Citation: Jitendra Daiya, Dinesh Kumar. S-function associated with fractional derivative and double Dirichlet average[J]. AIMS Mathematics, 2020, 5(2): 1372-1382. doi: 10.3934/math.2020094
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Fixed point theory plays a fundamental role in mathematics and applied science, such as optimization, mathematical models and economic theories. Also, this theory has been applied to show the existence and uniqueness of the solutions of differential equations, integral equations and many other branches of mathematics, see [1,2]. A prominent result in fixed point theory is the Banach contraction principle [3]. Since the appearance of this principle, there has been a lot of activity in this area. Bakhtin [4] in 1989 introduced the notion of a b-metric space (Bms). Shoaib et. al [5] proved certain fixed point results in rectangular metric spaces. Multivalued mappings in various types of metric spaces have been extensively studied by many researchers to establish fixed point results and their applications, see for instance [6,7,8,9,10,11,12].
In 1965, Zadeh [13] introduced the concept of a fuzzy set theory to deal with the unclear or inexplicit situations in daily life. Using this theory, Kramosil and Michálek [14] defined the concept of a fuzzy metric space (Fms). Grabiec [15] gave contractive mappings on a Fms and extended fixed point theorems of Banach and Edelstein in such a space. Successively, George and Veeramani [16] slightly modified the notion of a Fms introduced by Kramosil and Michálek [14] and then obtained a Hausdorff topology and a first countable topology on it. Many fixed point results have been established in a Fms. For instance, see [17,18,19,20,21,22,23,24,25] and the references therein. Recently, some coupled fuzzy fixed-point results on closed ball are established in fuzzy metric spaces [26]. The notion of generalized fuzzy metric spaces is studied in [27].
The notion of a fuzzy b-metric space (Fbms) was defined in [28]. The notion of a Hausdorff Fms is introduced in [29]. Fixed point theory for multivalued mapping in fuzzy metric spaces has been extended in many directions. For a multivalued mapping (Mvp) in a complete Fms, some fixed point results are establish in [30]. Some fixed point results for a Mvp in a Hausdorff fuzzy b-metric space (Hfbms) are proved in [31]. In this article, we prove some fixed point results for a Mvp using Geraghty type contractions in a Hfbms. Results in [31,32] and [30] turn out to be special cases of our results.
Throughout the article, ℧ refers to a non-empty set, N represents the set of natural numbers, R corresponds to the collection of real numbers, CB(℧) and ˆC0(℧) represent the collection of closed and bounded subsets and compact subsets of ℧, respectively.
Let us have a look at some core concepts that will be helpful for the proof of our main results.
Definition 1.1. [33] For a real number b≥1, the triplet (℧,Θfb,∗) is called a Fbms on ℧ if for all ψ1,ψ2,ψ3∈℧ and γ>0, the following axioms hold, where ∗ is a continuous t-norm and Θfb is a fuzzy set on ℧×℧×(0,∞):
[Fb1:] Θfb(ψ1,ψ2,γ)>0;
[Fb2:] Θfb(ψ1,ψ2,γ)=1 if and only if ψ1=ψ2;
[Fb3:] Θfb(ψ1,ψ2,γ)=Θfb(ψ2,ψ1,γ);
[Fb4:] Θfb(ψ1,ψ3,b(γ+β))≥Θfb(ψ1,ψ2,γ)∗Θfb(ψ2,ψ3,β) ∀γ,β≥0;
[Fb5:] Θfb(ψ1,ψ2,.):(0,∞)→[0,1] is left continuous.
The notion of a Fms in the sense of George and Veeramani [16] can be obtained by taking b=1 in the above definition.
Example 1.1. For a Bms (℧,Θb,∧), define a mapping Θfb:℧×℧×(0,∞)→[0,1] by
Θfb(ψ1,ψ2,γ)=γγ+db(ψ1,ψ1). |
Then (℧,Θfb,∧) is a Fbms.
Following Grabiec [15], the notions of G-Cauchyness and completeness are defined as follows:
Definition 1.2. [15]
(i) If for a sequence {ψn} in a Fbms (℧,Θfb,∗), there is ψ∈℧ such that
limn→∞Θfb(ψn,ψ,γ)=1,∀γ>0, |
then {ψn} is said to be convergent.
(ii) If for a sequence {ψn} in a Fbms (℧,Θfb,∗), limn→∞Θfb(ψn,ψn+q,γ)=1 then {ψn} is a G-Cauchy sequence for all γ>0 and positive integer q.
(iii) A Fbms is G-complete if every G-Cauchy sequence is convergent.
Definition 1.3. [30] Let B be any nonempty subset of a Fms (℧,Θfb,∗) and γ>0, then we define FΘfb(ϱ1,B,γ), the fuzzy distance between an element ϱ1∈℧ and the subset B, as follows:
FΘfb(ϱ1,B,γ)=sup{Θf(ϱ1,ϱ2,γ):ϱ2∈B}. |
Note that FΘfb(ϱ1,B,γ)=FΘfb(B,ϱ1,α).
Lemma 1.1. [31] Consider a Fbms (℧,Θfb,∗) and let CB(℧) be the collection of closed bounded subsets of ℧. If A∈CB(℧) then ψ∈A if and only if FΘfb(A,ψ,γ)=1∀γ>0.
Definition 1.4. [31] Let (℧,Θfb,∗) be a Fbms. Define HFΘfb on ˆC0(℧)׈C0(℧)×(0,∞) by
HFΘfb(A,B,γ)=min{ infψ∈AFΘfb(ψ,B,γ),infϱ∈BFΘfb(A,ϱ,γ)}, |
for all A,B∈^C0(℧) and γ>0.
For Geraghty type contractions, follow [33] to define a class FΘb of all functions β:[0,∞)→[0,1b) for b≥1, as
FΘb={β:[0,∞)→[0,1b)|limn→∞β(γn)=1b⇒limn→∞γn=0}. | (1.1) |
Lemma 1.2. [31] Let (℧,Θfb,∗) be a G-complete Fbms. If ψ,ϱ∈℧ and for a function β∈FΘfb
Θfb(ψ,ϱ,β(Θfb(ψ,ϱ,γ))γ)≥Θfb(ψ,ϱ,γ), |
then ψ=ϱ.
