Abstract
Authors
Stefan M. Soltuz
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
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Ş.M. Şoltuz, D. Otrocol, The convergence of Mann iteration with delay, Mathematical Sciences Research, v. 11 (2007) no. 3, pp. 390-393
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Mathematical Sciences Research
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Global Publishing Company
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1537-5978
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The convergence of Mann iteration with delay ŞTEFAN M. ŞOLTUZ and DIANA OTROCOL
Abstract
We show the convergence of Mann iteration with delay for various classes of non-Lipschitzian operators.
AMS Subject classification: 47 H 10 .
Key words: Mann iteration with delay, strongly pseudocontractive, strongly accretive map.
Key words: Mann iteration with delay, strongly pseudocontractive, strongly accretive map.
1 Introduction
Let XX be a real Banach space, BB be a nonempty, convex subset of XX, and T:B rarr BT: B \rightarrow B be an operator. Let u_(1)in Bu_{1} \in B be given and s > 0s>0 a fixed number. We consider the following iteration, to which we further refer as Mann iteration with delay, see [5]:
{:(1)u_(n+1)=(1-alpha_(n))u_(n-s)+alpha_(n)Tu_(n-s)",":}\begin{equation*}
u_{n+1}=\left(1-\alpha_{n}\right) u_{n-s}+\alpha_{n} T u_{n-s}, \tag{1}
\end{equation*}
the sequence {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfies
{:(2)lim_(n rarr oo)alpha_(n)=0","sum_(n=1)^(oo)alpha_(n)=oo.:}\begin{equation*}
\lim _{n \rightarrow \infty} \alpha_{n}=0, \sum_{n=1}^{\infty} \alpha_{n}=\infty . \tag{2}
\end{equation*}
We are inspired for such delays from economics and biology problems in which fixed point are required. Usually, TT is a contraction. It is wellknown that Mann iteration is desirable when TT is not a contraction.
The operator J:X rarr2^(X^(**))J: X \rightarrow 2^{X^{*}} given by Jx:={f inX^(**):(:x,f:)=||x||^(2),||f||=:}||x||},AA x in XJ x:=\left\{f \in X^{*}:\langle x, f\rangle=\|x\|^{2},\|f\|=\right. \|x\|\}, \forall x \in X, is called the normalized duality mapping. The Hahn-Banach theorem assures that Jx!=O/,AA x in XJ x \neq \emptyset, \forall x \in X. It is easy to see that we have (:j(x),y:) <= ||x||||y||,AA x,y in X,AA j(x)in J(x)\langle j(x), y\rangle \leq \|x\|\|y\|, \forall x, y \in X, \forall j(x) \in J(x).
Definition 1 Let XX be a real Banach space. Let BB be a nonempty subset. AA map T:B rarr BT: B \rightarrow B is called strongly pseudocontractive if there exists k in(0,1)k \in(0,1) and a j(x-y)in J(x-y)j(x-y) \in J(x-y) such that
{:(3)(:Tx-Ty","j(x-y):) <= (1-k)||x-y||^(2)","AA x","y in B.:}\begin{equation*}
\langle T x-T y, j(x-y)\rangle \leq(1-k)\|x-y\|^{2}, \forall x, y \in B . \tag{3}
\end{equation*}
A map S:B rarr BS: B \rightarrow B is called strongly accretive if there exists k in(0,1)k \in(0,1) and a j(x-y)in J(x-y)j(x-y) \in J(x-y) such that
{:(4)(:Sx-Sy","j(x-y):) >= k||x-y||^(2)","AA x","y in B.:}\begin{equation*}
\langle S x-S y, j(x-y)\rangle \geq k\|x-y\|^{2}, \forall x, y \in B . \tag{4}
\end{equation*}
In (3) when k=1k=1, then TT is called pseudocontractive. In (4) when k=0k=0, then SS is called accretive. Let us denote by II the identity map.
