In this chapter, an integral-type contraction condition is used to establish some common fuzzy FPs of FS-valued mappings involving Θ-contractions in a MS.

### Theorem 3.1

*Let* ((Upsilon , d)) *be a CMS and* (Phi , Psi :Upsilon rightarrow mathcal{F}(Upsilon )) *be two FMs*. *Suppose for each* (mu in Upsilon ), *there exist* (alpha _{Phi (mu )}), (alpha _{Psi (mu )}in (0,1]) *such that* ([Phi mu ]_{alpha _{Phi (mu )}}) *and* ([Psi mu ]_{alpha _{Psi (mu )}}) *are nonempty*, *and belong to* (operatorname{CB}(Upsilon )). *Assume that there are* (Theta in Xi ), (Delta in Pi ) *and* (kin (0,1)) *such that*

$$ int _{0}^{Theta (H ([Phi mu ]_{alpha _{Phi (mu )}}, [ Psi nu ]_{alpha _{Psi (nu )}} ) )} Delta (t),dt leq int _{0}^{ [Theta (d(mu , nu ) ) ]^{k}} Delta (t),dt $$

(2)

*for all* (mu , nu in Upsilon ) *with* (H ([Phi mu ]_{alpha _{Phi (mu )}}, [Psi nu ]_{alpha _{ Psi (nu )}} ) > 0). *Then*, *there is some* (zin Upsilon ) *such that* (zin [Phi z]_{alpha _{Phi (z)}}cap [Psi z]_{alpha _{Psi (z)}}).

### Proof

Let (mu _{0}in Upsilon ) be arbitrary. By hypothesis, there is (alpha _{Phi (mu _{0})}in (0,1]) so that ([Phi mu _{0}]_{alpha _{Phi (mu _{0})}}) is a nonempty, bounded, and closed subset of ϒ. Take (alpha _{Phi (mu _{0})}=alpha _{1}). Let (mu _{1}in [Phi mu _{0}]_{alpha _{Phi (mu _{0})}}). For this (mu _{1}), there is (alpha _{Psi (mu _{1})}in (0,1]) so that ([Psi mu _{1}]_{alpha _{Psi (mu _{1})}}) is a nonempty, bounded, and closed subset of ϒ. Due to Lemma 2.1,

$$ Theta bigl(dbigl(mu _{1}, [Psi mu _{1}]_{alpha _{Psi (mu _{1})}} bigr) bigr) leq Theta bigl(H bigl([Phi mu _{0}]_{alpha _{Phi ( mu _{0})}}, [Psi mu _{1}]_{alpha _{Psi (mu _{1})}} bigr) bigr). $$

(3)

From ((Theta _{1})), (2), and (3), we obtain

$$begin{aligned} int _{0}^{Theta (d(mu _{1}, [Psi mu _{1}]_{alpha _{Psi ( mu _{1})}}) )} Delta (t),dt & leq int _{0}^{Theta (H ([Phi mu _{0}]_{alpha _{Phi (mu _{0})}}, [Psi mu _{1}]_{ alpha _{Psi (mu _{1})}} ) )} Delta (t),dt \ & leq int _{0}^{ [Theta (d(mu _{0}, mu _{1}) ) ]^{k}} Delta (t),dt. end{aligned}$$

From ((Theta _{4})), we have

$$ Theta bigl(dbigl(mu _{1}, [Psi mu _{1}]_{alpha _{Psi (mu _{1})}} bigr) bigr) = inf_{nu in [Psi mu _{1}]_{alpha _{Psi ( mu _{1})}} } Theta (d(mu _{1}, nu ). $$

Thus,

$$ inf_{nu in [Psi mu _{1}]_{alpha _{Psi (mu _{1})}} } Theta (d(mu _{1}, nu ) leq bigl[ Theta bigl(d( mu _{0}, mu _{1}) bigr) bigr]^{k}. $$

(4)

Now, from (4), there is (mu _{2}in [Psi mu _{1}]_{alpha _{Psi (mu _{1})}}) such that

$$ int _{0}^{Theta (d(mu _{1}, mu _{2}) )}Delta (t),dt leq int _{0}^{ [Theta (d(mu _{0}, mu _{1}) ) ]^{k}}Delta (t),dt. $$

(5)