Lemma 1.3. [31] Let (^C0(℧),HFΘfb,∗) be a Hfbms where (Θfb,∗) is a Fbm on ℧. If for all A,B∈^C0(℧), for each ψ∈A and for γ>0 there exists ϱψ∈B, satisfying FΘfb(ψ,B,γ)=Θfb(ψ,ϱψ,γ), then
HFΘfb(A,B,γ)≤Θfb(ψ,ϱψ,γ). |
In this section, we develop some fixed point results by using the idea of a Hfbms. Furthermore, an example is also presented for a deeper understanding of our results.
Recall that, given a multivalued mapping Ξ:℧→ˆC0(℧), a point ψ is said to be a fixed point of Ξ if ψ∈Ξψ.
Theorem 2.1. Let (℧,Θfb,∗) be a G-complete Fbms and (^C0(℧),HFΘfb,∗) be a Hfbms. Let Ξ:℧→^C0(℧) be a Mvp satisfying
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)≥Θfb(ψ,ϱ,γ), | (2.1) |
for all ψ,ϱ∈℧, where β∈FΘfb as defined in (1.1). Then Ξ has a fixed point.
Proof. Choose {ψn} for ψ0∈℧ as follows: Let ψ1∈℧ such that ψ1∈Ξψ0 by the application of Lemma 1.3, we can choose ψ2∈Ξψ1 such that for all γ>0,
Θfb(ψ1,ψ2,γ)⩾ |
By induction, we have \psi_{r+1} \in \Xi \psi_{r} satisfying
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma)\; \forall\; r\; \in\mathbb{N}. \\ \end{equation*} |
By the application of (2.1) and Lemma 1.3, we have
\begin{align} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) &\geq \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma)\geq \Theta_{fb}\left( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) \\ & \geq \mathcal{H}_{F_{\Theta_{fb}}}\left( \Xi \psi_{r-2}, \Xi \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) \\ & \geq \Theta_{fb}\left( \psi_{r-2}, \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))}\right) \\ & \vdots \\ & \geq \mathcal{H}_{F_{\Theta_{fb}}}\left( \Xi \psi_{0}, \Xi \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{1}, \psi_{2}, \gamma))}\right)\\ & \geq \Theta_{fb}\left( \psi_{0}, \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\right) . \end{align} | (2.2) |
For any q \in \mathbb{N} , writing q(\frac{\gamma}{q}) = \frac{\gamma}{q}+\frac{\gamma}{q} + \ldots + \frac{\gamma}{q} and using [Fb4] repeatedly,
\begin{align*} &\quad\Theta_{fb}(\psi_{r}, \psi_{r+q}, \gamma)\\ &\geq \Theta_{fb} \left( \psi_{r}, \psi_{r+1}, \frac{\gamma}{qb}\right) * \Theta_{fb} \left( \psi_{r+1}, \psi_{r+2}, \frac{\gamma}{qb^{2}}\right) * \Theta_{fb} \left( \psi_{r+2}, \psi_{r+3}, \frac{\gamma}{qb^{3}}\right) * \ldots * \Theta_{fb} \left( \psi_{r+q-1}, \psi_{r+q}, \frac{\gamma}{qb^{q}}\right). \end{align*} |
Using (2.2) and [Fb5] , we get
\begin{align*} &\quad\Theta_{fb}(\psi_{r}, \psi_{r+q}, \gamma)\\ &\geq \Theta_{fb} \left( \psi_{0}, \psi_{1}, \frac{\gamma}{qb\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\right)\\ &*\Theta_{fb} \left( \psi_{0}, \psi_{1}, \frac{\gamma}{q b^{2}\beta(\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma))\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\right)\\ &*\Theta_{fb} \left( \psi_{0}, \psi_{1}, \frac{\gamma}{qb^{3}\beta(\Theta_{fb}(\psi_{r+1}, \psi_{r+2}, \gamma))\beta(\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\right) * \ldots *\\ &\Theta_{fb} \left( \psi_{0}, \psi_{1}, \frac{\gamma}{qb^{q}\beta(\Theta_{fb}(\psi_{r+q-2}, \psi_{r+q-1}, \gamma))\beta(\Theta_{fb}(\psi_{r+q-3}, \psi_{r+q-2}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\right). \end{align*} |
That is,
\begin{align*} &\quad\Theta_{fb}(\psi_{r}, \psi_{r+q}, \gamma)\\ &\geq \Theta_{fb} \left( \psi_{0}, \psi_{1}, \frac{b^{r-1}\gamma}{q}\right) * \Theta_{fb} \left( \psi_{0}, \psi_{1}, \frac{b^{r-1}\gamma}{q}\right)* \Theta_{fb} \left( \psi_{0}, \psi_{1}, \frac{b^{r-1}\gamma}{q}\right) * \ldots * \Theta_{fb} \left( \psi_{0}, \psi_{1}, \frac{b^{r-1}\gamma}{q}\right). \end{align*} |
Taking limit as r \rightarrow \infty , we get
\lim\limits_{n\rightarrow \infty}\Theta_{fb}(\psi_{r}, \psi_{r+q}, \gamma) = 1*1* \ldots * 1 = 1. |
Hence, \lbrace \psi_{r}\rbrace is G -Cauchy sequence. By the G -completeness of \mho , there exists \phi \in \mho such that
\begin{align*} \Theta_{fb}(\phi, \Xi \phi, \gamma)& \geq \Theta_{fb} \left( \phi, \psi_{r+1}, \frac{\gamma}{2b}\right)*\Theta_{fb} \left( \psi_{r+1}, \Xi \phi, \frac{\gamma}{{2b}}\right)\\ & \geq \Theta_{fb} \left( \phi, \psi_{r+1}, \frac{\gamma}{{2b}}\right)*\mathcal{H}_{F_{\Theta_{fb}}}\left( \Xi \psi_{r}, \Xi \phi, \frac{\gamma}{{2b}}\right)\\ &\geq \Theta_{fb} \left(\phi, \psi_{r+1}, \frac{\gamma}{{2b}}\right)*\Theta_{fb}\left( \psi_{r}, \phi, \frac{\gamma}{{2b}\beta(\Theta_{fb}(\psi_{r}, \phi, \gamma))}\right)\\ & \longrightarrow 1 \quad \text{as} \quad r \rightarrow \infty. \end{align*} |
By Lemma 1.1, it follows that \phi \in \Xi \phi . That is, \phi is a fixed point for \Xi .
