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Mira-Cristiana Anisiu
Institute of Mathematics, Cluj-Napoca, Romania
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M.-C. Anisiu, On fixed point theorems for mappings defined on spheres in metric spaces, Seminar on Mathematical Analysis, 1-6, Preprint, 91-7, Univ. Babeş-Bolyai Cluj-Napoca, 1991
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[1] J. Lopez-Gomez, |A fixed point therem for discontinuous operators, Glasnik Mat. 23 (1988) , 115-118
[2] L. Nova, G., Fixed point theorems for some discontinuous operators, Pacific, J. Math. 123 (1983), 189-196.
[3] I.A. Rus, Metrical fixed point theorems, University of Cluj-Napoca, Department of Mathematics, 1979 (mineographed)
"BABEŞ -- BOLYAI" UNIVERSITY
Faculty of Mathematics
Re search Seminars
Seminar on Mathematical Analysis
Preprint Nr.7, 1991, pp.95 - 100
ON FIXED POINT THEOREMS FOR MAPPINGS DEFINED ON BPHERES IN NETRIC SPACES
Mira-Cristina Anisiu
In the last years, the interest in metric fixed point theorems appeared again. New proofs were given [1, 2] for theorems of Hardy-Rogers, Ćirić - Reich - Rus.
In this note we study the case of mappings defined only on spheres, not on the entire metric space, following the well-known theorem:
THEOREM 1. Let ( X,d\mathrm{X}, \mathrm{d} ) be a complete metric space, f: bar(B)(2,r)rarr X\mathrm{f}: \overline{\mathrm{B}}(2, \mathrm{r}) \rightarrow X a kk-contraction, k in[0,1)k \in[0,1), i.e.
{:[(1)d(fx","fy) <= kd(x","y)" for all "x","y" in " bar(B)(z","r)=],[={x in X:d(x","z) <= r}.]:}\begin{align*}
& d(f x, f y) \leq k d(x, y) \text { for all } x, y \text { in } \bar{B}(z, r)= \tag{1}\\
& =\{x \in X: d(x, z) \leq r\} .
\end{align*}
If
{:(2)d(z","fz) <= Gamma(1-k)",":}\begin{equation*}
d(z, f z) \leq \Gamma(1-k), \tag{2}
\end{equation*}
then ff has a unique fixed point u in bar(B)(z,r)u \in \bar{B}(z, r).
Proof. The condition on fzf z implies the fact that f( bar(B)(z,r))= bar(B)(z,r)f(\bar{B}(z, r)) =\bar{B}(z, r). Indeed, let x in bar(B)(z,r)x \in \bar{B}(z, r). We estimate d(z,fx)d(z, f x) :
Proof. The condition on fzf z implies the fact that f( bar(B)(z,r))= bar(B)(z,r)f(\bar{B}(z, r)) =\bar{B}(z, r). Indeed, let x in bar(B)(z,r)x \in \bar{B}(z, r). We estimate d(z,fx)d(z, f x) :
d(z,fx) <= d(z,fz)+d(fz,fx) <= r(1-k)+kd(z,x^(˙)) <= r.d(z, f x) \leq d(z, f z)+d(f z, f x) \leq r(1-k)+k d(z, \dot{x}) \leq r .
It follows that f( bar(B)(z,x))sube bar(B)(z,r)f(\bar{B}(z, x)) \subseteq \bar{B}(z, r) and Banach theorem applies in this complete metric space.
This theorem extends easily to Hardy - Rogers contractions.