For this (mu _{2}) there is (alpha _{Phi (mu _{2})}in (0,1]) such that ([Phi mu _{2}]_{alpha _{Phi (mu _{2})}}) is a nonempty, bounded, and closed subset of ϒ. Due to Lemma 2.1,

$$ Theta bigl(dbigl(mu _{2}, [Phi mu _{2}]_{alpha _{Phi (mu _{2})}} bigr) bigr) leq Theta bigl(H bigl([Phi mu _{2}]_{alpha _{Phi ( mu _{2})}}, [Psi mu _{1}]_{alpha _{Psi (mu _{1})}} bigr) bigr). $$

(6)

From ((Theta _{1})), (2) and (6), we obtain

$$begin{aligned} int _{0}^{Theta (d(mu _{2}, [Phi mu _{2}]_{alpha _{Phi ( mu _{2})}}) )} Delta (t),dt & leq int _{0}^{Theta (H ([Phi mu _{2}]_{alpha _{Phi (mu _{2})}}, [Psi mu _{1}]_{ alpha _{Psi (mu _{1})}} ) )} Delta (t),dt \ & leq int _{0}^{ [Theta (d(mu _{1}, mu _{2}) ) ]^{k}} Delta (t),dt. end{aligned}$$

From ((Theta _{4})), we have

$$ Theta bigl(dbigl(mu _{1}, [Psi mu _{1}]_{alpha _{Psi (mu _{1})}} bigr) bigr) = inf_{nu in [Psi mu _{1}]_{alpha _{Psi ( mu _{1})}} } Theta (d(mu _{1}, nu ). $$

Thus,

$$ inf_{nu _{1}in [Phi mu _{2}]_{alpha _{Phi (mu _{2})}} } Theta (d(mu _{2}, nu _{1} ) leq bigl[Theta bigl(d(mu _{1}, mu _{2}) bigr) bigr]^{k}. $$

(7)

Now, from (7), there is (mu _{3}in [Phi mu _{2}]_{alpha _{Phi (mu _{2})}}) so that

$$ int _{0}^{Theta (d(mu _{2}, mu _{3}) )}Delta (t),dt leq int _{0}^{ [Theta (d(mu _{1}, mu _{2}) ) ]^{k}}Delta (t),dt. $$

(8)

Continuing this process, we generate a sequence ({mu _{n}}) in ϒ so that

$$ mu _{2n+1}in [Phi mu _{2n}]_{alpha _{Phi (mu _{2n})}} $$

and

$$ mu _{2n+2}in [Psi mu _{2n+1}]_{alpha _{Psi (mu _{2n+1})}} $$

with

$$ int _{0}^{Theta (d(mu _{2n+1}, mu _{2n+2}) )}Delta (t),dt leq int _{0}^{ [Theta (d(mu _{2n}, mu _{2n+1}) ) ]^{k}}Delta (t),dt $$

(9)

and

$$ int _{0}^{Theta (d(mu _{2n+2}, mu _{2n+3}) )}Delta (t),dt leq int _{0}^{ [Theta (d(mu _{2n+1}, mu _{2n+2}) ) ]^{k}}Delta (t),dt. $$

(10)

Combining (9) and (10), one writes

$$ int _{0}^{Theta (d(mu _{n}, mu _{n+1}) )}Delta (t),dt leq int _{0}^{ [Theta (d(mu _{n-1}, mu _{n}) ) ]^{k}}Delta (t),dt, $$

(11)

which further implies that

$$begin{aligned} int _{0}^{Theta (d(mu _{n}, mu _{n+1}) )}Delta (t),dt & leq int _{0}^{ [Theta (d(mu _{n-1}, mu _{n}) ) ]^{k}}Delta (t),dt \ & leq int _{0}^{ [Theta (d(mu _{n-2}, mu _{n-1}) ) ]^{k^{2}}}Delta (t),dt \ & leq int _{0}^{ [Theta (d(mu _{n-3}, mu _{n-2}) ) ]^{k^{3}}}Delta (t),dt \ &cdots \ & leq int _{0}^{ [Theta (d(mu _{0}, mu _{1}) ) ]^{k^{n}}}Delta (t),dt. end{aligned}$$

