η ( y ) d y 0 ψ 2 ( σ ( U s , V t ) , σ ( U s , q t ) , σ ( p s , V t ) , σ ( U s , p s ) , σ ( V t , q t ) , 1 2 { σ ( q t , U s ) + σ ( p s , V t ) } , 1 2 { σ ( U s , V t ) + σ ( p s , U s ) } ) η ( y ) d y , (27)

for all $s,t\in E$ , where ${\psi }_{1},{\psi }_{2}\in {\Psi }_{7}$ , ${\varphi }_{1}=\psi \left(e,e,e,e,e,e,e\right)$ , for $e\in \left[0,\infty \right)$ .

i: One of the four mappings p, q, U and V is continuous.

ii: (p, U) & (q, V) are sub compatible.

iii: The pairs $p\left(s\right)\subseteq V\left(s\right)$ and $q\left(s\right)\subseteq U\left(s\right)$ .

iv: Where $\eta :{R}^{+}\to {R}^{+}$ is Lebesgue-integrable mappings, which is sum able, non negative and such that for each $\in >0,{\int }_{0}^{\in }\eta \left(y\right)dy>0$ .

Then p, q, U and V have a unique common fixed point in E.

Proof: Consider arbitrary point ${e}_{0}\in E$ , we construct the sequence $\left\{{e}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ in E such that

$p{e}_{2n}=V{e}_{2n+1}={w}_{2n}$ and $q{e}_{2n+1}=U{e}_{2n+2}={w}_{n+1}$ , $n=0,1,2,\cdots$

Let ${r}_{n}=\sigma \left({w}_{n},{w}_{n+1}\right)$ , Substitution $s={e}_{2n}$ and $t={e}_{2n+1}$ in (27) we have

$\begin{array}{l}{\int }_{0}^{{\varphi }_{1}\left(\sigma \left(p{e}_{2n},q{e}_{2n+1}\right)\right)}\eta \left(y\right)\text{d}y={\int }_{0}^{{\varphi }_{1}\left(\sigma \left({e}_{2n+1},{e}_{2n+2}\right)\right)}\eta \left(y\right)\text{d}y\\ \le {\int }_{0}^{\begin{array}{l}{\psi }_{1}\left(\sigma \left(U{e}_{2n},V{e}_{2n+1}\right),\sigma \left(U{e}_{2n},q{e}_{2n+1}\right),\sigma \left(p{e}_{2n+1},V{e}_{2n+1}\right),\sigma \left(U{e}_{2n},p{e}_{2n}\right),\sigma \left(V{e}_{2n+1},q{e}_{2n+1}\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{2}\left\{\sigma \left(q{e}_{2n+1},U{e}_{2n}\right)+\sigma \left(p{e}_{2n},V{e}_{2n+1}\right)\right\},\frac{1}{2}\left\{\sigma \left(U{e}_{2n},V{e}_{2n+1}\right)+\sigma \left(p{e}_{2n},U{e}_{2n}\right)\right\}\right)\end{array}}\eta \left(y\right)\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-{\int }_{0}^{\begin{array}{l}{\psi }_{2}\left(\sigma \left(U{e}_{2n},V{e}_{2n+1}\right),\sigma \left(U{e}_{2n},q{e}_{2n+1}\right),\sigma \left(p{e}_{2n+1},V{e}_{2n+1}\right),\sigma \left(U{e}_{2n},p{e}_{2n}\right),\sigma \left(V{e}_{2n+1},q{e}_{2n+1}\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{2}\left\{\sigma \left(q{e}_{2n+1},U{e}_{2n}\right)+\sigma \left(p{e}_{2n},V{e}_{2n+1}\right)\right\},\frac{1}{2}\left\{\sigma \left(U{e}_{2n},V{e}_{2n+1}\right)+\sigma \left(p{e}_{2n},U{e}_{2n}\right)\right\}\right)\end{array}}\eta \left(y\right)\text{d}y\end{array}$

