Berinde Weak Contraction Principles

• Berinde weak contraction principles are a family of fixed-point conditions that relax Banach's strict inequality using comparison functions and altered distances.
• They employ methodologies such as weak φ-contractions, interpolative and derivative-type variants that guarantee Picard iteration convergence under completeness.
• Applications range from solving Fredholm integral equations to characterizing dynamical systems, emphasizing uniqueness and convergence in metric and Banach spaces.

Searching arXiv for the cited papers and closely related Berinde weak contraction work. arxiv_search.query({"^^^^^^^^2search_query2 OR ti:\2^^^^^^^^"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 PRESERVED_PLACEHOLDER_^^^^2search_query2^^^^-contraction, weak PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^-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 (&&&^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^&&&).

^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^. 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 PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2^^^^-contraction when

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^

Berinde’s 2^^^^2search_query2search_query2^^^^4 paper also uses the more specialized weak PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22^^^^-contraction form

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^3

with PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^4 and PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^5. When PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^6, this recovers the Banach contraction with constant PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^7 (&&&^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^&&&).

A second formulation, presented as the “classical Berinde weak contraction” in a normed-space setting, requires nonnegative constants PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^8 satisfying

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^9

and

$\varphi'(d(\cdot,\cdot))$^^^^2search_query2^^^^

When $\varphi'(d(\cdot,\cdot))$^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^, one recovers the Banach contraction $\varphi'(d(\cdot,\cdot))$2 (Abbas et al., 2022).

Formulation	Contractive condition	Limiting case
Weak $\varphi'(d(\cdot,\cdot))$3-contraction	$\varphi'(d(\cdot,\cdot))$4	Strict distance drop
Weak $\varphi'(d(\cdot,\cdot))$5-contraction	$\varphi'(d(\cdot,\cdot))$6	$\varphi'(d(\cdot,\cdot))$7 gives Banach
Classical Berinde weak contraction	$\varphi'(d(\cdot,\cdot))$8	$\varphi'(d(\cdot,\cdot))$9 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

$\varphi$^^^^2search_query2^^^^

is itself an interpolation between Banach’s and Kannan’s conditions (&&&^^^^2search_query2^^^^&&&).

2. Fixed-point conclusions and Picard behavior

The basic fixed-point conclusion attached to Berinde’s original weak $\varphi$^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^-contraction is that, on a complete metric space $\varphi$2, $\varphi$3 is a strong Picard operator: for every $\varphi$4, the Picard iteration $\varphi$5 converges to a fixed point of $\varphi$6 (&&&^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^&&&). 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 $\varphi$7 is complete, $\varphi$8 is induced by an altering function $\varphi$9, $(X,d)$^^^^2search_query2^^^^ is strictly subunitary and right-upper-semicontinuous in the Boyd–Wong sense, and $(X,d)$^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^ satisfies

$(X,d)$2

then $(X,d)$3 has exactly one fixed point and every Picard orbit converges to it; the map is a globally strong Picard operator (&&&^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^&&&).

An analogous global conclusion holds in the Suzuki–Berinde setting on Banach spaces. If $(X,d)$4 is a Banach space and the implication-type condition

$(X,d)$5

holds for every $(X,d)$6, then $(X,d)$7 has exactly one fixed point $(X,d)$8, and for every $(X,d)$9 the Picard iteration converges to $\varphi:[0,\infty)\to[0,\infty)$^^^^2search_query2^^^^ (Abbas et al., 2022).

A recurrent proof pattern runs through these results. One defines $\varphi:[0,\infty)\to[0,\infty)$^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^, 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 $\varphi:[0,\infty)\to[0,\infty)$2 and $\varphi:[0,\infty)\to[0,\infty)$3, the map $\varphi:[0,\infty)\to[0,\infty)$4 is an interpolative Berinde weak contraction if there exist constants

$\varphi:[0,\infty)\to[0,\infty)$5

such that for all $\varphi:[0,\infty)\to[0,\infty)$6,

$\varphi:[0,\infty)\to[0,\infty)$7

Here $\varphi:[0,\infty)\to[0,\infty)$8 denotes the fixed-point set. The derivative-type version replaces the metric argument by the positive derivative of a gauge function: if $\varphi:[0,\infty)\to[0,\infty)$9 is continuously differentiable and $\varphi(0)=0$^^^^2search_query2^^^^ for all $\varphi(0)=0$^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^, then $\varphi(0)=0$2 is called an interpolative Berinde weak operator of the derivative type if

$\varphi(0)=0$3

for all $\varphi(0)=0$4. In the special case $\varphi(0)=0$5, one recovers exactly the non-derivative interpolative Berinde condition (&&&^^^^2search_query2^^^^&&&).

The main theorem in this derivative-type setting assumes that $\varphi(0)=0$6 is a complete metric space which is $\varphi(0)=0$7-bounded, meaning

$\varphi(0)=0$8

Under the derivative-type interpolative inequality, $\varphi(0)=0$9 has exactly one fixed point $\varphi(t)>0$^^^^2search_query2^^^^, and for every $\varphi(t)>0$^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^, the Picard iteration $\varphi(t)>0$2 converges to $\varphi(t)>0$3 (&&&^^^^2search_query2^^^^&&&).

The proof proceeds from the estimate

$\varphi(t)>0$4

obtained by induction with $\varphi(t)>0$5. Summing over $\varphi(t)>0$6 and using the triangle inequality yields $\varphi(t)>0$7 as $\varphi(t)>0$8. Since $\varphi(t)>0$9 is strictly positive on $t>0$^^^^2search_query2^^^^, this forces $t>0$^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^, so $t>0$2 is Cauchy. Completeness gives a limit $t>0$3, and passing to the limit in the contractive inequality shows $t>0$4. Uniqueness is obtained by assuming two fixed points $t>0$5 and deriving

$t>0$6

which contradicts $t>0$7 in the proof outline given in the source (&&&^^^^2search_query2^^^^&&&).

