Common fixed point results for couples $(f,g)$ and $(S,T)$ satisfy strong common limit range property

1.   Introduction

In the area of fixed point theory the study of common fixed point for couple of mapping is a new research area (see ^[1,7] and the references therein). For hybrid pair the (E. A) property was introduced by Kamran ^[6] and established coincidence and fixed points results via hybrid strict contractive conditions. To obtain common fixed point results for hybrid type contractive condition, Liu et al. ^[8] extended this property to common (E. A) property for hybrid pairs of (set-valued and single) mappings. For mapping of single-valued Sintunavarat and Kumam ^[13] put together the notion of the common limit range $ (CLR) $ property and showed its dominance on the property (E. A). For the mapping of hybrid pair the common limit range property defined by Imdad et al. ^[5] for set-valued fixed point mapping in the semi-metric (symmetric) spaces.

Afrah ^[2] generalized this property for single hybrid pair of mapping and then the same author in ^[3] extended the $ CLR_g $ property for two hybrid pairs of of mapping. In fuzzy metric spaces, by means of $ CLRg $ property, Rold and Sintunavarat ^[14] established common fixed point results. In 2016, Shoaib and Sarwar ^[11] studied applications to two functional equations by using set-valued fixed point theorems for a pair of maps and made use of $ (CLR) $ property via generalized contractions. For generalized hybrid $ (F, \varphi) $-contractions Nashine et al.^[10] established common fixed point results and used the common limit range property. For more detail see(^{[4,12,15,16,17,18]}).

Motivated by above, using strong common limit range property, we derived set-valued common fixed point results in metric space for couples of maps. Using this property, established hybrid common fixed point results. Also explain the property by giving examples.

In the whole paper $ CB({\tilde N}) $ shows the class of all bounded and closed subsets of $ {\tilde N}, $ $ \mathbb{\textbf{R}^{+}} $ is the set of positive real numbers and $ \mathbb{\textbf{R}} $ the set of real numbers respectively.

Definition 1.1. ^[9] Maps $ f:\tilde{N} \rightarrow \tilde{N} $, $ S:\tilde{N} \rightarrow CB(\tilde{N}) $ are said to be occasionally S-weakly commuting if there exists $ \xi\in{\tilde N} $ such that $ f\xi\in S\xi $ and $ ff\xi\in fS\xi $.

Definition 1.2. ^[13] If for a sequence $ \{\xi_{n}\} $ in $ {\tilde N} $, $ \underset{n\rightarrow \infty}{\lim} f\xi_{n} = \underset{n\rightarrow \infty}{\lim}g\xi_{n} = fu $ for some $ u \in {\tilde N} $. Then, $ f, g: {\tilde N} \rightarrow {\tilde N} $ are said to have the common limit range property of $ f $ with respect to $ g $ (shortly, the $ (CLR_{f}) $-property w.r.t to $ g $).

The following two definitions can be found in ^[3].

Definition 1.3. If for a sequence $ \{\xi_{n}\} $ in $ {\tilde N} $ and $ \Omega_{1}\in CB({\tilde N}) $, $ \underset{n\rightarrow \infty}{\lim}f\xi_{n} = fu\in \Omega_{1} = \underset{n\rightarrow\infty}{\lim} S\xi_{n} $ for some $ u \in {\tilde N} $. Then, $ S:{\tilde N} \rightarrow CB({\tilde N}), $ $ f: {\tilde N} \rightarrow {\tilde N} $, over metric space $ ({\tilde N}, d) $ are said to have the common limit range property of $ f $ with respect to $ S $ (shortly, $ (CLR_{f}) $-property w.r.t to $ S $).

