#### On the convergence of a generalized modified Krasnoselskii iterative process for generalized strictly pseudocontractive mappings in uniformly convex Banach spaces

Saddeek Fixed Point Theory and Applications
On the convergence of a generalized modified Krasnoselskii iterative process for generalized strictly pseudocontractive mappings in uniformly convex Banach spaces
Ali Mohamed Saddeek
This paper aims to study the strong convergence of generalized modified Krasnoselskii iterative process for finding the minimum norm solutions of certain nonlinear equations with generalized strictly pseudocontractive, demiclosed, coercive, bounded, and potential mappings in uniformly convex Banach spaces. An application to nonlinear pseudomonotone equations is provided. The results extend and improve recent work in this direction.
generalized modified Krasnoselskii iterative; generalized strictly pseudocontractive mappings; minimum norm solutions; uniformly convex Banach spaces
1 Introduction and preliminaries
Let H be a real Hilbert space with norm · H and inner product (·, ·). Let C be a nonempty
closed and convex subset of H. Let T be a nonlinear mapping of H into itself. Let I denote
the identity mapping on H. Denote by F(T ) the set of fixed points of T .
Moreover, the symbols
and → stand for weak and strong convergence, respectively.
We say that T is generalized Lipschitzian iff there exists a nonnegative real valued
function r(x, y) satisfying supx,y∈H {r(x, y)} = λ < ∞ such that
Tx – Ty H ≤ r(x, y) x – y H ,
∀x, y ∈ H.
(.)
Recently, this class of mappings has been studied by Saddeek and Ahmed [], and Saddeek
[].
We say that T is generalized strictly pseudocontractive iff for each pair of points x, y in
H there exist nonnegative real valued functions ri(x, y), i = , , satisfying
sup
x,y∈H i=
such that
ri(x, y) = λ < ∞
Tx – Ty H ≤ r(x, y) x – y H + r(x, y) (I – T )(x) – (I – T )(y) H .
(.)
By letting r(x, y) = and r(x, y) = λ ∈ [, ) (resp., ri(x, y) = , i = , ) in (.), we may derive
the class of λ-strictly pseudocontractive (resp., pseudocontractive) mappings, which is due
to Browder and Petryshyn [].
The class of λ-strictly pseudocontractive mappings has been studied recently by various
authors (see, for example, [–]).
It worth noting that the class of generalized strictly pseudocontractive mappings
includes generalized Lipschizian mappings, λ-strictly pseudocontractive mappings,
λ-Lipschitzian mappings, pseudocontractive mappings, nonexpansive (or -strictly
pseudocontractive) mappings.
These mappings appear in nonlinear analysis and its applications.
Definition . For any x, y, z ∈ H the mapping T is said to be
(i) demiclosed at (see, for example, []) if Tx = whenever {xn} ⊂ H with xn
and Txn → , as n → ∞;
(ii) pseudomonotone (see, for example, []) if it is bounded and xn
x ∈ H and
x
lim sup(Txn, xn – x) ≤
n→∞
⇒
lim inf(Txn, xn – y) ≥ (Tx, x – y);
n→∞
(iii) coercive (see, for example, []) if
(Tx, x) ≥ ρ x H
x H ,
lim ρ(ξ ) = +∞;
ξ→+∞
T t(x + y), x + y – T (tx), x dt =
T (x + ty), y dt;
(iv) potential (see, for example, []) if
(v) hemicontinuous (see, for example, []) if
lim T (x + ty), z = (Tx, z);
t→
(vi) demicontinuous (see, for example, []) if
lim
xn–x H →
(Txn, y) = (Tx, y);
(Tx – Ty, x – y) ≥ α x – y pH ;
(vii) uniformly monotone (see, for example, []) if there exist p ≥ , α > such that
(viii) bounded Lipschitz continuous (see, for example, []) if there exist p ≥ , M >
such that
Tx – Ty H ≤ M
x H + y H p– x – y H .
