Multivalued Weak Contractions

• Multivalued weak contractions are a class of contractive principles for set-valued maps that replace uniform linear factors with weaker control mechanisms such as comparison functions and summability conditions.
• The theory employs selection-based schemes and alternating iterative mechanisms to achieve convergence to fixed points, endpoints, and invariant sets in both symmetric and oriented metric spaces.
• Extensions to quasi-pseudometric, ordered, and semi-metric frameworks highlight unique endpoint properties and clarify distinctions between fixed points and startpoints.

Multivalued weak contractions are contractive principles for set-valued mappings that relax classical Hausdorff-Lipschitz hypotheses while retaining existence theory for fixed points, endpoints, startpoints, or invariant families of sets. In the cited literature, the term covers several non-equivalent but structurally related schemes: Hausdorff-type weak contractions on complete metric spaces, dual weak contractions for pairs of multivalued maps, oriented weak contractions on quasi-pseudometric spaces, weak contractions valued in partially ordered groups, ordered-metric variants built from $\delta$ and $D$, and Matkowski-type weak contractions of hyperspace operators on semi-metric spaces (Cheng, 2011, Agyingi et al., 2018, Okamura, 30 Mar 2025). What unifies these formulations is the replacement of a uniform linear contraction factor by weaker control mechanisms such as comparison functions, order-sensitive distances, selection rules, or summability conditions.

1. Basic formulations and defining inequalities

A standard metric-space framework takes a multivalued map $T:X\to CB(X)$, where $CB(X)$ denotes the family of all nonempty, closed and bounded subsets of a complete metric space $(X,d)$, and equips $CB(X)$ with the Hausdorff metric

$$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$

In this setting, one notion of weak contractivity requires the existence of a compactly positive bivariate function $\psi:X\times X\to[0,+\infty)$ such that

$H(Tx,Ty)\le d(x,y)-\psi(x,y),$

while a generalized $p$-weak contraction is defined by

$D$0

with $D$1, $D$2 for $D$3, and

$D$4

These are the basic metric formulations used for single multivalued maps and for dual pairs of maps (Cheng, 2011).

A different metric-space formulation replaces global Hausdorff control by a localized selection principle. For a multifunction $D$5, one requires constants $D$6 with $D$7 such that for each $D$8 there exists $D$9 satisfying

$T:X\to CB(X)$0

Related variants use nearest-point selections,

$T:X\to CB(X)$1

or the weaker condition

$T:X\to CB(X)$2

This formulation is explicitly presented as weaker than Nadler’s Hausdorff contraction, because it imposes only a one-step decrease of point-to-set distances along selected orbits and makes no global Lipschitz demand in the Hausdorff metric (Pasicki, 2021).

In quasi-pseudometric spaces, the contractive inequality becomes oriented. If $T:X\to CB(X)$3 is a quasi-pseudometric space and $T:X\to CB(X)$4, then for a $T:X\to CB(X)$5-comparison function $T:X\to CB(X)$6 satisfying

$T:X\to CB(X)$7

together with the summability condition

$T:X\to CB(X)$8

$T:X\to CB(X)$9 is weakly contractive if for each $CB(X)$0 there exists $CB(X)$1 such that

$CB(X)$2

Here $CB(X)$3 is the directional Hausdorff quasi-pseudometric induced by $CB(X)$4 (Agyingi et al., 2018).

The term also appears in settings where Hausdorff distance is replaced altogether. In complete metric spaces $CB(X)$5 with multivalued maps $CB(X)$6, one uses

$CB(X)$7

together with

$CB(X)$8

$CB(X)$9

and the generalized weakly contractive inequality

$(X,d)$0

for $(X,d)$1, $(X,d)$2, $(X,d)$3 (Abkar et al., 2011).

This range of formulations shows that “multivalued weak contraction” is not a single canonical axiom but a family of contractive templates parameterized by the ambient geometry, the distance functional, and the target notion of solution.

2. Endpoints, startpoints, fixed points, and approximate endpoint properties

For a multivalued map $(X,d)$4, a fixed point is a point $(X,d)$5 with $(X,d)$6, whereas an endpoint is a point $(X,d)$7 with $(X,d)$8. The endpoint condition is therefore strictly stronger than fixed-point membership. In metric settings, an approximate endpoint property for a single map is commonly stated as

$(X,d)$9

and for a pair $CB(X)$0 as

$CB(X)$1

These quantities play a central role in endpoint existence theorems and in characterizations of when weakly contractive maps admit endpoints (Cheng, 2011).