Definition 1.4. If for sequences $ \{\xi_{n}\}, \{\zeta_{n}\} $ in $ {\tilde N} $ and $ \Omega_{1}, \Omega_{2}\in CB({\tilde N}) $, $ \underset{n\rightarrow \infty}{\lim} S\xi_{n} = \Omega_{1} $, $ \underset{n\rightarrow\infty}{\lim} T\zeta_{n} = \Omega_{2}, $ $ \underset{n\rightarrow \infty}{\lim} f\xi_{n} = \underset{n\rightarrow\infty}{\lim}g\zeta_{n} = fu\in \Omega_{1}\cap \Omega_{2} $, for some $ u \in {\tilde N}. $ Then, $ S, T: {\tilde N} \rightarrow CB({\tilde N}), $ $ f, g: {\tilde N} \rightarrow {\tilde N} $ on metric space $ ({\tilde N}, d) $ are said to have the common limit in the range of $ f $ with respect to $ S $ (shortly, $ (CLR_{f}) $-property w.r.t $ S $).

The below definition is new and it is a modification of Definition 1.3 for couples of functions.

Definition 1.5. Assume $ f, g: {\tilde N} \rightarrow {\tilde N} $ and $ S, T: {\tilde N} \rightarrow CB({\tilde N}) $ are functions defined on metric space $ ({\tilde N}, d) $. Then the couple $ (f, g) $ and the couple $ (S, T) $ are said to fulfil the common limit in the range of $ f $ with respect to $ (S, T) $ via $ g $ (shortly, $ (CLR_f) $-property with respect to $ (S, T) $ via $ g $) if there exist sequences $ \{\xi_{n}\} $ and $ \{\zeta_{n}\} $ in $ {\tilde N} $ and $ \Omega_{1}, \Omega_{2}\in CB({\tilde N}) $ such that, for some $ u \in {\tilde N} $ we have $ \underset{n\rightarrow \infty}{\lim} S\xi_{n} = \Omega_{1} $, $ \underset{n\rightarrow \infty}{\lim} T\zeta_{n} = \Omega_{2} $ and $ \underset{n\rightarrow \infty}{\lim} f\xi_{n} = \underset{n\rightarrow\infty}{\lim}g\zeta_{n} = fu\in \Omega_{1}\cap \Omega_{2}. $

Remark 1.6. Clearly, if $ f = g $ and $ S = T $ in Definition 1.4 then we reobtain Definition 1.3.

The below definition announces the so-called the strong common limit range property.

Definition 1.7. If the couples $ (f, g) $ and $ (S, T) $ satisfy the $ (CLR_f) $-property with respect to $ (S, T) $ via $ g $ and the $ (CLR_g) $ property with respect to $ (S, T) $ via $ f $ then we say the couples $ (f, g) $ and $ (S, T) $ fulfil the strong common limit range property.

Remark 1.8. $ f $ and $ S $ fulfil strong common limit range property if there exists $ \{\xi_{n}\} $ and $ \{\zeta_{n}\} $ in $ {\tilde N} $ and $ \Omega_{1}, \Omega_{2}\in CB({\tilde N}) $ such that, for some $ u \in {\tilde N} $ we have $ \underset{n\rightarrow \infty}{\lim} S\xi_{n} = \Omega_{1} $, $ \underset{n\rightarrow \infty}{\lim} S\zeta_{n} = \Omega_{2} $ and $ \underset{n\rightarrow \infty}{\lim} f\xi_{n} = \underset{n\rightarrow\infty}{\lim}f\zeta_{n} = fu\in \Omega_{1}\cap \Omega_{2}. $

Remark 1.9. Couples $ (f, g) $ and $ (S, T) $ said to be not strong common limit range property. if $ \underset{n\rightarrow\infty}{\lim} f\xi_{n} $ and $ \underset{n\rightarrow\infty}{\lim}g\zeta_{n} $ exist but not equal to $ fu $ or their does not exist $ \{\xi_n\}, $ $ \{\zeta_n\} $ such that $ \underset{n\rightarrow \infty}{\lim} f\xi_{n} = \underset{n\rightarrow\infty}{\lim}g\zeta_{n} = fu. $

Example 1.10. Let $ {\tilde N} = [0, \infty) $ with the usual metric. Define $ f, g:{\tilde N}\rightarrow {\tilde N} $ and $ S, T:{\tilde N}\times {\tilde N}\rightarrow CB({\tilde N}) $ by $ f(x) = 1+x $, $ g(x) = x^2 $, $ S(x) = [1, 2+2x], $ $ T(x) = [1, 2+\frac{3x}{4}], $ $ \forall x \in {\tilde N}. $

