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Picard Iterations for Solution of Nonlinear Equations in Certain Banach Spaces
Journal of Mathematical Analysis and Applications 245, 317᎐325 Ž2000. doi:10.1006rjmaa.2000.6718, available online at http:rrwww.idealibrary.com on
Picard Iterations for Solution of Nonlinear Equations in Certain Banach Spaces Chika Moore1 Department of Mathematics and Computer Science, Nnamdi Azikiwe Uni¨ ersity, P.M.B. 5025, Awka, Anambra State, Nigeria; and Abdus Salam International Center for Theoretical Physics, Trieste, Italy 2 E-mail: [email protected] Submitted by Charles E. Chidume Received December 7, 1998
Let E be a real uniformly smooth Banach space and let A: DŽ A. ; E ¬ E be locally Lipschitzian and strongly quasi-accretive. It is proved that a Picard recursion process converges strongly to the unique solution of the equation Ax s f, f g RŽ A., with the convergence being at least as fast as a geometric progression. Related results deal with the convergence of Picard iterations to the fixed point of locally Lipschitzian strong hemicontractions T and to the solutions of nonlinear equations of the forms x q Ax s f and x y Ax s f, where A is an accretive-type map. 䊚 2000 Academic Press Key Words: Žlocally. quasi-accretive; ŽPicard. iteration; hemicontraction; nonlinear equations; locally Lipschitzian; strong convergence.
1. INTRODUCTION Several iteration processes have been established for the constructive approximation of solutions to several classes of Žnonlinear. operator equations and several convergence results established using these iterative processes Žsee, e.g., w1᎐18x and the references cited therein .. The most classical of these processes seems to be the Picard process. It is a trite fact that if the Picard process converges then it is generally preferred to other global methods for the same problem. In fact, the Mann iterative scheme 1 This research was partially supported by Research Grant RGrMATHrAFrAC 97-210 from the Third World Academy of Sciences ŽTWAS.. 2 Regular associate of the Abdus Salam International Center for Theoretical Physics, Trieste, Italy.
317 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
318
CHIKA MOORE
Žsee, e.g., w13x. and the Ishikawa scheme Žsee, e.g., w11x. were developed specifically to tackle situations where the Picard iteration process fails or seems to have failed. Thus, the recent result of Chidume w6x that if E is an arbitrary real normed linear space and A: E ¬ E is strongly quasi-accretive and Lipschitzian, then a Picard-like iteration process converges strongly to the solution of the equation Ax s f is really interesting. Motivated by the said result w6x, it is our purpose in this paper to prove that if E is a real uniformly smooth Banach space and A: DŽ A. ; E ¬ E is a locally Lipschitz strongly quasi-accretive map with open domain DŽ A., then a Picard recursion process converges strongly to the unique solution of the equation Ax s f, f g RŽ A., the range of A, with convergence being at least as fast as a geometric progression. Let E be a real Banach space and let E* be its dual. For 1 - p - ⬁, the generalized duality mapping J p : E ¬ 2 E* is defined by J p x [ f * g E* : ² x, f *: s 5 x 5 p , 5 f * 5 s 5 x 5 py 1 4 , where ² ⭈ , ⭈ : denotes the generalized duality pairing between E and E*. It is known that J p is single-valued and uniformly continuous on bounded sets if E is uniformly smooth. We shall denote the single-valued duality map by j p . For p s 2, the duality map J s J 2 is called the normalized duality map. Recall that a map A: E ¬ E is said to be accreti¨ e if ᭙ x, y g DŽ A. ᭚ j p Ž x y y . g J p Ž x y y . such that
² Ax y Ay, j p Ž x y y .: G 0
Ž 1.
and is said to be strongly accreti¨ e if A y kI is accretive, where k g Ž0, 1. is a constant and I denotes the identity operator on E. Let S Ž A. s x g DŽ A. : Ax* s f 4 / ⭋ denote the solution set of the equation Ax s f. If S Ž A. / ⭋ and Ž1. holds for all x g DŽ A. and y g S Ž A., then A is said to be quasi-accreti¨ e. The notion of strongly quasi-accreti¨ e is similarly defined. A is said to be m-accreti¨ e if the operator Ž I q A. is surjective. Closely related to the class of accretive maps is the class of pseudocontractive maps. A map T : E ¬ E is said to be pseudocontracti¨ e if ᭙ x, y g DŽT . ᭚ j p Ž x y y . g J p Ž x y y . such that
² Ž I y T . x y Ž I y T . y, j p Ž x y y .: G 0.
