\|\cdot\|)\) is a Banach space and the implication-type condition
\[
v(r)\,\|x-Tx\|\le \|x-y\|
\quad\Longrightarrow\quad
\|\,b(x-y)+Tx-Ty\|\le r\,\|x-y\|
\]
holds for every \(x,y\in X\), then \(T\) has exactly one fixed point \(z\in X\), and for every \(x_0\in X\) the Picard iteration converges to \(z\) [2209.12554].

A recurrent proof pattern runs through these results. One defines \(x_{n+1}=Tx_n\), proves that successive distances decrease or satisfy a geometric estimate, obtains the Cauchy property, invokes completeness, and then shows that the limit is fixed. Uniqueness is typically deduced by applying the contractive inequality directly to two hypothetical fixed points.

## 3. Interpolative Berinde weak contractions and the derivative-type theorem

For a metric space \((X,d)\) and \(T:X\to X\), the map \(T\) is an interpolative Berinde weak contraction if there exist constants
\[
\lambda\in[0,1),\qquad \alpha\in(0,1)
\]
such that for all \(x,y\in X\setminus\Fix(T)\),
\[
d\bigl(Tx,Ty\bigr)\le \lambda\,\bigl[d(x,y)\bigr]^{\alpha}\bigl[d(x,Tx)\bigr]^{1-\alpha}.
\]
Here \(\Fix(T)\) denotes the fixed-point set. The derivative-type version replaces the metric argument by the positive derivative of a gauge function: if \(\varphi:\mathbb{R}^+\to\mathbb{R}^+\) is continuously differentiable and \(\varphi'(t)>0\) for all \(t>0\), then \(T\) is called an interpolative Berinde weak operator of the derivative type if
\[
\varphi'\bigl(d(Tx,Ty)\bigr)\le
\lambda\,
\Bigl[\varphi'\bigl(d(x,y)\bigr)\Bigr]^{\alpha}
\Bigl[\varphi'\bigl(d(x,Tx)\bigr)\Bigr]^{1-\alpha},
\]
for all \(x,y\in X\setminus\Fix(T)\). In the special case \(\varphi(t)=t\), one recovers exactly the non-derivative interpolative Berinde condition [2604.14157].

The main theorem in this derivative-type setting assumes that \((X,d)\) is a complete metric space which is \(d\)-bounded, meaning
\[
\delta_d(X)=\sup\{d(x,y):x,y\in X\}<\infty.
\]
Under the derivative-type interpolative inequality, \(T\) has exactly one fixed point \(x^*\in X\), and for every \(x\in X\), the Picard iteration \(x_{n+1}=T(x_n)\) converges to \(x^*\) [2604.14157].

The proof proceeds from the estimate
\[
\varphi'\bigl(d(x_n,x_{n+1})\bigr)\le \lambda^n\,\varphi'\bigl(d(x_0,x_1)\bigr)\le \lambda^n\,\varphi'\bigl(\delta_d(X)\bigr),
\]
obtained by induction with \(x_{n+1}=T(x_n)\). Summing over \(k=n,\dots,n+m-1\) and using the triangle inequality yields \(\varphi'(d(x_n,x_{n+m}))\to 0\) as \(n,m\to\infty\). Since \(\varphi'\) is strictly positive on \((0,\infty)\), this forces \(d(x_n,x_{n+m})\to 0\), so \(\{x_n\}\) is Cauchy. Completeness gives a limit \(x^*\), and passing to the limit in the contractive inequality shows \(x^*=T(x^*)\). Uniqueness is obtained by assuming two fixed points \(u\neq v\) and deriving
\[
\varphi'(d(u,v))\le \lambda\,[\varphi'(d(u,v))]^\alpha[\varphi'(0)]^{1-\alpha}=0,
\]
which contradicts \(u\neq v\) in the proof outline given in the source [2604.14157].

The paper situates this theorem between two earlier directions: Olatinwo introduced contractive definitions of the derivative type and a new characterization of the Banach contraction principle, while Ampadu et al. introduced derivative type contractions in multiplicative metric spaces. The derivative-type interpolative Berinde theorem is presented there as a unification and extension of those lines of work.

## 4. Altering metrics and the “almost cover” of Berinde’s theorem

The altering-metric approach replaces the original metric \(d\) by
\[
e(x,y)=\psi(d(x,y)),
\]
where \(\psi:[0,\infty)\to[0,\infty)\) is continuous and strictly increasing, with \(\psi(0)=0\) and \(\psi(t)>0\) for all \(t>0\). The resulting function \(e\) is symmetric and reflexive-sufficient, and although it need not satisfy the triangle-law, it preserves the ordering of distances and has the implication
\[
e(x_n,y_n)\to 0 \Longrightarrow d(x_n,y_n)\to 0.
\]
Turinici introduces
\[
M_1(x,y)=e(x,y),\quad
M_2(x,y)=\tfrac12\bigl[e(x,Tx)+e(y,Ty)\bigr],\quad
M_3(x,y)=\min\{e(x,Ty),e(Tx,y)\},
\]
and
\[
M(x,y)=\max\{M_1(x,y),M_2(x,y),M_3(x,y)\}.
\]
The contractive requirement becomes
\[
e(Tx,Ty)\le \phi(d(x,y))\,M(x,y),\qquad \forall x\neq y\in X,
\]
where \(\phi:[0,\infty)\to[0,1)\) is strictly subunitary and right-upper-semicontinuous in the Boyd–Wong sense [1302.4013].