Since (Theta in Xi ), we have at the limit (nrightarrow infty ),

$$ lim_{nrightarrow infty } Theta bigl(d(mu _{n}, mu _{n+1}) bigr) = 1. $$

(12)

Thus,

$$ lim_{nrightarrow infty } d(mu _{n}, mu _{n+1}) = 0^{+}, $$

(13)

by ((Theta _{2})). In view of ((Theta _{3})), there are (qin (0,1)) and (lin (0,infty ]) so that

$$ lim_{nrightarrow infty } frac{Theta (d(mu _{n}, mu _{n+1}) ) – 1}{[d(mu _{n}, mu _{n+1})]^{q}} = l. $$

(14)

*Case* 1. Let (l < infty ) and (frac{l}{2} = C > 0). Hence, there is (n_{0}in mathbb{N}) so that for all (n > n_{0}),

$$ bigglvert frac{Theta (d(mu _{n}, mu _{n+1}) ) – 1}{[d(mu _{n}, mu _{n+1})]^{q}} – l biggrvert leq C. $$

That is,

$$ frac{Theta (d(mu _{n}, mu _{n+1}) ) – 1}{[d(mu _{n}, mu _{n+1})]^{q}} geq l – C = C. $$

Then,

$$ nbigl[d(mu _{n}, mu _{n+1})bigr]^{q} leq Dn bigl[Theta bigl(d(mu _{n}, mu _{n+1}) bigr) – 1bigr], $$

where (D = frac{1}{C}).

*Case* 2. Suppose (l = infty ). Let (C>0) be a real. Easily, there is (n_{0}in mathbb{N}) so that

$$ C leq frac{Theta (d(mu _{n}, mu _{n+1}) ) – 1}{[d(mu _{n}, mu _{n+1})]^{q}} $$

for all (n > n_{0}). This implies that

$$ nbigl[d(mu _{n}, mu _{n+1})bigr]^{q} leq Dn bigl[Theta bigl(d(mu _{n}, mu _{n+1}) bigr) – 1bigr],quad n> n_{0}, $$

where (D = frac{1}{C}). In both cases, there are (D > 0) and (n_{0} in mathbb{N}) so that for all (n > n_{0}),

$$ nbigl[d(mu _{n}, mu _{n+1})bigr]^{q} leq Dn bigl[Theta bigl(d(mu _{n}, mu _{n+1}) bigr) – 1bigr]. $$

(15)

Now, we have

$$ nbigl[d(mu _{n}, mu _{n+1})bigr]^{q} leq Dn bigl(bigl[Theta bigl(d(mu _{0}, mu _{1}) bigr) bigr]^{k^{n}} – 1 bigr). $$

(16)

As (nrightarrow infty ), the above inequality yields that

$$ lim_{nrightarrow infty}nbigl[d(mu _{n}, mu _{n+1}) bigr]^{q} = 0. $$

Hence, there is an integer (n_{1}) so that for all (n > n_{1}),

$$ nbigl[d(mu _{n}, mu _{n+1})bigr]^{q} leq 1. $$

This implies that

$$ d(mu _{n}, mu _{n+1}) leq frac{1}{n^{frac{1}{q}}} $$

for all (n > n_{1}). Hence,

$$ int _{0}^{d(mu _{n}, mu _{n+1})}Delta (t),dt leq int _{0}^{ frac{1}{n^{1/q}}}Delta (t),dt $$

(17)

for all (n > n_{1}). Now, to prove that ({mu _{n}}) is a Cauchy sequence, suppose (m,nin mathbb{N}) such that (m > n > n_{1}). We have

$$begin{aligned} int _{0}^{d(mu _{n}, mu _{m})}Delta (t),dt &leq sum _{i=n}^{m-1} int _{0}^{d(mu _{i}, mu _{i+1})}Delta (t),dt \ & leq sum _{i=n}^{m-1} int _{0}^{frac{1}{i^{1/q}}}Delta (t),dt leq sum ^{infty}_{i=n} int _{0}^{frac{1}{i^{1/q}}}Delta (t),dt. end{aligned}$$

(18)