THEOREM 2. Let ( X,dX, d ) be a complete metric space, f: bar(B)(z,r)rarr X\mathbf{f}: \bar{B}(\mathbf{z}, \mathbf{r}) \rightarrow X a Hardy - Rogers contraction, i.e. there exists a_(i) >= 0a_{i} \geq 0, i= bar(1,5),A=sum_(i=1)^(5)a_(i) < 1i=\overline{1,5}, A=\sum_{i=1}^{5} a_{i}<1 such that
(3) d(fx,fy) <= a_(1)d(x,y)+a_(2)d(x,fx)+a_(3)d(y,fy)+a_(4)d(x_(y),fy)++a_(5)d(y,fx)d(f x, f y) \leq a_{1} d(x, y)+a_{2} d(x, f x)+a_{3} d(y, f y)+a_{4} d\left(x_{y}, f y\right)+ +a_{5} d(y, f x) for all x,yx, y in bar(B)(z,r)\bar{B}(z, r).
If
(4) d(z,fz) <= (2(1-A))/(2+A-a_(1))rd(z, f z) \leq \frac{2(1-A)}{2+A-a_{1}} r,
then ff has a unique fixed point u in bar(B)(2,r)u \in \bar{B}(2, r).
Proof. To prove that f( bar(B)(z,r))sub bar(B)(z,r)f(\bar{B}(z, r)) \subset \bar{B}(z, r), we use a symmetric form of (3) which is obtained evaluating also d(fy,fx)d(f y, f x) and adding,
(5) d(fx,fy) <= a_(1)d(xy)+(a_(2)+a_(3))/(2)[d(x,fx)+d(y,fy)]+d(f x, f y) \leq a_{1} d(x y)+\frac{a_{2}+a_{3}}{2}[d(x, f x)+d(y, f y)]+
THEOREM 2. Let ( X,dX, d ) be a complete metric space, f: bar(B)(z,r)rarr X\mathbf{f}: \bar{B}(\mathbf{z}, \mathbf{r}) \rightarrow X a Hardy - Rogers contraction, i.e. there exists a_(i) >= 0a_{i} \geq 0, i= bar(1,5),A=sum_(i=1)^(5)a_(i) < 1i=\overline{1,5}, A=\sum_{i=1}^{5} a_{i}<1 such that
(3) d(fx,fy) <= a_(1)d(x,y)+a_(2)d(x,fx)+a_(3)d(y,fy)+a_(4)d(x_(y),fy)++a_(5)d(y,fx)d(f x, f y) \leq a_{1} d(x, y)+a_{2} d(x, f x)+a_{3} d(y, f y)+a_{4} d\left(x_{y}, f y\right)+ +a_{5} d(y, f x) for all x,yx, y in bar(B)(z,r)\bar{B}(z, r).
If
(4) d(z,fz) <= (2(1-A))/(2+A-a_(1))rd(z, f z) \leq \frac{2(1-A)}{2+A-a_{1}} r,
then ff has a unique fixed point u in bar(B)(2,r)u \in \bar{B}(2, r).
Proof. To prove that f( bar(B)(z,r))sub bar(B)(z,r)f(\bar{B}(z, r)) \subset \bar{B}(z, r), we use a symmetric form of (3) which is obtained evaluating also d(fy,fx)d(f y, f x) and adding,
(5) d(fx,fy) <= a_(1)d(xy)+(a_(2)+a_(3))/(2)[d(x,fx)+d(y,fy)]+d(f x, f y) \leq a_{1} d(x y)+\frac{a_{2}+a_{3}}{2}[d(x, f x)+d(y, f y)]+
+(a_(4)+a_(5))/(2)[d(x,fy)+d(y,fx)]" for each "x,y" in "+\frac{a_{4}+a_{5}}{2}[d(x, f y)+d(y, f x)] \text { for each } x, y \text { in }
bar(B)(z,I)\bar{B}(z, I).
