Abstract
Let E be a real normed space with dual space \(E^{*}\) and let \(A:E\rightarrow2^{E^{*}}\) be any map. Let \(J:E\rightarrow2^{E^{*}}\) be the normalized duality map on E. A new class of mappings, Jpseudocontractive maps, is introduced and the notion of Jfixed points is used to prove that \(T:=(JA)\) is Jpseudocontractive if and only if A is monotone. In the case that E is a uniformly convex and uniformly smooth real Banach space with dual \(E^{*}\), \(T: E\rightarrow2^{E^{*}}\) is a bounded Jpseudocontractive map with a nonempty Jfixed point set, and \(JT :E\rightarrow2^{E^{*}}\) is maximal monotone, a sequence is constructed which converges strongly to a Jfixed point of T. As an immediate consequence of this result, an analog of a recent important result of Chidume for bounded maccretive maps is obtained in the case that \(A:E\rightarrow2^{E^{*}}\) is bounded maximal monotone, a result which complements the proximal point algorithm of Martinet and Rockafellar. Furthermore, this analog is applied to approximate solutions of Hammerstein integral equations and is also applied to convex optimization problems. Finally, the techniques of the proofs are of independent interest.
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1 Introduction
Let H be a real inner product space. A map \(A:H\rightarrow2^{H}\) is called monotone if for each \(x,y\in H\),
Monotone mappings were first studied in Hilbert spaces by Zarantonello [1], Minty [2], Kačurovskii [3] and a host of other authors. Interest in such mappings stems mainly from their usefulness in applications. In particular, monotone mappings appear in convex optimization theory. Consider, for example, the following:. Let \(g:H\rightarrow\mathbb{R}\cup\{\infty\}\) be a proper convex function. The subdifferential of g, \(\partial g:H\rightarrow2^{H}\), is defined for each \(x\in H\) by
It is easy to check that ∂g is a monotone operator on H, and that \(0\in\partial g(u)\) if and only if u is a minimizer of g. Setting \(\partial g\equiv A\), it follows that solving the inclusion \(0\in Au\), in this case, is solving for a minimizer of g.
Furthermore, the equation \(0\in Au\) when A is a monotone map from a real Hilbert space to itself also appears in evolution systems. Consider the evolution equation \(\frac{du}{dt} + Au=0\) where A is a monotone map from a real Hilbert space to itself. At an equilibrium state, \(\frac{du}{dt}=0\) so that \(Au=0\), whose solutions correspond to the equilibrium state of the dynamical system.
In particular, consider the following diffusion equation:
where Ω is an open subset of \({\mathbb{R}}^{n}\).
By a simple transformation, i.e., by setting \(v(t)=u(t,\cdot)\), where
is defined by \(v(t)(x)=u(t,x)\) and \(f(\varphi)(x)=g(\varphi(x))\), where
we see that equation (1.2) is equivalent to
where A is a nonlinear monotonetype mapping defined on \(L_{2}(\Omega )\). Setting f to be identically zero, at an equilibrium state (i.e., when the system becomes independent of time) we see that equation (1.3) reduces to
Thus, approximating zeros of equation (1.4) is equivalent to the approximation of solutions of the diffusion equation (1.2) at equilibrium state.
The notion of monotone mapping has been extended to real normed spaces. We now briefly examine two wellstudied extensions of Hilbert space monotonicity to arbitrary normed spaces.
1.1 Accretivetype mappings
Let E be a real normed space with dual space \(E^{*}\). A map \(J:E\rightarrow2^{E^{*}}\) defined by
is called the normalized duality map on E. We have with \(J^{1}=J^{*}\), \(JJ^{*}=I_{E^{*}}\) and \(J^{*}J =I_{E}\), where \(I_{E}\) and \(I_{E^{*}}\) are the identity mappings on E and \(E^{*}\), respectively.
A map \(A:E\rightarrow2^{E}\) is called accretive if for each \(x,y\in E\), there exists \(j(xy)\in J(xy)\) such that
A is called maccretive if, in addition, the graph of A is not properly contained in the graph of any other accretive operator. It is maccretive if and only if A is accretive and \(R(I+tA)=E\) for all \(t>0\).
In a Hilbert space, the normalized duality map is the identity map, and so, in this case, inequality (1.5) and inequality (1.1) coincide. Hence, accretivity is one extension of Hilbert space monotonicity to general normed spaces.
Accretive operators have been studied extensively by numerous mathematicians (see, e.g., the following monographs: Berinde [4], Browder [5], Chidume [6], Reich [7], and the references therein).
1.2 Monotonetype mappings in arbitrary normed spaces
Let E be a real normed space with dual \(E^{*}\). A map \(A:E\rightarrow 2^{E^{*}}\) is called monotone if for each \(x,y\in E\), the following inequality holds:
It is called maximal monotone if, in addition, the graph of A is not properly contained in the graph of any other monotone operator. Also, A is maximal monotone if and only if it is monotone and \(R(J+tA)=E^{*}\) for all \(t>0\).
It is obvious that monotonicity of a map defined from a normed space to its dual is another extension of Hilbert space monotonicity to general normed spaces.
The extension of the monotonicity condition from a Banach space into its dual has been the starting point for the development of nonlinear functional analysis…. The monotone mappings appear in a rather wide variety of contexts, since they can be found in many functional equations. Many of them appear also in calculus of variations, as subdifferential of convex functions (Pascali and Sburian [8], p.101).
Accretive mappings were introduced independently in 1967 by Browder [5] and Kato [9]. Interest in such mappings stems mainly from their firm connection with the existence theory for nonlinear equations of evolution in real Banach spaces. It is known (see, e.g., Zeidler [10]) that many physically significant problems can be modeled in terms of an initialvalue problem of the form
where A is a multivalued accretive map on an appropriate real Banach space. Typical examples of such evolution equations are found in models involving the heat, wave or Schrödinger equations (see, e.g., Browder [11], Zeidler [10]). Observe that in the model (1.7), if the solution u is independent of time (i.e., at the equilibrium state of the system), then \(\frac{du}{dt} = {0}\) and (1.7) reduces to
whose solutions then correspond to the equilibrium state of the system described by (1.7). Solutions of equation (1.8) can also represent solutions of partial differential equations (see, e.g., Benilan et al. [12], Khatibzadeh and Moroşanu [13], Khatibzadeh and Shokri [14], Showalter [15], Volpert [16], and so on).
In studying the equation \(0\in Au\), where A is a multivalued accretive operator on a Hilbert space H, Browder introduced an operator T defined by \(T:= IA\) where I is the identity map on H. He called such an operator pseudocontractive. It is clear that solutions of \(0\in Au\), if they exist, correspond to fixed points of T.
Within the past 35 years or so, methods for approximating solutions of equation (1.8) when A is an accretivetype operator have become a flourishing area of research for numerous mathematicians. Numerous convergence theorems have been published in various Banach spaces and under various continuity assumptions. Many important results have been proved, thanks to geometric properties of Banach spaces developed from the mid1980s to the early 1990s. The theory of approximation of solutions of the equation when A is of the accretivetype reached a level of maturity appropriate for an examination of its central themes. This resulted in the publication of several monographs which presented indepth coverage of the main ideas, concepts, and most important results on iterative algorithms for appropriation of fixed points of nonexpansive and pseudocontractive mappings and their generalizations, approximation of zeros of accretivetype operators; iterative algorithms for solutions of Hammerstein integral equations involving accretivetype mappings; iterative approximation of common fixed points (and common zeros) of families of these mappings; solutions of equilibrium problems; and so on (see, e.g., Agarwal et al. [17]; Berinde [4]; Chidume [6]; Reich [18]; Censor and Reich [19]; William and Shahzad [20], and the references therein). Typical of the results proved for solutions of equation (1.8) is the following theorem.
Theorem 1.1
(Chidume [21])
Let E be a uniformly smooth real Banach space with modulus of smoothness \(\rho_{E}\), and let \(A:E\rightarrow2^{E}\) be a multivalued bounded maccretive operator with \(D(A)=E\) such that the inclusion \(0\in Au\) has a solution. For arbitrary \(x_{1}\in E\), define a sequence \(\{x_{n}\}\) by
where \(\{\lambda_{n}\}\) and \(\{\theta_{n}\}\) are sequences in \((0,1)\) satisfying the following conditions:

(i)
\(\lim_{n\rightarrow\infty}\theta_{n} =0\), \(\{\theta_{n}\}\) is decreasing;

(ii)
\(\sum\lambda_{n}\theta_{n} = \infty\); \(\sum\rho_{E}(\lambda_{n}M_{1})<\infty\), for some constant \(M_{1} > 0\);

(iii)
\(\lim_{n\rightarrow\infty} \frac{ [\frac{\theta _{n1}}{\theta _{n}}1 ]}{\lambda_{n}\theta_{n}}=0\).
There exists a constant \(\gamma_{0} > 0\) such that \(\frac{\rho _{E}(\lambda_{n})}{\lambda_{n}}\leq\gamma_{0}\theta_{n}\). Then the sequence \(\{x_{n}\}\) converges strongly to a zero of A.
Unfortunately, developing algorithms for approximating solutions of equations of type (1.8) when \(A:E\rightarrow2^{E^{*}}\) is of monotone type has not been very fruitful. Part of the difficulty seems to be that all efforts made to apply directly the geometric properties of Banach spaces developed from the mid 1980s to the early 1990s proved abortive. Furthermore, the technique of converting the inclusion (1.8) into a fixed point problem for \(T:= IA : E\rightarrow E\) is not applicable since, in this case when A is monotone, A maps E into \(E^{*}\), and the identity map does not make sense.
Fortunately, Alber [22] (see also, Alber and Ryazantseva [23]) recently introduced a Lyapunov functional \(\phi:E\times E\rightarrow\mathbb{R}\), which signaled the beginning of the development of new geometric properties of Banach spaces which are appropriate for studying iterative methods for approximating solutions of (1.8) when \(A:E\rightarrow2^{E^{*}}\) is of monotone type. Geometric properties so far obtained have rekindled enormous research interest on iterative methods for approximating solutions of equation (1.8) where A is of monotone type, and other related problems (see, e.g., Alber [22]; Alber and GuerreDelabriere [24]; Chidume [21, 25]; Chidume et al. [26]; Diop et al.[27]; Moudafi [28], Moudafi and Tera [29]; Reich [30]; Sow et al. [31]; Takahashi [32]; Zegeye [33] and the references therein).
It is our purpose in this paper to apply the notion of Jfixed points (which has also been defined as a semifixed point (see, e.g., Zegeye [33]), a duality fixed point (see, e.g., Liu [34]) and, as far as we know, a new class of mappings called Jpseudocontractive maps introduced here to prove that \(T:=(JA)\) is Jpseudocontractive if and only if A is monotone; and in the case that E is a uniformly convex and a uniformly smooth real Banach space with dual \(E^{*}\), \(T: E\rightarrow2^{E^{*}}\) is a bounded Jpseudocontractive map with a nonempty Jfixed point set, and \(JT :E\rightarrow2^{E^{*}}\) is maximal monotone, a sequence is constructed which converges strongly to a Jfixed point of T. As an immediate application of this result, an analog of Theorem 1.1 for bounded maximal monotone maps is obtained, which is also a complement of the proximal point algorithm of Martinet [35] and Rockafellar [36], which has also been studied by numerous authors (see, e.g., Bruck [37]; Chidume [38]; Chidume [21]; Chidume and Djitte [39]; Kamimura and Takahashi [40]; Lehdili and Moudafi [41]; Reich [42]; Reich and Sabach [43, 44]; Solodov and Svaiter [45]; Xu [46] and the references therein). Furthermore, this analog is applied to approximate solutions of Hammerstein integral equations and is also applied to convex optimization problems. Finally, our techniques of proofs are of independent interest.
2 Preliminaries
Let E be a real normed linear space of dimension ≥2. The modulus of smoothness of E, \(\rho_{E}:[0,\infty )\rightarrow[0,\infty)\), is defined by
A normed linear space E is called uniformly smooth if
It is well known (see, e.g., Chidume [6], p.16, also Lindenstrauss and Tzafriri [47]) that \(\rho_{E}\) is nondecreasing. If there exist a constant \(c>0\) and a real number \(q>1\) such that \(\rho_{E}(\tau)\leq c\tau^{q}\), then E is said to be quniformly smooth. Typical examples of such spaces are the \(L_{p}\), \(\ell_{p}\), and \(W^{m}_{p}\) spaces for \(1< p<\infty\) where
A Banach space E is said to be strictly convex if
The modulus of convexity of E is the function \(\delta _{E}:(0,2]\rightarrow[0,1]\) defined by
The space E is uniformly convex if and only if \(\delta _{E}(\epsilon)>0\) for every \(\epsilon\in(0,2]\). It is also well known (see e.g., Chidume [6], p.34, Lindenstrauss and Tzafriri [47]) that \(\delta_{E}\) is nondecreasing. If there exist a constant \(c>0\) and a real number \(p>1\) such that \(\delta_{E}(\epsilon)\ge c\epsilon^{p}\), then E is said to be puniformly convex. Typical examples of such spaces are the \(L_{p}\), \(\ell_{p}\), and \(W^{m}_{p}\) spaces for \(1< p<\infty\) where
The norm of E is said to be Fréchet differentiable if, for each \(x\in S:= \{u\in E: \u\=1\}\),
exists and is attained uniformly for \(y\in E\).