In quasi-pseudometric spaces, the non-symmetry of $CB(X)$2 produces two oriented notions. A startpoint of $CB(X)$3 is defined by

$CB(X)$4

whereas an endpoint is defined by

$CB(X)$5

Because $CB(X)$6 and its conjugate $CB(X)$7 need not coincide, these conditions generally differ. In the symmetric case, however, the two notions collapse, and if $CB(X)$8 is closed and both directional equalities vanish, then one has $CB(X)$9, so the point is a fixed point in the usual sense (Agyingi et al., 2018).

Several papers emphasize uniqueness phenomena. In the ordered-group-valued metric setting, endpoints of a multivalued weak contraction are unique whenever they exist, so $$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$0 (Cheng, 2014). In the dual generalized weak contraction setting, one has

$$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$1

and if either $$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$2 or $$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$3 is single-valued, then endpoints and fixed points coincide (Cheng, 2011). In the ordered metric framework based on $$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$4 and $$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$5, uniqueness of a common end point is obtained under a comparative condition requiring the end points of the two mappings to be comparable in the underlying order (Nazir et al., 2023).

A recurrent source of confusion is the identification of endpoints with fixed points. The cited results systematically distinguish them: fixed-point existence can hold without singleton-valuedness at the solution, whereas endpoint theorems force the value set to collapse to exactly one point.

3. Existence theorems and iterative mechanisms in complete metric spaces

In complete metric spaces, one major line of results concerns pairs of multivalued maps. If $$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$6 form a duality of generalized weak contractions, meaning that

$$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$7

for

$$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$8

then under upper semicontinuity and sequence-wise boundedness assumptions on $$H(A,B)=\max\Big\{\sup_{a\in A}\inf_{b\in B}d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\Big\}.$$9, the pair has a common fixed point on a complete metric space. An analogous theorem holds for dual generalized $\psi:X\times X\to[0,+\infty)$0-weak contractions

$\psi:X\times X\to[0,+\infty)$1

provided $\psi:X\times X\to[0,+\infty)$2 is upper semicontinuous, satisfies $\psi:X\times X\to[0,+\infty)$3, $\psi:X\times X\to[0,+\infty)$4 for $\psi:X\times X\to[0,+\infty)$5, and

$\psi:X\times X\to[0,+\infty)$6

Common endpoints are obtained under corresponding approximate endpoint assumptions, and the endpoint is unique (Cheng, 2011).

The proof pattern is an alternating-selection scheme. Starting from $\psi:X\times X\to[0,+\infty)$7, one chooses points alternately in $\psi:X\times X\to[0,+\infty)$8 and $\psi:X\times X\to[0,+\infty)$9 and derives recursive estimates of the form

$H(Tx,Ty)\le d(x,y)-\psi(x,y),$0

for some $H(Tx,Ty)\le d(x,y)-\psi(x,y),$1. A technical lemma then yields that the sequence is Cauchy; completeness gives convergence; upper semicontinuity of the contractive control identifies the limit as a common fixed point or endpoint (Cheng, 2011).

A second metric-space direction, expressed without Hausdorff control, is the generalized weakly contractive pair theory based on $H(Tx,Ty)\le d(x,y)-\psi(x,y),$2 and $H(Tx,Ty)\le d(x,y)-\psi(x,y),$3. For multivalued mappings $H(Tx,Ty)\le d(x,y)-\psi(x,y),$4 on a complete metric space, the inequality

$H(Tx,Ty)\le d(x,y)-\psi(x,y),$5

implies the existence of a common end point. If the perturbation term $H(Tx,Ty)\le d(x,y)-\psi(x,y),$6 is removed, the common end point becomes unique. The proofs build a sequence $H(Tx,Ty)\le d(x,y)-\psi(x,y),$7 by alternating selections from $H(Tx,Ty)\le d(x,y)-\psi(x,y),$8 and $H(Tx,Ty)\le d(x,y)-\psi(x,y),$9, show that $p$0 is monotone decreasing, force its limit to be $p$1 through the strict decrement term $p$2, and then obtain a Cauchy orbit via the subadditivity of $p$3 and the eventual linear lower bound of $p$4 near $p$5 (Abkar et al., 2011).