Under completeness of \((X,d)\), this yields existence of exactly one fixed point and convergence of every Picard orbit. The source explicitly presents this theorem as one that “almost” covers Berinde’s weakly contractive metric-type result. The comparison is precise: Berinde’s weak \((a,A)\)-contraction requires only \(a\in[0,1)\), \(A\ge 0\), whereas the altering-metric theorem gains uniqueness by imposing a stronger global control condition, namely that the combined coefficient remains \(<1\) and satisfies the Boyd–Wong semicontinuity requirement [1302.4013].

The same paper also returns to the plain metric \(d\). If \(a,b\ge 0\) satisfy \(a+2b<1\) and \(K:[0,\infty)\to[0,\infty)\) is continuous at \(0\) with \(K(0)=0\), then
\[
d(Tx,Ty)\le
a\,d(x,y)+b\bigl[d(x,Tx)+d(y,Ty)\bigr]+K\bigl(d(Tx,y)\bigr)
\]
implies that \(T\) is a strong Picard operator in \((X,d)\). When the fixed-point set is a singleton, the paper also gives the explicit error estimate
\[
d(x_n,z)\le \frac{(a+b)^n}{1-(a+b)}\,d(x_0,Tx_0).
\]

## 5. Suzuki–Berinde contractions and completeness of Banach spaces

In the normed-space framework, a Suzuki–Berinde type contraction is defined from a parameter \(b\ge 0\) by setting
\[
A=\frac{b}{b+1},\qquad r=A,\quad 0\le r<1,
\]
and introducing the piecewise function
\[
v(r)=
\begin{cases}
\displaystyle \frac{r}{1-r}, & 0<r<\tfrac{\sqrt5-1}{2},\\[6pt]
\displaystyle \tfrac1{\sqrt2}, & \tfrac{\sqrt5-1}{2}\le r<\tfrac12,\\[6pt]
\displaystyle \frac{1+r}{1-r}, & \tfrac12\le r<1.
\end{cases}
\]
With \(\theta=r\), the defining implication is
\[
v(r)\,\|x-Tx\|\le \|x-y\|
\quad\Longrightarrow\quad
\|\,b(x-y)+Tx-Ty\|\le r\,\|x-y\|.
\]
This framework is presented as a large class of contractive mappings that unifies, generalizes and complements various known comparable results [2209.12554].

The fixed-point theorem states that if \((X,\|\cdot\|)\) is a Banach space and there are constants \(b\ge 0\) and \(\theta\in[0,b+1)\) with \(r=\theta\in[0,1)\), \(A=b/(b+1)\), and \(v\) as above, such that the implication holds for every \(x,y\in X\), then \(T\) has exactly one fixed point \(z\in X\), and for every \(x_0\in X\) the Picard iteration \(x_{n+1}=T(x_n)\) converges to \(z\) [2209.12554].

A distinctive feature of this theory is its completeness characterization. If \(\mathcal T_{b,r}\) denotes the family of self-maps satisfying the Suzuki–Berinde implication, then for a normed space \((X,\|\cdot\|)\) the following are equivalent: \(X\) is complete; for every \(r\in[0,1)\), every \(T\in\mathcal T_{b,r}\) has a fixed point; and there exists some \(r\in(0,1)\) such that every \(T\in\mathcal T_{b,r}\) has a fixed point [2209.12554]. This is stronger than a single fixed-point theorem: it turns the contractive class itself into a criterion for Banach-space completeness.

The same paper extends the theory to multivalued mappings \(T:X\to CB(X)\), where \(CB(X)\) is the family of nonempty closed bounded subsets of a Banach space. Using the Pompeiu–Hausdorff metric
\[
H(A,B)=\max\left\{\sup_{a\in A}d(a,B)\;;\;\sup_{b\in B}d(b,A)\right\},
\]
the multivalued implication
\[
v(r)\,d(x,T(x))\le \|x-y\|
\quad\Longrightarrow\quad
H\bigl(bx+T(x),\,by+T(y)\bigr)\le r\,\|x-y\|
\]
yields at least one fixed point \(z\in X\), meaning \(z\in T(z)\) [2209.12554].

The framework is also explicitly positioned as a unification device. By suitable specializations of \(b,r\), and \(v(r)\), the paper states that one recovers classical theorems of Banach, Suzuki, Edelstein, Berinde, Kannan, Chatterjea, and Ćirić. In particular, setting \(b=0\) recovers Suzuki’s generalized Banach contraction in the Banach-space theorem and Suzuki’s variant of Edelstein’s theorem in the compact setting.