Ž 2.
Observe that T is pseudocontractive if and only if A s Ž I y T . is accretive. ŽThe map T is said to be hemicontracti¨ e if and only if A s Ž I y T . is quasi-accretive.. Thus, the mapping theory of accretive maps is intimately tied to the fixed-point theory of pseudocontractions.
319
PICARD ITERATIONS
2. PRELIMINARIES Let E be a real normed linear space of dimension dim E G 2. The modulus of smoothness of E is defined by
E Ž . [ sup
½
5 x q y5 q 5 x y y5 2
y 1 : 5 x 5 s 1, 5 y 5 s ,
5
) 0.
If E Ž . ) 0 ᭙ ) 0 then E is said to be smooth. If there exist a constant c ) 0 and a real number 1 - q - ⬁ such that
E Ž . F c q then E is said to be q-uniformly smooth. Typical examples of such spaces are the Lebesgue L p , the sequence l p , and the Sobole¨ Wpm Ž1 - p - ⬁. spaces. In fact, for 1 - p F 2 these spaces are p-uniformly smooth and for p G 2 they are 2-uniformly smooth. E is said to be uniformly smooth if lim
ª0
E Ž .
s 0.
It is easy to see that every q-uniformly smooth space is uniformly smooth. If E is a real uniformly smooth Banach space, then Žsee, e.g., w18x. the following geometric inequality holds 5 x q y 5 2 F 5 x 5 2 q 2 ² y, j Ž x .: q D max 5 x 5 q 5 y 5 ,
½
C 2
5
E Ž 5 y 5. Ž 3.
for all x, y g E and some real positive constants D and C. If E is q-uniformly smooth, then the following geometric inequality also holds 5 x q y 5 q F 5 x 5 q q q ² y, j q Ž x .: q C q 5 y 5 q
Ž 4.
for all x, y g E and some constant C q G 1. 3. MAIN THEOREMS 3.1. Iterati¨ e solution of the equation Ax s f THEOREM 3.1. Let E be a real uniformly smooth Banach space and let A: DŽ A. ; E ¬ E be a locally Lipschitzian and strongly quasi-accreti¨ e operator with open domain DŽ A. in E such that the equation Ax s f has a solution
320
CHIKA MOORE
x* g DŽ A. for f g RŽ A. arbitrary but fixed. For some fixed g Ž0, 1. define A: DŽ A. ¬ E by A x [ x y Ž Ax y f . ᭙ x g DŽ A.. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the Picard sequence x n4 generated by x nq 1 s A x n ,
n G 0,
Ž 5.
remains in B and con¨ erges strongly to x*, with con¨ ergence being at least as fast as a geometric progression. Proof. Since A is locally Lipschitzian, there exists r ) 0 such that A is Lipschitz on B s Br Ž x*. [ x g E : 5 x y x* 5 F r 4 : DŽ A.. Let g Ž0, 1. and L G 1 denote the strong accretivity and Lipschitz constants of A, respectively. Pick x 0 g B arbitrary and choose ␣ g Ž0, 21L . sufficiently small such that
E Ž ␣ rL . ␣
r 2
-
½
4 D max Ž L q 1 . r ,
C 2
5
and define T : DŽ A. ¬ E by Tx [ x y Ax q f. Observe that Ax* s f if and only if Tx* s x*. Moreover, T is strongly hemicontractive with constant k [ Ž1 y . g Ž0, 1. and Lipschitz with constant L# [ Ž L q 1.. Let
½
[ min ␣ ,
2
2 Ž L q 1. q Ž L q 2. Ž 3 y k .
Observe that Ž1 y L. G Claim 1.
k
1
1 2
5
g Ž 0, 1 . .
and that L - 1.
x n g B ᭙ n G 0.