It should be noted that any demicontinuous mapping is hemicontinuous and every
uniformly monotone is monotone (i.e., (Tx – Ty, x – y) ≥ , ∀x, y ∈ H) and every monotone
hemicontinuous is pseudomonotone.
If T is uniformly monotone (resp. bounded Lipschitz continuous) with p = , then T is
called strongly monotone (resp. M-Lipschitzian).
For x ∈ C the Krasnoselskii iterative process (see, for example, []) starting at x is
defined by
xn+ = ( – τ )xn + τ Txn,
where τ ∈ (, ).
Recently, in a real Hilbert space setting, Saddeek and Ahmed [] proved that the
Krasnoselskii iterative sequence given by (.) converges weakly to a fixed point of T under
the basic assumptions that I – T is generalized Lipschitzian, demiclosed at , coercive,
bounded, and potential. Moreover, they also applied their result to the stationary
filtration problem with a discontinuous law.
However, the convergence in [] is in general not strong. Very recently, motivated and
inspired by the work in He and Zhu [], Saddeek [] introduced the following modified
Krasnoselskii iterative algorithm by the boundary method:
By replacing Tτ by T and taking h(xn) = , ∀n ≥ in (.), we can obtain (.).
Saddeek [] obtained some strong convergence theorems of the iterative algorithm (.)
for finding the minimum norm solutions of certain nonlinear operator equations.
The class of uniformly convex Banach spaces play an important role in both the
geometry of Banach spaces and relative topics in nonlinear functional analysis (see, for example,
[, ]).
Let X be a real Banach space with its dual X∗. Denote by ·, · the duality pairing between
X∗ and X. Let · X be a norm in X, and · X∗ be a norm in X∗.
A Banach space X is said to be strictly convex if x + y X < for every x, y ∈ X with
x X ≤ , y X ≤ and x = y.
A Banach space X is said to be uniformly convex if for every ε > , there exists an
increasing positive function δ(ε) with δ() = such that x X ≤ , y X ≤ with x – y X ≥ ε
imply x + y X ≤ ( – δ(ε)) for every x, y ∈ X.
It is well known that every Hilbert space is uniformly convex and every uniformly convex
Banach space is reflexive and strictly convex.
(.)
(.)
A Banach space X is said to have a Gateaux differentiable norm (see, for example, [],
p.) if for every x, y ∈ X with x X = , y X = the following limit exists:
lim [ x + ty X – x X ] .
t→+ t
X is said to have a uniformly Gateaux differentiable norm if for all y ∈ X with y X = , the
limit is attained uniformly for x X = .
Hilbert spaces, Lp (or lp) spaces, and Sobolev spaces Wp ( < p < ∞) are uniformly
convex and have a uniformly Gateaux differentiable norm.
The generalized duality mapping Jp, p > from X to X∗ is defined by
Jp(x) = x∗ ∈ X∗ : x∗, x = x pX , x∗ X∗ = x pX– ,
It is well known that (see, for example, [, ]) if the uniformly convex Banach space X is
a uniformly Gateaux differentiable norm, then Jp is single valued (we denote it by jp), one
to one and onto. In this case the inverse of jp will be denoted by jp–.
Definition . above can easily be stated for mappings T from C to X∗. The only change
here is that one replaces the inner product (·, ·) by the bilinear form ·, · .
Given a nonlinear mapping A of C into X∗. The variational inequality problem
associated with C and A is to find
x ∈ C : Ax – f , y – x ≥ ,
∀y ∈ C, f ∈ X∗.
(.)
The set of solutions of the variational inequality (.) is denoted by VI(C, A).
It is well known (see, for example, [, , ]) that if A is pseudomonotone and coercive,
then VI(C, A) is a nonempty, closed, and convex subset of X. Further, if A = jp – T , then
F˜ (jp, T ) = {x ∈ C : jpx = Tx} = A–. In addition, there exists also a unique element z =
projA–() ∈ VI(A–, jp), called the minimum norm solution of variational inequality (.)
(or the metric projection of the origin onto A–). If X = H, then jp = I and hence F˜ = F.
Example . Let be a bounded domain in Rn with Lipschitz continuous boundary.