Lemma 2 If XX is a real Banach space, then the following relation is true
{:(5)||x+y||^(2) <= ||x||^(2)+2(:y","j(x+y):)","AA x","y in X","AA j(x+y)in J(x+y).:}\begin{equation*}
\|x+y\|^{2} \leq\|x\|^{2}+2\langle y, j(x+y)\rangle, \forall x, y \in X, \forall j(x+y) \in J(x+y) . \tag{5}
\end{equation*}
Lemma 3 [6] Let (rho_(n))_(n)\left(\rho_{n}\right)_{n} be a nonnegative sequence which satisfies the following inequality
{:(6)rho_(n+1) <= (1-lambda_(n))rho_(n)+epsi_(n):}\begin{equation*}
\rho_{n+1} \leq\left(1-\lambda_{n}\right) \rho_{n}+\varepsilon_{n} \tag{6}
\end{equation*}
where lambda_(n)in(0,1),AA n inN,sum_(n=1)^(oo)lambda_(n)=oo\lambda_{n} \in(0,1), \forall n \in \mathbb{N}, \sum_{n=1}^{\infty} \lambda_{n}=\infty, and epsi_(n)=o(lambda_(n))\varepsilon_{n}=o\left(\lambda_{n}\right). Then lim_(n rarr oo)rho_(n)=0\lim _{n \rightarrow \infty} \rho_{n}=0.
Lemma 4 Let s >= 0s \geq 0 be a fixed number, {a_(n)}\left\{a_{n}\right\} a nonnegative sequence which satisfies the following inequality
Lemma 4 Let s >= 0s \geq 0 be a fixed number, {a_(n)}\left\{a_{n}\right\} a nonnegative sequence which satisfies the following inequality
{:(6)a_(n+1) <= (1-alpha_(n))a_(n-s)+sigma_(n):}\begin{equation*}
a_{n+1} \leq\left(1-\alpha_{n}\right) a_{n-s}+\sigma_{n} \tag{6}
\end{equation*}
where alpha_(n)in(0,1),AA n inN,sum_(n=1)^(oo)alpha_(n)=oo\alpha_{n} \in(0,1), \forall n \in \mathbb{N}, \sum_{n=1}^{\infty} \alpha_{n}=\infty, and sigma_(n)=o(alpha_(n))\sigma_{n}=o\left(\alpha_{n}\right). Then lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0.
Proof. Note that sequence {a_(n)}\left\{a_{n}\right\} is the reunion of s+1s+1 independent subsequences. If all such subsequences converges to zero then {a_(n)}\left\{a_{n}\right\} shall converge. The generic subsequence satisfies (6). Set rho_(n):=a_(n-s),lambda_(n):=alpha_(n),epsi_(n):=sigma_(n)\rho_{n}:=a_{n-s}, \lambda_{n}:=\alpha_{n}, \varepsilon_{n}:=\sigma_{n}, and use Lemma 3 to obtain lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0.
Proof. Note that sequence {a_(n)}\left\{a_{n}\right\} is the reunion of s+1s+1 independent subsequences. If all such subsequences converges to zero then {a_(n)}\left\{a_{n}\right\} shall converge. The generic subsequence satisfies (6). Set rho_(n):=a_(n-s),lambda_(n):=alpha_(n),epsi_(n):=sigma_(n)\rho_{n}:=a_{n-s}, \lambda_{n}:=\alpha_{n}, \varepsilon_{n}:=\sigma_{n}, and use Lemma 3 to obtain lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0.
2 Main result
Theorem 5 Let s >= 0s \geq 0 be a fixed number, XX a real Banach space with a uniformly convex dual X^(**),BX^{*}, B a nonempty closed convex bounded subset of XX, and T:B rarr BT: B \rightarrow B be a continuous strongly pseudocontractive mapping. Then the Mann iteration with delay {x_(n)}_(n=1)^(oo)\left\{x_{n}\right\}_{n=1}^{\infty} defined by (1) converges strongly to the unique fixed point of TT.