Suppose the contrary. Then ᭚ n g N such that x n g B but x nq1 f B. From 5 x n y x* 5 G 5 x nq1 y x* 5 y L 5 x n y x* 5 ) Ž 1 y L . r G
r 2
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and the inequality Ž3., we have 5 x nq 1 y x* 5 2 F 5 x n y x* 5 2 y 2 ² Ax n y Ax*, j Ž x n y x* .: q D max 5 x n y x* 5 q 5 Ax n y Ax* 5 ,
½
C 2
5
= E Ž 5 Ax n y Ax* 5 . F 5 x n y x* 5 2 y 2 5 x n y x* 5 2 q D max r Ž 1 q L . ,
½
F 5 x n y x* 5 2 y
r 2 2
q
C 2
5
E Ž Lr .
r 2 4
- 5 x n y x* 5 2 , a contradiction. Hence, the claim holds. We now prove that x n ª x*. Observe that the recursion formula x nq1 s A x n becomes x nq 1 s Ž 1 y . x n q Tx n ,
n G 0,
Ž 6.
and that x* s Ž 1 q . x* q Ž I y T y kI . x* y Ž 1 y k . x*.
Ž 7.
From Ž6. we obtain x n s Ž 1 q . x nq1 q Ž I y T y kI . x nq1 y Ž 1 y k . x n q Ž 2 y k . 2 Ž x n y Tx n . q Ž Tx nq1 y Tx n . . Combining Ž7. and Ž8. we get x n y x* s Ž 1 q . Ž x nq1 y x* . q Ž I y T y kI . x nq 1 y Ž I y T y kI . x* y Ž 1 y k . Ž x n y x* . q Ž 2 y k . 2 Ž x n y Tx n . q Ž Tx nq 1 y Tx n . .
Ž 8.
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CHIKA MOORE
As in w6x, we obtain 1 q Ž1 y k.
5 x nq 1 y x* 5 F
1q
q Ž 2 y k . 2 Ž L q 2 . q2 Ž L q 1 . Ž L q 2 . 5 x n y x* 5 2
F 1 y k q 2 Ž L q 1 . q Ž 3 y k . Ž L q 2 .
½
s 1y
ž
k 2
/
55 x
n
y x* 5
5 x n y x* 5 s ␦ 5 x n y x* 5
F ␦ 2 5 x ny 1 y x* 5 .. . F ␦ n 5 x 0 y x* 5 ª 0
as n ª ⬁,
where ␦ s Ž1 y 12 k . g Ž0, 1.. Hence, x n ª x* as n ª ⬁. Suppose that there exist y* g DŽT . s DŽ A., y* / x*, and Ty* s y*. Repeating the above argument with y* instead of x* yields x n ª y*. This contradiction yields the uniqueness of x* and completes the proof. Remark 3.1. If E is q-uniformly smooth, we have the following theorem. THEOREM 3.2. Let E be a real q-uniformly smooth Banach space and let A: DŽ A. ; E ¬ E be a locally Lipschitzian and strongly quasi-accreti¨ e operator with open domain DŽ A. in E such that the equation Ax s f has a solution x* g DŽ A. for f g RŽ A. arbitrary but fixed. For some fixed g Ž0, 1. define A: DŽ A. ¬ E by A x [ x y Ž Ax y f . ᭙ x g DŽ A.. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the Picard sequence x n4 generated by Ž5. remains in B and con¨ erges strongly to x*, with con¨ ergence being at least as fast as a geometric progression. Proof. Since A is locally Lipschitz, there is an r ) 0 such that A is Lipschitz on B s Br Ž x*. ; DŽ A.. Let k g Ž0, 1. and L G 1 denote the strong accretivity and Lipschitz constants of A, respectively. For arbitrary x g B, choose
s
k
ž / Lq C q
1r Ž qy1 .
g Ž 0, 1 .
since C q G 1, L G 1. Generate the sequence x n4 as in Ž5..
323
PICARD ITERATIONS
Claim 2.
x n g B ᭙ n G 0.
Suppose that x n g B for some n G 0. Then, using the inequality Ž4., we have 5 x nq 1 y x* 5 q s x n y x* y Ž Ax n y Ax* .
q
F 5 x n y x* 5 q y q ² Ax n y Ax*, j p Ž x n y x* .: q q C q 5 Ax n y Ax* 5 q F 1 y Ž qk y Lq C q qy1 . 5 x n y x* 5 q s 1 y Ž q y 1. k
1r Ž qy1 .
k
ž / Lq C q
5 x n y x* 5 q
F r q. Hence, since x 0 g B by our choice, it follows by induction that x n g B ᭙ n G 0. Set
s 1 y Ž q y 1. k
1r Ž qy1 . 1rq
k
ž / Lq C q
and observe that g Ž0, 1. since k-
ž
LC q1r q
Ž q y 1.
