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A Counterexample to a Theorem of Xu
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
213, 723]725 Ž1997.
AY975575
NOTE
A Counterexample to a Theorem of Xu ´ ´ Ljubomir Ciric Faculty of Mechanical Engineering, Uni¨ ersity of Belgrade, 27 Marta 80, Belgrade, 11000, Yugosla¨ ia Submitted by William Art Kirk Received August 19, 1996
In this note we shall construct a counterexample to the theorem of H.-K. Xu Ž J. Math. Anal. Appl. 167, 1992, 582]587.. Q 1997 Academic Press
Let X be a normed linear space and T be a selfmapping of X. The Mann iterative process w4x starting from x 0 is defined by x nq1 s Ž1 y a n . x n q a nTx n , n G 0, where a n4 satisfies certain conditions. The Ishikawa iteration process w3x is defined by x 0 g X, yn s Ž1 y bn . x n q bnTx n , x nq1 s Ž1 y a n . x n q a nTyn , n G 0, and the a n4 , bn4 satisfy certain conditions. The sequence of Mann iterations was studied by Rhoades in w8, 9x and several other authors. The Ishikawa iteration scheme was first used to establish the strong convergence for a pseudo-contractive selfmapping of a convex compact subset of a Hilbert space. Very soon both this and the Mann iterative process were used to establish the strong convergence of the corresponding iterations for certain contractive type mappings in Hilbert spaces and subsequently in more general Banach spaces w5, 6, 9x. One of the most general contractive-type definitions for which such theorems have been proved is that in w2x. Let X be a normed linear space and T a selfmapping of X. A selfmapping T is said to be a quasi-contracti¨ e if there exists a constant h, 0 F h - 1, such that, for each x, y g X d Ž Tx, Ty . F hM Ž x, y . ,
Ž 1.
723 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
724
NOTE
where M Ž x, y . s max 5 x y y 5 , 5 x y Tx 5 , 5 y y Ty 5 , 5 x y Ty 5 , 5 y y Tx 5 4 . Ž 2 . Rhoades proved the following: THEOREM w8, Theorem 7x. Let H be a Hilbert space and T be a quasi-contracti¨ e selfmapping of H. Then the Mann iterati¨ e process, with a n4 satisfying the conditions Ži. a0 s 1, Žii. 0 F a n F 1 for n ) 0, Žiii. Ýa n s ` and lim sup a n - 1 y h2 , con¨ erges to the unique fixed point of T. Chidume w1x extended the above result to L p spaces for 1 - p - `. In w9x Rhoades asked whether or not the above theorem could be extended to the Ishikawa process. Xu’s intention was to answer that question by the following theorem. THEOREM w11x. Let C be a nonempty closed con¨ ex subset of Banach space X and T : C ª C a quasi-contraction. Suppose a n ) 0 for all n and Ý a n s `. Then the sequence x n4 defined by x0 g C yn g co Ž
n x i is0
.,
nG0
Ž 4.
x nq 1 s Ž 1 y a n . x n q a nTyn ,
nG0
Ž 5.
4
j
Ž 3.
n Tx i is0
4
con¨ erges strongly to the unique fixed point z of T. The purpose of this note is to give a counterexample showing that, contrary to Xu’s theorem, the process Ž3. ] Ž5. does not converge to the unique fixed point of T. Let C be the closed interval w0, 1x with 5 x y y 5 s < x y y <. We define the selfmapping T by Tx s xr2. Clearly C is the compact convex subset of the reals and T has the unique fixed point z s 0. Consider the sequence x n4 defined by x 0 g C _ 04 , yn s x nq 1 s
1 2 1 2
x0 q xn q
1 2 1 2
Tx 0 s Tyn s
3 4 1 2
x0 xn q
for all n G 0, 3 16
x0 ,
n G 0.
Clearly, x nq 1 ) Ž3r16. x 0 for all n G 0 and so the sequence x n4 cannot converge to 0, the unique fixed point of T.
725
NOTE
The proof of the Lemma in w11x is correct but the inequality diam Ž Bn . s max sup Ž 5 x n y Tx j 5 : j G n . , sup Ž 5 x n y Ty j 5 : j G n . 4 in w11, p. 585x is incorrect. However, the result can be recovered if in the process Ž1. ] Ž3., the given sequence yn4 is replaced by n
n
yn g co x i 4 isk n j Tx i 4 isk n ,
ž
/
where k n4 is a nondecreasing sequence of positive integers such that k n F n and lim nª` k n s q`. REFERENCES 1.. C. E. Chidume, Fixed point iterations for certain classes of nonlinear mappings, Appl. Anal. 27 Ž1988., 19]26. ´ ´ A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 2. L. B. Ciric, 45 Ž1974., 267]273. 3. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 Ž1974., 147]150. 4. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 Ž1953., 506]510. 5. S. Naimpally and K. L. Singh, Extensions of some fixed point theorems of Rhoades, J. Math. Anal. Appl. 96 Ž1983., 437]446. 6. L. Qihou, On Naimpally and Singh’s open questions, J. Math. Anal. Appl. 124 Ž1987., 159]164. 7. L. Qihou, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, J. Math. Anal. Appl. 146 Ž1990., 301]305. 8. B. E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. 196 Ž1974., 161]176. 9. B. E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 Ž1976., 741]750. 10. B. E. Rhoades, Convergence of an Ishikawa-type iteration scheme for a generalized contraction, J. Math. Anal. Appl. 185 Ž1994., 350]355. 11. H.-K. Xu, A note on the Ishikawa iteration scheme, J. Math. Anal. Appl. 167 Ž1992., 582]587.
