On the convergence of a generalized modified Krasnoselskii iterative ...

Saddeek Fixed Point Theory and Applications (2016) 2016:60 DOI 10.1186/s13663-016-0549-9

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On the convergence of a generalized modified Krasnoselskii iterative process for generalized strictly pseudocontractive mappings in uniformly convex Banach spaces Ali Mohamed Saddeek* *

Correspondence: [email protected] Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt

Abstract 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. MSC: 47H05; 47H10 Keywords: 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 – TyH ≤ r(x, y)x – yH ,

∀x, y ∈ H.

(.)

Recently, this class of mappings has been studied by Saddeek and Ahmed [], and Saddeek []. For r(x, y) = λ ∈ (, ) (resp., r(x, y) = ) such mappings are said to be λ-contractive (resp., nonexpansive) mappings. If r(x, y) = λ > , then the class of generalized Lipschitzian mappings coincide with the class of λ-Lipschitzian mappings. © 2016 Saddeek. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Saddeek Fixed Point Theory and Applications (2016) 2016:60

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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



ri (x, y) = λ < ∞

i=

such that  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 x and Txn → , as n → ∞; (ii) pseudomonotone (see, for example, []) if it is bounded and xn x ∈ H and lim sup(Txn , xn – x) ≤  n→∞

⇒

lim inf(Txn , xn – y) ≥ (Tx, x – y); n→∞

(iii) coercive (see, for example, []) if (Tx, x) ≥ ρ xH xH ,

lim ρ(ξ ) = +∞;

ξ →+∞

(iv) potential (see, for example, []) if



T t(x + y), x + y – T(tx), x dt =





T(x + ty), y dt;



(v) hemicontinuous (see, for example, []) if lim T(x + ty), z = (Tx, z);

t→

(vi) demicontinuous (see, for example, []) if lim

xn –xH →

(Txn , y) = (Tx, y);

(vii) uniformly monotone (see, for example, []) if there exist p ≥ , α >  such that p

(Tx – Ty, x – y) ≥ αx – yH ;


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(viii) bounded Lipschitz continuous (see, for example, []) if there exist p ≥ , M >  such that p– Tx – TyH ≤ M xH + yH x – yH . 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 ,

n ≥ ,

(.)

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: xn+ =  – τ h(xn ) xn + τ Tτ xn ,

n ≥ ,

(.)

where x = x ∈ C, τ ∈ (, ), Tτ = ( – τ )I + τ T and h :→ [, ] is a function defined by

h(x) = inf α ∈ [, ] : αx ∈ C ,

∀x ∈ C.

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 + yX <  for every x, y ∈ X with xX ≤ , yX ≤  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 xX ≤ , yX ≤  with x – yX ≥ ε imply x + yX ≤ ( – 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.


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A Banach space X is said to have a Gateaux differentiable norm (see, for example, [], p.) if for every x, y ∈ X with xX = , yX =  the following limit exists: lim+

t→

[x + tyX – xX ] . t

X is said to have a uniformly Gateaux differentiable norm if for all y ∈ X with yX = , the limit is attained uniformly for xX = . 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 p p–

Jp (x) = x∗ ∈ X ∗ : x∗ , x = xX , x∗ X ∗ = xX ,

∀x ∈ X.

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 ˜ p , T) = {x ∈ C : jp x = Tx} = A– . In addition, there exists also a unique element z = F(j 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. ˚ p() ( ), X ∗ = Wq(–) ( ). The p-Laplacian is the Let us consider p ≥ , p + q = , and X = W ˚ p() ( ) → Wq(–) ( ), p u = div(|∇u|p– ∇u) for u ∈ W ˚ p() ( ). mapping –p : W It is well known that the p-Laplacian is in fact the generalized duality mapping jp (more

˚ p() ( ). specifically, jp = –p ), i.e., jp u, v = |∇u|p– (∇u, ∇u) dx, ∀u, v ∈ W From [], p., we have

jp u – jp v, 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 p– jp u – jp v, w ≤ Mu – v () wW˚ () ( ) , ˚ ( ) uW ˚ () ( ) + vW ˚ () ( ) W p

p

p

p


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or jp u – jp vW (–) ( ) = q

sup ˚ p() ( ) w∈W

|jp u – jp v, w| wW˚ () ( ) p

≤ Mu – vW˚ () ( ) uW˚ () ( ) + vW˚ () ( ) p

p

p

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. ˚ p() ( ) is the subgradient of  up () , it follows that jp is Since jp u ∈ Wq(–) ( ), ∀u ∈ W p ˚ Wp ( )