Consider the sequences $ \{\varsigma_{n}\} = \{1+\frac{1}{n}\}, $ $ \{\zeta_{n}\} = \{2+\frac{1}{n}\}, $

Clearly $ \underset{n\rightarrow \infty}{\lim}S(\varsigma_{n}) = [1,4] = \Omega_1, $

$ \underset{n\rightarrow\infty}{\lim}T(\zeta_{n}) = [1, \frac{7}{2}] = \Omega_2, $ $ \underset{n\rightarrow\infty}{\lim}f(\varsigma_{n}) = \underset{n\rightarrow\infty}{\lim}g(\zeta_{n}) = 2 = f(1) = g(2)\in \Omega_1\cap\Omega_2. $

Therefore couples the $ (f, g) $ and $ (S, T) $ fulfil the strong common limit range property.

Example 1.11. Let $ {\tilde N} = [0, 10) $ with the usual metric. Define $ f, g:{\tilde N}\rightarrow {\tilde N} $ and $ S, T:{\tilde N}\times {\tilde N}\rightarrow CB({\tilde N}) $ by $ f(x) = 1 $, $ g(x) = 2 $, $ S(x) = [2x, 2+2x], $ $ T(x) = [x, 2+\frac{3x}{4}], $ $ \forall x \in {\tilde N}. $

For any choice of $ \{\varsigma_{n}\} $ and $ \{\zeta_{n}\}, $ the couples the $ (f, g) $ and $ (S, T) $ does not hold the strong common limit range property.

Now, we provide definitions defined for set-valued mappings in a metric space $ ({\tilde N}, d) $. Defined the function $ H:CB({\tilde N})\times CB({\tilde N})\rightarrow \mathbb{\textbf{R}^{+}} $ for $ {\tilde N}_{1}, {\tilde N}_{2} \in CB({\tilde N}) $ by

where

and

Lemma 1.12. ^[19] Let $ ({\tilde N}, d) $ be a metric space. For any $ {\tilde N}_{1}, {\tilde N}_{2} \in CB({\tilde N}). $ We have

$ d(\xi, {\tilde N}_{2})\leq H({\tilde N}_{1}, {\tilde N}_{2}) $, for all $ \xi \in {\tilde N}_{1} $.

Definition 1.13. ^[20] Let $ \eta:{\tilde N}\rightarrow{\tilde N} $ and $ \lambda:{\tilde N}\rightarrow CB({\tilde N}) $ be a mapping. Then $ \eta $ are known occasionally weakly $ \lambda $- commuting $ \Leftrightarrow $ there exist $ x $ in $ {\tilde N} $ such that $ \eta\eta x\in \lambda\eta x $ for $ \eta x\in \lambda x $.

Theorem 1.14. ^[3] Let $ f, g:{\tilde N}\rightarrow {\tilde N} $ and $ S, T: {\tilde N}\rightarrow CB({\tilde N}) $ be a maps on metric space $ ({\tilde N}, d) $ satisfying the following condition.

(a)The pairs $ (S, f) $ and $ (T, g) $ have common limit range property $ (CLR_f) $,

(b)for all $ x, y\in {\tilde N} $

where,

Here, $ p\geq1 $, and $ \varphi:[0, \infty)\rightarrow [0, \infty) $ is continuous monotone increasing function such that $ \varphi(0) = 0 $ and $ \varphi(t) < t $ for all $ t > 0 $. If $ f({\tilde N}) $ and $ g({\tilde N}) $ are closed subsets of $ {\tilde N} $, then we have the following:

$(A_1)$ $ (f, S) $ have coincidence point.

$(A_2)$ $ (g, T) $ have coincidence point.

$(A_3)$ $ f $ and $ S $ has a common fixed point, if $ ffv = fv $ and $ f $ and $ S $ are weakly compatible at $ v $;

$(A_4)$ $ g $ and $ T $ has a common fixed point, if $ ffv = fv $ and $ g $ and $ T $ are weakly compatible at $ v $;

$(A_5)$ if $ (A_3) $ and $ (A_4) $ holds. Then $ g $, $ f $, $ T $ and $ S $ have a common fixed point.