The selection-based theory gives yet another existence mechanism. Under

$p$6

one constructs a sequence $p$7 with geometric decay of successive increments, hence a Cauchy sequence. If

$p$8

which holds, for instance, when $p$9 is closed, then $D$00 has a fixed point. Variants using nearest-point selections or the weaker one-step condition

$D$01

yield Theorems 1.5 and 1.7, the latter requiring compactness of $D$02 (Pasicki, 2021).

These metric results share a common architecture: the contractive hypothesis is used to manufacture an orbit whose defect from being fixed or singleton-valued decays to zero; completeness then converts asymptotic smallness into existence of a solution.

4. Oriented weak contractions on quasi-pseudometric spaces

The quasi-pseudometric setting replaces symmetry by orientation. A quasi-pseudometric on a nonempty set $D$03 is a map $D$04 satisfying $D$05 and

$D$06

If $D$07 implies $D$08, then $D$09 is a $D$10-quasi-metric. Its conjugate is

$D$11

and its symmetrization is

$D$12

The topology $D$13 is generated by the forward balls

$D$14

Completeness can be formulated as left $D$15-completeness, Smyth completeness, or bicompleteness, depending on whether one requires $D$16-convergence, $D$17-convergence, or completeness of $D$18 (Agyingi et al., 2018).

For set-valued maps, the directional Hausdorff quasi-pseudometric is

$D$19

where $D$20 and $D$21. A startpoint of $D$22 satisfies $D$23, and an endpoint satisfies $D$24. The main existence theorem states that if $D$25 is left $D$26-complete and $D$27 is weakly contractive in the sense that for each $D$28 there exists $D$29 with

$D$30

for a $D$31-comparison function $D$32, then $D$33 has a startpoint. Under the dual inequality

$D$34

$D$35 ունի an endpoint. Under a symmetric Hausdorff contraction formulated in $D$36, one obtains a genuine fixed point (Agyingi et al., 2018).

The proof is driven by a summability mechanism rather than a global Lipschitz constant. Starting from $D$37, one recursively selects $D$38 and obtains

$D$39

Hence $D$40 is nonincreasing and converges to $D$41. Summing the inequalities yields

$D$42

and property (72) of $D$43 then implies

$D$44

which is exactly the left $D$45-Cauchy condition. Left $D$46-completeness gives a $D$47-limit $D$48, and lower semicontinuity of the oriented Hausdorff functional

$D$49

yields $D$50 (Agyingi et al., 2018).

An explicit example uses

$D$51

with

$D$52

$D$53, and

$D$54

This satisfies the weak contraction condition, and the constructed startpoint is $D$55 (Agyingi et al., 2018).

The quasi-pseudometric theory makes explicit that asymmetry changes both the contractive inequality and the solution concept. In this sense, startpoint theory is not a cosmetic variation of fixed-point theory but an orientation-sensitive extension of it.

5. Ordered, algebra-valued, and semi-metric extensions

One extension replaces scalar distances by values in a partially ordered group $D$56. A metric valued in $D$57 is a symmetric map $D$58 satisfying positivity and the triangle inequality in the order of $D$59. The convergence structure is controlled by an auxiliary relation $D$60 with axioms (t1)–(t5), and completeness is defined through Cauchy sequences in this ordered sense. In this environment, a multivalued $D$61-weak contraction requires that for all $D$62 and for each $D$63 there exists $D$64 such that

$D$65

where $D$66 whenever $D$67. The map must also satisfy the $D$68-condition

$D$69

The principal result states that such a map on a complete metric space valued in a partially ordered group has a unique endpoint if and only if it has the approximate endpoint property. A global weak contraction on a complete regular space has a unique endpoint without separately assuming approximate endpoint property (Cheng, 2014).

Ordered metric spaces produce another generalization. In a complete ordered LI metric space or ordered $D$70 metric space, with multivalued maps $D$71 that are partially dominated or partially dominating, the generalized $D$72-weak contractive condition

$D$73

is imposed for comparable pairs $D$74, with

$D$75

Under the stated sequential regularity assumptions on $D$76 and $D$77, there exists $D$78 such that

$D$79

If the end points of $D$80 and $D$81 are comparable under the order, the common end point is unique (Nazir et al., 2023).