Proof. Corollary 1 of [2] assures the existence of a fixed point. The uniqueness of the fixed point comes from (3). Because X^(**)X^{*} is uniformly convex the duality map is singled valued (see, e.g., [1]). Let x^(**)x^{*} be the fixed point of TT. Using (1), (3) and Lemma 2 we get
{:[||u_(n+1)-x^(**)||^(2)=||(1-alpha_(n))(u_(n-s)-x^(**))+alpha_(n)(Tu_(n-s)-Tx^(**))||^(2)],[ <= (1-alpha_(n))^(2)||u_(n-s)-x^(**)||^(2)+2alpha_(n)(:Tu_(n-s)-Tx^(**),J(u_(n+1)-x^(**)):)],[=(1-alpha_(n))^(2)||u_(n-s)-x^(**)||^(2)+2alpha_(n)(:Tu_(n-s)-Tx^(**),J(u_(n-s)-x^(**)):)+],[+2alpha_(n)(:Tu_(n-s)-Tx^(**),J(u_(n+1)-x^(**))-J(u_(n-s)-x^(**)):)],[ <= (1-alpha_(n))^(2)||u_(n-s)-x^(**)||^(2)+2alpha_(n)(1-k)||u_(n-s)-x^(**)||^(2)+2alpha_(n)sigma_(n)]:}\begin{aligned}
\left\|u_{n+1}-x^{*}\right\|^{2} & =\left\|\left(1-\alpha_{n}\right)\left(u_{n-s}-x^{*}\right)+\alpha_{n}\left(T u_{n-s}-T x^{*}\right)\right\|^{2} \\
& \leq\left(1-\alpha_{n}\right)^{2}\left\|u_{n-s}-x^{*}\right\|^{2}+2 \alpha_{n}\left\langle T u_{n-s}-T x^{*}, J\left(u_{n+1}-x^{*}\right)\right\rangle \\
& =\left(1-\alpha_{n}\right)^{2}\left\|u_{n-s}-x^{*}\right\|^{2}+2 \alpha_{n}\left\langle T u_{n-s}-T x^{*}, J\left(u_{n-s}-x^{*}\right)\right\rangle+ \\
& +2 \alpha_{n}\left\langle T u_{n-s}-T x^{*}, J\left(u_{n+1}-x^{*}\right)-J\left(u_{n-s}-x^{*}\right)\right\rangle \\
& \leq\left(1-\alpha_{n}\right)^{2}\left\|u_{n-s}-x^{*}\right\|^{2}+2 \alpha_{n}(1-k)\left\|u_{n-s}-x^{*}\right\|^{2}+2 \alpha_{n} \sigma_{n}
\end{aligned}
where
sigma_(n)=(:Tu_(n-s)-Tx^(**),J(u_(n+1)-x^(**))-J(u_(n-s)-x^(**)):).\sigma_{n}=\left\langle T u_{n-s}-T x^{*}, J\left(u_{n+1}-x^{*}\right)-J\left(u_{n-s}-x^{*}\right)\right\rangle .
Now we shall show sigma_(n)rarr0\sigma_{n} \rightarrow 0 as n rarr oon \rightarrow \infty. Observe that (||Tu_(n-s)-Tx^(**)||)_(n)\left(\left\|T u_{n-s}-T x^{*}\right\|\right)_{n} is bounded. We prove now that
{:(7)J(u_(n+1)-x^(**))-J(u_(n-s)-x^(**))rarr0" as "n rarr oo:}\begin{equation*}
J\left(u_{n+1}-x^{*}\right)-J\left(u_{n-s}-x^{*}\right) \rightarrow 0 \text { as } n \rightarrow \infty \tag{7}
\end{equation*}
Proposition 12.3 on page 115, of [2] assures that, when X^(**)X^{*} is uniformly convex, then JJ is uniformly continuous on every bounded set of XX. To prove (7) it is sufficient to see that
||u_(n+1)-u_(n-s)||=alpha_(n)||u_(n-s)-Tu_(n-s)|| <= alpha_(n)(||u_(n-s)||+||Tu_(n-s)||) <= 2alpha_(n)M rarr0\left\|u_{n+1}-u_{n-s}\right\|=\alpha_{n}\left\|u_{n-s}-T u_{n-s}\right\| \leq \alpha_{n}\left(\left\|u_{n-s}\right\|+\left\|T u_{n-s}\right\|\right) \leq 2 \alpha_{n} M \rightarrow 0