Definition 1.15. ^[17] Let $ F_{s} $ denotes the class of all mapping $ F_1:\mathbb{\textbf{R}^{+}}\rightarrow \mathbb{\textbf{R}} $, with the below conditions

(1) $ F_1 $ is strictly increasing and continuous;

(2) $ \underset{n\rightarrow \infty}{\lim}{\alpha_{n}} = 0 $ if and only if $ \underset{n\rightarrow \infty}{\lim}F_1({\alpha_{n}}) = -\infty $;

(3) For $ \{\alpha_{n}\}\subset \mathbb{\textbf{R}^{+}} $, $ \underset{n\rightarrow \infty}{\lim}{\alpha_{n}} = 0 $, there exists $ q\in(0, 1), $ such that $ \underset{\alpha\rightarrow 0^+}{\lim}({\alpha_{n}})^{q}F_1({\alpha_{n}}) = 0. $

Thoroughly in this section $ \Phi $ denote the below class

Theorem 1.16. Let $ f, g:{\tilde N}\rightarrow {\tilde N} $ and $ S, T: {\tilde N}\rightarrow CB({\tilde N}) $ be a maps on metric space $ ({\tilde N}, d) $. Suppose that $ (S, g) $ and $ (T, f) $ have strong common limit range property. Furthermore assume that

where, $ H^p(Tx, Sy) > 0 $ and

Here, $ \tau\in \mathbb{\textbf{R}^{+}}, $ $ \alpha+\beta+\gamma+\sigma+\eta < 1, $ $ p\geq1 $, $ F\in F_s $ and $ \varphi\in \Phi $. Then the below assumption holds.

$(A_1)$ $ (f, T) $ have coincidence point.

$(A_2)$ $ (g, S) $ have coincidence point.

$(A_3)$ $ T $ and $ f $ has a common fixed point, if $ ffv = fv $ and $ f $ is occasionally $ T $-weakly commuting at $ v $;

$(A_4)$ $ g $ and $ S $ has a common fixed point, if $ ggu = gu $ and $ g $ is occasionally $ S $-weakly commuting at $ w $;

$(A_5)$ if $ (A_3) $ and $ (A_4) $ holds. Then $ g $, $ f $, $ T $ and $ S $ have a common fixed point.

Proof. Since $ f, g: {\tilde N} \rightarrow {\tilde N} $ and $ T, S: {\tilde N} \rightarrow CB({\tilde N}) $ have strong common limit range property, therefore there exists a sequence $ \{\xi_{n}\} $ and $ \{\zeta_{n}\} $ in $ {\tilde N} $ and $ \Omega_{1}, \Omega_{2}\in CB({\tilde N}) $ such that,

for some $ u, v\in {\tilde N}. $

Now, we show $ gv\in Sv $, suppose $ gv\notin Sv. $ then putting $ x = \xi_{n}, y = v $ in inequality $ (1.2) $, we have

where,

By taking limit to $ \Delta $, we have

Apply limit over (1.3) and by using (1.4), we have

Which implies that

Using definitions of $ F $ and $ \varphi $, we have

But $ \alpha < 1 $ and using Lemma 1.12

Which is contradiction. Hence, $ gv\in Sv $.

Again from (1.2), we have

where,

By taking limit to $ \Delta $, we have

By taking limit to (1.6) and using (1.7), we have

Which implies that

Using definitions of $ F $ and $ \varphi $, we have

But $ \alpha < 1 $ and using Lemma 1.12

Which is contradiction. Hence, $ fu\in Tu $. Since $ ffv = fv $ and $ fv \in Tfv $, therefore $ \gamma = f\gamma\in T\gamma $.

Similarly $ \gamma = g\gamma\in S\gamma $. $ (A_5) $ hold obviously.

Example 1.17. Let $ {\tilde N} = [0, \infty) $ is a metric(w.r.t) the usual metric. $ S, T:{\tilde N}\rightarrow CB({\tilde N}) $, $ f, g:{\tilde N}\rightarrow {\tilde N} $ and $ \varphi:\mathbb{\textbf{R}}^{+}\rightarrow \mathbb {\textbf{R}}^{+} $ define by $ S(x) = [1, 5+2\beta x], $ $ T(x) = [1, 5+\beta x], $ $ f(x) = 4x $, $ g(x) = 2x $ $ \forall x \in {\tilde N} $ and $ \varphi(t) = \beta t, $ $ 0 < \beta < 1 $.