$\begin{array}{l}{\int }_{0}^{\varphi \left(\sigma \left({w}_{2n},{w}_{2n+1}\right)\right)}\eta \left(y\right)\text{d}y\\ \le {\int }_{0}^{\begin{array}{l}{\psi }_{1}\left(\sigma \left({w}_{2n-1},{w}_{2n}\right),\sigma \left({w}_{2n-1},{w}_{2n+1}\right),\sigma \left({w}_{2n+1},{w}_{2n}\right),\sigma \left({w}_{2n-1},{w}_{2n}\right),\sigma \left({w}_{2n},{w}_{2n+1}\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{2}\left\{\sigma \left({w}_{2n+1},{w}_{2n}\right)+\sigma \left({w}_{2n},{w}_{2n}\right)\right\},\frac{1}{2}\sigma \left({w}_{2n-1},{w}_{2n}\right)+\sigma \left({w}_{2n},{w}_{2n-1}\right)\right)\end{array}}\eta \left(y\right)\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-{\int }_{0}^{\begin{array}{l}{\psi }_{2}\left(\sigma \left({w}_{2n-1},{w}_{2n}\right),\sigma \left({w}_{2n-1},{w}_{2n+1}\right),\sigma \left({w}_{2n+1},{w}_{2n}\right),\sigma \left({w}_{2n-1},{w}_{2n}\right),\sigma \left({w}_{2n},{w}_{2n+1}\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{2}\left\{\sigma \left({w}_{2n+1},{w}_{2n}\right)+\sigma \left({w}_{2n},{w}_{2n}\right)\right\},\frac{1}{2}\sigma \left({w}_{2n-1},{w}_{2n}\right)+\sigma \left({w}_{2n},{w}_{2n-1}\right)\right)\end{array}}\eta \left(y\right)\text{d}y\end{array}$

$\begin{array}{c}{\int }_{0}^{{\varphi }_{1}\left({r}_{2n}\right)}\eta \left(y\right)\text{d}y\le {\int }_{0}^{{\psi }_{1}\left({r}_{2n-1},{r}_{2n-1}+{r}_{2n},{r}_{2n,}{r}_{2n-1},{r}_{2n},\frac{1}{2}\left\{{r}_{2n}\right\},\frac{1}{2}\left\{{r}_{2n-1}+{r}_{2n-1}\right\}\right)}\eta \left(y\right)\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{0}^{{\psi }_{2}\left({r}_{2n-1},{r}_{2n-1}+{r}_{2n},{r}_{2n,}{r}_{2n-1},{r}_{2n},\frac{1}{2}\left\{{r}_{2n}\right\},\frac{1}{2}\left\{{r}_{2n-1}+{r}_{2n-1}\right\}\right)}\eta \left(y\right)\text{d}y\end{array}$

If ${r}_{2n+1}\le {r}_{2n}$ then ${r}_{2n+1}+{r}_{2n}\le 2{r}_{2n}$ and

$\begin{array}{c}{\int }_{0}^{{\varphi }_{1}\left({r}_{2n}\right)}\eta \left(y\right)\text{d}y\le {\int }_{0}^{{\psi }_{1}\left({r}_{2n-1},2{r}_{2n},{r}_{2n,}{r}_{2n-1},{r}_{2n},\frac{1}{2}\left\{{r}_{2n}\right\},{r}_{2n-1}\right)}\eta \left(y\right)\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{0}^{{\psi }_{2}\left({r}_{2n-1},2{r}_{2n},{r}_{2n,}{r}_{2n-1},{r}_{2n},\frac{1}{2}\left\{{r}_{2n}\right\},{r}_{2n-1}\right)}\eta \left(y\right)\text{d}y\\ <{\int }_{0}^{{\varphi }_{1}\left({r}_{2n}\right)}\eta \left(y\right)\text{d}y.\end{array}$ (28)

Thus we arrive at a contradiction. Hence ${r}_{2n}\le {r}_{2n-1}$ , similarly by substituting $s={r}_{2n+2},t={r}_{2n+1}$ in (27) we can prove that, ${r}_{2n+1}\le {r}_{2n}$ , for $n=0,1,2,\cdots$ . Thus ${r}_{n+1}\le {r}_{n}$ , for $n=0,1,2,\cdots$ . Hence the sequence $\left\{{r}_{n}\right\}$ is sequence of positive real numbers, which is decreasing and converges to $r\in R$ .