Remark 2.1.
(1) If we take \beta(\Theta_{fb}(\psi, \varrho, \gamma)) = k with bk < 1 , we get Theorem 3.1 of [31].
(2) By setting \hat{C_{0}} (\mho) = \mho the mapping \Xi \colon \mho \rightarrow \hat{C_{0}} (\mho) becomes a single valued and we get Theorem 3.1 of [32]. Notice that when \Xi is a singlevalued map, \Xi\psi becomes a singleton set and the fact that \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \gamma) = \Theta_{fb}(\Xi\psi, \Xi\varrho, \gamma) indicates that the fixed point will be unique as proved in [32].
(3) Set b = 1 and \hat{C_{0}} (\mho) = \mho and let k\in (0, 1) be such that \beta(\Theta_{fb}(\psi, \varrho, \gamma)) = k then we get the main result of [15].
The next example illustrates Theorem 2.1.
Example 2.1. Let \mho = \left[0, 1\right] and define a G -complete Fbms by
\Theta_{fb}(\psi, \varrho, \gamma) = \dfrac{\gamma}{\gamma+(\psi-\varrho)^{2}} , |
with b\geq1. For \beta\in\mathbb{F}_{b} , define a mapping \Xi \colon \mho \rightarrow \hat{C_{0}} (\mho) by
\begin{equation*} \Xi\psi = \begin{cases}\lbrace0\rbrace &\text {if } \psi = 0, \\ \lbrace 0, \frac{\sqrt{\beta(\Theta_{fb}(\psi, \varrho, \gamma))}\psi}{2}\rbrace & \text {otherwise.} \end{cases} \\ \end{equation*} |
For \psi = \varrho ,
\begin{align*} \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma) = 1 = & \Theta_{fb}(\psi, \varrho, \gamma). \end{align*} |
If \psi\neq \varrho , then following cases arise.
For \psi = 0 and \varrho\in(0, 1] , we have
\begin{align*} &\quad\mathcal{H}_{F_{\Theta_{fb}}}(\Xi0, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma) \\ & = \min \lbrace \ \inf\limits_{a \in \Xi0} F_{\Theta_{fb}}( a, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma), \inf\limits_{b \in \Xi\varrho} F_{\Theta_{fb}}( \Xi0, b, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma)) \rbrace\\ & = \min \Bigg\lbrace \inf\limits_{a \in \Xi0} F_{\Theta_{fb}}\left( a, \left\lbrace 0, \frac{\sqrt{\beta(\Theta_{fb}(\psi, \varrho, \gamma))}\varrho}{2}\right\rbrace , \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma\right), \inf\limits_{b \in \Xi\varrho} F_{\Theta_{fb}}\left( \left\lbrace 0\right\rbrace , b, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma\right) \Bigg\rbrace \\ & = \min \Biggl \lbrace \inf \left\lbrace F_{\Theta_{fb}}\left( 0, \lbrace0, \frac{\sqrt{\beta(\Theta_{fb}(\psi, \varrho, \gamma))}\varrho}{2} \rbrace , \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma\right) \right\rbrace, \\ &\quad\quad\quad\inf \left\lbrace F_{\Theta_{fb}}\left( \lbrace 0\rbrace , 0, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma\right), F_{\Theta_{fb}}\left( \lbrace 0\rbrace , \frac{\sqrt{\beta(\Theta_{fb}(\psi, \varrho, \gamma))}\varrho}{2}, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma\right) \right\rbrace \Biggr \rbrace \\ & = \min \Biggl \lbrace \inf \left\lbrace \sup \left\lbrace F_{\Theta_{fb}}( 0, 0, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma), F_{\Theta_{fb}}( 0, \frac{\sqrt{\beta(\Theta_{fb}(\psi, \varrho, \gamma))}\varrho}{2}, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma)\right\rbrace \right\rbrace, \\ &\quad\quad\quad \inf \left\lbrace F_{\Theta_{fb}}( 0, 0, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma), F_{\Theta_{fb}}( 0, \frac{\sqrt{\beta(\Theta_{fb}(\psi, \varrho, \gamma))}y}{2}, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma)\right\rbrace \Biggr\rbrace\\ & = \min \Biggl \lbrace \inf \left\lbrace \sup \left\lbrace 1, \dfrac{\gamma}{\gamma+\frac{\varrho^{2}}{4}}\right\rbrace \right\rbrace, \inf \left\lbrace 1, \dfrac{\gamma}{\gamma+\frac{\varrho^{2}}{4}}\right\rbrace \Biggr\rbrace\\ & = \min \Biggl \lbrace \inf \left\lbrace 1 \right\rbrace, \dfrac{\gamma}{\gamma+\frac{\varrho^{2}}{4}} \Biggr\rbrace = \min \left\lbrace 1, \dfrac{\gamma}{\gamma+\frac{\varrho^{2}}{4}} \right\rbrace = \dfrac{\gamma}{\gamma+\frac{\varrho^{2}}{4}}. \end{align*} |
It follows that
\mathcal{H}_{F_{\Theta_{fb}}}(\Xi0, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma) > \Theta_{fb}(0, \varrho, \gamma) = \dfrac{\gamma}{\gamma+\varrho^{2}}. |
For \psi and \varrho\in(0, 1] , after simplification we have
\begin{align*} \mathcal{H}_{F_{\Theta_{fb}}}(S(\psi), \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma)& = \min\left\lbrace \sup\left\lbrace \dfrac{\gamma}{\gamma+\frac{\psi^{2}}{4}}, \dfrac{\gamma}{\gamma+\frac{(\psi-\varrho)^{2}}{4}}\right\rbrace , \sup\left\lbrace \dfrac{\gamma}{\gamma+\frac{\varrho^{2}}{4}}, \dfrac{\gamma}{\gamma+\frac{(\psi-\varrho)^{2}}{4}}\right\rbrace\right\rbrace \\ & \geq\dfrac{\gamma}{\gamma+\frac{(\psi-\varrho)^{2}}{4}} > \dfrac{\gamma}{\gamma+(\psi-\varrho)^{2}} = \Theta_{fb}(\psi, \varrho, \gamma). \end{align*} |
Thus, for all cases, we have
\mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma) \geq \Theta_{fb}(\psi, \varrho, \gamma). |
Since all conditions of Theorem 2.1 are satisfied and 0 is a fixed point of \Xi .