Consider the sequences $ \{\xi_{n}\} = \{1+\frac{1}{n}\}, $ $ \{\zeta_{n}\} = \{2+\frac{1}{n}\}, $

Now, $ \underset{n\rightarrow\infty}{\lim}S(\xi_{n}) = [1, 5+2\beta] $ $ \underset{n\rightarrow\infty}{\lim}T(\zeta_{n}) = [1, 5+2\beta], \underset{n\rightarrow\infty}{\lim}g(\zeta_{n}) = \underset{n\rightarrow\infty}{\lim}f(\xi_{n}) = 4 = f(1) = g(2)\in [1, 5+\beta]\cap[1, 5+2\beta]. $

Therefore couples $ (f, g) $ and $ (S, T) $ satisfy the strong common limit range. Now

By taking natural logarithm on both sides we conclude by Theorem 1.20 that $ C(S, f)\neq \emptyset $ and $ C(T, g)\neq \emptyset. $ where $ C(S, f) $ represent coincidence point of $ S $ and $ f. $

Example 1.18. Let $ {\tilde N} = (-11, 11) $ with the usual metric. Define $ S, T:{\tilde N}\times {\tilde N}\rightarrow CB({\tilde N}) $, $ f, g:{\tilde N}\rightarrow {\tilde N}, $ $ \varphi:\mathbb{\textbf{R}^{+}}\rightarrow \mathbb{\textbf{R}^{+}} $ and $ F:\mathbb{\textbf{R}^{+}}\rightarrow \mathbb{\textbf{R}} $ by $ S(x) = [-6, 2+\frac{\beta x}{4}], $ $ T(x) = [-6, 2+\frac{\beta}{6} x], $ $ f(x) = \frac{x}{2} $, $ g(x) = \frac{x}{3} $ $ \forall x \in {\tilde N}, $ $ \varphi(t) = \beta t, $ $ 0 < \beta < 1 $ and $ F(t) = ln(t) $.

Consider the sequences $ \{\xi_{n}\} = \{2+\frac{1}{n}\}, $ $ \{\zeta_{n}\} = \{3+\frac{1}{n}\}, $

Now, $ \underset{n\rightarrow\infty}{\lim}S(\xi_{n}) = [-6, 2+\frac{\beta}{2}] $ $ \underset{n\rightarrow\infty}{\lim}T(\zeta_{n}) = [-6, 2+\frac{\beta}{2}], \underset{n\rightarrow\infty}{\lim}g(\zeta_{n}) = \underset{n\rightarrow\infty}{\lim}f(\xi_{n}) = 1 = f(2) = g(3)\in [-6, 2+\frac{\beta}{2}]\cap[-6, 2+\frac{\beta}{2}]. $

Therefore couples $ (f, g) $ and $ (S, T) $ satisfy the strong common limit range. Now

Now,

Taking logarithm on both sides and $ p = 1 $, we conclude that all the other condition of Theorem 1.16 are satisfied. Therefore $ (f, g) $ and $ (S, T) $ have coincidence point point.

Remark 1.19. From above examples it is clear that

● Theorem 1.14 is not applicable to Example 1.18 because $ f({\tilde N}) = (-\frac{11}{2}, \frac{11}{2}) $ nor $ g({\tilde N}) = (\frac{-11}{3}, \frac{11}{3}) $ are closed.

Theorem 1.20. Let $ f, g:{\tilde N}\rightarrow {\tilde N} $ and $ S, T: {\tilde N}\rightarrow CB({\tilde N}) $ are mapping on metric space $ ({\tilde N}, d) $. Furthermore assume that $ (S, g) $ and $ (T, f) $ have strong common limit range property and

where, $ H^p(Tx, Sy) > 0 $ and

Here, $ \tau\in \mathbb{\textbf{R}^{+}} $, $ p\geq1 $, $ F\in F_s $ and $ \varphi\in \Phi $. Then the below condition holds.