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 an+ ≤ ( – γn )an + bn + cn , where γn ⊂ (, ),

∞

n= γn

∀n ≥ ,

= ∞, lim supn→∞

bn γn

≤ , 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∗  ∗ +  y∗ , j– x∗ – y , p X X

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: jp jp xn+ =  – τ h(xn ) jp xn + τ Tτ xn ,

n ≥ ,

(.)

j

where x = x ∈ C, τ ∈ (, ), Tτp = ( – 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: p

p

Tx – TyX ∗ ≤ r (x, y)jp x – jp yX ∗ p + r (x, y)(jp – T)(x) – (jp – T)(y)X ∗ ,

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.


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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 –  jp x – τ Tx,

∀x ∈ C,

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: p p sup r +  – h(x) r = λ < ∞,

p ≥ .

x,y∈C

Suppose that the constant α appearing in (.) is as follows: p– –p x – yX , α = sup x – yX +  sup xX x,y∈C

p ≥ .

x∈C

Then the iterative sequence {xn } generated by algorithm (.) with ∞ n= h(xn ) = ∞ and – – ¯ , j ), x = proj  < τ = min{, λM }, converges strongly to x¯ ∈ VI(Sh(¯ – p S  (), where Sh(¯x)  = x) h(¯x)

˜ x)jp , Tτjp ). F(h(¯

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 ,

n ≥ ,

(.)

where S = {x ∈ C : F(x) ≤ F(x )}, R = supx∈S xX , and F : X → (–∞, ∞] is a real function defined as follows:  Sh(x) (tx), x dt, ∀x ∈ X. (.) F(x) = 

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 Sh(x ) xn+ + t(xn – xn+ ) – Sh(x ) (xn )p ∗ n n X p ≤ r jp xn+ + t(xn – xn+ ) – jp xn X ∗


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p + r (jp – Sh(xn ) ) xn+ + t(xn – xn+ ) – (jp – Sh(xn ) )(xn )X ∗ p ≤ r jp xn+ + t(xn – xn+ ) – jp (xn )X ∗ + r  – τ – h(xn ) jp xn+ + t(xn – xn+ ) – jp (xn )X ∗ p + τ T xn+ + t(xn – xn+ ) – T(xn )X ∗ p ≤ ( – t)p Mp r +  – h(xn ) r p(p–) p xn – xn+ X × xn+ + t(xn – xn+ )X + xn X p = ( – t)p Mp r +  – h(xn ) r p(p–) p × xn+ + t(xn – xn+ )X – xn X + xn X xn – xn+ X p ≤ ( – t)p Mp r +  – h(xn ) r p(p–) p xn – xn+ X × ( – t)xn – xn+ X + xn X p ≤ Mp r +  – h(xn ) r p(p–) p xn – xn+ X for t ∈ [, ]. × xn – xn+ X + R Hence Sh(x ) xn+ + t(xn – xn+ ) – Sh(x ) (xn )p ∗ n n X p– ≤ Mλ xn – xn+ X + R xn – xn+ X .

(.)

This implies that Sh(x ) xn+ + t(xn – xn+ ) – Sh(x ) (xn ), xn – xn+ n n p– xn – xn+ X . ≤ Mλ xn – xn+ X + R

(.)

Since Sh(x) is potential and jp is uniformly monotone, by (.), (.), and (.) it follows that



Sh(xn ) (txn ), xn – Sh(xn ) (txn+ ), xn+ dt

F(xn ) – F(xn+ ) = 

Sh(xn ) xn+ + t(xn – xn+ ) , xn – xn+ dt



= 



Sh(xn ) xn+ + t(xn – xn+ ) , xn – xn+ dt

= 



Sh(xn ) (xn ), xn – xn+ dt + Sh(xn ) (xn ), xn – xn+

–

 

Sh(x

≥–

n)

xn+ + t(xn – xn+ ) – Sh(xn ) (xn ), xn – xn+ dt



+ Sh(xn ) (xn ), xn – xn+ p– xn – xn+ X ≥ –Mλ xn – xn+ X + R  + jp xn – jp xn+ , xn – xn+ τ p– α p ≥ –Mλ xn – xn+ X + R xn – xn+ X + xn – xn+ X , τ


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which together with the restriction on α implies that p– xn – xn+ X , F(xn ) – F(xn+ ) ≥ μ xn – xn+ X + R

μ=

 – Mλ > . τ

(.)