Let $m=\underset{n\to \infty }{\mathrm{lim}}\frac{1}{2}d\left({w}_{n},{w}_{n+2}\right)$ . Taking $n\to \infty$ in (27) we have

$\begin{array}{c}{\int }_{0}^{{\varphi }_{1}\left({r}_{2n}\right)}\eta \left(y\right)\text{d}y\le {\int }_{0}^{{\psi }_{1}\left(r,r,r,r,r,r,r\right)}\eta \left(y\right)\text{d}y-{\int }_{0}^{{\psi }_{2}\left(r,r,r,r,r,r,m\right)}\eta \left(y\right)\text{d}y\\ \le {\int }_{0}^{{\varphi }_{1}\left(r\right)}\eta \left(y\right)\text{d}y-{\int }_{0}^{{\psi }_{2}\left(r,r,r,r,r,r,m\right)}\eta \left(y\right)\text{d}y.\end{array}$

$\begin{array}{l}\text{Thus}\text{\hspace{0.17em}}{\psi }_{2}\left(r,r,r,r,r,r,m\right)=0\text{\hspace{0.17em}}\text{So}\text{\hspace{0.17em}}\text{that}\text{\hspace{0.17em}}r=m=0.\\ \text{Hence}\text{\hspace{0.17em}}\underset{n\to \infty }{\mathrm{lim}}d\left({y}_{n},{y}_{n+1}\right)=0\end{array}$ (29)

In view of (29), to prove sequence $\left\{{w}_{n}\right\}$ is a Cauchy sequence it is sufficient to prove the subsequence $\left\{{w}_{2n}\right\}$ of sequence $\left\{{w}_{n}\right\}$ is a Cauchy sequence. If $\left\{{w}_{2n}\right\}$ is not a Cauchy sequence there exist $\in \text{\hspace{0.17em}}>0$ & sequence of natural numbers $\left\{2m\left(k\right)\right\}$ & $\left\{2n\left(k\right)\right\}$ which are monotone increasing such that $n\left(k\right)>m\left(k\right)$ .

$\sigma \left({w}_{2m\left(k\right)},{w}_{2n\left(k\right)}\right)\ge \text{\hspace{0.17em}}\in \text{\hspace{0.17em}}\text{\hspace{0.17em}}&\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left({w}_{2m\left(k\right)},{w}_{2n\left(k\right)-2}\right)<\text{\hspace{0.17em}}\in .$ (30)

Then from (29) we have

$\begin{array}{l}\in \text{\hspace{0.17em}}<\sigma \left({w}_{2m\left(k\right)},{w}_{2n\left(k\right)}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\le \sigma \left({w}_{2m\left(k\right)},{w}_{2n\left(k\right)-2}\right)+\sigma \left({w}_{2n\left(k\right)-1},{w}_{2n\left(k\right)-2}\right)+\sigma \left({w}_{2n\left(k\right)-1},{w}_{2n\left(k\right)}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}\in +\sigma \left({w}_{2n\left(k\right)-1},{w}_{2n\left(k\right)-2}\right)+\sigma \left({w}_{2n\left(k\right)-1},{w}_{2n\left(k\right)}\right).\end{array}$ (31)

Taking $k\to \infty$ and using (29) we have

$\underset{n\to \infty }{\mathrm{lim}}\sigma \left({w}_{2m\left(k\right)},{w}_{2n\left(k\right)}\right)=\text{\hspace{0.17em}}\in .$ (32)

Taking $k\to \infty$ using (29) & (30) in

$|\sigma \left({w}_{2m\left(k\right)},{w}_{2n\left(k\right)+1}\right)-\sigma \left({w}_{2m\left(k\right)},{w}_{2n\left(k\right)}\right)|\le \sigma \left({w}_{2n\left(k\right)},{w}_{2n\left(k\right)+1}\right).$ (33)

We get $\underset{n\to \infty }{\mathrm{lim}}\sigma \left({w}_{2n\left(k\right)+1},{w}_{2m\left(k\right)}\right)=\text{\hspace{0.17em}}\in .$ (34)

Letting $k\to \infty$ and from Equations (29) & (30) in

$|\sigma \left({w}_{2m\left(k\right)-1},{w}_{2n\left(k\right)}\right)-\sigma \left({w}_{2m\left(k\right)},{w}_{2n\left(k\right)}\right)|\le \sigma \left({w}_{2m\left(k\right)},{w}_{2m\left(k\right)-1}\right).$

We get $\underset{k\to \infty }{\mathrm{lim}}\sigma \left({w}_{2m\left(k\right)},{w}_{2m\left(k\right)-1}\right)=\text{\hspace{0.17em}}\in .$ (35)

Putting $s={x}_{2m\left(k\right)},t={x}_{2n\left(k\right)-1}$ in (27), for all $k=1,2,3,\cdots$ we obtain Taking $k\to \infty$ & using (29), (30), (32), (33) & (35) we get