Let xx be in bar(B)(z,r)\bar{B}(z, r), hence d(x,z) <= rd(x, z) \leq r. We estimate
Let xx be in bar(B)(z,r)\bar{B}(z, r), hence d(x,z) <= rd(x, z) \leq r. We estimate
{:[d(z","fx) <= d(z","fz)+d(fz","fx) <= ],[ <= d(z","fz)+a_(1)d(z","x)+(a_(2)+a_(3))/(2)[d(z","fz)+d(x","z)+],[+d(z","fx)]+(a_(4)+a_(5))/(2)[d(z","fx)+d(x","z)+d(z","fz)]=],[=(1+(a_(2)+a_(3)+a_(4)+a_(5))/(2))d(z","fz)+(a_(1)+(a_(2)+a_(3)+a_(4)+a_(5))/(2))],[*d(x","z)+(a_(2)+a_(3)+a_(4)+a_(5))/(2)d(z","fx)=],[=(1+(A-a_(1))/(2))d(z","fz)+(A+a_(1))/(2)d(x","z)+(A-a_(1))/(2)d(z","fx)]:}\begin{aligned}
& d(z, f x) \leq d(z, f z)+d(f z, f x) \leq \\
& \leq d(z, f z)+a_{1} d(z, x)+\frac{a_{2}+a_{3}}{2}[d(z, f z)+d(x, z)+ \\
& +d(z, f x)]+\frac{a_{4}+a_{5}}{2}[d(z, f x)+d(x, z)+d(z, f z)]= \\
& =\left(1+\frac{a_{2}+a_{3}+a_{4}+a_{5}}{2}\right) d(z, f z)+\left(a_{1}+\frac{a_{2}+a_{3}+a_{4}+a_{5}}{2}\right) \\
& \cdot d(x, z)+\frac{a_{2}+a_{3}+a_{4}+a_{5}}{2} d(z, f x)= \\
& =\left(1+\frac{A-a_{1}}{2}\right) d(z, f z)+\frac{A+a_{1}}{2} d(x, z)+\frac{A-a_{1}}{2} d(z, f x)
\end{aligned}
It follows
(1-(A-a_(1))/(2))d(z,fx) <= (1+(A-a_(1))/(2))d(z,fz)+(A+a_(1))/(2)d(x,z) <=\left(1-\frac{A-a_{1}}{2}\right) d(z, f x) \leq\left(1+\frac{A-a_{1}}{2}\right) d(z, f z)+\frac{A+a_{1}}{2} d(x, z) \leq
<= (1-A)r+(A+a_(1))/(2)r=(1-(A-a_(1))/(2))r\leq(1-A) r+\frac{A+a_{1}}{2} r=\left(1-\frac{A-a_{1}}{2}\right) r.
Dividing by 1-(A-a_(1))/(2) > 01-\frac{A-a_{1}}{2}>0 we obtain d(z,fx) <= rd(z, f x) \leq r and Hardy
(1-(A-a_(1))/(2))d(z,fx) <= (1+(A-a_(1))/(2))d(z,fz)+(A+a_(1))/(2)d(x,z) <=\left(1-\frac{A-a_{1}}{2}\right) d(z, f x) \leq\left(1+\frac{A-a_{1}}{2}\right) d(z, f z)+\frac{A+a_{1}}{2} d(x, z) \leq
<= (1-A)r+(A+a_(1))/(2)r=(1-(A-a_(1))/(2))r\leq(1-A) r+\frac{A+a_{1}}{2} r=\left(1-\frac{A-a_{1}}{2}\right) r.
Dividing by 1-(A-a_(1))/(2) > 01-\frac{A-a_{1}}{2}>0 we obtain d(z,fx) <= rd(z, f x) \leq r and Hardy
- Rogers theorem applies and it assures the existence and uniqueness of the fixed point uu.
Ramark 1. For the Ciric - Reich - Rus contractions ( a_(1)=a_(1),a_(2)^(')=a_(3)^(')=b_(1),a_(4)=a_(5)=0,a+2b < 1a_{1}=a_{1}, a_{2}^{\prime}=a_{3}^{\prime}=b_{1}, a_{4}=a_{5}=0, a+2 b<1 ), condition (4) becomes:
d(z,fz) <= (1-a-2b)/(1+b)rd(z, f z) \leq \frac{1-a-2 b}{1+b} r
For the Kannan contractions (a_(1)=a_(4)=a_(5)=0,a_(2)=a_(3)=b < :} < (1)/(2)\left(a_{1}=a_{4}=a_{5}=0, a_{2}=a_{3}=b<\right. <\frac{1}{2} ), condition (4) writes
d(z,fz) <= (1-2b)/(1+b)r.d(z, f z) \leq \frac{1-2 b}{1+b} r .