Since (0< q<1), the series (sum_{i=n}^{infty}int _{0}^{frac{1}{i^{1/q}}} Delta (t),dt) converges. When (n,mrightarrow infty ), we obtain (d(mu _{n}, mu _{m}) rightarrow 0). Hence, ({mu _{n}}) is a Cauchy sequence in ((Upsilon , d)). Since ϒ is complete, there is (zin Upsilon ) so that (lim_{nrightarrow infty}mu _{n}rightarrow z). Now, we will show that (zin [Psi z]_{alpha _{Psi (z)}}). On the contrary, suppose that (znotin [Psi z]_{alpha _{Psi (z)}}), then there are (p in mathbb{N}) and a sequence ({mu _{n_{t}}}) of ({mu _{n}}) such that (d(mu _{n_{t+1}}, [Psi z]_{alpha _{Psi (z)}} ) > 0) (forall n_{t} geq p). By using ((Theta _{1})) and Lemma 2.1, we have

$$ Theta bigl[dbigl(mu _{n_{t+1}}, [Psi z]_{alpha _{Psi (z)}} bigr) bigr] leq Theta bigl[H bigl([Phi mu _{2n_{t}}]_{alpha _{ Phi (mu _{2n_{t}})}}, [Psi z]_{alpha _{Psi (z)}} bigr) bigr]. $$

(19)

Now, from (2) and (19), we have

$$begin{aligned} int ^{Theta [d(mu _{n_{t+1}}, [Psi z]_{alpha _{Psi (z)}} ) ]}_{0}Delta (t),dt & leq int ^{Theta [H ([Phi mu _{2n_{t}}]_{alpha _{Phi (mu _{2n_{t}})}}, [Psi z]_{alpha _{ Psi (z)}} ) ]}_{0}Delta (t),dt \ & leq int ^{ [Theta (d(mu _{2n_{t}}, z) ) ]^{k}}_{0} Delta (t),dt. end{aligned}$$

Letting (trightarrow infty ), then by using the continuity of Θ, the above inequality implies that

$$ int ^{Theta [d(z, [Psi z]_{alpha _{Psi (z)}} ) ]}_{0} Delta (t),dt leq 0. $$

That is,

$$ Theta bigl[dbigl(z, [Psi z]_{alpha _{Psi (z)}} bigr) bigr] leq 0. $$

Hence, (zin [Psi z]_{alpha _{Psi (z)}}). Similarly, (zin [Phi z]_{alpha _{Phi (z)}}). Thus, (zin [Phi z]_{alpha _{Phi (z)}}cap [Psi z]_{alpha _{Psi (z)}} ). □

### Example 3.1

Let (Upsilon = [0,infty )) and define (d: Upsilon times Upsilon rightarrow mathbb{R}_{+}) by

$$ d(mu ,nu )= vert mu -nu vert . $$

Define two mappings (Phi , Psi : Upsilon rightarrow mathcal{F}(Upsilon )) for (alpha in [0,1 ]) as

$$ Phi (mu ) (t)= textstylebegin{cases} alpha , & text{if } 0leq t leq 2mu , \ frac{alpha}{2}, & text{if } 2mu < t leq 6mu , \ frac{alpha}{6}, & text{if } 6mu < t leq 10mu , \ frac{alpha}{10}, & text{if } 10mu < t < infty , end{cases} $$

and

$$ Psi (mu ) (t)= textstylebegin{cases} alpha , & text{if } 0leq t leq 3mu , \ frac{alpha}{3}, & text{if } 3mu < t leq 6mu , \ frac{alpha}{6}, & text{if } 6mu < t leq 9mu , \ frac{alpha}{9}, & text{if } 9mu < t < infty . end{cases} $$

The *α*-level sets are

$$begin{aligned}& [Phi mu ]_{alpha} = [0,2mu ],\& [Psi mu ]_{alpha} = [0,3mu ]. end{aligned}$$

Consider (Theta (t) = 2^{sqrt[k]{t}}). Then, there is some (k=frac{1}{sqrt{3}}in (0,1 )) such that

$$ int _{0}^{Theta (H ([Phi mu ]_{alpha _{Phi (mu )}}, [ Psi nu ]_{alpha _{Psi (nu )}} ) )} Delta (t),dt leq int _{0}^{ [Theta (d(mu , nu ) ) ]^{k}} Delta (t),dt $$

for all (mu , nu in Upsilon ) with (H ([Phi mu ]_{alpha _{Phi (mu )}}, [Psi nu ]_{alpha _{ Psi (nu )}} ) > 0). Hence, Theorem 3.1 can be applied to find (0in Upsilon ) such that (0in [Phi 0]_{alpha} cap [Psi 0]_{alpha}).

