162 | P a g e
SOME FIXED POINT THEOREMS ON PRODUCT
SPACES BY THE SETTING OF RHOADES
Sanjay Kumar Gupta
Department of Mathematics,
RustamJi Institute of Technology (India)
ABSTRACT
Rhoades discussed a number of fixed point theorems dealing with contractive conditions with rational expressions. In an analogous manner we define mappings on product spaces which satisfy such contractive like conditions in the first variable, and generalize the result of Nadler to such mappings. Here we discuss only those conditions which involve a single mapping. In the Nadler’s result we enlarge the class of mappings by Rhoades type contractive conditions in the first variable and the class of metric spaces Z by the class of uniform spaces.
Keywords:
Complete Metric Space, Fixed Point Property, Locally Compact Space, Product Space ,
Uniformly Continuous Mapping.
I. INTRODUCTION
The fixed point property (f. p. p.) is not necessarily preserved under the Cartesian product of spaces [2,3]. It is preserved when the maps f: XxZXxZ have special contraction properties. Nadler’s results are in this direction. Nadler’s main results are as follows:
1.1 Theorem
Let (X, d) be a metric space. Let Ai: XX be a function with at least one fixed point ai for each i=1, 2,
……….., and let A0:XX be a contraction mapping with fixed point a0. If the sequence {Ai} converges
uniformly to A0, then the sequence {ai} converges to a0.
1.2 Theorem
Let (X, d) be a locally compact metric space, let Ai: XX be a contraction mapping with fixed point ai for each
i=1,2,…….. If the sequence {Ai} converges pointwise to A0, then the sequence {ai} converges to a0.
1.3 Theorem
Let (X, d) be a complete metric space, Z a metric space which has the f.p.p. and f: XxZXxZ be a contraction in the first variable.
(a) If f is uniformly continuous, then f has a fixed point.
(b) If (X, d) is locally compact, f is continuous, then f has a fixed point.
163 | P a g e
for all xX, where 1 is the projection of XxZ on X along Z. (m)', 1m10; will denote the condition (m)' inRhoades [7] with the modification that constant or functions that appear in (m)' depend on z.
II. SOME DEFINITIONS FROM RHOADES [7]
Let (X, d) be a complete metric space and f:X X be mapping. For x X, let 0(x)={x, f(x), f 2 (x), ……} be the orbit of x under f. Consider the following conditions on f and (X, d):
(1)' (Dass and Gupta) – There exist numbers , 0, + 1 and for each x, x* X, x*0 (x) such that
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(6)' (Jain and Dixit) – There exist αi , βi 0,α1 + 2α3 + 2α4 + β1 + β2 + β3 + 2β5 < 1, α2 + β1 + β4 + β5 <1 and
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(7)' (Sharma and Bajaj) – There exist a number β, 0 < β < ½ and for each x, x* X, x*0 (x) such that
164 | P a g e
(8)' (Dass) – There exist numbers αi , βj >0 with1
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Now we prove the following results:
2.1 Theorem
Let (X, d) be a complete metric space, Z a uniform space in which sequences are adequate which has the f. p. p. and let f:XxZXxZ be a mapping.
(a) If f is uniformly continuous such that for each zZ, fz(3)', then f has a fixed point.
(b) If X is locally compact, f is continuous such that for each zZ, fz(3)', then f has a fixed point.
Proof: We prove (a) and (b) simultaneously: Step I: We define a sequence {tn} in X as follows:
For a fixed x0 in X,
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z165 | P a g e
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Since hn0 as n, this inequality shows that {tn} is a Cauchy sequence. Since X is a complete metric space,
there exists a point p1 in X such that tnp1
Step II: we show that p1 is a unique fixed point of fz.
Since fz(3)'. We have (taking x=tn, x*=p1)
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d(p1, fz(p1))a. d(p1,fz(p1)
which is possible only when d(p1, fz(p1))=0 or p1=fz(p1), i.e., p1 is a fixed point of fz.