Theorem 2.2. Let (\mho, \Theta_{fb}, *) be a G -complete Fbms with b\geqslant 1 and (\hat{C_{0}} (\mho), \mathcal{H}_{F_{\Theta_{fb}}}, *) be a Hfbms. Let \Xi \colon \mho \rightarrow \hat{C_{0}} (\mho) be a Mvp satisfying
\begin{equation} \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma) \geq \min \left\lbrace \dfrac{F_{\Theta_{fb}}( \varrho, \Xi\varrho, \gamma) \left[ 1+F_{\Theta_{fb}}( \psi, \Xi\psi, \gamma) \right] }{1+\Theta_{fb}( \psi, \varrho, \gamma) }, \Theta_{fb}( \psi, \varrho, \gamma) \right\rbrace, \end{equation} | (2.3) |
for all \psi, \varrho \in \mho , where \beta\in \mathbb{F}_{\Theta_{fb}} as given in (1.1). Then \Xi has a fixed point.
Proof. Choose \lbrace \psi_{n}\rbrace for \psi_{0} \in \mho as follows: Let \psi_{1} \in \mho such that \psi_{1} \in \Xi\psi_{0} . By the application of Lemma 1.3 we can choose \psi_{2} \in \Xi \psi_{1} such that
\begin{equation*} \Theta_{fb}(\psi_{1}, \psi_{2}, \gamma)\geqslant\mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{0}, \Xi \psi_{1}, \gamma), \quad \forall\; \gamma > 0.\\ \end{equation*} |
By induction, we have \psi_{r+1} \in \Xi \psi_{r} satisfying
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma), \quad \forall\; r\; \in\mathbb{N}. \\ \end{equation*} |
By the application of (2.3) and Lemma 1.3 we have
\begin{align*} &\quad\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) \geq \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma) \notag\\ &\geq \min \Biggl\lbrace \dfrac{F_{\Theta_{fb}}\left( \psi_{r}, \Xi \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) \left[ 1+F_{\Theta_{fb}}\left( \psi_{r-1}, \Xi \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) \right] }{1+\Theta_{fb}\left( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right)}, \notag\\ &\quad \quad\quad\Theta_{fb}\left( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) \Biggr\rbrace \notag\\ &\geq \min \Biggl\lbrace \dfrac{\Theta_{fb}\left( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) \left[ 1+\Theta_{fb}\left( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) \right] }{1+\Theta_{fb}\left( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) }, \notag\\ &\quad\quad\quad\quad \Theta_{fb}\left( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) \Biggr\rbrace, \notag\\ &\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) \geq \min \left\lbrace \Theta_{fb}\left( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right), \Theta_{fb}\left( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right)\right\rbrace. \end{align*} | (2.4) |
If
\begin{align*} &\quad \min \Bigg\lbrace \Theta_{fb}\left( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right), \Theta_{fb}\left( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right) \Bigg\rbrace\\ & = \Theta_{fb}\left( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\right), \end{align*} |
then (2.4) implies
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) \geq \Theta_{fb}\Bigl( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). \end{equation*} |
The result is obvious by Lemma 1.2.
If
\begin{align*} &\quad\min \Bigg\lbrace \Theta_{fb}\Bigl( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigr( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigg\rbrace \\ & = \Theta_{fb}\Bigr( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigl), \end{align*} |
then from (2.4) we have
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)& \geq \Theta_{fb}\Big( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Big)\\ &\geq \Theta_{fb}\Big( \psi_{r-2}, \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))}\Big) \\ & \vdots \\ &\geqslant \Theta_{fb}\Big( \psi_{0}, \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma)}\Big). \end{align*} |
The rest of the proof can be done by proceeding same as in Theorem 2.1.
Remark 2.2.
(1) If we take \beta(\Theta_{fb}(\psi, \varrho, \gamma)) = k with bk < 1 , we get Theorem 3.2 of [31].
(2) By taking b = 1 and for some 0 < k < 1 setting \beta(\Theta_{fb}(\psi, \varrho, \gamma)) = k in Theorem 2.2, we get the main result of [30].
Theorem 2.3. Let (\mho, \Theta_{fb}, *) be a G -complete Fbms with b\geqslant 1 and (\hat{C_{0}} (\mho), \mathcal{H}_{F_{\Theta_{fb}}}, *) be a Hfbms. Let \Xi \colon \mho \rightarrow \hat{C}_{0}(\mho) be a Mvp satisfying
\begin{align} &\quad\mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma) \\ &\geq \min \left\lbrace \dfrac{F_{\Theta_{fb}}( \varrho, \Xi\varrho, \gamma) \left[ 1+F_{\Theta_{fb}}( \psi, \Xi\psi, \gamma)+F_{\Theta_{fb}}( \varrho, \Xi\psi, \gamma) \right] }{2+\Theta_{fb}( \psi, \varrho, \gamma) }, \Theta_{fb}( \psi, \varrho, \gamma) \right\rbrace \end{align} | (2.5) |
for all \psi, \varrho \in \mho , where \beta\in \mathbb{F}_{\Theta_{fb}} , the class of functions defined in (1.1). Then \Xi has a fixed point.