Let us consider p ≥ , p + q = , and X = W˚ p()( ), X∗ = Wq(–)( ). The p-Laplacian is the
mapping – p : W˚ p()( ) → Wq(–)( ), pu = div(|∇u|p–∇u) for u ∈ W˚ p()( ).
It is well known that the p-Laplacian is in fact the generalized duality mapping jp (more
specifically, jp = – p), i.e., jpu, v = |∇u|p–(∇u, ∇u) dx, ∀u, v ∈ W˚ p()( ).
From [], p., we have
jpu – jpv, u – v =
|∇u|p–∇u – |∇v|p–∇v , ∇(u – v) dx
≥ M
|∇u – ∇v|p dx for some M > ,
which implies that jp is uniformly monotone.
By [], p., we have
jpu – jpv, w
≤ M u – v W˚ p()( )
u W˚ p()( ) + v W˚ p()( ) p– w W˚ p()( ),
or
jpu – jpv Wq(–)( ) =
sup
w∈W˚ p()( )
| jpu – jpv, w |
w W˚ p()( )
this shows that jp is bounded Lipschitz continuous.
The generalized duality mapping jp = – p is bounded, demicontinuous (and hence
hemicontinuous) and monotone, and hence jp is pseudomonotone.
From the definition of jp, it follows that jp is coercive.
Since jpu ∈ Wq(–)( ), ∀u ∈ W˚ p()( ) is the subgradient of p u pW˚ p()( ), it follows that jp is
potential.
Since jp is pseudomonotone and coercive (it is surjective), then jp is demiclosed at (see
Saddeek [] for an explanation).
The mapping jp is generalized strictly pseudocontractive with r(x, y) = .
The following two lemmas play an important role in the sequel.
Lemma . ([]) Let {an}, {bn}, and {cn} be nonnegative real sequences satisfying
n∞= γn = ∞, lim supn→∞ bγnn ≤ , and
n∞= cn < ∞. Then limn→∞ an = .
Lemma . ([]) Let X be a real uniformly convex Banach space with a uniformly
Gateaux differentiable norm, and let X∗ be its dual. Then, for all x∗, y∗ ∈ X∗, the following
inequality holds:
x∗ + y∗ X∗ ≤ x∗ X∗ + y∗, jp–x∗ – y , y ∈ X,
where jp– is the inverse of the duality mapping jp.
Let us now generalize the algorithm (.) for a pair of mappings as follows:
jpxn+ = – τ h(xn) jpxn + τ Tτjp xn,
where x = x ∈ C, τ ∈ (, ), Tτjp = ( – τ )jp + τ T , T : C → X∗ is a suitable mapping, and
jp : X → X∗ is the generalized duality mapping.
This algorithm can also be regarded as a modification of algorithm () in []. We shall
call this algorithm the generalized modified Krasnoselskii iterative algorithm.
In the case when X is uniformly convex Banach space, the generalized strictly
pseudocontractive mapping (.) can be written as follows:
Tx – Ty pX∗ ≤ r(x, y) jpx – jpy pX∗
+ r(x, y) (jp – T )(x) – (jp – T )(y) pX∗ , p ∈ [, ∞),
(.)
where r(x, y) and r(x, y) satisfy the same conditions as above.
Obviously, (.) and (.) reduce to (.) and (.), respectively, when X is a Hilbert space.
The main purpose of this paper is to extend the results in [] to uniformly convex Banach
spaces and to generalized modified iterative processes with generalized strictly
pseudocontractive mappings.
2 Main results
Now we are ready to state and prove the results of this paper.
Theorem . Let X be a real uniformly convex Banach space with a uniformly Gateaux
differentiable norm and X∗ be its dual. Let C be a nonempty closed convex subset of X. Let
jp : X → X∗ be the generalized duality mapping and let T : C → X∗ be a bounded Lipschitz
continuous nonlinear mapping. Define Sh(x) : C → X∗ by
Sh(x)x = h(x) + τ – jpx – τ Tx,
where the function h(x) is defined as above and τ ∈ (, ).