where M=s u p{||u_(n-s)||,||Tu_(n-s)||}M=\sup \left\{\left\|u_{n-s}\right\|,\left\|T u_{n-s}\right\|\right\}. The sequences (u_(n-s))_(n),(Tu_(n-s))_(n)\left(u_{n-s}\right)_{n},\left(T u_{n-s}\right)_{n} are bounded being in the bounded set BB. Hence (7) holds. Then
{:[(8)||u_(n+1)-x^(**)||^(2) <= ((1-alpha_(n))^(2)+2alpha_(n)(1-k))||u_(n-s)-x^(**)||^(2)+2alpha_(n)sigma_(n)],[=(1-2alpha_(n)+alpha_(n)^(2)+2alpha_(n)-2alpha_(n)k)||u_(n-s)-x^(**)||^(2)+2alpha_(n)sigma_(n)],[=(1+alpha_(n)^(2)-2alpha_(n)k)||u_(n-s)-x^(**)||^(2)+2alpha_(n)sigma_(n)]:}\begin{align*}
\left\|u_{n+1}-x^{*}\right\|^{2} & \leq\left(\left(1-\alpha_{n}\right)^{2}+2 \alpha_{n}(1-k)\right)\left\|u_{n-s}-x^{*}\right\|^{2}+2 \alpha_{n} \sigma_{n} \tag{8}\\
& =\left(1-2 \alpha_{n}+\alpha_{n}^{2}+2 \alpha_{n}-2 \alpha_{n} k\right)\left\|u_{n-s}-x^{*}\right\|^{2}+2 \alpha_{n} \sigma_{n} \\
& =\left(1+\alpha_{n}^{2}-2 \alpha_{n} k\right)\left\|u_{n-s}-x^{*}\right\|^{2}+2 \alpha_{n} \sigma_{n}
\end{align*}
The condition lim_(n rarr oo)alpha_(n)=0\lim _{n \rightarrow \infty} \alpha_{n}=0 implies the existence of an n_(0)n_{0} such that for all n >= n_(0)n \geq n_{0} we have
{:(9)alpha_(n) < k:}\begin{equation*}
\alpha_{n}<k \tag{9}
\end{equation*}
Substituting (9) into (8) we get 1+alpha_(n)^(2)-2alpha_(n)k < 1-alpha_(n)k1+\alpha_{n}^{2}-2 \alpha_{n} k<1-\alpha_{n} k. Finally, the above inequality yields
||u_(n+1)-x^(**)||^(2) <= (1-alpha_(n)k)||u_(n-s)-x^(**)||^(2)+2alpha_(n)sigma_(n)\left\|u_{n+1}-x^{*}\right\|^{2} \leq\left(1-\alpha_{n} k\right)\left\|u_{n-s}-x^{*}\right\|^{2}+2 \alpha_{n} \sigma_{n}
Setting a_(n)=||u_(n-s)-x^(**)||^(2),lambda_(n)=alpha_(n)k in(0,1)a_{n}=\left\|u_{n-s}-x^{*}\right\|^{2}, \lambda_{n}=\alpha_{n} k \in(0,1), and using Lemma 4, we obtain lim_(n rarr oo)a_(n)=lim_(n rarr oo)||u_(n-s)-x^(**)||^(2)=0\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty}\left\|u_{n-s}-x^{*}\right\|^{2}=0 i.e.
lim_(n rarr oo)||u_(n-s)-x^(**)||=0\lim _{n \rightarrow \infty}\left\|u_{n-s}-x^{*}\right\|=0
3 The accretive and strongly accretive cases
Let II denote the identity map.
Remark 6 The operator TT is a (strongly) pseudocontractive map if and only if (I-T)(I-T) is a (strongly) accretive map.
Remark 6 The operator TT is a (strongly) pseudocontractive map if and only if (I-T)(I-T) is a (strongly) accretive map.
Remark 7
- Let T,S:X rarr XT, S: X \rightarrow X, and f in Xf \in X be given. A fixed point for the map Tx=f+(I-S)x,AA x in XT x=f+(I-S) x, \forall x \in X is a solution for Sx=fS x=f.
- Let f in Xf \in X be a given point. If SS is an accretive map then T=f-ST=f-S is a strongly pseudocontractive map.
