Ugwunnadi et al. Fixed Point Theory and Applications 2014, 2014:231
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RESEARCH
Open Access
Strong convergence theorem for
quasi-Bregman strictly pseudocontractive
mappings and equilibrium problems in
Banach spaces
Godwin C Ugwunnadi1,2 , Bashir Ali3* , Ibrahim Idris3 and Maaruf S Minjibir3
*
Correspondence:
[email protected]
3
Department of Mathematical
Sciences, Bayero University Kano,
P.M.B. 3011, Kano, Nigeria
Full list of author information is
available at the end of the article
Abstract
In this paper, we introduce a new iterative scheme by a hybrid method and prove a
strong convergence theorem of a common element in the set of ﬁxed points of a
ﬁnite family of closed quasi-Bregman strictly pseudocontractive mappings and
common solutions to a system of equilibrium problems in reﬂexive Banach space. Our
results extend important recent results announced by many authors.
MSC: 47H09; 47J25
Keywords: Bregman distance; quasi-Bregman strictly pseudocontractive map; ﬁxed
point
1 Introduction
Let E be a real Banach space and C a nonempty closed convex subset of E. The normalized
∗
duality map from E to E (E∗ is the dual space of E) denoted by J is deﬁned by
J(x) = f ∈ E∗ : x, f = x = f  .
Let T : C → C be a map, a point x ∈ C is called a ﬁxed point of T if Tx = x, and the set of
all ﬁxed points of T is denoted by F(T). The mapping T is called L-Lipschitzian or simply
Lipschitz if there exists L > , such that Tx – Ty ≤ Lx – y, ∀x, y ∈ C and if L = , then
the map T is called nonexpansive.
Let g : C × C → R be a bifunction. The equilibrium problem with respect to g is to ﬁnd
z ∈ C such that g(z, y) ≥ ,
∀y ∈ C.
The set of solution of equilibrium problem is denoted by EP(g). Thus
EP(g) := z ∈ C : g(z, y) ≥ , ∀y ∈ C .
Numerous problems in physics, optimization and economics reduce to ﬁnding a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium
©2014 Ugwunnadi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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problem in Hilbert spaces; see for example Blum and Oettli [], Combettes and Hirstoaga
[]. Recently, Tada and Takahashi [, ] and Takahashi and Takahashi [] obtain weak and
strong convergence theorems for ﬁnding a common element of the set of solutions of an
equilibrium problem and set of ﬁxed points of a nonexpansive mapping in Hilbert space.
In particular, Takahashi and Zembayashi [] established a strong convergence theorem
for ﬁnding a common element of the two sets by using the hybrid method introduced
in Nakajo and Takahashi []. They also proved such a strong convergence theorem in a
uniformly convex and uniformly smooth Banach space.
Reich and Sabach [] and Kassay et al. [] proved some convergence theorems for the
solution of some equilibrium and variational inequality problems in the setting of reﬂexive
Banach spaces.
Let φ : E × E → [, ∞) denote the Lyapunov functional deﬁned by
φ(x, y) = x – x, Jy + y ,
∀x, y ∈ E.
A mapping T : C → C is said to be quasi-φ strictly pseudocontractive, see [], if F(T) = ∅
and there exists a constant k ∈ (, ] such that
φ(p, Tx) ≤ φ(p, x) + kφ(x, Tx),
∀x ∈ C and p ∈ F(T).
Let E be a real reﬂexive Banach space with norm · and E∗ the dual space of E. Throughout this paper, we shall assume f : E → (–∞, +∞] is a proper, lower semi-continuous and
convex function. We denote by dom f := {x ∈ E : f (x) < +∞} the domain of f .
Let x ∈ int dom f ; the subdiﬀerential of f at x is the convex set deﬁned by
∂f (x) = x∗ ∈ E∗ : f (x) + x∗ , y – x ≤ f (y), ∀y ∈ E ,
where the Fenchel conjugate of f is the function f ∗ : E∗ → (–∞, +∞] deﬁned by
f ∗ x∗ = sup x∗ , x – f (x) : x ∈ E .