For the Banach contractions ( a_(1)=k,a_(i)=0,i= bar(2,5)a_{1}=k, a_{i}=0, i=\overline{2,5} ),
condition (4) becomes (2).
Similar results hold for the more general contractions defined by Rus [3].
condition (4) becomes (2).
Similar results hold for the more general contractions defined by Rus [3].
Let :R_(+)^(5)rarrR_(+): \mathbf{R}_{+}^{5} \rightarrow \mathbf{R}_{+}be a continuous function such that
(a) if r_(i) <= s_(i),i= bar(1,5)r_{i} \leq s_{i}, i=\overline{1,5}, then varphi(r) <= varphi(s)\varphi(r) \leq \varphi(s);
(b) phi(r) < r\phi(r)<r for each r > 0r>0, where phi(r)=varphi(r,r,r,r,r)\phi(r)=\varphi(r, r, r, r, r);
(c) r-phi(r)rarr oor-\phi(r) \rightarrow \infty for r rarr oor \rightarrow \infty.
(a) if r_(i) <= s_(i),i= bar(1,5)r_{i} \leq s_{i}, i=\overline{1,5}, then varphi(r) <= varphi(s)\varphi(r) \leq \varphi(s);
(b) phi(r) < r\phi(r)<r for each r > 0r>0, where phi(r)=varphi(r,r,r,r,r)\phi(r)=\varphi(r, r, r, r, r);
(c) r-phi(r)rarr oor-\phi(r) \rightarrow \infty for r rarr oor \rightarrow \infty.
For ( X,dX, d ) a metric space, a function f: bar(B)(z,r)rarr Xf: \bar{B}(z, r) \rightarrow X is a pp -
contraction if
(6) d(fx,fy) <= varphi(d(x,y)),d(x,fx),d(y,fy),d(x,fy),d(y,fx))d(f x, f y) \leq \varphi(d(x, y)), d(x, f x), d(y, f y), d(x, f y), d(y, f x)) for all x,y in bar(B)(z,r)x, y \in \bar{B}(z, r).
contraction if
(6) d(fx,fy) <= varphi(d(x,y)),d(x,fx),d(y,fy),d(x,fy),d(y,fx))d(f x, f y) \leq \varphi(d(x, y)), d(x, f x), d(y, f y), d(x, f y), d(y, f x)) for all x,y in bar(B)(z,r)x, y \in \bar{B}(z, r).
Let r_(z)r_{z} be equal to sup {r inR_(+):r-phi(r) <= d(z,fz)}\left\{r \in R_{+}: r-\phi(r) \leq d(z, f z)\right\} which is finite because r=0r=0 is in the considered set (which is hence nonvoid) and varphi\varphi is continuous. We obtain a generalization of the fixed point theorem of Rus [3] which is given for. contractions defined on the entire space XX.
THEOREM 3. Let (X,d)(X, d) be a complete metric space and ff : bar(B)(Z,r)rarrX\overline{\mathrm{B}}(\mathrm{Z}, \mathrm{r}) \rightarrow \mathrm{X} be a - contraction. If
(7) quad d(z,fz) <= r-phi(r_(z))\quad d(z, f z) \leq r-\phi\left(r_{z}\right),
then ff has a unique fixed point u in bar(B)(z,x),u=lim_(n rarr oo)f^(n)zu \in \bar{B}(z, x), u=\lim _{n \rightarrow \infty} f^{n} z and d(f^(n)z,x) <= phi^(n)(r_(z))d\left(f^{n} z, x\right) \leq \phi^{n}\left(r_{z}\right).