For \(q>1\), let \(J_{q}\) denote the generalized duality mapping from E to \(2^{E^{\ast}}\) defined by
where \(\langle\cdot,\cdot\rangle\) denotes the generalized duality pairing. \(J_{2}\) is called the normalized duality mapping and is denoted by J. It is well known that if E is smooth, then \(J_{q}\) is singlevalued.
In the sequel, we shall need the following definitions and results. Let E be a smooth real Banach space with dual \(E^{*}\). The Lyapounov functional \(\phi:E\times E\to\mathbb{R}\), defined by
where J is the normalized duality mapping from E into \(E^{*}\) will play a central role in the sequel. It was introduced by Alber and has been studied by Alber [22], Alber and GuerreDelabriere [24], Kamimura and Takahashi [48], Reich [18], and a host of other authors. If \(E=H\), a real Hilbert space, then equation (2.1) reduces to \(\phi(x,y)=\xy\^{2}\) for \(x,y\in H\). It is obvious from the definition of the function ϕ that
Define a map \(V:X\times X^{*}\to\mathbb{R}\) by
Then it is easy to see that
Lemma 2.1
(Alber and Ryazantseva [23])
Let X be a reflexive strictly convex and smooth Banach space with \(X^{*}\) as its dual. Then
for all \(x\in X\) and \(x^{*},y^{*}\in X^{*}\).
Lemma 2.2
(Alber and Ryazantseva [23], p.50)
Let X be a reflexive strictly convex and smooth Banach space with \(X^{*}\) as its dual. Let \(W:X\times X\rightarrow\mathbb{R}^{1}\) be defined by \(W(x,y)=\frac {1}{2}\phi(y,x)\). Then
i.e.,
and also
for all \(x, y, z\in X\).
Lemma 2.3
(Alber and Ryazantseva [23], p.45)
Let X be a uniformly convex Banach space. Then, for any \(R>0\) and any \(x, y\in X\) such that \(\x\\le R\), \(\y\\le R\), the following inequality holds:
where \(c_{2}=2\max\{1,R\}\), \(1< L<1.7\).
Define
Lemma 2.4
(Alber and Ryazantseva [23], p.46)
Let X be a uniformly smooth and strictly convex Banach space. Then for any \(R>0\) and any \(x, y\in X\) such that \(\x\\le R\), \(\y\\le R\) the following inequality holds:
where \(c_{2}=2\max\{1,R\}\), \(1< L<1.7\).
Let \(E^{*}\) be a real strictly convex dual Banach space with a Fréchet differentiable norm. Let \(A:E\rightarrow2^{E^{*}}\) be a maximal monotone operator with no monotone extension. Let \(z\in E^{*}\) be fixed. Then for every \(\lambda>0\), there exists a unique \(x_{\lambda}\in E\) such that \(Jx_{\lambda}+\lambda Ax_{\lambda}\ni z\) (see Reich [7], p. 342). Setting \(J_{\lambda}z=x_{\lambda}\), we have the resolvent \(J_{\lambda}:=(J+\lambda A)^{1} :E^{*}\rightarrow E\) of A for every \(\lambda>0\). The following is a celebrated result of Reich.
Lemma 2.5
(Reich, [7]; see also, Kido, [49])
Let \(E^{*}\) be a strictly convex dual Banach space with a Fréchet differentiable norm, and let A be a maximal monotone operator from E to \(E^{*}\) such that \(A^{1}0\ne\emptyset\). Let \(z\in E^{*}\) be arbitrary but fixed. For each \(\lambda>0\) there exists a unique \(x_{\lambda}\in E\) such that \(Jx_{\lambda}+ \lambda Ax_{\lambda}\ni z\). Furthermore, \(x_{\lambda}\) converges strongly to a unique \(p\in A^{1}0\).
Lemma 2.6
From Lemma 2.5, setting \(\lambda_{n}:=\frac{1}{\theta_{n}}\) where \(\theta_{n} \rightarrow0\) as \(n\rightarrow\infty\), \(z=Jv\) for some \(v\in E\), and \(y_{n}:= (J+\frac{1}{\theta_{n}}A )^{1}z\), we obtain
where \(A:E\rightarrow E^{*}\) is maximal monotone.
Remark 1
Let \(R>0\) such that \(\v\\le R\), \(\y_{n}\\le R\) for all \(n\ge1\). We observe that equation (2.7) yields
Taking the duality pairing of the LHS of this equation with \(y_{n1}y_{n}\), applying CauchySchwarz, and using (2.8), we obtain
It follows that if E is uniformly convex and uniformly smooth, using Lemma 2.3 we obtain
which gives, using equation (2.6),
Similarly, using Lemma 2.4, we obtain
Remark 2
In puniformly convex spaces, we have (see, e.g., Chidume [6], p.34), for some constant \(c>0\),
From inequality (2.9), using inequality (2.12), we obtain
which gives
Also, we have from Lemma 2.4 that
Again, using inequality (2.12), we obtain
which gives
Lemma 2.7
(Kamimura and Takahashi [48])
Let X be a real smooth and uniformly convex Banach space, and let \(\{ x_{n}\}\) and \(\{y_{n}\}\) be two sequences of X. If either \(\{x_{n}\}\) or \(\{ y_{n}\}\) is bounded and \(\phi(x_{n},y_{n})\to0\) as \(n\to\infty\), then \(\ x_{n}y_{n}\ \to0\) as \(n\to\infty\).
Lemma 2.8
(Xu [50])
Let \(\{a_{n}\}_{n=1}^{\infty}\) be a sequence of nonnegative real numbers satisfying the following relation:
where \(\{\sigma_{n}\}_{n=0}^{\infty}\), \(\{b_{n}\}_{n=1}^{\infty}\), and \(\{ c_{n}\}_{n=1}^{\infty}\) satisfy the conditions:

(i)
\(\{\sigma_{n}\}_{n=1}^{\infty}\subset[0,1]\), \(\sum_{n=1}^{\infty}\sigma_{n}=\infty\), or equivalently, \(\prod_{n=1}^{\infty}(1\sigma_{n})=0\);

(ii)
\(\limsup_{n\rightarrow\infty}b_{n}\le0\);