Proof. Choose \lbrace \psi_{n}\rbrace for \psi_{0} \in \mho as follows: Let \psi_{1} \in \mho such that \psi_{1} \in \Xi\psi_{0} . by the application of Lemma 1.3 we can choose \psi_{2} \in \Xi \psi_{1} such that
\begin{equation*} \Theta_{fb}(\psi_{1}, \psi_{2}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{0}, \Xi \psi_{1}, \gamma), \quad \forall\; \gamma > 0.\\ \end{equation*} |
By induction, we have \psi_{r+1} \in \Xi \psi_{r} satisfying
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma), \quad \forall\; r\; \in\mathbb{N}. \\ \end{equation*} |
By the application of (2.5) and Lemma 1.3, we have
\begin{align} &\quad\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) \geq \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma)\\ &\geq \min \Biggl\lbrace \dfrac{F_{\Theta_{fb}}( \psi_{r}, \Xi \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 1+F_{\Theta_{fb}}( \psi_{r-1}, \Xi \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})+F_{\Theta_{fb}}( \psi_{r}, \Xi \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right] }{2+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \\ &\quad\quad\quad \Theta_{fb}\Bigl( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Biggr\rbrace \\ &\geq \min \Biggl\lbrace \dfrac{\Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 1+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})+\Theta_{fb}( \psi_{r}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right] }{2+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \\ &\quad\quad\quad \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \Biggr\rbrace \\ &\geq \min \left\lbrace \dfrac{\Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 1+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})+1 \right] }{2+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right\rbrace \\ &\geq \min \left\lbrace \dfrac{\Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 2+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right] }{2+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right\rbrace \\ &\geq \min \left\lbrace \Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right\rbrace. \end{align} | (2.6) |
If
\begin{align*} &\quad\min \Bigg\lbrace \Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \Bigg\rbrace\\ & = \Theta_{fb}\Bigg( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigg), \end{align*} |
then (2.6) implies
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) \geq \Theta_{fb}\Bigl( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{equation*} |
and the proof follows by Lemma 1.2.
If
\begin{align*} &\quad\min \left\lbrace \Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right\rbrace\\ & = \Theta_{fb}\Bigl( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr).\nonumber \end{align*} |
Then from (2.6) we have
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) &\geq \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})\\ &\geqslant \ldots \geqslant \Theta_{fb}\Biggl( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\Biggr). \end{align*} |
The rest of the proof is same as in Theorem 2.1.
Remark 2.3. Theorem 3.3 of [31] becomes a special csae of the above theorem by setting \beta(\Theta_{fb}(\psi, \varrho, \gamma)) = k where k is chosen such that bk < 1 .
Theorem 2.4. Let (\mho, \Theta_{fb}, *) be a G -complete Fbms with b\geqslant 1 and (\hat{C_{0}} (\mho), \mathcal{H}_{F_{\Theta_{fb}}}, *) be a Hfbms. Let \Xi \colon \mho \rightarrow \hat{C}_{0}(\mho) be a multivalued mapping satisfying
\begin{align} &\quad\mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma) \\ &\geq \min \Biggl\lbrace \dfrac{F_{\Theta_{fb}}( \psi, \Xi\psi, \gamma) \left[ 1+F_{\Theta_{fb}}( \varrho, \Xi\varrho, \gamma) \right] }{1+F_{\Theta_{fb}}( \Xi\psi, \Xi\varrho, \gamma) }, \dfrac{F_{\Theta_{fb}}( \psi, \Xi\varrho, \gamma) \left[ 1+F_{\Theta_{fb}}( \psi, \Xi\psi, \gamma) \right] }{1+\Theta_{fb}( \psi, \varrho, \gamma) }, \\ &\quad\quad\quad \dfrac{F_{\Theta_{fb}}( \psi, \Xi\psi, \gamma) \left[ 2+F_{\Theta_{fb}}( \psi, \Xi\varrho, \gamma) \right] }{1+\Theta_{fb}( \psi, \Xi\varrho, \gamma)+F_{\Theta_{fb}}( \varrho, \Xi\psi, \gamma) }, \Theta_{fb}( \psi, \varrho, \gamma) \Biggr \rbrace, \end{align} | (2.7) |
for all \psi, \varrho \in \mho , where \beta \in \mathbb{F}_{fb} , the class of functions defined in (1.1). Then \Xi has a fixed point.
Proof. In the same way as Theorem 2.1, we have
\begin{equation*} \Theta_{fb}(\psi_{1}, \psi_{2}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{0}, \Xi \psi_{1}, \gamma), \quad \forall\; \gamma > 0. \end{equation*} |
By induction, we obtain \psi_{r+1} \in \Xi \psi_{r} satisfying
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma), \quad \forall\; n\; \in\mathbb{N}. \end{equation*} |
Now, by (2.7) together with Lemma 1.3, we have
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)& \geq \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma) \\ &\geq \min \Biggl\lbrace \dfrac{F_{\Theta_{fb}}( \psi_{r-1}, \Xi \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 1+F_{\Theta_{fb}}( \psi_{r}, S_{\psi_{r}}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right] }{1+F_{\Theta_{fb}}( \Xi \psi_{r-1}, \Xi \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \\ &\quad \quad \dfrac{F_{\Theta_{fb}}( \psi_{r}, \Xi \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 1+F_{\Theta_{fb}}( \psi_{r-1}, \Xi \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right] }{1+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \\ &\quad \quad\dfrac{F_{\Theta_{fb}}( \psi_{r-1}, \Xi \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 2+F_{\Theta_{fb}}( \psi_{r-1}, \Xi \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right] }{1+F_{\Theta_{fb}}( \psi_{r-1}, \Xi \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})+F_{\Theta_{fb}}( \psi_{r}, \Xi \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \\ &\quad \quad \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \Biggr\rbrace\\ &\geq \min \Biggl\lbrace \dfrac{\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 1+\Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right] }{1+\Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \\ &\quad \quad \dfrac{\Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 1+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right] }{1+\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \\ & \quad \quad \dfrac{\Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \left[ 2+\Theta_{fb}( \psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right] }{1+\Theta_{fb}( \psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})+\Theta_{fb}( \psi_{r}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) }, \\ & \quad\quad \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \Biggr\rbrace, \\ \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geq & \min \left\lbrace \Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \right\rbrace.\end{align*} | (2.8) |
If
\begin{align*} &\quad\min \Bigg\lbrace \Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \Bigg\rbrace \\ & = \Theta_{fb}\Bigr( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigl), \end{align*} |
then (2.8) implies
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) \geq \Theta_{fb}\Bigl( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). \end{equation*} |
Then the proof follows by Lemma 1.2.
If
\begin{align*} &\quad\min \Bigg\lbrace \Theta_{fb}( \psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \Bigg\rbrace \\ & = \Theta_{fb}\Bigl( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then from (2.6) we have
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) &\geq \Theta_{fb}\Bigl( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & \vdots\\ & \geqslant \Theta_{fb}\bigl( \psi_{0}, \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\Bigr). \end{align*} |
The rest of the proof is similar as in Theorem 2.1.
