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

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

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,

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

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

Thus,

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

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,

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

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

Thus,

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

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

and

with

and

Combining (9) and (10), one writes

which further implies that

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

Thus,

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

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}),

That is,

Then,

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

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

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}),

Now, we have

As (nrightarrow infty ), the above inequality yields that

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

This implies that

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

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

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

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

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

That is,

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

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

and

The α-level sets are

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

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

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 ),

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

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 ),

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

and

Then,

and

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

 □

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

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

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

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

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

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

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,

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.,

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

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 ). □