# Common fixed-point results of fuzzy mappings and applications on stochastic Volterra integral equations – Journal of Inequalities and Applications

#### ByShazia Kanwal, Mohammed Shehu Shagari, Hassen Aydi, Aiman Mukheimer and Thabet Abdeljawad

Aug 19, 2022

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 metriclinear 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 ). □