Remark 2.4. Again by taking \beta(\Theta_{fb}(\psi, \varrho, \gamma)) = k with kb < 1 , we get Theorem 3.4 of [31].
Theorem 2.5. Let (\mho, \Theta_{fb}, *) be a G -complete Fbms with b\geqslant 1 and (\hat{C_{0}} (\mho), \mathcal{H}_{F_{\Theta_{fb}}}, *) be a Hfbms. Let \Xi \colon \mho \rightarrow \hat{C}_{0}(\mho) be a Mvp satisfying
\begin{align} &\quad\mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma)\\ &\geq \dfrac{ \min \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \gamma). \Theta_{fb}(\psi, \varrho, \gamma) , \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma). \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\varrho, \gamma) \rbrace )}{ \max \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma), \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\varrho, \gamma)\rbrace}, \end{align} | (2.9) |
for all \psi, \varrho \in \mho , where \beta \in \mathbb{F}_{fb}. Then \Xi has a fixed point.
Proof. In the same way as Theorem 2.1, we have
\begin{equation*} \Theta_{fb}(\psi_{1}, \psi_{2}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{0}, \Xi \psi_{1}, \gamma), \quad \forall\; \gamma > 0. \end{equation*} |
By induction we have \psi_{r+1} \in \Xi \psi_{r} satisfying
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma), \quad \forall\; n\; \in\mathbb{N}. \\ \end{equation*} |
Now by (2.7) together with Lemma 1.3 and some obvious simplification step, we have
\begin{align} &\quad\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) \geq \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{r-1}, \Xi \psi_{r}, \gamma) \\ &\geq \dfrac{\min \Biggl\lbrace \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}).\Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}). \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})\Biggr \rbrace}{\max \lbrace \Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}), \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \rbrace}\\ &\geq \dfrac{\Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}). \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})}{\max \lbrace \Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}), \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \rbrace} \end{align} | (2.10) |
If
\begin{align*} &\quad\max \lbrace \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then (2.10) implies
\Theta_{fb}(\psi_{r}, \psi_{r+1}, t)\geq \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) |
Then the proof follows by Lemma 1.2.
If
\begin{align*} &\quad\max \lbrace \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \rbrace\\ & = \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then from (2.10) we have
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) &\geq \Theta_{fb}\Bigr( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & \vdots\\ & \geqslant \Theta_{fb}\Bigl( \psi_{0}, \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\Bigr). \end{align*} |
The remaining proof follows in the same way as in Theorem 2.1.
Theorem 2.6. Let (\mho, \Theta_{fb}, *) be a G -complete Fbms with b\geqslant 1 and (\hat{C_{0}} (\mho), \mathcal{H}_{F_{\Theta_{fb}}}, *) be a Hfbms. Let \Xi \colon \mho \rightarrow \hat{C}_{0}(\mho) be a Mvp satisfying
\begin{equation} \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma)\geq \Gamma_{1}(\psi, \varrho, \gamma)* \Gamma_{2}(\psi, \varrho, \gamma), \end{equation} | (2.11) |
where,
\begin{align} \left. \begin{cases} \Gamma_{1}(\psi, \varrho, \gamma)& = \min \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \gamma) , \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma), \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\varrho, \gamma), \Theta_{fb}(\psi, \varrho, \gamma) \rbrace\\ \Gamma_{2}(\psi, \varrho, \gamma)& = \max \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\varrho, \gamma) , \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \varrho, \gamma) \rbrace \end{cases} \right\}, \end{align} | (2.12) |
for all \psi, \varrho \in \mho , and \beta \in \mathbb{F}_{fb}. Then \Xi has a fixed point.
Proof. In the same way as Theorem 2.1, we have
\begin{equation*} \Theta_{fb}(\psi_{1}, \psi_{2}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{0}, \Xi \psi_{1}, \gamma), \quad \forall\; \gamma > 0.\\ \end{equation*} |
By induction we have \psi_{r+1} \in \Xi \psi_{r} satisfying
\begin{align} \mathcal{H}_{F_{\Theta_{fb}}}(\psi_{r}, \psi_{r+1}, \gamma)& = F_{\theta}(\Xi\psi_{r-1}, \Xi\psi_{r}, \gamma)\\ &\geq \Gamma_{1}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)* \Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \end{align} | (2.13) |
Now,
\begin{align} &\quad\Gamma_{1}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & = \min \Biggl \lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\Xi\psi_{r-1}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \\ &\quad\quad\quad \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Biggr \rbrace \\ & = \min \Biggl \lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \\ & \quad\quad\quad \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Biggr \rbrace. \\&\quad\Gamma_{1}(\psi_{r-1}, \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \\ & = \min \Biggl\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \biggr\rbrace. \end{align} | (2.14) |
\begin{align} &\quad\Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigl) \\ & = \max \Bigl\lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\Xi\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Bigr\rbrace\\ & = \max \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigr\rbrace \\ & = \max \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , 1 \Bigr\rbrace. \notag \\ \\ & \ \ \ \ \ \ \ \ \ \Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) = 1. \end{align} | (2.15) |
Using (2.14) and (2.15) in (2.13) we have
\begin{align} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)&\geq \min \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigr\rbrace *1, \\ \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)&\geq \min \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigr\rbrace. \end{align} | (2.16) |
If
\begin{align*} &\min \Bigg\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigg\rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigl), \end{align*} |
then (2.16) implies
\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geq \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigl) |
Then the proof follows by Lemma 1.2
If
\begin{align*} &\quad\min \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigr\rbrace\\ & = \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then from (2.16), we have
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) &\geq \Theta_{fb}\Bigl( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & \vdots \\ &\geqslant \Theta_{fb}\Bigl( \psi_{0}, \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\Big). \end{align*} |
The remaining proof is similar as in Theorem 2.1.
Remark 2.5. If we set \hat{C_{0}} (\mho) = \mho the map \Xi becomes a singlevalued and we get Theorem 3.11 of [32]. Again as stated in Remark 2.1, the corresponding fixed point will be unique.