Assume that Sh(x) is demiclosed at , coercive, potential, bounded, and generalized strictly
pseudocontractive in the sense of (.), here ri = ri(x, y), i = , , satisfy the following
condition:
sup r + – h(x) pr = λ p < ∞,
x,y∈C
p ≥ .
Suppose that the constant α appearing in (.) is as follows:
α = sup
x,y∈C
x – y X + sup x X
x∈C
p–
x – y X–p,
p ≥ .
Then the iterative sequence {xn} generated by algorithm (.) with n∞= h(xn) = ∞ and
< τ = min{, λ M }, converges strongly to x¯ ∈ VI(Sh–(x¯), jp), x¯ = projSh–(x¯)(), where Sh–(x¯) =
jp
F˜(h(x¯)jp, Tτ ).
Proof First observe that {xn} is well defined because Sh(x) is bounded and λ < ∞. Next, we
show that the sequence {xn} is bounded. Since Sh(x) is coercive, it is sufficient (see proof of
Theorem . in []) to show that
(.)
(.)
{xn} ⊂ S,
xn X ≤ R,
p
Sh(xn) xn+ + t(xn – xn+) – Sh(xn)(xn) X∗
p
≤ r jp xn+ + t(xn – xn+) – jpxn X∗
where S = {x ∈ C : F(x) ≤ F(x)}, R = supx∈S x X , and F : X → (–∞, ∞] is a real
function defined as follows:
From the definition of S, it follows immediately that x ∈ S. Suppose, for n ≥ , that
xn ∈ S. We now claim that xn+ ∈ S. Indeed, from (.), the bounded Lipschitz continuity
of jp, T , and the definition of Sh(x), we obtain
+ r (jp – Sh(xn)) xn+ + t(xn – xn+) – (jp – Sh(xn))(xn) pX∗
p
≤ r jp xn+ + t(xn – xn+) – jp(xn) X∗
+ r – τ – h(xn) jp xn+ + t(xn – xn+) – jp(xn) X∗
+ τ T xn+ + t(xn – xn+) – T(xn) X∗ p
≤ ( – t)pMp r + – h(xn) pr
× xn+ + t(xn – xn+) X + xn X p(p–) xn – xn+ pX
= ( – t)pMp r + – h(xn) pr
× xn+ + t(xn – xn+) X – xn X + xn X p(p–) xn – xn+ pX
≤ ( – t)pMp r + – h(xn) pr
× ( – t) xn – xn+ X + xn X p(p–) xn – xn+ pX
≤ Mp r + – h(xn) pr
× xn – xn+ X + R p(p–) xn – xn+ pX for t ∈ [,].
Hence
This implies that
Sh(xn) xn+ + t(xn – xn+) – Sh(xn)(xn) pX∗
≤ Mλ xn – xn+ X + R p– xn – xn+ X.
(.)
Sh(xn) xn+ + t(xn – xn+) – Sh(xn)(xn),xn – xn+
≤ Mλ xn – xn+ X + R p– xn – xn+ X.
≥ –Mλ xn – xn+ X + R p– xn – xn+ X + ατ xn – xn+ pX,
which together with the restriction on α implies that
F(xn) – F(xn+) ≥ μ xn – xn+ X + R p– xn – xn+ X ,
μ = τ – Mλ > .
(.)
Therefore, F(xn+) ≤ F(xn) ≤ F(x), which implies that xn+ ∈ S. Thus, by mathematical
induction we get xn ∈ S for all n ≥ . This shows that xn is bounded. This, together with
the definition of jp, the boundedness of Sh(xn), and (.), (.), implies that the sequences
{Sh(xn)(xn)}, {jp(xn)}, {Tτjp (xn)}, and {F(xn)} are also bounded.
Further, it follows from (.) that the sequence {F(xn)} is monotonically decreasing and
therefore convergent. Consequently, from (.), we have
(.)
(.)
(.)
(.)
(.)
(.)
(.)