(7) quad d(z,fz) <= r-phi(r_(z))\quad d(z, f z) \leq r-\phi\left(r_{z}\right),
then ff has a unique fixed point u in bar(B)(z,x),u=lim_(n rarr oo)f^(n)zu \in \bar{B}(z, x), u=\lim _{n \rightarrow \infty} f^{n} z and d(f^(n)z,x) <= phi^(n)(r_(z))d\left(f^{n} z, x\right) \leq \phi^{n}\left(r_{z}\right).
Proof. If fz=zf z=z, the conclusion is obvious.
Let fz!=zf z \neq z; one has then r_(z) > 0r_{z}>0. Let z_(0)=z;z_(1)=fzz_{0}=z ; z_{1}=f z satisfies
Let fz!=zf z \neq z; one has then r_(z) > 0r_{z}>0. Let z_(0)=z;z_(1)=fzz_{0}=z ; z_{1}=f z satisfies
d(z_(0),z_(1)) <= r-phi(r_(2)) <= r_(1)d\left(z_{0}, z_{1}\right) \leq r-\phi\left(r_{2}\right) \leq r_{1}
hence z_(1)in bar(B)(z,r)z_{1} \in \bar{B}(z, r). Suppose z_(n)=f^(n)z in bar(B)(z,r)z_{n}=f^{n} z \in \bar{B}(z, r). Then
d(z,z_(n+1)) <= d(z,fz)+d(fz,f^(n+1)z) <= .d\left(z, z_{n+1}\right) \leq d(z, f z)+d\left(f z, f^{n+1} z\right) \leq .
<= d(z,fz)+varphi(d(z,f^(n)z),d(z,fz),d(f^(n)z,f^(n+1)z),( bar(d))(z,f^(n+1)z),d(f^(n)z,fz))≤≤d(z,fz)+phi(diamO_(f)(z,n+1)),O_(f)(z,n+1)={z,fz,dots,f^(n+1)z}\leq d(z, f z)+\varphi\left(d\left(z, f^{n} z\right), d(z, f z), d\left(f^{n} z, f^{n+1} z\right), \bar{d}\left(z, f^{n+1} z\right), d\left(f^{n} z, f z\right)\right) \leq \leq d(z, f z)+\phi\left(\operatorname{diam} O_{f}(z, n+1)\right), O_{f}(z, n+1)=\left\{z, f z, \ldots, f^{n+1} z\right\}.
Because of (6), diam o_(f)(z,n+1)o_{f}(z, n+1) is larger than d(f^(i)z,f^(j)z)d\left(f^{i} z, f^{j} z\right), i,j >= 1i, j \geq 1, so there exists p <= n+1p \leq n+1 such that diam O_(f)(z,n+1)==d(z,f^(P)z){O_{f}}(z, n+1)= =d\left(z, f^{P} z\right).
Then
diamO_(f)(z,n+1)=d(z,f^(P)z) <= d(z,fz)+d(fz,f^(P)z) <=\operatorname{diam} O_{f}(z, n+1)=d\left(z, f^{P} z\right) \leq d(z, f z)+d\left(f z, f^{P} z\right) \leq
<= d(z,fz)+phi(dianrho_(f)(z,n+1))quad\leq d(z, f z)+\phi\left(\operatorname{dian} \rho_{f}(z, n+1)\right) \quad and
diamO_(f)(z,n+1)-phi(diamO_(f)(z,n+1)) <= d(z,fz)\operatorname{diam} O_{f}(z, n+1)-\phi\left(\operatorname{diam} O_{f}(z, n+1)\right) \leq d(z, f z),
hence diam rho_(f)(z,n+1) <= r_(z)\rho_{f}(z, n+1) \leq r_{z}.
It follows d(z,z_(n+1)) <= d(z,fz)+phi(x_(z)) <= rd\left(z, z_{n+1}\right) \leq d(z, f z)+\phi\left(x_{z}\right) \leq r.