(iii)
\(c_{n}\ge0\) (\(n\ge0\)), \(\sum_{n=1}^{\infty}c_{n}<\infty\).
Then \(\lim_{n\rightarrow\infty}a_{n}=0\).
Definition 2.9
(Jfixed point)
Let E be an arbitrary normed space and \(E^{*}\) be its dual. Let \(T:E\rightarrow2^{E^{*}}\) be any mapping. A point \(x\in E\) will be called a Jfixed point of T if and only if there exists \(\eta\in Tx\) such that \(\eta\in Jx\).
Remark 3
The notion of Jfixed points, as far as we know, was first introduced by Zegeye [33] who called a point \(x^{*}\in E\) such that \(Tx^{*}=Jx^{*}\), a semifixed point of T. Later, Liu [34] called such a point a duality fixed point of T.
3 Main results
We introduce the following definition.
Definition 3.1
(Jpseudocontractive mappings)
Let E be a normed space. A mapping \(T:E\rightarrow2^{E^{*}}\) is called Jpseudocontractive if for every \(x, y\in E\),
Example 1
If \(E=H\), a real Hilbert space, then J is the identity map on H. Consequently, every pseudocontractive map on H is Jpseudocontractive.
For our next example, we need the following characterization of the normalized duality map on \(l_{p}\), \(1< p<\infty\).
In \(l_{p}\) spaces, \(1< p<\infty\), for arbitrary \(x\in l_{p}\), \(x=(x_{1},x_{2},x_{3},\ldots)\),
(see, e.g., Alber and Ryazantseva [23], p.36).
Example 2
Let \(1< q< p<\infty\) and let \(\lambda\in\mathbb{R}\) be arbitrary. Define \(T:l_{p}\rightarrow l_{q}\) by
Then (i) T is Jpseudocontractive, (ii) \(x_{\lambda}:=(\lambda,0,0,\ldots)\) is a Jfixed point of T.
Remark 4
We observe that, assuming existence, a zero of a monotone mapping \(A:E\rightarrow2^{E^{*}}\) corresponds to a Jfixed point of a Jpseudocontractive mapping, T.
The following lemma asserts that \(A:E\rightarrow2^{E^{*}}\) is monotone if and only if \(T:=(JA):E\rightarrow2^{E^{*}}\) is Jpseudocontractive.
Lemma 3.2
Let E be an arbitrary real normed space and \(E^{*}\) be its dual space. Let \(A:E\rightarrow2^{E^{*}}\) be any mapping. Then A is monotone if and only if \(T:=(JA): E\rightarrow2^{E^{*}}\) is Jpseudocontractive.
Proof
Let \(x, y\in E\) be arbitrary. Suppose A is monotone. Then, for every \(\mu_{x} \in Ax\), \(\mu_{y}\in Ay\), \(jx\in Jx\), \(jy\in Jy\), \(\tau _{x}\in Tx\), \(\tau_{y}\in Ty\), such that \(\tau_{x}=jx\mu_{x}\), \(\tau _{y}=jy\mu_{y}\), we have
Hence, T is Jpseudocontractive.
Conversely, suppose \(T:= (JA)\) is Jpseudocontractive, we prove \(A:= JT\) is monotone. For all \(x, y\in E\), let \(\mu_{x}\in Ax\), \(\mu_{y}\in Ay\). Then \(\mu_{x}=jx\zeta_{x}\) and \(\mu_{y}=jy\zeta_{y}\) for some \(\zeta _{x}\in Tx\), \(\zeta_{y}\in Ty\), \(jx\in Jx\), and \(jy\in Jy\). We have
Hence, A is monotone. □
We now prove the following lemma, which will be crucial in the sequel.
Lemma 3.3
Let E be a smooth real Banach space with dual \(E^{*}\). Let \(\phi :E\times E\to\mathbb{R}\) be the Lyapounov functional. Then
Proof
Let \(x, y\in E\), we have
But,
so that
and substituting in (3.1), the result follows. □
In Theorem 3.4 below, the sequence \(\{\lambda_{n}\}_{n=1}^{\infty}\subset(0,1)\) satisfies the following conditions:

(i)
\(\sum_{n=1}^{\infty}\lambda_{n}=\infty\);

(ii)
\(\lambda_{n}M_{0}^{*}\le\gamma_{0}\theta_{n}\); \(\delta ^{1}_{E}(\lambda _{n}M_{0}^{*}) \leq\gamma_{0}\theta_{n}\),
for all \(n\ge1\) and for some constants \(M_{0}^{*}>0\), \(\gamma_{0}>0\).
Theorem 3.4
Let E be a uniformly convex and uniformly smooth real Banach space and let \(E^{*}\) be its dual. Let \(T:E\to2^{E^{*}}\) be a multivalued Jpseudocontractive and bounded map. Suppose \(F_{E}^{J}(T):=\{v\in E: Jv\in Tx\}\ne\emptyset\). For arbitrary \(u\in E\), define a sequence \(\{x_{n}\}\) iteratively by: \(x_{1}\in E\),
Then the sequence \(\{x_{n}\}\) is bounded.
Proof
Since \(F_{E}^{J}(T)\ne\emptyset\), let \(x^{*}\in F_{E}^{J}(T)\). Then there exists \(r>0\) such that \(\max \{\phi(x^{*},u), \phi(x^{*},x_{1})\}\le\frac{r}{8}\). Let \(B:=\{x\in E: \phi(x^{*},x)\le r\}\), and since T is bounded, we define:
Let \(M:=\max\{M_{2}M_{0}, c_{2}M_{0}, c_{2}M_{1}\}\), and
where \(c_{2}\) is the constant in Lemma 2.3. We show that \(\phi(x^{*},x_{n})\le r\) for all \(n\ge1\). We proceed by induction. Clearly, \(\phi(x^{*},x_{1})\le r\). Suppose \(\phi(x^{*},x_{n})\le r\) for some \(n\ge1\). We show \(\phi(x^{*},x_{n+1})\le r\). Suppose this is not the case, then \(\phi(x^{*},x_{n+1})>r\). Observe that
From Lemma 2.3 and the recurrence relation (3.2), we have
We hence obtain
Using inequality (2.5) with \(y^{*}=\lambda_{n} [Jx_{n}\eta _{n}+\theta_{n}(Jx_{n}Ju) ]\), we obtain using also inequality (3.4)
Since T is Jpseudocontractive, so that \((JT)\) is monotone, and using the recursion formula, we have
We have from Lemma 2.2
Substituting this in inequality (3.5), we obtain
This is a contradiction. Hence, \(\{x_{n}\}_{n=1}^{\infty}\) is bounded. □
In Theorem 3.5 below, \(\lambda_{n}\) and \(\theta_{n}\) are real sequences in \((0,1)\) satisfying the following conditions:

(i)
\(\sum_{n=1}^{\infty}\lambda_{n}\theta_{n}=\infty\),

(ii)
\(\lambda_{n}M_{0}^{*}\le\gamma_{0}\theta_{n}\); \(\delta ^{1}_{E}(\lambda_{n}M_{0}^{*}) \leq\gamma_{0}\theta_{n}\),