Theorem 2.7. Let (\mho, \Theta_{fb}, *) be a G -complete Fbms with b\geqslant 1 and (\hat{C_{0}} (\mho), \mathcal{H}_{F_{\Theta_{fb}}}, *) be a Hfbms. Let \Xi \colon \mho \rightarrow \hat{C}_{0}(\mho) be a Mvp satisfying
\begin{align} \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma)\geq \dfrac{\Gamma_{1}(\psi, \varrho, \gamma)* \Gamma_{2}(\psi, \varrho, \gamma)}{\Gamma_{3}(\psi, \varrho, \gamma)}, \end{align} | (2.17) |
where
\begin{align} \left. \begin{cases} \Gamma_{1}(\psi, \varrho, \gamma)& = \min \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \gamma). \Theta_{fb}(\psi, \varrho, \gamma) , \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma). \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\varrho, \gamma) \rbrace\\ \Gamma_{2}(\psi, \varrho, \gamma)& = \max \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma). \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\varrho, \gamma) , \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\psi, \gamma))^{2}\rbrace\\ \Gamma_{3}(\psi, \varrho, \gamma)& = \max \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma), \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\varrho, \gamma)\rbrace \end{cases} \right\}, \end{align} | (2.18) |
for all \psi, \varrho \in \mho , and \beta \in \mathbb{F}_{fb}. Then \Xi has a fixed point.
Proof. In the same way as Theorem 2.1, we have
\begin{equation*} \Theta_{fb}(\psi_{1}, \psi_{2}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{0}, \Xi \psi_{1}, \gamma), \quad \forall\; \gamma > 0. \end{equation*} |
By induction we have \psi_{r+1} \in \Xi \psi_{r} satisfying
\begin{align} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)& = \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi_{r-1}, \Xi\psi_{r}, \gamma)\\ &\geq \dfrac{\Gamma_{1}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})* \Gamma_{2}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})}{\Gamma_{3}(\psi, \varrho, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})}. \end{align} | (2.19) |
\begin{align} &\quad\Gamma_{1}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \\ & = \min \Biggl \lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\Xi\psi_{r-1}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr).F_{\theta}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \\ &\quad\quad\quad \mathcal{H}_{F_{\Theta_{fb}}}(\psi_{r-1}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}). \mathcal{H}_{F_{\Theta_{fb}}}(\psi_{r}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \Biggr \rbrace \\ & = \min \Biggl \lbrace \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}).\Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \\ &\quad\quad\quad\Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Biggr \rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr).\Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). \end{align} | (2.20) |
Similarly,
\begin{align} &\quad\Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & = \max \Biggl \lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \\ &\quad\quad\quad \Bigl(\mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Bigr)^{2} \Biggr \rbrace\\ & = \max \Biggl \lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \\ &\quad\quad\quad \Bigl(\Theta_{fb}\Bigl(\psi_{r}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Bigr)^{2} \Biggr \rbrace\\ & = \max \Biggl\lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , 1 \Biggr\rbrace. \end{align} |
It follows that
\begin{equation} \Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) = 1. \end{equation} | (2.21) |
\begin{align} &\quad\Gamma_{3}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & = \max\Big\lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Big), \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Big\rbrace \\ & = \max \Big\lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Big) \Big\rbrace. \end{align} | (2.22) |
Using (2.20), (2.21) and (2.22) in (2.19), we have
\begin{align} \Theta_{fb}(\psi_{r}, \psi_{r+1}, t)\geq \dfrac{ \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}).\Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})}{\max \lbrace \Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}), \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \rbrace}. \end{align} | (2.23) |
If
\begin{align*} &\quad\max \Bigg \lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigg\rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then (2.23) implies
\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geq \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). |
It is obvious by Lemma 1.2.
If
\begin{align*} &\quad\max \Bigg\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigg\rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then from (2.23), we have
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geq \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). \end{equation*} |
Continuing in this way, we will get
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, t)&\geq \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(F_{\theta}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & \vdots\\ & \geq \Theta_{fb}\Bigl(\psi_{0}, \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma)).\beta(\Theta_{fb}(\psi_{n-2}, \psi_{r-1}, \gamma))\ldots\beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\Bigr). \end{align*} |
The rest of the proof follows in the same way as in Theorem 2.1.
Remark 2.6. By setting \hat{C_{_0}} (\mho) = \mho , the mapping \Xi \colon \mho \rightarrow \hat{C_{_0}} (\mho) becomes a self (singlevalued) mapping and we get Theorem 3.13 of [32].
An application of Theorem 2.1 is presented here. Recall that the space of all continuous realvalued functions on [0, 1] is denoted by C([0, 1], \mathbb{R}) . Now set \mho = C([0, 1], \mathbb{R}) and define the G -complete Fbm on \mho by
\Theta_{fb} (\psi, \varrho, \gamma) = e^{-\dfrac{ {\sup\limits_{u \in[0, 1]}}\vert \psi(u)-\varrho(u) \vert^{2}} {\gamma}}, \quad \forall \;\; \gamma > 0 \;\; \text{and} \; \; \psi, \varrho \in \mho. |
Consider
\begin{equation} \psi(u)\in \int_0^u G(u, v, \psi(v))dv + h(u) \quad \text{for all} \;\; u, v \in[0, 1], \quad \text{whereas}\quad h, \psi\in C([0, 1], \mathbb{R}). \end{equation} | (3.1) |
Here G \colon [0, 1]\times[0, 1] \times \mathbb{R}\rightarrow P_{cv}(\mathbb{R}) is multivalued function and P_{cv}(\mathbb{R}) represents the collections of convex and compact subsets of \mathbb{R} . Moreover, for each \psi in C([0, 1], \mathbb{R}) the operator G(\cdot, \cdot, \psi) is lower semi-continuous.