(.)
lim
n→∞
xn – xn+ X = .
lim jpxn – jpxn+ X∗ = .
n→∞
lim
n→∞
lim
k→∞
Sh(xn)xn X∗ = .
xnk – x¯ X → σx¯ .
Hence, by the bounded Lipschitz continuity of jp, we obtain
Therefore, by (.) and the definition of Sh(x), we then have
Let x¯ be a weak limit point of {xn}, then there exists a subsequence {xnk } of {xn} such that
Since Sh(x) is demiclosed at , it follows from (.) and (.) that Sh(x¯)x¯ = , and hence
x¯ ∈ Sh–(x¯).
Now, we show that
lim sup Sh(xn)xn, xn+ – x˜ ≤ ,
n→∞
∀x˜ ∈ Sh–(x˜).
Using (.) and the definition of Sh(x), we get
Sh(xn)(xn), xn+ – x˜ ≤ Sh(xn)(xn), xn+ – xn + τ – jp(xn) – jp(xn+), xn – x˜
≤ Sh(xn)(xn) X∗ xn – xn+ X
+ τ – jpxn – jpxn+ X∗ xn – x˜ X .
Taking the lim sup as n → ∞ in (.) and using (.), (.), and (.) yield the desired
inequality (.).
Now, let us show that
lim sup –jp(x¯), xn+ – x¯ ≤ ,
n→∞
where x¯ is the metric projection of the origin onto Sh–(x¯).
Let {xnk } be a subsequence of {xn} such that xnk+ → x˜ ∈ Sh–(x˜) and
lim sup –jp(x¯), xn+ – x¯ = lim sup –jp(x¯), xnk+ – x¯ .
n→∞ k→∞
It follows from Kato [] that
lim sup –jp(x¯), xn+ – x¯ = –jp(x¯), x˜ – x¯ ≤ .
n→∞
This proves the desired inequality (.), and, hence by (.) and (.), we obtain
x¯ ∈ Sh–(x¯) ∩ VI Sh–(x¯), jp .
Now, we prove that jpxn → jpx¯ as n → ∞.
By using (.) and Lemma ., we get
jpxn+ – jpx¯ X∗ =
jp
– τ h(xn) (jpxn – jpx¯) + τ Tτ xn – h(xn)jpx¯ X∗
≤ – τ h(xn) jpxn – jpx¯ X∗ + τ Tτjp xn – h(xn)jpx¯, xn+ – x¯
≤ – τ h(xn) jpxn – jpx¯ X∗ + τ Tτjp xn – h(xn)jpxn, xn+ – x¯
+ h(xn) –jp(x¯), xn+ – x¯ + τ h(xn) jp(xn) X∗ xn+ – x¯ X .
(.)
(.)
(.)
(.)
(.)
(.)
jp
Set γn = τ h(xn)( – τ h(xn)), an = jpxn – jpx¯ X∗ , bn = τ [ Tτ xn – h(xn)jpxn, xn+ – x¯ +
h(xn) –jp(x¯), xn+ – x¯ ], and cn = τ h(xn) jp(xn) X∗ xn+ – x¯ X .
Then inequality (.) becomes
From n∞= h(xn) = ∞, and (.), it follows that n∞= γn = ∞, lim supn→∞ bγnn ≤ , and
n∞= cn < ∞. Consequently, applying Lemma . to (.), we conclude that
lim jpxn – jpx¯ X∗ = .
n→∞
Finally, we show that xn → x¯ as n → ∞.
From the uniform monotonicity of jp, we have
xn – x¯ pX ≤ α jpxn – jpx¯, xn – x¯
≤ α jpxn – jpx¯ X∗ xn X + x¯ X .
Letting n → ∞ in (.) and using (.) and the boundedness of {xn}, we obtain xn → x¯,
as n → ∞. This completes the proof.
An immediate consequence of Theorem . is the following corollary.