(iii)
\(\frac{\delta^{1}_{E} (\frac{\theta_{n1}\theta _{n}}{\theta_{n}}K )}{\lambda_{n}\theta_{n}} \rightarrow0\), \(\frac{\delta ^{1}_{E^{*}} (\frac{\theta_{n1}\theta_{n}}{\theta_{n}}K )}{\lambda_{n}\theta_{n}} \rightarrow0\), as \(n\rightarrow\infty\),

(iv)
\(\frac{1}{2} (\frac{\theta_{n1}\theta_{n}}{\theta _{n}}K )\in(0,1)\),
for some constants \(M_{0}^{*}>0\), and \(\gamma_{0}>0\); where \(\delta_{E}: (0,\infty)\rightarrow(0,\infty)\) is the modulus of convexity of E and \(K>0\) is as defined in Lemma 2.3.
Theorem 3.5
Let E be a uniformly convex and uniformly smooth real Banach space and let \(E^{*}\) be its dual. Let \(T:E\to2^{E^{*}}\) be a Jpseudocontractive and bounded map such that \((JT)\) is maximal monotone. Suppose \(F_{E}^{J}(T)=\{v\in E: Jv\in Tv\}\ne\emptyset\). For arbitrary \(x_{1}, u\in E\), define a sequence \(\{x_{n}\}\) iteratively by:
where \(\{\lambda_{n}\}\) and \(\{\theta_{n}\}\) are sequences in \((0,1)\) satisfying conditions (i)(iv) above. Then the sequence \(\{x_{n}\}\) converges strongly to a Jfixed point of T.
Proof
Setting \(y^{*}=\lambda_{n} [Jx_{n}\eta_{n}+\theta _{n}(Jx_{n}Ju) ]\in E^{*}\), applying inequality (2.5) and using Lemma 3.3, we compute as follows:
But we have from Lemma 2.6, \(y_{n}=J^{1} [\tau _{n}\theta _{n}(Jy_{n}Ju) ]\) for some \(\tau_{n}\in Ty_{n}\) and thus obtain
Hence, substituting this in inequality (3.7) and using Lemma 3.3, we obtain
Furthermore, using Lemma 2.2, we obtain
Substituting this inequality in inequality (3.8), we thus have
Estimating the underlined terms, we obtain
We thus have
Now, setting
and
inequality (3.11) becomes
It now follows from Lemma 2.8 that \(\phi (y_{n1},x_{n})\rightarrow0\) as \(n\rightarrow\infty\). From Lemma 2.7, we have \(\x_{n}y_{n1}\\rightarrow0\) and since \(y_{n}\rightarrow y^{*}\in(JT)^{1}0\), we obtain \(x_{n}\rightarrow y^{*}\in(JT)^{1}0\). This completes the proof. □
Example 3
We have (see, e.g., [23], p.47) for \(p>1\), \(q>1\), \(X=L^{p}\), \(X^{*}=L^{q}\),
and so obtain
The prototypes for our theorems are the following:
In particular, without loss of generality, let \(r=p\). Then one can choose \(a:=\frac{1}{(p+1)}\) and \(b:= \min \{\frac{1}{2K},\frac {1}{2p(p+1)} \}\).
We now verify that, with these prototypes, the conditions (i)(iii) of Theorem 3.5 are satisfied. Clearly (i) and the first part of (ii) are easily verified.
For the second part of condition (ii), we have
For condition (iii), we have
Similarly, we obtain
Finally, for condition (iv), we have
This completes the verification.
Remark 5
We remark, following Lindenstrauss and Tzafriri [47], that in applications, we do not often use the precise value of the modulus of convexity but only a power type estimate from below.
A uniformly convex space X has modulus of convexity of power type p if, for some \(0< K<\infty\), \(\delta_{X}(\epsilon)\ge K\epsilon^{p}\). For instance, \(L_{p}\) spaces have modulus of convexity of power type 2, for \(1< p\le2\), and of power type p, for \(p>2\) (see, e.g., [47], p.63). We observe that the condition for modulus of convexity of power type p corresponds to that of puniformly convex spaces. However, we see that \(L_{p}\) spaces are puniformly convex, for \(1< p< 2\), and are 2uniformly convex, for \(p\ge2\). These lead us to prove the following corollary of Theorem 3.4, which will be crucial in several applications.
Corollary 3.6
For \(p>1\), \(q>1\), let E be a puniformly convex and quniformly smooth real Banach space and let \(E^{*}\) be its dual. Let \(T:E\to E^{*}\) be a Jpseudocontractive and bounded map. Suppose \(F_{E}^{J}(T):=\{u^{*}\in E: Tu^{*}=Ju^{*}\}\ne\emptyset\). For arbitrary \(x_{1}, u\in E\), define a sequence \(\{x_{n}\}\) iteratively by:
where \(\{\lambda_{n}\}\) and \(\{\theta_{n}\}\) are sequences in \((0,1)\) satisfying conditions (i)(iii) of Theorem 3.4. Then the sequence \(\{x_{n}\}\) converges strongly to a Jfixed point of T.
Proof
We observe, for puniformly convex space, using Remark 2, that conditions (i)(iv) of Theorem 3.5 reduce to:
 (i)^{∗} :

\(\lambda_{n}\le\gamma_{0}\theta_{n}\),
 (ii)^{∗} :

\(\sum_{n=1}^{\infty}\lambda_{n}\theta_{n}=\infty\),
 (iii)^{∗} :