The sequence {z_(n)}_(n in N)\left\{z_{n}\right\}_{n \in N} is well-defined for f: bar(B)(z,r)rarr Xf: \bar{B}(z, r) \rightarrow X.
We shall prove now that this sequence converges. Let n >= 1n \geq 1.
diamO_(f)(z,n+1)=d(z,f^(P)z) <= d(z,fz)+d(fz,f^(P)z) <=\operatorname{diam} O_{f}(z, n+1)=d\left(z, f^{P} z\right) \leq d(z, f z)+d\left(f z, f^{P} z\right) \leq
<= d(z,fz)+phi(dianrho_(f)(z,n+1))quad\leq d(z, f z)+\phi\left(\operatorname{dian} \rho_{f}(z, n+1)\right) \quad and
diamO_(f)(z,n+1)-phi(diamO_(f)(z,n+1)) <= d(z,fz)\operatorname{diam} O_{f}(z, n+1)-\phi\left(\operatorname{diam} O_{f}(z, n+1)\right) \leq d(z, f z),
hence diam rho_(f)(z,n+1) <= r_(z)\rho_{f}(z, n+1) \leq r_{z}.
It follows d(z,z_(n+1)) <= d(z,fz)+phi(x_(z)) <= rd\left(z, z_{n+1}\right) \leq d(z, f z)+\phi\left(x_{z}\right) \leq r.
The sequence {z_(n)}_(n in N)\left\{z_{n}\right\}_{n \in N} is well-defined for f: bar(B)(z,r)rarr Xf: \bar{B}(z, r) \rightarrow X.
We shall prove now that this sequence converges. Let n >= 1n \geq 1.
We estimate:
{:[d(z_(n),z_(n+p)) <= Phi(d(z_(n-1),z_(n+p-1)),d(z_(n-1),z_(n)),d(z_(n+p-1),z_(n+p)):}],[{:d(z_(n-1),z_(n+p)),d(z_(n+p-1),z_(n))) <= phi(" diam "o_(f)(z_(n-1),p+1)) <= ],[ <= phi^(2)(" diam "o_(f)(z_(n-2),p+2)) <= dots <= phi^(n)(" diam "o_(f)(z,p+n)) <= phi^(n)(r_(z))]:}\begin{aligned}
& d\left(z_{n}, z_{n+p}\right) \leq \Phi\left(d\left(z_{n-1}, z_{n+p-1}\right), d\left(z_{n-1}, z_{n}\right), d\left(z_{n+p-1}, z_{n+p}\right)\right. \\
& \left.d\left(z_{n-1}, z_{n+p}\right), d\left(z_{n+p-1}, z_{n}\right)\right) \leq \phi\left(\text { diam } o_{f}\left(z_{n-1}, p+1\right)\right) \leq \\
& \leq \phi^{2}\left(\text { diam } o_{f}\left(z_{n-2}, p+2\right)\right) \leq \ldots \leq \phi^{n}\left(\text { diam } o_{f}(z, p+n)\right) \leq \phi^{n}\left(r_{z}\right)
\end{aligned}
But r_(z) > 0r_{z}>0, so phi^(n)(r_(z))rarr0(n rarr oo)\phi^{n}\left(r_{z}\right) \rightarrow 0(n \rightarrow \infty); indeed, the sequence is descreasing and bounded and if its limit were a * 0 , from the continuity of phi\phi it would follow a=phi(a)a=\phi(a), contradicting the property (b).