$\begin{array}{c}{\int }_{0}^{{\varphi }_{1}\left(\in \right)}\eta \left(y\right)\text{d}y\le {\int }_{0}^{{\psi }_{1}\left(\in ,\in ,\in ,0,0,\frac{1}{2}\left[\in +\in \right],\frac{1}{2}\in \right)}\eta \left(y\right)\text{d}y-{\int }_{0}^{{\psi }_{2}\left(\in ,\in ,\in ,0,0,\frac{1}{2}\left[\in +\in \right],\frac{1}{2}\in \right)}\eta \left(y\right)\text{d}y\\ <{\int }_{0}^{{\psi }_{1}\left(\in ,\in ,\in ,\in ,\in ,\in ,\in \right)}\eta \left(y\right)\text{d}y={\int }_{0}^{{\varphi }_{1}\left(\in \right)}\eta \left(y\right)\text{d}y.\end{array}$

This is contradiction. Hence $\left\{{w}_{2n}\right\}$ is a Cauchy sequence and is convergent. Since E is complete there exist $z\in E$ such that as $n\to \infty$ we have ${w}_{n}\to z$ .

Case I: Assume that U is continuous then $Up{e}_{2n}\to Uz$ , ${U}^{2}{e}_{2n}\to Uz.$ Since (p, U) is sub compatible, we have $pU{e}_{2n}\to Uz.$

Step I: Substituting $s=U{e}_{2n},t={e}_{2n+1}$ in (27), we have

$\begin{array}{l}{\int }_{0}^{{\varphi }_{1}\sigma \left(pU{e}_{2n},q{e}_{2n+1}\right)}\eta \left(y\right)\text{d}y\\ \le {\int }_{0}^{\begin{array}{l}{\psi }_{1}\left(\sigma \left({U}^{2}{e}_{2n},V{e}_{2n+1}\right),\sigma \left({U}^{2}{e}_{2n},q{e}_{2n+1}\right),\sigma \left(pU{e}_{2n},V{e}_{2n+1}\right),\sigma \left({U}^{2}{e}_{2n},pU{e}_{2n}\right),\sigma \left(V{e}_{2n+1},q{e}_{2n+1}\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{2}\left\{\sigma \left(q{e}_{2n+1},{U}^{2}{e}_{2n}\right)+\sigma \left(pU{e}_{2n},V{e}_{2n+1}\right)\right\},\frac{1}{2}\left\{\sigma \left({U}^{2}{e}_{2n},V{e}_{2n+1}\right)+\sigma \left(pU{e}_{2n},{U}^{2}{e}_{2n}\right)\right\}\right)\end{array}}\eta \left(y\right)\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-{\int }_{0}^{\begin{array}{l}{\psi }_{2}\left(\sigma \left({U}^{2}{e}_{2n},V{e}_{2n+1}\right),\sigma \left({U}^{2}{e}_{2n},q{e}_{2n+1}\right),\sigma \left(pU{e}_{2n},V{e}_{2n+1}\right),\sigma \left({U}^{2}{e}_{2n},pU{e}_{2n}\right),\sigma \left(V{e}_{2n+1},q{e}_{2n+1}\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{2}\left\{\sigma \left(q{e}_{2n+1},{U}^{2}{e}_{2n}\right)+\sigma \left(pU{e}_{2n},V{e}_{2n+1}\right)\right\},\frac{1}{2}\left\{\sigma \left({U}^{2}{e}_{2n},V{e}_{2n+1}\right)+\sigma \left(pU{e}_{2n},{U}^{2}{e}_{2n}\right)\right\}\right)\end{array}}\eta \left(y\right)\text{d}y,\end{array}$

$\begin{array}{l}{\int }_{0}^{{\varphi }_{1}\left(\sigma \left(Uz,z\right)\right)}\eta \left(y\right)\text{d}y\\ \le {\int }_{0}^{{\psi }_{1}\left(\sigma \left(Uz,z\right),\sigma \left(Uz,z\right),\sigma \left(Uz,z\right),\sigma \left(Uz,Uz\right),\sigma \left(z,z\right),\frac{1}{2}\left\{\sigma \left(z,Uz\right)+\sigma \left(Uz,z\right)\right\},\frac{1}{2}\left\{\sigma \left(Uz,z\right)+\sigma \left(Uz,Uz\right)\right\}\right)}\eta \left(y\right)\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-{\int }_{0}^{{\psi }_{2}\left(\sigma \left(Uz,z\right),\sigma \left(Uz,z\right),\sigma \left(Uz,z\right),\sigma \left(Uz,Uz\right),\sigma \left(z,z\right),\frac{1}{2}\left\{\sigma \left(z,Uz\right)+\sigma \left(Uz,z\right)\right\},\frac{1}{2}\left\{\sigma \left(Uz,z\right)+\sigma \left(Uz,Uz\right)\right\}\right)}\eta \left(y\right)\text{d}y\\ \le {\int }_{0}^{{\psi }_{1}\left(\sigma \left(Uz,z\right),\sigma \left(Uz,z\right),\sigma \left(Uz,z\right),0,0,\sigma \left(Uz,z\right),\frac{1}{2}\left\{\sigma \left(Uz,z\right)\right\}\right)}\eta \left(y\right)\text{d}y\\ \le {\int }_{0}^{{\varphi }_{1}\left(\sigma \left(Uz,z\right)\right)}\eta \left(y\right)\text{d}y.\end{array}$