$(A_1)$ $ (f, T) $ have coincidence point.

$(A_2)$ $ (g, S) $ have coincidence point.

$(A_3)$ $ T $ and $ f $ has a common fixed point, if $ ffv = fv $ and $ f $ is occasionally $ T $-weakly commuting at $ v $;

$(A_4)$ $ S $ and $ g $ has a common fixed point, if $ ggw = gw $ and $ g $ is occasionally $ S $-weakly commuting at $ w $;

$(A_5)$ if $ (A_3) $ and $ (A_4) $ holds. Then $ f $, $ S $, $ g $ and $ T $ have a common fixed point.

Proof. Since $ f, g: {\tilde N} \rightarrow {\tilde N} $ and $ T, S: {\tilde N} \rightarrow CB({\tilde N}) $ have strong $ (CLR) $-property, therefore there exists a sequence $ \{\xi_{n}\} $ and $ \{\zeta_{n}\} $ in $ {\tilde N} $ and $ \Omega_{1}, \Omega_{2}\in CB({\tilde N}) $ such that,

for some $ u, v\in {\tilde N}. $ Now, we show $ gv\in Sv $, suppose $ gv\notin Sv, $ by putting $ x = \xi_{n}, y = v $ in inequality $ (1.22) $, we have

Taking limit to $ \Delta $, we have

By taking limit over (1.9), and using (1.10), we have

Which implies that

Using definitions of $ F $ and $ \varphi $, we have

But $ \alpha < 1 $ and using Lemma 1.12

which is contradiction. Hence, $ gv\in Sv $.

We show $ fu\in Tu $, suppose $ fu\notin Tu. $ then putting $ x = u, y = \zeta_{n} $ in inequality $ (1.22) $, we have

By taking limit over $ \Delta $, we have

By taking limit over (1.12) and using (1.13), we have

Which implies that

Using definitions of $ F $ and $ \varphi $, we have

But $ \alpha < 1 $ and using Lemma 1.12

Which is contradiction. Hence, $ fu\in Tu $.

Succeeding the parallel line of Theorem 1.16, we can achieved that $ S, T, f $ and $ g $ have common coupled fixed point.

If $ S = T $ and $ f = g $ in Theorem 1.16, by using Remark 1.6, we have

Corollary 1.21. Let $ f:{\tilde N}\rightarrow {\tilde N} $ and $ S: {\tilde N}\rightarrow CB({\tilde N}) $ be a maps on metric space $ ({\tilde N}, d) $. Suppose $ (S, f) $ have strong common limit range property. Furthermore

where, $ H(Sx, Sy) > 0 $ and

Here, $ \tau\in \mathbb{\textbf{R}^{+}}, $ $ \alpha+\beta+\gamma+\sigma+\eta < 1, $ $ p\geq1 $, $ F\in F_s $ and $ \varphi\in \Phi $. Then the below condition holds.

$(A_1)$ $ (f, S) $ have coincidence point.

$(A_2)$ $ f $ and $ S $ has a common fixed point, if $ ffv = fv $ and $ f $ is occasionally $ S $-weakly commuting at $ v $; Then $ f $, $ S $ have a common fixed point.

If $ f = g $ and $ S = T $ in Theorem 1.20 by using Remark 1.6, we have

Corollary 1.22. Let $ f:{\tilde N}\rightarrow {\tilde N} $ and $ S: {\tilde N}\rightarrow CB({\tilde N}) $ be a maps on metric space $ ({\tilde N}, d) $. Suppose $ (S, f) $ have strong common limit range property. Furthermore

where, $ H(Sx, Sy) > 0 $ and

Here, $ \tau\in \mathbb{\textbf{R}^{+}} $, $ p\geq1 $, $ F\in F_s $ and $ \varphi\in \Phi $. Then the below assumption holds.

$(A_1)$ $ (f, S) $ have coincidence point.

$(A_2)$ $ f $ and $ S $ has a common fixed point, if $ ffv = fv $ and $ f $ is occasionally $ S $-weakly commuting at $ v $; Then $ f $, $ S $ have a common fixed point.