\((\frac{\theta_{n1}\theta_{n}}{\theta_{n}} )^{1/p}\rightarrow0\), \(\frac{M^{*} (\frac{\theta_{n1}\theta _{n}}{\theta_{n}} )^{1/p}}{\lambda_{n}\theta_{n}}\rightarrow0\), \(\frac{ (\lambda _{n}^{(1/p)}M_{0}^{**} )}{\theta_{n}} \rightarrow0\), as \(n\rightarrow\infty\), for some \(M_{0}^{**}, M^{*}>0\),
and for puniformly convex spaces, we have from (3.3), using equation (2.12),
Following the proof of Theorem 3.5, we have from inequality (3.9), using (3.13):
Now, setting
and
It now follows from Lemma 2.8 that \(\phi (y_{n1},x_{n})\rightarrow0\) as \(n\rightarrow\infty\). From Lemma 2.7, we have \(\x_{n}y_{n1}\\rightarrow0\), and since \(y_{n}\rightarrow y^{*}\in(JT)^{1}0\), this completes the proof. □
Example 4
Real sequences that satisfy the conditions (i)^{∗}(iv)^{∗} in Corollary 3.6 are the following:
For example, one can choose \(a:=\frac{1}{(p+1)}\) and \(b:= \frac {1}{2p(p+1)}\). We now check these prototypes.
Clearly conditions (i)^{∗}(ii)^{∗} are satisfied. We verify condition (iii)^{∗}. Using the fact that \((1+x)^{s}\le1+sx\), for \(x>1\) and \(0< s<1\), we have
Also,
and
4 Application to zeros of maximal monotone maps
Corollary 4.1
Let E be a uniformly convex and uniformly smooth real Banach space and let \(E^{*}\) be its dual. Let \(A:E\to2^{E^{*}}\) be a multivalued maximal monotone and bounded map such that \(A^{1}0\ne\emptyset\). For fixed \(u, x_{1}\in E\), let a sequence \(\{x_{n}\}\) be iteratively defined by
where \(\{\lambda_{n}\}\) and \(\{\theta_{n}\}\) are sequences in \((0,1)\). Then the sequence \(\{x_{n}\}\) converges strongly to a zero of A.
Proof
Recall that A is monotone if and only if \(T=(JA)\) is Jpseudocontractive and that zeros of A correspond to Jfixed points of T.. Now, if we replace A by \(JT\) in equation (4.1), the equation reduces to (3.6) and hence the proof follows. □
5 Complement to proximal point algorithm
The proximal point algorithm of Martinet [35] and Rockafellar [36] was introduced to approximate a solution of \(0\in Au\) where A is the subdifferential of some convex functional defined on a real Hilbert space. A solution of this inclusion gives the minimizers of the convex functional. Let E be a real normed space with dual space, \(E^{*}\) and \(f:E\rightarrow\mathbb{R}\) be a convex functional. The subdifferential of f, \(\partial f:E\rightarrow2^{E^{*}}\) at \(u\in E\), is defined as follows:
It is well known that ∂f is a maximal monotone map on E and that \(0\in(\partial f)(u)\) if and only if u is a minimizer of f. Following this, the proximal point algorithm has been studied for minimizers of f in real Banach spaces more general than Hilbert spaces.
Rockafellar [36] proved that the proximal point algorithm defined as follows:
where \(\lambda_{k}>0\) is a regularizing parameter; converges weakly to a solution of \(0\in Au\) where A is the subdifferential of a convex functional on a Hilbert space provided a solution exists. He then asked if the proximal point algorithm always converge strongly.
This was resolved in the negative by Güler [51] who produced a proper closed convex function g in the infinite dimensional Hilbert space \(l_{2}\) for which the proximal point algorithm converges weakly but not strongly (see also Bauschke et al. [52]). Several authors modified the proximal point algorithm to obtain strong convergence (see, e.g., Bruck [37]; Kamimura and Takahashi [40]; Lehdili and Moudafi [41]; Reich [42]; Solodov and Svaiter [45]; Xu [46]). We remark that in every one of these modifications, the recursion formula developed involved either the computation of \((I+\lambda_{k} A)^{1}(x_{k})\) at each point of the iteration process or the construction, at each iteration, of two subsets of the space, intersecting them and projecting the initial vector onto the intersection. As far as we know, the first iteration process to approximate a solution of \(0\in Au\) in real Banach spaces more general than Hilbert spaces and which does not involve either of these setbacks was given by Chidume and Djitte [39] who proved a special case of Theorem 1.1 in which the space E is a 2uniformly smooth real Banach space. These spaces include \(L_{p}\) spaces, \(2\le p<\infty\), but do not include \(L_{p}\) spaces, \(1< p<2\). This result of Chidume and Djitte has recently been proved in uniformly convex and uniformly smooth real Banach spaces (which include \(L_{p}\) spaces, \(1< p<\infty\)) (Chidume (Theorem 1.1) above).
Corollary 4.1 of this paper is an analog of Theorem 1.1 for maximal monotone maps when \(A:E\rightarrow2^{E^{*}}\) is a maximal monotone and bounded map, a result which complements the proximal point algorithm, under this setting, in the sense that it yields strong convergence to a solution of \(0\in Au\) and without requiring either the computation of \((J+\lambda A)^{1}(z_{n})\) at each iteration process, or the construction of two subsets of E, and projection of the initial vector onto their intersection, at each stage of the iteration process.
6 Application to solutions of Hammerstein integral equations
Definition 6.1
Let \(\Omega\subset{\mathbb{R}}^{n}\) be bounded. Let \(k:\Omega\times \Omega\to\mathbb{R}\) and \(f:\Omega\times\mathbb{R} \to\mathbb{R}\) be measurable realvalued functions. An integral equation (generally nonlinear) of Hammersteintype has the form
where the unknown function u and inhomogeneous function w lie in a Banach space E of measurable realvalued functions.
By a simple transformation (6.1) can put in the form
which, without loss of generality can be written as
Interest in Hammerstein integral equations stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear part posses Green’s function can, as a rule, be transformed into the form (6.1) (see, e.g., Pascali and Sburian [8], p.164).
Among the first early results on the approximation of solution of Hammerstein equations is the following result of Brézis and Browder.
Theorem 6.2
(Brézis and Browder [53])
Let H be a separable Hilbert space and C be a closed subspace of H. Let \(K:H \to C\) be a bounded continuous monotone operator and \(F:C\to H\) be anglebounded and weakly compact mapping. For a giving \(f\in C\), consider the Hammerstein equation
and its nth Galerkin approximation given by
where \(K_{n}=P_{n}^{*}KP_{n}:H\to C\) and \(F_{n}=P_{n}FP_{n}^{*}:C_{n} \to H\), where the symbols have their usual meanings (see [8]). Then, for each \(n\in\mathbb{N}\), the Galerkin approximation (6.5) admits a unique solution \(u_{n}\) in \(C_{n}\) and \(\{u_{n}\}\) converges strongly in H to the unique solution \(u\in C\) of the equation (6.4) where \(K_{n}=P_{n}^{*}KP_{n}:H\to C\) and \(F_{n}=P_{n}FP_{n}^{*}:C_{n} \to H\), where the symbols have their usual meanings (see [53]). Then, for each \(n\in\mathbb{N}\), the Galerkin approximation (6.5) admits a unique solution \(u_{n}\) in \(C_{n}\) and \(\{u_{n}\}\) converges strongly in H to the unique solution \(u\in C\) of the equation (6.4).
It is obvious that if an iterative algorithm can be developed for the approximation of solutions of equation of Hammersteintype (6.3), this will certainly be preferred.
Attempts have been made to approximate solutions of equations of Hammersteintype using Manntype iteration scheme. However, the results obtained were not satisfactory (see, e.g., [54]). The recurrence formulas used in early attempts involved \(K^{1}\) which is also required to be strongly monotone, and this, apart from limiting the class of mappings to which such iterative schemes are applicable, it is also not convenient in applications. Part of the difficulty is the fact that the composition of two monotone operators need not to be monotone.
The first satisfactory results on iterative methods for approximating solutions of Hammerstein equations in real Banach spaces more general Hilbert spaces, as far as we know, were obtained by Chidume and Zegeye [55–57]. For the case of real Hilbert space H, for \(F,K:H \to H\), they defined an auxiliary map on the Cartesian product \(E:=H\times H\), \(T:E\to E\) by
We note that
With this, they were able to obtain strong convergence of an iterative scheme defined in the Cartesian product space E to a solution of Hammerstein equation (6.3). The method of proof used by Chidume and Zegeye provided the clue to the establishment of the following couple explicit algorithm for computing a solution of the equation \(u+KFu=0\) in the original space X. With initial vectors \(u_{0}, v_{0}\in X\), sequences \(\{u_{n}\}\) and \(\{v_{n}\}\) in X are defined iteratively as follows:
where \(\alpha_{n}\) is a sequence in \((0,1)\) satisfying appropriate conditions.
Some typical results obtained using the recursion formulas described above in approximating solutions of nonlinear Hammerstein equations involving monotone maps in Hilbert spaces can be found in [57, 58].
In real Banach space X more general than Hilbert spaces, where \(F,K: X\rightarrow X\) are of accretivetype, Chidume and Zegeye considered an operator \(A:E\rightarrow E\) where \(E:= X\times X\) and were able to successfully approximate solutions of Hammerstein equations using recursion formulas described above. These schemes have now been employed by Chidume and other authors to approximate solutions of Hammerstein equations in various Banach spaces under various continuity assumptions (see, e.g., [27, 31, 55–71]). This success has not carried over to the case of monotonetype mappings in Banach spaces where K and F map a space into its dual. In this section, we introduce a new iterative scheme and prove that a sequence of our scheme converges strongly to a solution of a Hammerstein equation under this setting. For this purpose, we begin with the following preliminaries and lemmas.
We now prove the following lemmas.
Lemma 6.3
Let X, Y be real uniformly convex and uniformly smooth spaces. Let \(E=X\times Y\) with the norm \(\z\_{E}=(\u\^{q}_{X} + \v\^{q}_{Y})^{\frac {1}{q}}\), for arbitrary \(z=[u,v]\in E\). Let \(E^{*}=X^{*}\times Y^{*}\) denote the dual space of E. For arbitrary \(x=[x_{1},x_{2}]\in E\), define the map \(j_{q}^{E}:E\rightarrow E^{*}\) by
so that for arbitrary \(z_{1}=[u_{1},v_{1}]\), \(z_{2}=[u_{2},v_{2}]\) in E, the duality pairing \(\langle\cdot,\cdot\rangle\) is given by
Then