Ž qy1 .rq
/
᭙1 - q - ⬁.
Hence, 5 x n y x* 5 F n 5 x 0 y x* 5 so that x n ª x* as n ª ⬁, completing the proof. Remark 3.2. In the special setting of Hilbert spaces, we have the following result which follows readily from Theorem 3.2 on setting q s 2, C q s 1. COROLLARY 3.3. Let H be a real Hilbert space and let A: DŽ A. ; H ¬ H be a locally Lipchitz and strongly quasi-accreti¨ e operator with open domain DŽ A. ; H such that Ax s f has a solution x* g DŽ A. for f g RŽ A.. Set [ kLy2 , where k g Ž0, 1. and L G 1 are the strong accreti¨ ity and Lipschitz constants of A, respecti¨ ely. Generate the sequence x n4 by x nq1 s A x n , where A: DŽ A. ¬ E is defined by A x [ x y Ž Ax y f . ᭙ x g DŽ A.. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the Picard sequence x n4 remains in B and con¨ erges strongly to x*.
324
CHIKA MOORE
3.2. Iterati¨ e Solution of the Equation x q Ax s f If A is locally Lipschitzian Žwith constant L ) 0. and quasi-accretive, then the operator G s I q A is locally Lipschitzian Žwith constant L# s L q 1. and strongly quasi-accretive. Hence, we have the following corollary to Theorem 3.1. COROLLARY 3.4. Let A be a locally Lipschitzian and quasi-accreti¨ e operator with open domain DŽ A. in E such that the equation x q Ax s f has a solution x* g DŽ A.. Let the sequence x n4 be generated by x nq1 s A x n , where A: DŽ A. ¬ E is defined by A x [ x y Ž x q Ax y f . ᭙ x g DŽ A.. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the sequence x n4 remains in B and con¨ erges strongly to x*. 3.3. Iteration of Fixed Points of Strong Hemicontractions COROLLARY 3.5. Let E be a real uniformly smooth Banach space and let T : DŽT . ; E ¬ E be a locally Lipschitzian and strongly hemicontracti¨ e mapping with open domain DŽT . in E and a fixed point x* g DŽT .. For some fixed g Ž0, 1. define T: DŽT . ¬ E by T x [ x y Ž I y T . x ᭙ x g DŽT .. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the Picard sequence x n4 generated by x nq 1 s T x n ,
n G 0,
Ž 9.
remains in B and con¨ erges strongly to x*, with con¨ ergence being at least as fast as a geometric progression. Proof. Observe that A s Ž I y T . is strongly quasi-accretive and locally Lipschitzian. Thus, the corollary follows easily from Theorem 3.1. Remark 3.3. Ži. While convergence rates in theorems similar to our theorems are of the order O Ž ny1 r2 ., the convergence rate in all our theorems is at least as fast as a geometric progression. Žii. Our theorems easily extend to the case where A is locally strongly quasiaccretive and locally Lipschitzian. In such a case, it is clear that there exists r ) 0 such that A is strongly quasi-accretive and Lipschitzian in Br Ž x*.. ACKNOWLEDGMENTS The author is grateful to the referee for the valuable suggestion that helped immensely in improving this paper. This research was carried out while the author was visiting the Abdus Salam International Center for Theoretical Physics as an Associate; a generous grant from the Swedish International Development Cooperation Agency ŽSIDA. made the visit possible. The author is grateful to both.