Theorem 1.23. Let $ f, g:{\tilde N}\rightarrow {\tilde N} $ and $ S, T: {\tilde N}\rightarrow CB({\tilde N}) $ be a maps on metric space $ ({\tilde N}, d) $. Suppose that $ (S, g) $ and $ (T, f) $ have strong common limit range property. Moreover assume that

where, $ 0 < \alpha < 1 $. Then

$(A_1)$ $ (f, T) $ have coincidence point.

$(A_2)$ $ (g, S) $ have coincidence point.

$(A_3)$ $ f $ and $ T $ has a common fixed point, if $ ffv = fv $ and $ f $ is occasionally $ T $-weakly commuting at $ v $;

$(A_4)$ $ g $ and $ S $ has a common fixed point, if $ ggu = gu $ and $ g $ is occasionally $ S $-weakly commuting at $ w $;

$(A_5)$ if $ (A_3) $ and $ (A_4) $ holds. Then $ f $, $ g $, $ S $ and $ T $ have a common fixed point.

Proof. Since $ f, g: {\tilde N} \rightarrow {\tilde N} $ and $ T, S: {\tilde N} \rightarrow CB({\tilde N}) $ have strong $ (CLR) $-property, therefore there exists a sequence $ \{\xi_{n}\} $ and $ \{\zeta_{n}\} $ in $ {\tilde N} $ and $ \Omega_{1}, \Omega_{2}\in CB({\tilde N}) $ such that,

for some $ u, v\in {\tilde N}. $ Now, we show $ gv\in Sv $, suppose $ gv\notin Sv, $ by putting $ x = \xi_{n}, y = u $ in inequality $ (1.14) $, we have

Applying limit, we have

By using Lemma 1.12

which is possible if $ gv\in Sv $.

We show $ fu\in Tu $, suppose $ fu\notin Tu. $ then putting $ x = u, y = \zeta_{n} $ in inequality $ (1.19) $, we have

Taking limit, we have

Using Lemma 1.12, we have

Which is possible only if $ fu\in Tu $.

After succeeding the similar lines of Theorem 1.16 we can obtained that $ S, T, f $ and $ g $ have common coupled fixed point.

Theorem 1.24. Let $ f, g:{\tilde N}\rightarrow {\tilde N} $ and $ S, T: {\tilde N}\rightarrow CB({\tilde N}) $ be a maps on metric space $ ({\tilde N}, d) $. Suppose that $ (S, g) $ and $ (T, f) $ have strong common limit range property. Moreover assume that

Where $ \Delta^* = d^{p}(fx, Sy)+d^{p}(gy, Tx) $, $ 0 < \alpha < 1 $. Then

$(A_1)$ $ (f, T) $ have coincidence point.

$(A_2)$ $ (g, S) $ have coincidence point.

$(A_3)$ $ T $ and $ f $ has a common fixed point, if $ ffv = fv $ and $ f $ is occasionally $ T $-weakly commuting at $ v $;

$(A_4)$ $ S $ and $ g $ has a common fixed point, if $ ggu = gu $ and $ g $ is occasionally $ S $-weakly commuting at $ w $;

$(A_5)$ if $ (A_3) $ and $ (A_4) $ holds. Then $ g $, $ f $, $ S $ and $ T $ have a common fixed point.

Succeeding the steps of Theorem 1.23 we can obtained that $ S, T, f $ and $ g $ have common coupled fixed point.

If $ f = g $, $ T = S $ in Theorem 1.24 by using Remark 1.6, we get

Corollary 1.25. Let $ f:{\tilde N}\rightarrow {\tilde N} $ and $ S: {\tilde N}\rightarrow CB({\tilde N}) $ be a maps on metric space $ ({\tilde N}, d) $. Suppose that $ (S, f) $ have strong common limit range property. Furthermore

Where $ \Delta^* = d^{p}(fx, Sy)+d^{p}(gy, Tx) $, $ 0 < \alpha < 1 $. Then

$(A_1)$ $ (f, S) $ have coincidence point.

$(A_2)$ $ S $ and $ f $ has a common fixed point, if $ ffv = fv $ and $ f $ is occasionally $ S $-weakly commuting at $ v $. Then $ f $, $ S $ have a common fixed point.