(a)
E is uniformly smooth and uniformly convex,

(b)
\(j_{q}^{E}\) is singlevalued duality mapping on E.
Proof
(a) Let \(p>1\), \(q>1\). Let \(x=[x_{1},x_{2}]\), \(y=[y_{1},y_{2}]\) be arbitrary elements of E. Using Condition (iii)′ of Corollary 2^{r} in [72], we have
where \(g^{*}_{1}\), \(g^{*}_{2}\) are strictly increasing continuous and convex functions on \(\mathbb{R}^{+}\) and \(g^{*}_{1}(0)=g^{*}_{2}(0)=0\). It follows that
where \(g^{*}(\xy\)=g^{*}_{1}(\x_{1}y_{1}\) + g^{*}_{2}(\x_{2}y_{2}\)\). Hence the result follows from Corollary 2′ that E is uniformly smooth.
Also, using condition (iii) of Corollary 3 in [72], we have
where \(g_{1}\), \(g_{2}\) are strictly increasing continuous and convex functions on \(\mathbb{R}^{+}\) and \(g_{1}(0)=g_{2}(0)=0\). It follows that
where \(g(\xy\)=g_{1}(\x_{1}y_{1}\) + g_{2}(\x_{2}y_{2}\)\). Hence the result follows from Corollary 3 that E is uniformly convex. Since E is uniformly smooth, it is smooth and hence any duality mapping on E is singlevalued.
(b) For arbitrary \(x=[x_{1},x_{2}]\in E\), let \(j_{q}^{E}(x)=j_{q}^{E}[x_{1},x_{1}] = \psi_{q}\). Then \(\psi_{q}=[j_{q}^{X}(x_{1}),j_{q}^{Y}(x_{2})]\in E^{*}\). We have, for \(p>1\) such that \(1/p + 1/q=1\),
Hence, \(\\psi\_{E^{*}}=\x\_{E}^{q1}\). Furthermore,
Hence, \(j_{q}^{E}\) is a singlevalued normalized duality mapping on E. □
The following lemma will be needed in the following.
Lemma 6.4
(Browder [73])
Let X be a strictly convex reflexive Banach space with a strictly convex conjugate space \(X^{*}\), \(T_{1}\) a maximal monotone mapping from X to \(X^{*}\), \(T_{2}\) a hemicontinuous monotone mapping of all of X into \(X^{*}\) which carries bounded subsets of X into bounded subsets of \(X^{*}\). Then the mapping \(T=T_{1}+T_{2}\) is a maximal monotone map of X into \(X^{*}\).
Using Lemma 6.4, we prove the following important lemma which will be used in the sequel.
Lemma 6.5
Let E be a Banach space. Let \(F:E\rightarrow E^{*}\) and \(K:E^{*}\rightarrow E\) be bounded and maximal monotone mappings with \(D(F)=D(K)=E\). Let \(T:E\times E^{*}\rightarrow E^{*}\times E\) be defined by
then the mapping \(A:=(JT)\) is maximal monotone.
Proof
We show that the mapping \(A=(JT):E\times E^{*}\rightarrow E^{*}\times E\) defined as
is maximal monotone. Let \(S,T:E\times E^{*}\rightarrow E^{*}\times E\) be defined as
Then \(A=S+T\). It suffices to show S, T are maximal monotone.
Observe that S is monotone. Let \(h=[h_{1},h_{2}]\in E^{*}\times E\). Since F, K are maximal monotone, take \(u=(J+\lambda F)^{1}h_{1}\) and \(v=(J_{*}+\lambda K)^{1}h_{2}\). Then \((J+\lambda S)w=h\), where \(w=[u,v]\). Hence, S is maximal monotone.
Clearly, T is bounded and monotone. Furthermore it is continuous. Hence, it is hemicontinuous. Therefore by Lemma 6.4, \(A=S+T\) is maximal monotone. □
Lemma 6.6
Let E be a uniformly convex and uniformly smooth real Banach space. Let \(F:E\rightarrow E^{*}\) and \(K:E^{*}\rightarrow E\) be monotone mappings with \(D(F)=D(K)=E\). Let \(T:E\times E^{*}\rightarrow E^{*}\times E\) be defined by \(T[u,v]=[JuFu+v,J_{*}vKvu]\) for all \((u,v)\in E\times E^{*}\), then T is Jpseudocontractive. Moreover, if the Hammerstein equation \(u+KFu=0\) has a solution in E, then \(u^{*}\) is a solution of \(u+KFu=0\) if and only if \((u^{*},v^{*})\in F_{E}^{J}(T)\), where \(v^{*}=Fu^{*}\).
Proof
Using the monotonicity of F and K, we easily obtain \(\langle Tw_{1}Tw_{2},w_{1}w_{2}\rangle\le\langle Jw_{1}Jw_{2},w_{1}w_{2}\rangle\) for all \(w_{1}=[u_{1},v_{1}], w_{2}=[u_{2},v_{2}]\in E\times E^{*}\).
Moreover, we observe that
□
We now prove the following theorem.
Theorem 6.7
Let E be a uniformly smooth and uniformly convex real Banach space and \(F:E\rightarrow E^{*}\), \(K:E^{*}\rightarrow E\) be maximal monotone and bounded maps, respectively. For \((x_{1},y_{1}), (u_{1},v_{1})\in E\times E^{*}\), define the sequences \(\{u_{n}\}\) and \(\{v_{n}\}\) in E and \(E^{*}\) respectively, by
Assume that the equation \(u+KFu=0\) has a solution. Then the sequences \(\{u_{n}\} _{n=1}^{\infty}\) and \(\{v_{n}\}_{n=1}^{\infty}\) converge strongly to \(u^{*}\) and \(v^{*}\), respectively, where \(u^{*}\) is the solution of \(u+KFu=0\) with \(v^{*}=Fu^{*}\).
Proof
From Lemma 6.6 we see that \(T:E\times E^{*}\rightarrow E^{*}\times E\) defined by \(T[u,v]=[JuFu+v,J_{*}vKvu]\) for all \((u,v)\in E\times E^{*}\) is Jpseudocontractive, and \(A:=(JT)\) is maximal monotone.
Applying Theorem 3.4 where \(X=E\times E^{*}\), from Lemma 6.3, X is uniformly convex and uniformly smooth. We obtain (6.8) and (6.9) and the proof follows. □
7 Application to convex optimization problem
The following lemma is well known (see, e.g., [74], p.23, for similar proof in the Hilbert space case).