### Corollary 3.1

*Let* ((Upsilon , d)) *be a CMS and* (Phi , Psi :Upsilon rightarrow mathcal{F}(Upsilon )) *be two fuzzy maps*. *Suppose for each* (mu in Upsilon ), *there exist* (alpha _{Phi (mu )}), (alpha _{Psi (mu )}in (0,1]) *such that* ([Phi mu ]_{alpha _{Phi (mu )}}) *and* ([Psi mu ]_{alpha _{Psi (mu )}}) *are nonempty*, *and belong to* (operatorname{CB}(Upsilon )). *Assume that there are* (Theta in Xi ) *and* (kin (0,1)) *so that*

$$ Theta bigl(H bigl([Phi mu ]_{alpha _{Phi (mu )}}, [Psi nu ]_{ alpha _{Psi (nu )}} bigr) bigr) leq bigl[Theta bigl(d( mu , nu ) bigr) bigr]^{k} $$

(20)

*for all* (mu , nu in Upsilon ) *with* (H ([Phi mu ]_{alpha _{Phi (mu )}}, [Psi nu ]_{alpha _{ Psi (nu )}} ) > 0). *Then*, *there is* (zin Upsilon ) *so that* (zin [Phi z]_{alpha _{Phi (z)}}cap [Psi z]_{alpha _{Psi (z)}}).

### Proof

By letting (Delta (t)equiv 1) in Theorem 3.1, we will obtain the required result. □

### Theorem 3.2

*Let* ((Upsilon , d)) *be a CMS and* (Phi :Upsilon rightarrow mathcal{F}(Upsilon )) *be a FM*. *Suppose for each* (mu in Upsilon ), *there is* (alpha _{Phi (mu )}in (0,1]) *such that* ([Phi mu ]_{alpha _{Phi (mu )}}) *is nonempty*, *and belongs to* (operatorname{CB}(Upsilon )). *If there are* (Theta in Xi ), (Delta in Pi ) *and* (kin (0,1)) *so that for all* (mu , nu in Upsilon ),

$$ int _{0}^{Theta (H ([Phi mu ]_{alpha _{Phi (mu )}}, [ Phi nu ]_{alpha _{Phi (nu )}} ) )} Delta (t),dt leq int _{0}^{ [Theta (d(mu , nu ) ) ]^{k}} Delta (t),dt $$

(21)

*with* (H ([Phi mu ]_{alpha _{Phi (mu )}}, [Phi nu ]_{alpha _{ Phi (nu )}} ) > 0), *then there is* (zin Upsilon ) *so that* (zin [Phi z]_{alpha _{Phi (z)}}).

### Corollary 3.2

*Let* ((Upsilon , d)) *be a CMS and* (Phi :Upsilon rightarrow mathcal{F}(Upsilon )) *be a FM*. *Suppose for each* (mu in Upsilon ), *there are* (alpha _{Phi (mu )}in (0,1]) *such that* ([Phi mu ]_{alpha _{Phi (mu )}}) *is nonempty*, *and belong to* (operatorname{CB}(Upsilon )). *If there are* (Theta in Xi ) *and* (kin (0,1)) *so that*

$$ Theta bigl(H bigl([Phi mu ]_{alpha _{Phi (mu )}}, [Phi nu ]_{ alpha _{Phi (nu )}} bigr) bigr) leq bigl[Theta bigl(d( mu , nu ) bigr) bigr]^{k} $$

(22)

*for all* (mu , nu in Upsilon ) *with* (H ([Phi mu ]_{alpha _{Phi (mu )}}, [Phi nu ]_{alpha _{ Phi (nu )}} ) > 0), *then there is* (zin Upsilon ) *so that* (zin [Phi z]_{alpha _{Phi (z)}}).

### Proof

Put (Delta (t)=1) in Theorem 3.2 to obtain the required result. □

Now, we will establish common FP results.