213, 723]725 Ž1997.
AY975575
NOTE
A Counterexample to a Theorem of Xu ´ ´ Ljubomir Ciric Faculty of Mechanical Engineering, Uni¨ ersity of Belgrade, 27 Marta 80, Belgrade, 11000, Yugosla¨ ia Submitted by William Art Kirk Received August 19, 1996
In this note we shall construct a counterexample to the theorem of H.-K. Xu Ž J. Math. Anal. Appl. 167, 1992, 582]587.. Q 1997 Academic Press
Let X be a normed linear space and T be a selfmapping of X. The Mann iterative process w4x starting from x 0 is defined by x nq1 s Ž1 y a n . x n q a nTx n , n G 0, where a n4 satisfies certain conditions. The Ishikawa iteration process w3x is defined by x 0 g X, yn s Ž1 y bn . x n q bnTx n , x nq1 s Ž1 y a n . x n q a nTyn , n G 0, and the a n4 , bn4 satisfy certain conditions. The sequence of Mann iterations was studied by Rhoades in w8, 9x and several other authors. The Ishikawa iteration scheme was first used to establish the strong convergence for a pseudo-contractive selfmapping of a convex compact subset of a Hilbert space. Very soon both this and the Mann iterative process were used to establish the strong convergence of the corresponding iterations for certain contractive type mappings in Hilbert spaces and subsequently in more general Banach spaces w5, 6, 9x. One of the most general contractive-type definitions for which such theorems have been proved is that in w2x. Let X be a normed linear space and T a selfmapping of X. A selfmapping T is said to be a quasi-contracti¨ e if there exists a constant h, 0 F h - 1, such that, for each x, y g X d Ž Tx, Ty . F hM Ž x, y . ,
Ž 1.
723 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
724
NOTE
where M Ž x, y . s max 5 x y y 5 , 5 x y Tx 5 , 5 y y Ty 5 , 5 x y Ty 5 , 5 y y Tx 5 4 . Ž 2 . Rhoades proved the following: THEOREM w8, Theorem 7x. Let H be a Hilbert space and T be a quasi-contracti¨ e selfmapping of H. Then the Mann iterati¨ e process, with a n4 satisfying the conditions Ži. a0 s 1, Žii. 0 F a n F 1 for n ) 0, Žiii. Ýa n s ` and lim sup a n - 1 y h2 , con¨ erges to the unique fixed point of T. Chidume w1x extended the above result to L p spaces for 1 - p - `. In w9x Rhoades asked whether or not the above theorem could be extended to the Ishikawa process. Xu’s intention was to answer that question by the following theorem. THEOREM w11x. Let C be a nonempty closed con¨ ex subset of Banach space X and T : C ª C a quasi-contraction. Suppose a n ) 0 for all n and Ý a n s `. Then the sequence x n4 defined by x0 g C yn g co Ž
n x i is0
.,
nG0
Ž 4.
x nq 1 s Ž 1 y a n . x n q a nTyn ,
nG0
Ž 5.
4
j
Ž 3.
n Tx i is0
4
con¨ erges strongly to the unique fixed point z of T. The purpose of this note is to give a counterexample showing that, contrary to Xu’s theorem, the process Ž3. ] Ž5. does not converge to the unique fixed point of T. Let C be the closed interval w0, 1x with 5 x y y 5 s < x y y <. We define the selfmapping T by Tx s xr2. Clearly C is the compact convex subset of the reals and T has the unique fixed point z s 0. Consider the sequence x n4 defined by x 0 g C _ 04 , yn s x nq 1 s
1 2 1 2
x0 q xn q
1 2 1 2
Tx 0 s Tyn s
3 4 1 2
x0 xn q
for all n G 0, 3 16
x0 ,
n G 0.
Clearly, x nq 1 ) Ž3r16. x 0 for all n G 0 and so the sequence x n4 cannot converge to 0, the unique fixed point of T.
725
NOTE
The proof of the Lemma in w11x is correct but the inequality diam Ž Bn . s max sup Ž 5 x n y Tx j 5 : j G n . , sup Ž 5 x n y Ty j 5 : j G n . 4 in w11, p. 585x is incorrect. However, the result can be recovered if in the process Ž1. ] Ž3., the given sequence yn4 is replaced by n
n
yn g co x i 4 isk n j Tx i 4 isk n ,
ž
/
where k n4 is a nondecreasing sequence of positive integers such that k n F n and lim nª` k n s q`. REFERENCES 1.. C. E. Chidume, Fixed point iterations for certain classes of nonlinear mappings, Appl. Anal. 27 Ž1988., 19]26. ´ ´ A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 2. L. B. Ciric, 45 Ž1974., 267]273. 3. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 Ž1974., 147]150. 4. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 Ž1953., 506]510. 5. S. Naimpally and K. L. Singh, Extensions of some fixed point theorems of Rhoades, J. Math. Anal. Appl. 96 Ž1983., 437]446. 6. L. Qihou, On Naimpally and Singh’s open questions, J. Math. Anal. Appl. 124 Ž1987., 159]164. 7. L. Qihou, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, J. Math. Anal. Appl. 146 Ž1990., 301]305. 8. B. E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. 196 Ž1974., 161]176. 9. B. E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 Ž1976., 741]750. 10. B. E. Rhoades, Convergence of an Ishikawa-type iteration scheme for a generalized contraction, J. Math. Anal. Appl. 185 Ž1994., 350]355. 11. H.-K. Xu, A note on the Ishikawa iteration scheme, J. Math. Anal. Appl. 167 Ž1992., 582]587.