Lemma 7.1
Let X be a normed space. Let \(f:X\rightarrow\mathbb{R}\) be a convex function that is bounded on bounded subsets of X. Then the subdifferential, \(\partial f:X\rightarrow2^{X^{*}}\) is bounded on bounded subsets of E.
We now prove the following strong convergence theorem.
Theorem 7.2
Let E be a uniformly convex and uniformly smooth real Banach space with dual \(E^{*}\). Let \(f:E\rightarrow(\infty,\infty]\) be a lower semicontinuously Frèchet differentiable convex and bounded functional such that \((\partial f)^{1}0\ne\emptyset\). For given \(u,x_{1}\in E\), let \(\{x_{n}\}\) be generated by the algorithm
Then \(\{x_{n}\}\) converges strongly to some \(x^{*}\in(\partial f)^{1}0\).
Proof
Since f is convex and bounded, we see that ∂f is bounded. By Rockafellar [75, 76] (see also, e.g., Minty [2], Moreau [77]), we see that \((\partial f)\) is maximal monotone mapping from \(E^{*}\) into E and \(0\in(\partial f)^{1}v\) if and only if \(f(v)=\min_{x\in E}f(x)\). Since f is convex and bounded, from Lemma 7.1 we see that ∂f is bounded, hence, the conclusion follows from Corollary 4.1. □
Remark 6
The analytical representations of duality mappings are known in a number of Banach spaces. For instance, in the spaces \(L^{p}(G)\) and \(W^{p}_{m}(G)\), \(p \in(1,\infty)\) we have, respectively,
and
where \(p^{1}+q^{1}=1\). (See, e.g., Alber and Ryazantseva [23], p.36.)
8 Conclusion
Let E be a uniformly convex and uniformly smooth real Banach space with dual \(E^{*}\). Approximation of zeros of accretivetype maps of E to itself, assuming existence, has been studied extensively within the past 40 years or so (see, e.g., Agarwal et al. [17]; Berinde [4]; Chidume [6]; Reich [18]; Censor and Reich [19]; William and Shahzad [20], and the references therein). The key tool for this study has been the study of fixed points of pseudocontractivetype maps.
Unfortunately, for approximating zeros of monotonetype maps from E to \(E^{*}\), the normal fixed point technique is not applicable. This motivated the study of the notion of Jpseudocontractive maps introduced in this paper. The main result of this paper is Theorem 3.5 which provides an easily applicable iterative sequence that converges strongly to a Jfixed point of T, where \(T:E\rightarrow2^{E^{*}}\) is a Jpseudocontractive and bounded map such that \(JT\) is maximal monotone. The two parameters in the recursion formula of the theorem, \(\theta_{n}\) and \(\lambda_{n}\), are easily chosen in any possible application of the theorem (see Example 4 above).
The theorem is, in particular, applicable in \(L_{p}\) and \(l_{p}\) spaces, \(1< p<\infty\). In these spaces, the normalized duality maps J and \(J^{1}\) which appear in the recursion formula of the theorem are precisely known (see Remark 6 above).
Consequently, while the proof of the theorem is very technical and nontrivial, with the simple choices of the iteration parameters and the exact explicit formula for J and \(J^{1}\), the recursion formula of the theorem which does not involve the resolvent operator, \((J+\lambda A)^{1}\), is extremely attractive and user friendly.
Theorem 3.5 is applicable in numerous situations. In this paper, it has been applied to approximate a zero of a bounded maximal monotone map \(A:E\rightarrow2^{E^{*}}\) with \(A^{1}(0)\ne\emptyset\).
Furthermore, the theorem complements the proximal point algorithm by providing strong convergence to a zero of a maximal monotone operator A and without involving the resolvent \(J_{r}:=(J+rA)^{1}\) in the recursion formula. In addition, it is applied to approximate solutions of Hammerstein integral equations and also to approximate solutions of convex optimization problems. Theorem 3.5 continues to be applicable in approximating solutions of nonlinear equations. It has recently been applied to approximate a common zero of an infinite family of Jnonexpansive maps, \(T_{i}: E\rightarrow2^{E^{*}}\), \(i\ge1\) (see Chidume et al. [78]). In the case that \(E=H\) is a real Hilbert space, the result obtained in Chidume et al. [78] is a significant improvement of important known results. We strongly believed that the results of this paper will continue to be applied to approximate solutions of equilibrium problems in nonlinear operator theory.
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Chidume, C.E., Idu, K.O. Approximation of zeros of bounded maximal monotone mappings, solutions of Hammerstein integral equations and convex minimization problems. Fixed Point Theory Appl 2016, 97 (2016). https://doi.org/10.1186/s1366301605828
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DOI: https://doi.org/10.1186/s1366301605828