Corollary . Let X = H be a real Hilbert space, and let C be a nonempty closed convex
subset of H. Let T : C → H be an M-Lipschitzian mapping. Define Sˆh(x) : C → H by
Sˆh(x)x = h(x)x – Tτ x,
where Tτ = ( – τ )I + τ T . Let Sˆh(x), τ , and h(x) be as in Theorem . and p = (i.e., jp = I,
α = , and supx,y∈C[r + ( – h(x))r] = (λ ) < ∞). Then the sequence {xn} defined by
Corollary . Except for the M-Lipschitzian condition for the mapping T , let all the other
assumptions of Corollary . be satisfied and r = . Then the sequence {xn} defined by
(.) with n∞= h(xn) = ∞, < τ = min{, λ }, and supx,y∈C[r(x, y)] = (λ) < ∞, converges
strongly to x¯ = projSh–(x¯)().
Remark . All conditions imposed in Theorem . on the mapping Sh(x) are
quintessential to prove the main theorem, more precisely for the existence solution of Sh(x)x = , and
to ensure the strong convergence of the generalized modified Krasnoselski iterative
algorithm.
3 Application to nonlinear pseudomonotone equations
In this section, we study nonlinear equations for pseudomonotone mappings; that is; we
seek x ∈ C such that
(.)
(.)
(.)
Ax = f , f ∈ X∗,
where A : C → X∗ is a nonlinear pseudomonotone mapping.
To ensure the existence of solutions of (.), we shall assume that A is pseudomonotone
and coercive on W˚ p()( ) ( < p < ∞) (see, for example, []). Such nonlinear equations
occur, in particular, in descriptions of a stabilized filtration and in problems of finding the
equilibria of soft shells (see, for example, [, ]).
Theorem . Besides the assumptions on A, let A be potential and satisfy the following
condition:
Ax – Ay X∗ ≤ jpx – jpy X∗ ,
∀x, y ∈ C.
Then the sequence {xn} generated by x = x ∈ C,
jpxn+ = jpxn – τ A(xn) – f ,
where < τ = min{, M }, converges strongly to the minimum norm solution of equation (.),
provided that n∞= h(xn) = ∞.
that sh(x) is potential.
of Sh(x), we get
Proof Define Sh(x) : C → X∗ by Sh(x)x = Ax – f , ∀x ∈ C. Since (.) has at least one solution,
then Sh–(x) = φ. On the other hand, condition (.) with the bounded Lipschitz continuity
of jp clearly imply that A is bounded Lipschitz continuous and the potentiality of jp imply
Now, we show that condition (.) is implied by (.). Indeed by (.) and the definition
Sh(x)x – Sh(x)y pX∗ = Ax – Ay pX∗ ≤ jpx – jpy pX∗ .
Hence Sh(x) satisfies condition (.) with r(x, y) = , r(x, y) = , and λ = .
Finally, the pseudomonotonicity of A implies that Sh(x) is demiclosed at can be proved
by proceeding as in the proof of Theorem . of []. Now we apply Theorem . to yield
the desired result.
Remark . If we set X = H (i.e., jp = I and p = ), then the condition (.) reduces to the
M-Lipschitzian condition of the operator A. Hence from Theorem . we obtain
Theorem . of [], which in turn is a generalization of Theorem of [].
4 Conclusion
In this work, we introduce a generalized modified Krasnoselskii iterative process
involving a pair of a generalized strictly pseudocontractive mapping and a generalized duality
mapping and prove some strong convergence theorems of the proposed iterative process
to the minimum norm solutions of certain nonlinear equations in the framework of
uniformly convex Banach spaces. These results improve and generalize recent work in this
direction.
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author have read and approved the final manuscript.
Acknowledgements
The author would like to thank the editor and the reviewers for their valuable suggestions and comments.