For the integral inclusion given in (3.1), define a multivalued operator S: \mho\rightarrow \hat{C_{0}} (\mho) by
S\psi(u) = \Biggl\lbrace w \in \mho : w \in \int_0^u G(u, v, \psi(v))dv + h(u), \quad u \in [0, 1]\Biggr\rbrace. |
Now for arbitrary \psi \in (C([0, 1], \mathbb{R}) , denote G_\psi(u, v) = G(u, v, \psi(v)) where u, v\in [0, 1] . For the multivalued map G_\psi : [0, 1]\times [0, 1]\rightarrow P_{cv}(\mathbb{R}) , by Michael selection theorem [34], there exists a continuous selection g_\psi :[0, 1] \times [0, 1]\rightarrow \mathbb{R} such that g_\psi(u, v)\in G_\psi(u, v) for each u, v\in [0, 1] . It follows that
\int_0^u g_\psi(u, v)dv + h(u) \in S\psi(u). |
Since g_\psi is continuous on [0, 1]\times [0, 1] and h is continuous on [0, 1] , therefore both g_\psi and h are bounded realvalued functions. It follows that, the operator S\psi is nonempty and S\psi \in \hat{C_{0}} (\mho) .
With the above setting, the upcoming outcome shows the existence of a solution of the integral inclusion (3.1).
Theorem 3.1. Let \mho = C([0, 1], \mathbb{R}) and define the multivalued operator S: \mho \rightarrow \hat{C_{0}} (\mho) by
S\psi(u) = \Biggl\lbrace w \in \mho : w \in \int_0^u G(u, v, \psi(v))dv + h(u), \quad u \in [0, 1]\Biggr\rbrace, |
where h:[0, 1]\rightarrow \mathbb{R} is continuous and the map G:[0, 1]\times [0, 1]\times \mathbb{R}\rightarrow P_{cv}(\mathbb{R}) is defined in such a way that for every \psi\in C([0, 1], \mathbb{R}) , the operator G(\cdot, \cdot, \psi) is lower semi-continuous. Assume further that the given terms are satisfied:
(i) There exists a continuous mapping f \colon [0, 1]\times [0, 1]\rightarrow [0, \infty) such that
\mathcal{H}_{F_{\Theta_{fb}}}( G(u, v, \psi(v))-G(u, v, \varrho(v))\leq f^{2}(u, v)\vert \psi(v)-\varrho(v)\vert^{2}, |
for each \psi, \varrho \in \mho and u, v\in [0, 1] .
(ii) There exists \beta \in \mathbb{F}_{\Theta 2} , such that
{ \sup\limits_{u \in[0, 1]}}\int_0^u f^{2}(u, v)dv \leq \beta({\Theta_{fb}}(\psi, \varrho, \gamma)) . |
Then (3.1) has a solution in \mho .
Proof. We will show that the operator S satisfies the conditions of Theorem 2.1. In particular we prove (2.1) as follows:
Let \psi, \varrho \in \mho be such that q\in S\psi . As stated earlier, by selection theorem there is g_\psi(u, v)\in G_\psi(u, v) = G(u, v, \psi(v)) for u, v \in [0, 1] such that
q(u) = \int_0^u g_\psi(u, v)dv+h(u), \quad u\in [0, 1]. |
Further, the condition (ⅰ) ensures that there is some g(u, v)\in G_\varrho(u, v) such that
\vert g_\psi(u, v)-g(u, v)\le f^{2}(u, v)\vert \psi(v)-\varrho(v)\vert^{2}, \quad\forall\, u, v\in [0, 1]. |
Now consider the multivalued operator T defined as follows:
T(u, v) = G_\varrho(u, v)\cap \left\lbrace w\in \mathbb{R} : \left\vert g_\psi(u, v)-w\right\vert\le f^{2}(u, v)\vert \psi(v)-\varrho(v)\vert^{2}\right\rbrace. |
Since, by construction, T is lower semi-continuous, it follows again by the selection theorem that there is continuous function g_\varrho(u, v):[0, 1]\times [0, 1]\rightarrow \mathbb{R} such that for each u, v \in [0, 1] , g_\varrho(u, v) \in T(u, v) .
Thus, we have
r(u) = \int_0^u g_\varrho(u, v)dv+h(u)\, \in \int_0^u G(u, v, \varrho(v))dv+h(u), \quad u\in [0, 1]. |
Therefore, for each u\in [0, 1] we get
\begin{align*} e^{-\dfrac{ {\sup\limits_{t \in[0, 1]}}\vert q(u)- r(u)) \vert^{2}}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}& \geq e^{-\dfrac{ {\sup\limits_{u \in[0, 1]}}\int_0^u \vert g_{\psi}(u, v)-g_{\varrho}(u, v) \vert^{2} dv}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}\\ & \geq e^{-\dfrac{ {\sup\limits_{u \in[0, 1]}}\int_0^u f^{2}(u, v) \vert \psi(v)-\varrho(v) \vert^{2} dv}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}\\ & \geq e^{-\dfrac{\vert \psi(v)-\varrho(v) \vert^{2} {\sup\limits_{u \in[0, 1]}}\int_0^u f^{2}(u, v) dv}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}\\ & \geq e^{-\dfrac{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\vert \psi(v)-\varrho(v) \vert^{2}}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}\\ & = e^{-\dfrac{\vert \psi(v)-\varrho(v) \vert^{2}}{\gamma}}\\ & \geq e^{-\dfrac{ {\sup\limits_{v \in[0, 1]}}\vert \psi(v)-\varrho(v) \vert^{2}}{\gamma}}\\ & = \Theta_{fb}(\psi, \varrho, \gamma). \end{align*} |
This implies that,
\Theta_{fb}(q, r, \beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma)\geq \Theta_{fb}(\psi, \varrho, \gamma). |
Interchanging the roles of \psi and \varrho , we get
\mathcal{H}_{F_{\Theta_{fb}}}(S\psi, S\varrho, \beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma) \geq \Theta_{fb}( \psi, \varrho, \gamma). |
Hence, by Theorem 2.1, the operator S has a fixed point which in turn proves the existence of a solution of integral inclusion (3.1).
In the present work, in the setting of a Hausdorff Fbms, some fixed fixed point results for multivalued mappings are established. The main result, that is, Theorem 2.1 shows that a multivalued mapping satisfying Geraghty type contractions on G -complete Hfbms has a fixed point. Example 2.1 illustrates the main result. Some other interesting fixed point theorems are also proved for the multivalued mappings satisfying certain contraction condition on G -complete Hfbms. The results proved in [30,31,32] turn out to be special cases of the results established in this work. For the significance of our results, an application is presented to prove the existence of solution of an integral inclusion.
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work grant code: 22UQU4331214DSR02
The authors declare that they have no conflict of interest.
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