Suppose p2 is another fixed point of fz such that p1p2 then,
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166 | P a g e
Step III: Let a mapping F:ZX be such that F(z)=p1 is the unique fixed point of fz. Now, let z0Z and let
1
}
{
z
i i be a sequence in Z which converges to z0. Then by the hypothesis in (a), the sequence 1
}
{
f
zi i converges uniformly to fz0 and hence by the Theorem 1.1, the sequence 1
)}
(
{
F
z
i i converges toF(z0), under the assumptions of (b), we may apply Theorem 1.2, to conclude that the sequence
1
)}
(
{
F
z
i iconverges to F(z0). Hence in either case this proves that F is continuous on Z. Also 1f(F(z), z)=fz(F(z))=F(z),
because F(z) is a fixed point of fz. Next, let G:ZZ be defined by setting G(z)= 2f(F(z),z). Then G is a
continuous map of Z to itself. Since Z has the f.p.p., there exists a point pZ such that G(p)=p, then the point (F(p), p) is such that 1f(F(p), p) =F(p) and 2f(F(p),p)= G(p) = p. Therefore (F(p), p) is a fixed point of f in
XxZ.
2.2 Theorem
Let (X, d) be a complete metric space, Z a uniform space in which sequences are adequate which has the f. p. p. and let f:XxZXxZ be a mapping.
(a) If f is uniformly continuous such that for each zZ, fz satisfies any one of the conditions (2)', (5)', (6)',
(7)', (8)', (9)' and (10)', then f has a fixed point.
(b) If (X, d) is locally compact, f is continuous such that for each zZ, fz satisfies any one of the
conditions (2)', (5)', (6)', (7)', (8)', (9)' and (10)', then f has a fixed point. Proof: We prove (a) and (b) simultaneously:
We define a sequence tn(z)= tn in X as follows:
For a fixed x0 in X and any zZ,
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167 | P a g e
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Here we note that if the function f: XxZXxZ is such that fz(5)' then by using similar arguments, we can show
that
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Similarly, if f is such that fz(6) '
, then we can obtain, the condition
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(
1
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1
d
t
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t
t
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n n
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Likewise, if f is such that fz(7)', then we obtain
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t
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t
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n n
n…….(4)
Now, if f is such that fz(9)', then we can obtain
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t
t
1q
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t
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1d
n n
n…….(5)
If f is such that fz(10)', then we can obtain
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t
1q
2d
t
0t
1d
n n
n…….(6)
Finally, if the function f is such that fz(8) '
, then to obtain a condition of the type above, we proceed as follows:
Define
g
1
f
zm, then we have…….(7)
168 | P a g e
…….(8)Adding (7) and (8) above, we get
….(9)
where,
2
,
2
,
,
2
,
5 4 53 2 4 1 3 3 2 2 1 1
and
with
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1
2
2
2
1
i i
In the equation (9), we apply similar procedure described above for the equation (1) and if mapping g1 referred
as fz then we can obtain
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(
1
)
,
(
0 15 4 2 1
5 4 3 2
1
d
t
t
t
t
d
n n
n
…….(10)
Clearly according to conditions (2)' , (5)' , (6)' , (7)' , (9)' , (10)' and (8)' we obtain equations (1), (2), (3), (4), (5), (6) and (10) respectively. However, in each of these cases we see that {tn} is a Cauchy sequence in X. However,
by the completeness of X, there is a point p1 in X such that tn converges to p1. We can easily see that p1 is a
unique fixed point of fz. By the help of step-III of the above theorem 2.1, we can conclude the theorem 2.2.
III. CONCLUSION
We observe that condition (1)', (4)' are stronger than (3)', therefore the above Theorem 2.1 has two corollaries corresponding to each of these two conditions. We also observe that condition (4)' is stronger than conditions (6)' and (8)' therefore the Theorem 2.2 has one corollary corresponding to (4)'. This paper is extension of Nadler’s result according to contractive conditions of Rhoades [7].
REFERENCES
[1] G. Bredon, Some examples of fixed point property, Pacific J. Math. 38 (1971), 571-573
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[2] R.F. Brown, On some old problems of fixed point theory, Rocky Mountain J. Math. 4 (1974), 3-14 [3] E.R. Fadell, Recent results in fixed point theory of continuous maps, Bull. Amer. Math. Soc. 76 (1970),10-29
[4] W. Lopez, An example in the fixed point theory of polyhedra, Bull. Amer. Math. Soc. 73 (1967), 922-924 [5] S.B. Nadler Jr., Sequence of contractions and fixed points, Pacific J. Math. 27 (1968), 579-585.
[6] B.E. Rhoades, A comparison of various definitions of contractive mappings,Trans. Amer. Math. Soc.226(1977),257-290
[7] B.E. Rhoades, Proving fixed point theorem using general principles, Indian J. Pure App. Math. 27(8) (1996), 741-770