## 6. Examples and application to Fredholm integral equations

The derivative-type interpolative framework is illustrated on \(X=[0,1]\) with the usual metric \(d(x,y)=|x-y|\), the map
\[
T(x)=x^{2/3},
\]
and
\[
\varphi(t)=t^3,\qquad \varphi'(t)=3t^2.
\]
For all \(x,y\in(0,1]\),
\[
3\,|x^{2/3}-y^{2/3}|^{2}
\le
\tfrac12 \bigl[3\,|x-y|^2\bigr]^{1/2}
\bigl[3\,|x-x^{2/3}|^2\bigr]^{1/2},
\]
so the derivative-type interpolative condition holds with \(\lambda=\alpha=\tfrac12\). All hypotheses of the theorem are satisfied, and the unique fixed point is \(x^*=0\) [2604.14157].

The same paper applies the theorem to the nonlinear Fredholm integral equation
\[
u(t)=v(t)+\int_0^1 K\bigl(t,s,u(s)\bigr)\,ds.
\]
Let \(Y=C([0,1])\) with the sup-metric
\[
p(u,v)=\max_{t\in[0,1]}|u(t)-v(t)|,
\]
and define
\[
(Fu)(t)=v(t)+\int_0^1 K\bigl(t,s,u(s)\bigr)\,ds.
\]
If \(K\) satisfies, for all \(u,v\in Y\) and \(s,t\in[0,1]\),
\[
\varphi'\!\bigl(|K(t,s,u(s))-K(t,s,v(s))|\bigr)
\le
\lambda
\Bigl[\varphi'(|u(s)-v(s)|)\Bigr]^\alpha
\Bigl[\varphi'(|u(s)-Fu(s)|)\Bigr]^{1-\alpha},
\]
then one obtains
\[
\varphi'\bigl(p(Fu,Fv)\bigr)\le
\lambda\,
\Bigl[\varphi'(p(u,v))\Bigr]^\alpha
\Bigl[\varphi'(p(u,Fu))\Bigr]^{1-\alpha}.
\]
Hence \(F\) is an interpolative Berinde derivative-type contraction on the complete space \(Y\), and there is a unique continuous solution \(u\in Y\) of the integral equation [2604.14157].

The source also states why this derivative reformulation can be useful in applications. The new feature is the replacement of all occurrences of \(d(\cdot,\cdot)\) by the positive derivative \(\varphi'(d(\cdot,\cdot))\). This allows one to handle mappings that are not Lipschitz in the ordinary sense but whose “rate of change” can be controlled via \(\varphi'\). The paper further notes that, in applications such as the Fredholm equation, it is often easier to verify a bound on \(\varphi'(|K(u)-K(v)|)\) than on \(|K(u)-K(v)|\) itself.

## 7. Broader interpretations and relation to dynamical weak contraction

The supplied literature also presents a continuous-time notion of weak contraction that is explicitly compared with Berinde’s metric fixed-point theory. For a time-varying system
\[
\dot x=f(t,x),\qquad t\ge 0,\ x\in C\subseteq \mathbb R^n,
\]
with \(C\) convex and \(\mu\) the matrix measure induced by a norm, the system is weakly contracting on \(C\) if
\[
\mu(D_x f(t,x))\le 0
\]
for all \((t,x)\in\mathbb R_+\times C\). Equivalently, along any two trajectories \(x(t),y(t)\),
\[
\|x(t)-y(t)\|\le \|x(0)-y(0)\|,\qquad t\ge 0.
\]
Thus distances are non-increasing but need not decrease strictly [2005.09774].

For autonomous systems \( \dot x=f(x)\) on a convex invariant \(C\), twice continuously differentiable and satisfying \(\mu(Df(x))\le 0\) on \(C\), the cited result gives a dichotomy: either there exists at least one equilibrium \(x^*\in C\), in which case every trajectory in \(C\) is bounded; or no equilibrium lies in \(C\), in which case every trajectory in \(C\) is unbounded. Under additional hypotheses, such as piecewise real-analyticity in a weighted \(\ell_1\) or \(\ell_\infty\) setting, or strict negativity of \(\mu(Df(x^*))\) at an equilibrium, one obtains convergence to an equilibrium. For doubly-contracting systems, every trajectory converges exponentially to a unique equilibrium in a subspace of equilibrium points [2005.09774].

The source explicitly interprets this as a dynamic analogue of Berinde’s weak contraction principle: \(\mu(Df)\le 0\) yields either no equilibrium and unbounded trajectories, or existence of an equilibrium and boundedness of all trajectories. This suggests a conceptual parallel rather than an identity of theories. In both settings, strict contraction is relaxed to non-expansion, and additional structure is what restores uniqueness or full asymptotic convergence.

Across these developments, the Berinde weak contraction principle is less a single inequality than a family of contractive paradigms. The family includes direct metric estimates, implication-based norm inequalities, altering-metric control schemes, and derivative-gauge formulations; the common outcome is that fixed-point existence, uniqueness, or iterative convergence can still be established under hypotheses that are weaker or differently structured than the classical Banach contraction.

Source: https://www.emergentmind.com/topics/berinde-weak-contraction-principles