It follows that {z_(n)}⊏ bar(B)(z,r)\left\{z_{n}\right\} \sqsubset \bar{B}(z, r) is a cauchy sequence, hence it converges to a limit uu in bar(B)(z,r)\bar{B}(z, r), which is a fixed point for f. Indeed, suppose u**u * fu. Then, for n in Nn \in N
d(u,fu) <= d(u,f^(n)z)+d(f^(n)z,fu) <=d(u, f u) \leq d\left(u, f^{n} z\right)+d\left(f^{n} z, f u\right) \leq
<= d(u,f^(n)z)+varphi(d(f^(n-1)z,u),d(f^(n-1)z,f^(n)z),d(u,fu),d(f^(n-1)z,fu):}\leq d\left(u, f^{n} z\right)+\varphi\left(d\left(f^{n-1} z, u\right), d\left(f^{n-1} z, f^{n} z\right), d(u, f u), d\left(f^{n-1} z, f u\right)\right., {:d(u,f^(n)z)}\left.d\left(u, f^{n} z\right)\right\}
and for n rarr oon \rightarrow \infty
d(u,fu) <= varphi(0,0,d(u,fu),d(u,fu),0) <= phi(d(u,fu)) < d(u,fu)d(u, f u) \leq \varphi(0,0, d(u, f u), d(u, f u), 0) \leq \phi(d(u, f u))<d(u, f u),
which im a contradiction.
The uniquaness of the fixed point can be aasily established
in a similar way.
mamark 2. Condition (3) in Theorem 2 corresponds to a varphi\varphi contraction with
d(u,fu) <= d(u,f^(n)z)+d(f^(n)z,fu) <=d(u, f u) \leq d\left(u, f^{n} z\right)+d\left(f^{n} z, f u\right) \leq
<= d(u,f^(n)z)+varphi(d(f^(n-1)z,u),d(f^(n-1)z,f^(n)z),d(u,fu),d(f^(n-1)z,fu):}\leq d\left(u, f^{n} z\right)+\varphi\left(d\left(f^{n-1} z, u\right), d\left(f^{n-1} z, f^{n} z\right), d(u, f u), d\left(f^{n-1} z, f u\right)\right., {:d(u,f^(n)z)}\left.d\left(u, f^{n} z\right)\right\}
and for n rarr oon \rightarrow \infty
d(u,fu) <= varphi(0,0,d(u,fu),d(u,fu),0) <= phi(d(u,fu)) < d(u,fu)d(u, f u) \leq \varphi(0,0, d(u, f u), d(u, f u), 0) \leq \phi(d(u, f u))<d(u, f u),
which im a contradiction.
The uniquaness of the fixed point can be aasily established
in a similar way.
mamark 2. Condition (3) in Theorem 2 corresponds to a varphi\varphi contraction with
varphi(x_(1),dots,x_(5))=sum_(i=1)^(5)a_(i)x_(i),quada_(i) >= 0,i bar(1=1,5),lambda=sum_(i=1)^(5)a_(i) < 1.\varphi\left(x_{1}, \ldots, x_{5}\right)=\sum_{i=1}^{5} a_{i} x_{i}, \quad a_{i} \geq 0, i \overline{1=1,5}, \lambda=\sum_{i=1}^{5} a_{i}<1 .
In this case r_(z)=s u p{r >= 0;r-Ar <= d(z,fz)}=(1)/(1-A)d(z,fz)r_{z}=\sup \{r \geq 0 ; r-A r \leq d(z, f z)\}=\frac{1}{1-A} d(z, f z) and condition (7) in Theorem 3 becomes
{:[d(z","fz) <= r-(A)/(1-A)d(z","fz)","" i.e. "],[d(z","fz) <= (1-A)r.]:}\begin{aligned}
& d(z, f z) \leq r-\frac{A}{1-A} d(z, f z), \text { i.e. } \\
& d(z, f z) \leq(1-A) r .
\end{aligned}
This condition is less restrictive than (4), but when condition (4) is applicable the sequence of succesive approximations starting from each xx in bar(B)(z,r)\bar{B}(z, r) converges to the fixed point uu.
REFEREWCES
- J.Lopez - Gomez, A fixed point theorem for discontinuous operators, Glasnik Mat. 23 (1988), 115-118.
- L.Nova, G., Fixed point theorems for some discontinuous operators, Pacific J. Math. 123 (1986), 189-196.
- I.A.Rus, Metrical fixed point theorems, University of Cluj Napoca, Department of Mathematics, 1979 (mimeographed).
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