Consider Mann iteration with delay and set Tx=f+(I-S)xT x=f+(I-S) x to obtain
{:(10)u_(n+1)=(1-alpha_(n))u_(n-s)+alpha_(n)(f+(I-S)u_(n-s)):}\begin{equation*}
u_{n+1}=\left(1-\alpha_{n}\right) u_{n-s}+\alpha_{n}\left(f+(I-S) u_{n-s}\right) \tag{10}
\end{equation*}
Remarks 6 and 7 and Theorem 5 lead to the following results.
Corollary 8 Let XX be a real Banach space with a uniformly convex dual X^(**)X^{*}, and S:X rarr XS: X \rightarrow X a continuous and strongly accretive map with (I-S)(X)(I-S)(X) bounded, {alpha_(n)}\left\{\alpha_{n}\right\} satisfies (2), and u_(0)=x_(0)in Xu_{0}=x_{0} \in X, then, the Mann iteration with delay (10) converges to the solution of Sx=fS x=f.
Corollary 8 Let XX be a real Banach space with a uniformly convex dual X^(**)X^{*}, and S:X rarr XS: X \rightarrow X a continuous and strongly accretive map with (I-S)(X)(I-S)(X) bounded, {alpha_(n)}\left\{\alpha_{n}\right\} satisfies (2), and u_(0)=x_(0)in Xu_{0}=x_{0} \in X, then, the Mann iteration with delay (10) converges to the solution of Sx=fS x=f.
Let SS be an accretive operator. The operator Tx=f-SxT x=f-S x is strongly pseudocontractive, for a given f in Xf \in X. A solution for Tx=xT x=x becomes a solution for x+Sx=fx+S x=f. Consider Mann iteration with delay, set Tx:=f-SxT x:=f-S x such that
{:(11)u_(n+1)=(1-alpha_(n))u_(n-s)+alpha_(n)(f-Su_(n-s)):}\begin{equation*}
u_{n+1}=\left(1-\alpha_{n}\right) u_{n-s}+\alpha_{n}\left(f-S u_{n-s}\right) \tag{11}
\end{equation*}
Again, using the Remarks 6 and 7 and Theorem 5 we obtain the following result.
Corollary 9 Let XX be a real Banach space with a uniformly convex dual X^(**)X^{*}, and BB a nonempty, convex, closed subset of XX. Let S:B rarr BS: B \rightarrow B be a continuous and accretive operator with (I-S)(X)(I-S)(X) bounded, {alpha_(n)}\left\{\alpha_{n}\right\} satisfies (2). Then, the Mann iteration with delay (11) converges to the solution of x+Sx=fx+S x=f.
Corollary 9 Let XX be a real Banach space with a uniformly convex dual X^(**)X^{*}, and BB a nonempty, convex, closed subset of XX. Let S:B rarr BS: B \rightarrow B be a continuous and accretive operator with (I-S)(X)(I-S)(X) bounded, {alpha_(n)}\left\{\alpha_{n}\right\} satisfies (2). Then, the Mann iteration with delay (11) converges to the solution of x+Sx=fx+S x=f.
References
[1] C.E. Chidume, Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc., 120, 2, 1994, 545-551.
[2] K. Deimling, Zeros of accretive operators, Manucripta Math. 13 (1974), 283288.
[3] Z. Haiyun, J. Yuting, Approximation on fixed points on strongly psedocontractive maps without Lipschitz assumption, Proc. Amer. Math. Soc., 125, 6, 1997, 1705-1709.
[4] S. Ishikawa, Fixed Points by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147-150.
[5] W. R. Mann, Mean Value in Iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
[6] X. Weng, Fixed Point Iteration for Local Strictly Pseudocontractive Mapping, Proc. Amer. Math. Soc. 113 (1991), 727-731.
[2] K. Deimling, Zeros of accretive operators, Manucripta Math. 13 (1974), 283288.
[3] Z. Haiyun, J. Yuting, Approximation on fixed points on strongly psedocontractive maps without Lipschitz assumption, Proc. Amer. Math. Soc., 125, 6, 1997, 1705-1709.
[4] S. Ishikawa, Fixed Points by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147-150.
[5] W. R. Mann, Mean Value in Iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
[6] X. Weng, Fixed Point Iteration for Local Strictly Pseudocontractive Mapping, Proc. Amer. Math. Soc. 113 (1991), 727-731.