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 $t>0$8 by

$t>0$9

where $T:X\to X$^^^^2search_query2^^^^ is continuous and strictly increasing, with $T:X\to X$^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^ and $T:X\to X$2 for all $T:X\to X$3. The resulting function $T:X\to X$4 is symmetric and reflexive-sufficient, and although it need not satisfy the triangle-law, it preserves the ordering of distances and has the implication

$T:X\to X$5

Turinici introduces

$T:X\to X$6

and

$T:X\to X$7

The contractive requirement becomes

$T:X\to X$8

where $T:X\to X$9 is strictly subunitary and right-upper-semicontinuous in the Boyd–Wong sense (&&&^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^&&&).

Under completeness of PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2search_query2^^^^, 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 PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^-contraction requires only PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query22^^^^, PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2^^^^3, whereas the altering-metric theorem gains uniqueness by imposing a stronger global control condition, namely that the combined coefficient remains PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2^^^^4 and satisfies the Boyd–Wong semicontinuity requirement (&&&^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^&&&).

The same paper also returns to the plain metric PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2^^^^5. If PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2^^^^6 satisfy PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2^^^^7 and PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2^^^^8 is continuous at PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2^^^^9 with PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2search_query2^^^^, then

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^

implies that PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\22^^^^ is a strong Picard operator in PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^3. When the fixed-point set is a singleton, the paper also gives the explicit error estimate

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^4

5. Suzuki–Berinde contractions and completeness of Banach spaces

In the normed-space framework, a Suzuki–Berinde type contraction is defined from a parameter PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^5 by setting

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^6

and introducing the piecewise function

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^7

With PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^8, the defining implication is

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^9

This framework is presented as a large class of contractive mappings that unifies, generalizes and complements various known comparable results (Abbas et al., 2022).

The fixed-point theorem states that if PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22search_query2^^^^ is a Banach space and there are constants PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\222^^^^ with PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22^^^^3, PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22^^^^4, and PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22^^^^5 as above, such that the implication holds for every PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22^^^^6, then PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22^^^^7 has exactly one fixed point PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22^^^^8, and for every PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\22^^^^9 the Picard iteration PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2max_results2search_query2^^^^ converges to PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2max_results2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^ (Abbas et al., 2022).

A distinctive feature of this theory is its completeness characterization. If PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^32 denotes the family of self-maps satisfying the Suzuki–Berinde implication, then for a normed space PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^33 the following are equivalent: PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^34 is complete; for every PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^35, every PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^36 has a fixed point; and there exists some PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^37 such that every PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^38 has a fixed point (Abbas et al., 2022). 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 PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^39, where PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2sort_by2search_query2^^^^ is the family of nonempty closed bounded subsets of a Banach space. Using the Pompeiu–Hausdorff metric

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2sort_by2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^

the multivalued implication

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^42

yields at least one fixed point PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^43, meaning PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^44 (Abbas et al., 2022).

The framework is also explicitly positioned as a unification device. By suitable specializations of PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^45, and PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^46, the paper states that one recovers classical theorems of Banach, Suzuki, Edelstein, Berinde, Kannan, Chatterjea, and Ćirić. In particular, setting PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^47 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 PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^48 with the usual metric PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^49, the map

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2submittedDate2search_query2^^^^

and

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2submittedDate2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^

For all PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^52,

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^53

so the derivative-type interpolative condition holds with PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^54. All hypotheses of the theorem are satisfied, and the unique fixed point is PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^55 (&&&^^^^2search_query2^^^^&&&).

The same paper applies the theorem to the nonlinear Fredholm integral equation

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^56

Let PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^57 with the sup-metric

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^58

and define

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^59

If PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2sort_order2search_query2^^^^ satisfies, for all PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2sort_order2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^62,

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^63

then one obtains

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^64

Hence PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^65 is an interpolative Berinde derivative-type contraction on the complete space PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^66, and there is a unique continuous solution PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^67 of the integral equation (&&&^^^^2search_query2^^^^&&&).

The source also states why this derivative reformulation can be useful in applications. The new feature is the replacement of all occurrences of PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^68 by the positive derivative PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^69. This allows one to handle mappings that are not Lipschitz in the ordinary sense but whose “rate of change” can be controlled via PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2descending2search_query2^^^^. The paper further notes that, in applications such as the Fredholm equation, it is often easier to verify a bound on PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2descending2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^ than on PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^72 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

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^73

with PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^74 convex and PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^75 the matrix measure induced by a norm, the system is weakly contracting on PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^76 if

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^77

for all PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^78. Equivalently, along any two trajectories PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^79,

PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^8^^^^2search_query2^^^^

Thus distances are non-increasing but need not decrease strictly (&&&^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^9&&&).

For autonomous systems PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^8^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^ on a convex invariant PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^82, twice continuously differentiable and satisfying PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^83 on PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^84, the cited result gives a dichotomy: either there exists at least one equilibrium PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^85, in which case every trajectory in PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^86 is bounded; or no equilibrium lies in PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^87, in which case every trajectory in PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^88 is unbounded. Under additional hypotheses, such as piecewise real-analyticity in a weighted PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^89 or PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^9^^^^2search_query2^^^^ setting, or strict negativity of PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^9^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^ 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 (&&&^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^9&&&).

The source explicitly interprets this as a dynamic analogue of Berinde’s weak contraction principle: PRESERVED_PLACEHOLDER_^^^^2id:(Ampadu, 22 Mar 2026) OR ti:\2^^^^92 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.