### Theorem 3.3

*Let* ((Upsilon , d)) *be a CMS and* (A, B:Upsilon rightarrow mathcal{CB}(Upsilon )) *be two multivalued maps*. *Assume that there are* (Theta in Xi ), (Delta in Pi ) *and* (kin (0,1)) *so that for all* (mu , nu in Upsilon ),

$$ int ^{Theta (H (Amu , Bnu ) )}_{0}Delta (t),dt leq int ^{ [Theta (d(mu , nu ) ) ]^{k}}_{0} Delta (t),dt $$

(23)

*with* (H (Amu , Bnu ) > 0). *Then*, *there is some* (zin Upsilon ) *such that* (zin Azcap Bz).

### Proof

Consider (alpha :Upsilon rightarrow (0,1]). Let (Phi ,Psi :Upsilon rightarrow mathcal{F}(Upsilon )) be two fuzzy maps defined by

$$ Phi (mu ) (t)= textstylebegin{cases} alpha (t), & text{if } t in Amu , \ 0, & text{if } t notin Amu end{cases} $$

and

$$ Psi (mu ) (t)= textstylebegin{cases} alpha (t), & text{if } t in Bmu , \ 0, & text{if } t notin Bmu . end{cases} $$

Then,

$$ [Phi mu ]_{alpha (mu )} = bigl{ t : Phi (mu ) (t) geq alpha ( mu ) bigr} = Amu $$

and

$$ [Psi mu ]_{alpha (mu )} = bigl{ t : Psi (mu ) (t) geq alpha ( mu ) bigr} = Bmu . $$

Thus, Theorem 3.1 can be applied to obtain (zin Upsilon ) so that

$$ zin [Phi z]_{alpha (z)}cap [Psi z]_{alpha (z)} = Azcap Bz. $$

□

### Corollary 3.3

*Let* ((Upsilon , d)) *be a CMS and* (A, B:Upsilon rightarrow mathcal{CB}(Upsilon )) *be two multivalued maps*. *If there are* (Theta in Xi ) *and* (kin (0,1)) *so that*

$$ Theta bigl(H (Amu , Bnu ) bigr) leq bigl[ Theta bigl(d(mu , nu ) bigr) bigr]^{k} $$

(24)

*for all* (mu , nu in Upsilon ) *with* (H (Amu , Bnu ) > 0), *then there is some* (zin Upsilon ) *so that* (zin Azcap Bz).

### Proof

By considering (Delta (t) = 1) in Theorem 3.3, we will obtain the required result. □

### Corollary 3.4

*Let* ((Upsilon , d)) *be a CMS and* (A:Upsilon rightarrow mathcal{CB}(Upsilon )) *be a multivalued map*. *If there are* (Theta in Xi ), (Delta in Pi ) *and* (kin (0,1)) *so that*

$$ int ^{Theta (H (Amu , Anu ) )}_{0}Delta (t),dt leq int ^{ [Theta (d(mu , nu ) ) ]^{k}}_{0} Delta (t),dt $$

(25)

*for all* (mu , nu in Upsilon ) *with* (H (Amu , Anu ) > 0), *then there is* (zin Upsilon ) *so that* (zin Azcap Bz).

### Theorem 3.4

*Let* ((Upsilon , d)) *be a complete metric*–*linear space and* (Phi , Psi :Upsilon rightarrow mathcal{F}(Upsilon )) *be two fuzzy maps*. *If there are* (Theta in Xi ), (Delta in Pi ) *and* (kin (0,1)) *so that*

$$ int _{0}^{Theta (d_{infty} (Phi (mu ), Psi (nu ) ) )} Delta (t),dt leq int _{0}^{ [Theta (p( mu , nu ) ) ]^{k}} Delta (t),dt $$

(26)

*for all* (mu , nu in Upsilon ) *with* (d_{infty} (Phi (mu ), Psi (nu ) ) > 0), *then there is* (zin Upsilon ) *so that* ({z}subset Phi (z)) *and* ({z}subset Psi (z) ).