It is contradiction if $Uz\ne z$ . Hence $Uz=z.$

Step II: Substituting $s=z,t={e}_{2n+1}$ in (27) and taking limit as n tends to infinity we get $pz=z$ .

Step III: We know that $z=pz\in p\left(e\right)\subseteq V\left(e\right)$ then there exist $u\in E$ such that $z=Vu$ . Substituting $s={e}_{2n},t=u$ in (27) we get $qz=z$ . Hence $qz=z=Vz$ and $qVu=Vqu$ , which gives $qz=Vz$ .

Step IV: Substituting $s=z,t=z$ in (27) we have $qz=z$ so that $q\left(z\right)=z=Vz$ . Hence p, q, U & V have a common fixed point z in E.

Case II: Assume that U is continuous then ${p}^{2}{e}_{2n}\to pz,$ $pU{e}_{2n}\to pz$ . Similarly we can prove that z is common fixed point of p, q, U & V. When q or V is continuous, then the uniqueness of common fixed point follows easily from (27).

Example: Let $E=\left[0,1\right]$ with the usual metric $\sigma \left(s,t\right)=\frac{1}{2}|s-t|$ . Define $p,q,U,V:E\to E$ such that $ps=\frac{s}{4}$ , $qt=\frac{t}{4}$ , $Us=s$ , $Vt=t$ .

Let ${\psi }_{1}\left({y}_{1},{y}_{2},{y}_{3},{y}_{4},{y}_{5},{y}_{6},{y}_{7}\right)=\mathrm{max}\left({y}_{1},{y}_{2},{y}_{3},{y}_{4},{y}_{5},{y}_{6},{y}_{7}\right),$ $\phi \left(y\right)=2y$ ,

${\psi }_{2}=\frac{1}{4}{\psi }_{1}$ then ${\psi }_{1}\left(y\right)=y,\forall y\in \left[0,\infty \right)$

$\begin{array}{l}{|\frac{s}{4}-\frac{t}{4}|}^{2}\le \frac{1}{4}\mathrm{max}\left\{\sigma \left(s,t\right),\sigma \left(s,\frac{t}{4}\right),\sigma \left(\frac{s}{4},t\right),\sigma \left(s,\frac{s}{4}\right),\sigma \left(t,\frac{t}{4}\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{2}\left\{\sigma \left(\frac{t}{4},s\right)+\sigma \left(\frac{s}{4},t\right)\right\},\frac{1}{2}\left\{\sigma \left(s,t\right)+\sigma \left(\frac{s}{4},s\right)\right\}\right\}\end{array}$

For all $s,t\in E$ , it follows that the condition (27).

Let $\left\{{e}_{n}\right\}$ be a sequence in E such that $p{e}_{n}\to z$ & $U{e}_{n}\to z$ for some z in E. Then z = 0, $\sigma \left(pU{e}_{n},Up{e}_{n}\right)\to 0$ . Hence $\left\{p,U\right\}$ is sub compatible. We have common fixed point in E.

3. Conclusion

In this paper, we proved the fixed point theorem for four sub compatible maps under a contractive condition of integral type. These results can be extended to any directions and can also be extended to fixed point theory of non-expansive multi-valued mappings.

Acknowledgements

The authors would like to give their sincere thanks to the editor and the anonymous referees for their valuable comments and useful suggestions in improving the article.

Cite this paper

Mishra, V.N., Wadkar, B.R., Bhardwaj, R., Khan, I.A. and Singh, B. (2017) Common Fixed Point Theorems in Metric Space by Altering Distance Function. Advances in Pure Mathematics, 7, 335-344. https://doi.org/10.4236/apm.2017.76020

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