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 jp {Sh(xn ) (xn )}, {jp (xn )}, {Tτ (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 xn – xn+ X = .

(.)

n→∞

Hence, by the bounded Lipschitz continuity of jp , we obtain lim jp xn – jp xn+ X ∗ = .

(.)

n→∞

Therefore, by (.) and the definition of Sh(x) , we then have lim Sh(xn ) xn X ∗ = .

(.)

n→∞

Let x¯ be a weak limit point of {xn }, then there exists a subsequence {xnk } of {xn } such that lim xnk – x¯ X → σx¯ .

(.)

k→∞

Since Sh(x) is demiclosed at , it follows from (.) and (.) that Sh(¯x) x¯ = , and hence – x¯ ∈ Sh(¯ x) .

(.)

Now, we show that lim supSh(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 – jp xn – jp xn+ 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) .

(.)


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– 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 jp xn → jp x¯ as n → ∞. By using (.) and Lemma ., we get j  jp xn+ – jp x¯ X ∗ =  – τ h(xn ) (jp xn – jp x¯ ) + τ Tτp xn – h(xn )jp x¯ X ∗  j ≤  – τ h(xn ) jp xn – jp x¯ X ∗ + τ Tτp xn – h(xn )jp x¯ , xn+ – x¯  jp ≤  – τ h(xn ) jp xn – jp x¯ X ∗ + τ Tτ xn – h(xn )jp xn , 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 = jp xn – jp x¯ X ∗ , bn = τ [Tτ xn – h(xn )jp xn , xn+ – x¯ + h(xn )–jp (¯x), xn+ – x¯ ], and cn = τ h(xn )jp (xn )X ∗ xn+ – x¯ X . Then inequality (.) becomes an+ ≤ ( – γn )an + bn + cn ,

∀n ≥ .

= ∞, and (.), it follows that ∞ n= γn = ∞, lim supn→∞ n= cn < ∞. Consequently, applying Lemma . to (.), we conclude that

From ∞

∞

n= h(xn )

lim jp xn – jp x¯ X ∗ = .

n→∞

(.) bn γn

≤ , and

(.)

Finally, we show that xn → x¯ as n → ∞. From the uniform monotonicity of jp , we have p

 jp xn – jp x¯ , xn – x¯ α  ≤ jp xn – jp x¯ X ∗ xn X + ¯xX . α

xn – 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 Sˆ h(x) : C → H by Sˆ h(x) x = h(x)x – Tτ x,

∀x ∈ C,


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where Tτ = ( – τ )I + τ T. Let Sˆ h(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 xn+ =  – τ h(xn ) xn + τ Tτ xn , with

∞

n= h(xn )

n ≥ ,

(.)

– – ¯ = projS–  (), where Sˆ h(¯ = ∞ converges strongly to x¯ ∈ VI(Sˆ h(¯ x) , I), x x)  = h(¯x)

˜ x)I, Tτ ). F(h(¯

A special case of Corollary . is the following theorem due to Saddeek [], who proved it under the condition that T is generalized Lipschitzian, which in turn, is a generalization of Theorem  of Saddeek and Ahmed []. 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¯ = projS–  (). h(¯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 ˚ p() ( ) ( < p < ∞) (see, for example, []). Such nonlinear equations and coercive on W 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 – AyX ∗ ≤ jp x – jp yX ∗ ,

∀x, y ∈ C.

(.)

Then the sequence {xn } generated by x = x ∈ C, jp xn+ = jp xn – τ A(xn ) – f ,

n ≥ ,

(.)

where  < τ = min{, M }, converges strongly to the minimum norm solution of equation (.), provided that ∞ n= h(xn ) = ∞.


Page 11 of 12

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 that sh(x) is potential. Now, we show that condition (.) is implied by (.). Indeed by (.) and the definition of Sh(x) , we get p

p

p

Sh(x) x – Sh(x) yX ∗ = Ax – AyX ∗ ≤ jp x – jp yX ∗ . 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. Received: 25 August 2015 Accepted: 25 April 2016 References 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¸ tiin¸t. Univ. ‘Ovidius’ Constan¸ta 18(1), 219-228 (2010) 10. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)


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