### Proof

Consider (mu in Upsilon ), Lemma 2.2 implies that there is some (nu in Upsilon ) such that (nu in [Phi mu ]_{1}). Also, we can find (win Upsilon ) such that (win [Psi mu ]_{1}). Hence, for each (mu in Upsilon ), ([Phi mu ]_{alpha (mu )}) and ([Psi mu ]_{alpha (mu )}) are nonempty, and belong to (operatorname{CB}(Upsilon )). Since (alpha (mu ) = alpha (nu ) = 1), one writes

$$ H bigl([Phi mu ]_{alpha (mu )}, [Psi nu ]_{alpha (nu )} bigr) leq d_{infty} bigl(Phi (mu ), Psi (nu ) bigr) $$

∀ (mu ,nu in Upsilon ). Since Θ is nondecreasing, one obtains

$$begin{aligned} Theta bigl(H bigl([Phi mu ]_{alpha (mu )}, [Psi nu ]_{ alpha (nu )} bigr) bigr) & leq Theta bigl(d_{infty} bigl( Phi (mu ), Psi (nu ) bigr) bigr) \ & leq bigl[Theta bigl(p (mu ,nu ) bigr) bigr]^{k} end{aligned}$$

for all (mu ,nu in Upsilon ). This implies that

$$begin{aligned} int _{0}^{Theta (H ([Phi mu ]_{alpha (mu )}, [Psi nu ]_{alpha (nu )} ) )} Delta (t),dt & leq int _{0}^{ Theta (d_{infty} (Phi (mu ), Psi (nu ) ) )} Delta (t),dt \ & leq int _{0}^{ [Theta (p (mu ,nu ) ) ]^{k}} Delta (t),dt end{aligned}$$

for all (mu ,nu in Upsilon ). Now, since ([Phi mu ]_{1} subseteq [Phi mu ]_{alpha}) for any (alpha in (0,1]), and so (d (mu , [Phi mu ]_{alpha} ) leq d (mu , [ Phi mu ]_{1} )), for every (alpha in (0,1]). Thus, we have (p (mu , Phi (mu ) ) leq d (mu , [Phi mu ]_{1} )). Similarly, (p (mu , Psi (mu ) ) leq d (mu , [Psi mu ]_{1} )).

Moreover,

$$ int _{0}^{Theta (H ([Phi mu ]_{1}, [Psi nu ]_{1} ) )} Delta (t),dt leq int _{0}^{ [Theta (d (mu ,nu ) ) ]^{k}} Delta (t),dt. $$

Due to Theorem 3.1, we obtain (zin Upsilon ), so that (zin [Phi z]_{1}cap [Psi z]_{1}), i.e., ({z}subset Phi (z) text{and} {z}subset Psi (z)). □

Here, we consider that Ψ̂ is the set-valued mapping induced from FM (Psi :Upsilon rightarrow mathcal{F}(Upsilon )), i.e.,

$$ widehat{Psi}mu = Bigl{ nu : Psi (mu ) (nu ) = max_{t in Upsilon} Psi (mu ) (t) Bigr} . $$

### Corollary 3.5

*Let* ((Upsilon , d)) *be a CMS*. *Consider two fuzzy maps* (Phi , Psi :Upsilon rightarrow mathcal{F}(Upsilon )) *such that* ∀ (mu in Upsilon ), (widehat{Phi}(mu )), (widehat{Psi}(mu )) *are nonempty*, *and belong to* (operatorname{CB}(Upsilon )). *Suppose there are* (kin (0,1)), (Delta in Pi ) *and* (Theta in Xi ) *such that*

$$ int _{0}^{Theta (H (widehat{Phi}(mu ), widehat{Psi}( nu ) ) )}Delta (t),dt leq int _{0}^{ [Theta (d(mu , nu ) ) ]^{k}}Delta (t),dt $$

*for all* (mu ,nu in Upsilon ) *with* (H (widehat{Phi}(mu ), widehat{Psi}(nu ) ) > 0). *Then*, *there is a point* (uin Upsilon ) *such that* (Phi (u)(u) geq Phi (u)(mu )) *and* (Psi (u)(u) geq Psi (u)(mu )) (forall mu in Upsilon ).

### Proof

From Theorem 3.3, we obtain (uin Upsilon ) such that (uin widehat{Phi}(u)cap widehat{Psi}(u)). Then, by Lemma 2.2, we have (Phi (u)(u) geq Phi (u)(mu )) and (Psi (u)(u) geq Psi (u)(mu )) (forall mu in Upsilon ). □

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

##### Disclaimer:

This article is autogenerated using RSS feeds and has not been created or edited by OA JF.

Click here for Source link (https://www.springeropen.com/)