A third direction treats multivalued weak contractions at the hyperspace level over semi-metric spaces. A semi-metric is symmetric and positive but need not satisfy the triangle inequality. Regularity is encoded through the basic triangle function

$D$82

where

$D$83

For bounded subsets, the Hausdorff–Pompeiu distance is

$D$84

If $D$85 are $D$86-contractions and $D$87 is a right-continuous comparison function, then

$D$88

For graph-directed systems, the induced operator $D$89 on $D$90 satisfies

$D$91

so $D$92 is weakly contractive on the hyperspace. The main theorem yields a unique compact $D$93-invariant family

$D$94

and constructs it as

$D$95

where $D$96 is a singleton $D$97-subinvariant seed (Okamura, 30 Mar 2025).

These extensions show that weak contractivity is robust under substantial changes of ambient structure: non-symmetric distances, ordered convergence, vector-valued metrics, and absence of the triangle inequality can all be accommodated by modifying the defect functional and the convergence mechanism.

6. Relation to classical contraction theory, scope, and recurrent misunderstandings

The classical benchmark is Nadler’s theorem: for a complete metric space and $D$98,

$D$99

implies fixed-point existence. Several of the cited papers position multivalued weak contractions explicitly as generalizations of this metric-Hausdorff model. In the quasi-pseudometric startpoint theory, the difference from Nadler is described in three ways: orientation, because $T:X\to CB(X)$00 is not symmetric and one uses directional $T:X\to CB(X)$01; weak contraction, because the hypothesis compares $T:X\to CB(X)$02 to $T:X\to CB(X)$03 for selected $T:X\to CB(X)$04 instead of imposing a uniform Hausdorff contraction between $T:X\to CB(X)$05 and $T:X\to CB(X)$06; and existence via summability, because the summability property of $T:X\to CB(X)$07 replaces a global contraction factor (Agyingi et al., 2018).

The selection-based metric theory makes the contrast even sharper. Nadler’s Hausdorff contraction implies the localized one-step condition

$T:X\to CB(X)$08

but the converse fails: the paper gives an example of a multifunction on $T:X\to CB(X)$09 satisfying this selection condition while violating every uniform Hausdorff Lipschitz bound. This is why the condition is described as strictly weaker than Nadler’s (Pasicki, 2021).

A second common misunderstanding is to assume that every multivalued weak contraction is formulated through the Hausdorff metric. This is false in several directions covered by the literature. The ordered-metric endpoint theory of (Nazir et al., 2023) uses $T:X\to CB(X)$10 and $T:X\to CB(X)$11 rather than Hausdorff distance. The common endpoint theory of (Abkar et al., 2011) also uses $T:X\to CB(X)$12, $T:X\to CB(X)$13, and the mixed quantities $T:X\to CB(X)$14 and $T:X\to CB(X)$15. The graph-directed semi-metric theory of (Okamura, 30 Mar 2025) does use a hyperspace Hausdorff–Pompeiu distance, but only for an induced operator on families of sets; it explicitly does not address general point-to-set multivalued contractions $T:X\to CB(X)$16 with a Hausdorff-type contraction condition.

A third recurrent issue concerns solution concepts. In symmetric metric spaces, fixed points, endpoints, and the collapse of the value set at the solution can often be linked. In non-symmetric spaces, however, startpoints and endpoints reflect genuinely different orientations and need not coincide. Likewise, approximate endpoint properties characterize existence in some frameworks but not in others; for example, in ordered-group-valued metrics they are equivalent to endpoint existence under the $T:X\to CB(X)$17-condition, while in global weak contraction settings stronger hypotheses can force endpoint existence directly (Cheng, 2014).

Taken together, these results place multivalued weak contractions within a broad fixed-point and endpoint program. The unifying theme is the replacement of rigid global contractive constants by weaker comparison devices adapted to the structure of the space and the desired notion of solution. The resulting theory encompasses common fixed points of dual mappings, unique endpoints, oriented startpoints, and unique compact invariant families, while preserving a recognizably contraction-based mechanism throughout (Cheng, 2011, Okamura, 30 Mar 2025).