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REFERENCES 1. F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 Ž1967., 197᎐228. 2. C. E. Chidume, Iterative approximation of Lipschitz strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 99 Ž1987., 283᎐288. 3. C. E. Chidume, Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc. 120 Ž1994., 545᎐551. 4. C. E. Chidume, Global iteration schemes for strongly pseudocontractive maps, Proc. Amer. Math. Soc. 126 Ž1998., 2641᎐2649. 5. C. E. Chidume, Iterative solution of nonlinear equations of strongly accretive type, Math. Nachr. 189 Ž1998., 46᎐60. 6. C. E. Chidume, Picard iteration for strongly quasi-accretive and strongly hemicontractive Lipschitz maps in arbitrary real normed linear spaces, Proc. Amer. Math. Soc., to appear. 7. C. E. Chidume and M. O. Osilike, Ishikawa iteration process for nonlinear Lipschitz strongly accretive mappings, J. Math. Anal. Appl. 192 Ž1995., 727᎐741. 8. C. E. Chidume and M. O. Osilike, Nonlinear accretive and pseudocontractive operator equations in Banach spaces, Nonlinear Anal. 31 Ž1998., 779᎐789. 9. L. Deng, Iteration process for nonlinear Lipschitzian strongly accretive mappings in L p spaces, J. Math. Anal. Appl. 188 Ž1994., 128᎐140. 10. L. Deng, On Chidume’s open questions, J. Math. Anal. Appl. 174 Ž1993., 441᎐449. 11. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 Ž1974., 147᎐150. 12. L. S. Liu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 Ž1994., 114᎐125. 13. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 Ž1953., 506᎐510. 14. C. Moore, The solution by iteration of nonlinear equations involving psi-strongly accretive operators in Banach spaces, Nonlinear Anal. 37 Ž1999., 125᎐138. 15. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 85 Ž1980., 257᎐292. 16. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 Ž1991., 1127᎐1138. 17. Y. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 Ž1998., 91᎐101. 18. Z. B. Xu and G. F. Roach, Characteristic inequalities for uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl. 157 Ž1991., 189᎐210.
Picard Iterations for Solution of Nonlinear Equations in Certain Banach Spaces Chika Moore1 Department of Mathematics and Computer Science, Nnamdi Azikiwe Uni¨ ersity, P.M.B. 5025, Awka, Anambra State, Nigeria; and Abdus Salam International Center for Theoretical Physics, Trieste, Italy 2 E-mail: [email protected] Submitted by Charles E. Chidume Received December 7, 1998
Let E be a real uniformly smooth Banach space and let A: DŽ A. ; E ¬ E be locally Lipschitzian and strongly quasi-accretive. It is proved that a Picard recursion process converges strongly to the unique solution of the equation Ax s f, f g RŽ A., with the convergence being at least as fast as a geometric progression. Related results deal with the convergence of Picard iterations to the fixed point of locally Lipschitzian strong hemicontractions T and to the solutions of nonlinear equations of the forms x q Ax s f and x y Ax s f, where A is an accretive-type map. 䊚 2000 Academic Press Key Words: Žlocally. quasi-accretive; ŽPicard. iteration; hemicontraction; nonlinear equations; locally Lipschitzian; strong convergence.
1. INTRODUCTION Several iteration processes have been established for the constructive approximation of solutions to several classes of Žnonlinear. operator equations and several convergence results established using these iterative processes Žsee, e.g., w1᎐18x and the references cited therein .. The most classical of these processes seems to be the Picard process. It is a trite fact that if the Picard process converges then it is generally preferred to other global methods for the same problem. In fact, the Mann iterative scheme 1 This research was partially supported by Research Grant RGrMATHrAFrAC 97-210 from the Third World Academy of Sciences ŽTWAS.. 2 Regular associate of the Abdus Salam International Center for Theoretical Physics, Trieste, Italy.
317 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
318
CHIKA MOORE
Žsee, e.g., w13x. and the Ishikawa scheme Žsee, e.g., w11x. were developed specifically to tackle situations where the Picard iteration process fails or seems to have failed. Thus, the recent result of Chidume w6x that if E is an arbitrary real normed linear space and A: E ¬ E is strongly quasi-accretive and Lipschitzian, then a Picard-like iteration process converges strongly to the solution of the equation Ax s f is really interesting. Motivated by the said result w6x, it is our purpose in this paper to prove that if E is a real uniformly smooth Banach space and A: DŽ A. ; E ¬ E is a locally Lipschitz strongly quasi-accretive map with open domain DŽ A., then a Picard recursion process converges strongly to the unique solution of the equation Ax s f, f g RŽ A., the range of A, with convergence being at least as fast as a geometric progression. Let E be a real Banach space and let E* be its dual. For 1 - p - ⬁, the generalized duality mapping J p : E ¬ 2 E* is defined by J p x [ f * g E* : ² x, f *: s 5 x 5 p , 5 f * 5 s 5 x 5 py 1 4 , where ² ⭈ , ⭈ : denotes the generalized duality pairing between E and E*. It is known that J p is single-valued and uniformly continuous on bounded sets if E is uniformly smooth. We shall denote the single-valued duality map by j p . For p s 2, the duality map J s J 2 is called the normalized duality map. Recall that a map A: E ¬ E is said to be accreti¨ e if ᭙ x, y g DŽ A. ᭚ j p Ž x y y . g J p Ž x y y . such that
² Ax y Ay, j p Ž x y y .: G 0
Ž 1.