1. Saddeek , AM , Ahmed, SA : Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces . Arch. Math. 44 , 273 - 281 ( 2008 )
2. Saddeek , AM: A strong convergence theorem for a modified Krasnoselskii iteration method and its application to seepage theory in Hilbert spaces . J. Egypt. Math. Soc . 22 , 476 - 480 ( 2014 )
3. Browder , FE , Petryshyn, WV : Construction of fixed points of nonlinear mappings in Hilbert spaces . J. Math. Anal. Appl . 20 , 197 - 228 ( 1967 )
4. Marino , G , Xu, HK : Weak and strong convergence theorems for k-strict pseudocontractions in Hilbert spaces . J. Math. Anal. Appl . 329 , 336 - 349 ( 2007 )
5. Zhou , H: Convergence theorems of fixed points for k-strict pseudocontractions in Hilbert spaces . Nonlinear Anal . 69 , 456 - 462 ( 2008 )
6. Enyi , CD , Iyiola, OS: A new iterative scheme for common solution of equilibrium problems, variational inequalities and fixed point of k-strictly pseudocontractive mappings in Hilbert spaces . Br. J. Math. Comput. Sci. 4 ( 4 ), 512 - 527 ( 2014 )
7. Hao , Y: A strong convergence theorem on generalized equilibrium problems and strictly pseudocontractive mappings . Proc. Est. Acad. Sci . 60 ( 1 ), 12 - 24 ( 2011 )
8. He , Z: Strong convergence of the new modified composite iterative method for strict pseudocontractions in Hilbert spaces . Note Mat . 31 ( 2 ), 67 - 78 ( 2011 )
9. Li , M , Yao, Y: Strong convergence of an iterative algorithm for λ-strictly pseudocontractive mappings in Hilbert spaces. An. S¸tiint¸ . Univ. 'Ovidius' Constant¸a 18 ( 1 ), 219 - 228 ( 2010 )
10. Goebel , K , Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge ( 1990 )
11. Zeidler , E: Nonlinear Functional Analysis and Its Applications . II/B. Nonlinear Monotone Operators. Springer, Berlin ( 1990 )
12. Lions , JL: Quelques Methodes de Resolution des Problemes aux Limites Nonlineaires. Dunod/Gauthier-Villars, Paris ( 1969 )
13. Gajewski , H , Groger, K , Zacharias, K : Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie Verlag, Berlin ( 1974 )
14. Krasnoselskii , MA: Two observations about the method of successive approximations . Usp. Mat. Nauk 10 , 123 - 127 ( 1955 )
15. He , S , Zhu, W: A modified Mann iteration by boundary point method for finding minimum-norm fixed point of nonexpansive mappings . Abstr. Appl. Anal . 2013 , Article ID 768595 ( 2013 )
16. Benyamini , Y , Lindenstrauss, J : Geometric Nonlinear Functional Analysis , vol. 1. Amer. Math. Soc. Colloq. Publ. , vol. 48 . Am. Math. Soc. , Providence ( 2000 )
17. Diestel , J : Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, vol. 92 . Springer, New York ( 1984 )
18. Reich, S: On the asymptotic behavior of nonlinear semigroups and the range of accretive operators . J. Math. Anal. Appl . 79 ( 1 ), 123 - 126 ( 1981 )
19. Takahashi , W: Nonlinear Functional Analysis - Fixed Point Theory and Its Applications . Yokohama Publishers, Yokohama ( 2000 )
20. Alber , YI: Metric and generalized projection operators in Banach spaces . In: Kartsatos, A (ed.) Properties and Applications: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type , pp. 15 - 50 . Dekker, New York ( 1996 )
21. Kato , T: Nonlinear semigroups and evolution equations . J. Math. Soc. Jpn . 19 , 508 - 520 ( 1957 )
22. Ciarlet , P: The Finite Element Method for Elliptic Problems . North-Holland, New York ( 1978 )
23. Liu, LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces . J. Math. Anal. Appl . 194 , 114 - 125 ( 1995 )
24. Cioranescu , I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Mathematics and Its Applications , vol. 62 . Kluwer Academic, Dordrecht ( 1990 )
25. Badriev , IB , Shagidullin, RR : Investigating one dimensional equations of the soft shell statistical condition and algorithm of their solution . Izv . Vysš. Ucˇebn. Zaved., Mat. 6 , 8 - 16 ( 1992 )
26. Lapin , AV: On the research of some problems of nonlinear filtration theory . Ž. Vycˇisl. Mat. Mat. Fiz . 19 ( 3 ), 689 - 700 ( 1979 )