If $ T = S $, $ f = g $ in Theorem 1.23. From Remark 1.6, we have

Corollary 1.26. Let $ f:{\tilde N}\rightarrow {\tilde N} $ and $ S: {\tilde N}\rightarrow CB({\tilde N}) $ be a maps on metric space $ ({\tilde N}, d) $. Suppose $ (S, f) $ have strong common limit range property. Furthermore

where, $ 0 < \alpha < 1 $. Then

$(A_1)$ $ (f, S) $ have coincidence point.

$(A_2)$ $ f $ and $ S $ has a common fixed point, if $ ffv = fv $ and $ f $ is occasionally $ S $-weakly commuting at $ v $. Then $ S $, $ f $ have a common fixed point.

2.   Application to system of integral and differential equation

Now, we study solutions of 2nd kind general nonlinear system of Fredholm integral equations given by

Let $ {\tilde N} = C[p, q] $ be the set of all continuous function defined on $ [p, q] $. Define

$ d:{\tilde N}\times {\tilde N}\rightarrow \mathbb{\textbf{R}^{+}} $, by

Where $ \|x\| = \sup\{|x(t)|:t\in[p, q]\} $. Then $ ({\tilde N}, d) $ is a complete $ d $ metric space on $ {\tilde N} $. We give the following theorem.

Theorem 2.1. Assume that the following assumptions hold

$(A_1)$ $ Q_{j}:[p, q]\times[p, q] \times \mathbb{\textbf{R}^{+}}\rightarrow \mathbb{\textbf{R}^{+}} $, for $ j = 1, 2 $ and $ \phi: \mathbb{\textbf{R}^{+}}\rightarrow \mathbb{\textbf{R}^{+}} $ is continuous;

$(A_2)$ there exist a continuous function $ G:[p, q]\times[p, q]\rightarrow [0, \infty) $ such that,

for each $ t, s\in[p, q] $, $ 0 < \gamma < 1 $,

$(A_3) $ $ \sup_{t, s\in[p, q]}\int^q_p |G(t, s)|ds\leq e^{-\tau} $ for $ \tau > 0 $.

Then the system of integral equations 2.1 has a common solution in $ C([p, q]) $.

Proof. Define $ S, T:C([p, q])\rightarrow C([p, q]) $ by,

Now we have,

By taking natural log to both side, we have

Define $ f(x) = g(x) = x $, $ F(t) = ln(t), $ $ \varphi(t) = \gamma t, $ and $ p = 1 $ then by Theorem 1.20 the system (2.1) has a common solution in $ {\tilde N} $.

With the help of Theorem 1.20, one can also solve the following coupled system of nonlinear fractional ordered differential equations given by

Where $ \hat{g_1}, \hat{g_2}:[0, 1]\times [0, \infty)\rightarrow [0, \infty) $. Then the equivalent system of integral equations corresponding to (2.2) is given by

Where $ \mathcal{G}(t, s) $ is the Green's function

and continuous on $ [0, 1]\times[0, 1] $. Moreover $ \sup\limits_{t\in[0, 1]}\int_0^1|\mathcal{G}(t, s)|ds\leq 1 $. Further, using $ Q(t, s, x(s)) = \mathcal{G}(t, s)\hat{g_1}(v(s) $ etc. Then the system (2.3)become

Clearly by Theorem 1.20 the System(2.4) has a solution, which is the corresponding solution of the system of nonlinear fractional differential equation(2.2).

3.   Conclusion

In this work, we introduced strong common limit range property for couples $ (f, g) $ and $ (S, T) $ to relaxed the conditions of completeness (closedness), the containment of the range of the mappings, convexity of the underline space and continuity of the mappings and by means of this new concept we established common fixed point results for hybrid pair via $ (F, \varphi) $-contraction and rational type contraction conditions. Further, using the established results existence of solution to the system of integral and differential equations are also discussed. We provided example where the main theorem is applicable but relevant classic result in literature fail to have a common fixed point.

Acknowledgments

The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

Conflict of interest

The authors declare that they have no competing interest.