and is said to be strongly accreti¨ e if A y kI is accretive, where k g Ž0, 1. is a constant and I denotes the identity operator on E. Let S Ž A. s x g DŽ A. : Ax* s f 4 / ⭋ denote the solution set of the equation Ax s f. If S Ž A. / ⭋ and Ž1. holds for all x g DŽ A. and y g S Ž A., then A is said to be quasi-accreti¨ e. The notion of strongly quasi-accreti¨ e is similarly defined. A is said to be m-accreti¨ e if the operator Ž I q A. is surjective. Closely related to the class of accretive maps is the class of pseudocontractive maps. A map T : E ¬ E is said to be pseudocontracti¨ e if ᭙ x, y g DŽT . ᭚ j p Ž x y y . g J p Ž x y y . such that
² Ž I y T . x y Ž I y T . y, j p Ž x y y .: G 0.
Ž 2.
Observe that T is pseudocontractive if and only if A s Ž I y T . is accretive. ŽThe map T is said to be hemicontracti¨ e if and only if A s Ž I y T . is quasi-accretive.. Thus, the mapping theory of accretive maps is intimately tied to the fixed-point theory of pseudocontractions.
319
PICARD ITERATIONS
2. PRELIMINARIES Let E be a real normed linear space of dimension dim E G 2. The modulus of smoothness of E is defined by
E Ž . [ sup
½
5 x q y5 q 5 x y y5 2
y 1 : 5 x 5 s 1, 5 y 5 s ,
5
) 0.
If E Ž . ) 0 ᭙ ) 0 then E is said to be smooth. If there exist a constant c ) 0 and a real number 1 - q - ⬁ such that
E Ž . F c q then E is said to be q-uniformly smooth. Typical examples of such spaces are the Lebesgue L p , the sequence l p , and the Sobole¨ Wpm Ž1 - p - ⬁. spaces. In fact, for 1 - p F 2 these spaces are p-uniformly smooth and for p G 2 they are 2-uniformly smooth. E is said to be uniformly smooth if lim
ª0
E Ž .
s 0.
It is easy to see that every q-uniformly smooth space is uniformly smooth. If E is a real uniformly smooth Banach space, then Žsee, e.g., w18x. the following geometric inequality holds 5 x q y 5 2 F 5 x 5 2 q 2 ² y, j Ž x .: q D max 5 x 5 q 5 y 5 ,
½
C 2
5
E Ž 5 y 5. Ž 3.
for all x, y g E and some real positive constants D and C. If E is q-uniformly smooth, then the following geometric inequality also holds 5 x q y 5 q F 5 x 5 q q q ² y, j q Ž x .: q C q 5 y 5 q
Ž 4.
for all x, y g E and some constant C q G 1. 3. MAIN THEOREMS 3.1. Iterati¨ e solution of the equation Ax s f THEOREM 3.1. Let E be a real uniformly smooth Banach space and let A: DŽ A. ; E ¬ E be a locally Lipschitzian and strongly quasi-accreti¨ e operator with open domain DŽ A. in E such that the equation Ax s f has a solution
320
CHIKA MOORE
x* g DŽ A. for f g RŽ A. arbitrary but fixed. For some fixed g Ž0, 1. define A: DŽ A. ¬ E by A x [ x y Ž Ax y f . ᭙ x g DŽ A.. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the Picard sequence x n4 generated by x nq 1 s A x n ,
n G 0,
Ž 5.
remains in B and con¨ erges strongly to x*, with con¨ ergence being at least as fast as a geometric progression. Proof. Since A is locally Lipschitzian, there exists r ) 0 such that A is Lipschitz on B s Br Ž x*. [ x g E : 5 x y x* 5 F r 4 : DŽ A.. Let g Ž0, 1. and L G 1 denote the strong accretivity and Lipschitz constants of A, respectively. Pick x 0 g B arbitrary and choose ␣ g Ž0, 21L . sufficiently small such that
E Ž ␣ rL . ␣
r 2
-
½
4 D max Ž L q 1 . r ,
C 2
5
and define T : DŽ A. ¬ E by Tx [ x y Ax q f. Observe that Ax* s f if and only if Tx* s x*. Moreover, T is strongly hemicontractive with constant k [ Ž1 y . g Ž0, 1. and Lipschitz with constant L# [ Ž L q 1.. Let
½
[ min ␣ ,
2
2 Ž L q 1. q Ž L q 2. Ž 3 y k .
Observe that Ž1 y L. G Claim 1.
k
1
1 2
5
g Ž 0, 1 . .
and that L - 1.
x n g B ᭙ n G 0.
Suppose the contrary. Then ᭚ n g N such that x n g B but x nq1 f B. From 5 x n y x* 5 G 5 x nq1 y x* 5 y L 5 x n y x* 5 ) Ž 1 y L . r G
r 2
321
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and the inequality Ž3., we have 5 x nq 1 y x* 5 2 F 5 x n y x* 5 2 y 2 ² Ax n y Ax*, j Ž x n y x* .: q D max 5 x n y x* 5 q 5 Ax n y Ax* 5 ,
½
C 2
5
= E Ž 5 Ax n y Ax* 5 . F 5 x n y x* 5 2 y 2 5 x n y x* 5 2 q D max r Ž 1 q L . ,
½
F 5 x n y x* 5 2 y
r 2 2
q
C 2
5
E Ž Lr .
r 2 4
- 5 x n y x* 5 2 , a contradiction. Hence, the claim holds. We now prove that x n ª x*. Observe that the recursion formula x nq1 s A x n becomes x nq 1 s Ž 1 y . x n q Tx n ,
n G 0,
Ž 6.
and that x* s Ž 1 q . x* q Ž I y T y kI . x* y Ž 1 y k . x*.
Ž 7.
From Ž6. we obtain x n s Ž 1 q . x nq1 q Ž I y T y kI . x nq1 y Ž 1 y k . x n q Ž 2 y k . 2 Ž x n y Tx n . q Ž Tx nq1 y Tx n . . Combining Ž7. and Ž8. we get x n y x* s Ž 1 q . Ž x nq1 y x* . q Ž I y T y kI . x nq 1 y Ž I y T y kI . x* y Ž 1 y k . Ž x n y x* . q Ž 2 y k . 2 Ž x n y Tx n . q Ž Tx nq 1 y Tx n . .
Ž 8.
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CHIKA MOORE
As in w6x, we obtain 1 q Ž1 y k.
5 x nq 1 y x* 5 F
1q
q Ž 2 y k . 2 Ž L q 2 . q2 Ž L q 1 . Ž L q 2 . 5 x n y x* 5 2
F 1 y k q 2 Ž L q 1 . q Ž 3 y k . Ž L q 2 .
½
s 1y
ž
k 2
/
55 x
n
y x* 5
5 x n y x* 5 s ␦ 5 x n y x* 5
F ␦ 2 5 x ny 1 y x* 5 .. . F ␦ n 5 x 0 y x* 5 ª 0
as n ª ⬁,
where ␦ s Ž1 y 12 k . g Ž0, 1.. Hence, x n ª x* as n ª ⬁. Suppose that there exist y* g DŽT . s DŽ A., y* / x*, and Ty* s y*. Repeating the above argument with y* instead of x* yields x n ª y*. This contradiction yields the uniqueness of x* and completes the proof. Remark 3.1. If E is q-uniformly smooth, we have the following theorem. THEOREM 3.2. Let E be a real q-uniformly smooth Banach space and let A: DŽ A. ; E ¬ E be a locally Lipschitzian and strongly quasi-accreti¨ e operator with open domain DŽ A. in E such that the equation Ax s f has a solution x* g DŽ A. for f g RŽ A. arbitrary but fixed. For some fixed g Ž0, 1. define A: DŽ A. ¬ E by A x [ x y Ž Ax y f . ᭙ x g DŽ A.. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the Picard sequence x n4 generated by Ž5. remains in B and con¨ erges strongly to x*, with con¨ ergence being at least as fast as a geometric progression. Proof. Since A is locally Lipschitz, there is an r ) 0 such that A is Lipschitz on B s Br Ž x*. ; DŽ A.. Let k g Ž0, 1. and L G 1 denote the strong accretivity and Lipschitz constants of A, respectively. For arbitrary x g B, choose
s
k
ž / Lq C q
1r Ž qy1 .
g Ž 0, 1 .
since C q G 1, L G 1. Generate the sequence x n4 as in Ž5..
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Claim 2.
x n g B ᭙ n G 0.
Suppose that x n g B for some n G 0. Then, using the inequality Ž4., we have 5 x nq 1 y x* 5 q s x n y x* y Ž Ax n y Ax* .
q
F 5 x n y x* 5 q y q ² Ax n y Ax*, j p Ž x n y x* .: q q C q 5 Ax n y Ax* 5 q F 1 y Ž qk y Lq C q qy1 . 5 x n y x* 5 q s 1 y Ž q y 1. k
1r Ž qy1 .
k
ž / Lq C q
5 x n y x* 5 q
F r q. Hence, since x 0 g B by our choice, it follows by induction that x n g B ᭙ n G 0. Set
s 1 y Ž q y 1. k
1r Ž qy1 . 1rq
k
ž / Lq C q
and observe that g Ž0, 1. since k-
ž
LC q1r q
Ž q y 1.
Ž qy1 .rq
/
᭙1 - q - ⬁.
Hence, 5 x n y x* 5 F n 5 x 0 y x* 5 so that x n ª x* as n ª ⬁, completing the proof. Remark 3.2. In the special setting of Hilbert spaces, we have the following result which follows readily from Theorem 3.2 on setting q s 2, C q s 1. COROLLARY 3.3. Let H be a real Hilbert space and let A: DŽ A. ; H ¬ H be a locally Lipchitz and strongly quasi-accreti¨ e operator with open domain DŽ A. ; H such that Ax s f has a solution x* g DŽ A. for f g RŽ A.. Set [ kLy2 , where k g Ž0, 1. and L G 1 are the strong accreti¨ ity and Lipschitz constants of A, respecti¨ ely. Generate the sequence x n4 by x nq1 s A x n , where A: DŽ A. ¬ E is defined by A x [ x y Ž Ax y f . ᭙ x g DŽ A.. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the Picard sequence x n4 remains in B and con¨ erges strongly to x*.
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CHIKA MOORE
3.2. Iterati¨ e Solution of the Equation x q Ax s f If A is locally Lipschitzian Žwith constant L ) 0. and quasi-accretive, then the operator G s I q A is locally Lipschitzian Žwith constant L# s L q 1. and strongly quasi-accretive. Hence, we have the following corollary to Theorem 3.1. COROLLARY 3.4. Let A be a locally Lipschitzian and quasi-accreti¨ e operator with open domain DŽ A. in E such that the equation x q Ax s f has a solution x* g DŽ A.. Let the sequence x n4 be generated by x nq1 s A x n , where A: DŽ A. ¬ E is defined by A x [ x y Ž x q Ax y f . ᭙ x g DŽ A.. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the sequence x n4 remains in B and con¨ erges strongly to x*. 3.3. Iteration of Fixed Points of Strong Hemicontractions COROLLARY 3.5. Let E be a real uniformly smooth Banach space and let T : DŽT . ; E ¬ E be a locally Lipschitzian and strongly hemicontracti¨ e mapping with open domain DŽT . in E and a fixed point x* g DŽT .. For some fixed g Ž0, 1. define T: DŽT . ¬ E by T x [ x y Ž I y T . x ᭙ x g DŽT .. Then there exists a neighbourhood B of x* such that, starting with an arbitrary x 0 g B, the Picard sequence x n4 generated by x nq 1 s T x n ,
n G 0,
Ž 9.
remains in B and con¨ erges strongly to x*, with con¨ ergence being at least as fast as a geometric progression. Proof. Observe that A s Ž I y T . is strongly quasi-accretive and locally Lipschitzian. Thus, the corollary follows easily from Theorem 3.1. Remark 3.3. Ži. While convergence rates in theorems similar to our theorems are of the order O Ž ny1 r2 ., the convergence rate in all our theorems is at least as fast as a geometric progression. Žii. Our theorems easily extend to the case where A is locally strongly quasiaccretive and locally Lipschitzian. In such a case, it is clear that there exists r ) 0 such that A is strongly quasi-accretive and Lipschitzian in Br Ž x*.. ACKNOWLEDGMENTS The author is grateful to the referee for the valuable suggestion that helped immensely in improving this paper. This research was carried out while the author was visiting the Abdus Salam International Center for Theoretical Physics as an Associate; a generous grant from the Swedish International Development Cooperation Agency ŽSIDA. made the visit possible. The author is grateful to both.
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