We know that the Young-Fenchel inequality holds:
∗ x , x ≤ f (x) + f ∗ x∗ ,
∀x ∈ E, x∗ ∈ E∗ .
A function f on E is coercive [] if the sublevel set of f is bounded; equivalently,
lim f (x) = +∞.
x→+∞
A function f on E is said be strongly coercive [] if
lim
x→+∞
f (x)
= +∞.
x
For any x ∈ int dom f and y ∈ E, the right-hand derivative of f at x in the direction y is
deﬁned by
f ◦ (x, y) := lim+
t→
f (x + ty) – f (x)
.
t
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(x)
The function f is said to be Gâteaux diﬀerentiable at x if limt→+ f (x+ty)–f
exists for any y.
t
◦
In this case, f (x, y) coincides with ∇f (x), the value of the gradient ∇f of f at x. The function f is said to be Gâteaux diﬀerentiable if it is Gâteaux diﬀerentiable for any x ∈ int dom f .
The function f is said to be Fréchet diﬀerentiable at x if this limit is attained uniformly in
y = . Finally, f is said to be uniformly Fréchet diﬀerentiable on a subset C of E if the
limit is attained uniformly for x ∈ C and y = . It is well known that if f is Gâteaux differentiable (resp. Fréchet diﬀerentiable) on int dom f , then f is continuous and its Gâteaux
derivative ∇f is norm-to-weak∗ continuous (resp. continuous) on int dom f (see also [,
]). We will need the following results.
Lemma . [] If f : E → R is uniformly Fréchet diﬀerentiable and bounded on bounded
subsets of E, then ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E∗ .
Deﬁnition . [] The function f is said to be:
(i) essentially smooth, if ∂f is both locally bounded and single-valued on its domain;
(ii) essentially strictly convex, if (∂f )– is locally bounded on its domain and f is strictly
convex on every convex subset of dom ∂f ;
(iii) Legendre, if it is both essentially smooth and essentially strictly convex.
Remark . Let E be a reﬂexive Banach space. Then we have:
(i) f is essentially smooth if and only if f ∗ is essentially strictly convex (see [],
Theorem .);
(ii) (∂f )– = ∂f ∗ (see []);
(iii) f is Legendre if and only if f ∗ is Legendre (see [], Corollary .);
(iv) if f is Legendre, then ∇f is a bijection satisfying ∇f = (∇f ∗ )– ,
ran ∇f = dom ∇f ∗ = int dom f ∗ and ran ∇f ∗ = dom f = int dom f (see [],
Theorem .).
Examples of Legendre functions were given in [, ]. One important and interesting
Legendre function is p · p ( < p < ∞) when E is a smooth and strictly convex Banach
space. In this case the gradient ∇f of f is coincident with the generalized duality mapping
of E, i.e., ∇f = Jp ( < p < ∞). In particular, ∇f = I the identity mapping in Hilbert spaces.
In the rest of this paper, we always assume that f : E → (–∞, +∞] is Legendre.
Let f : E → (–∞, +∞] be a convex and Gâteaux diﬀerentiable function. The function
Df : dom f × int dom f → [, +∞), deﬁned as follows:
Df (y, x) := f (y) – f (x) – ∇f (x), y – x ,
(.)
is called the Bregman distance with respect to f (see []). It is obvious from the deﬁnition
of Df that
Df (z, x) = Df (z, y) + Df (y, x) + ∇f (y) – ∇f (x), z – y .
(.)
Recall that the Bregman projection [] of x ∈ int dom f onto the nonempty, closed, and
f
convex set C ⊂ dom f is the necessarily unique vector PC (x) ∈ C satisfying
f
Df PC (x), x = inf Df (y, x) : y ∈ C .
Concerning the Bregman projection, the following are well known.
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Lemma . [] Let C be a nonempty, closed, and convex subset of a reﬂexive Banach
space E. Let f : E → R be a Gâteaux diﬀerentiable and totally convex function and let
x ∈ E. Then:
f
(a) z = PC (x) if and only if ∇f (x) – ∇f (z), y – z ≤ , ∀y ∈ C;
f
f
(b) Df (y, PC (x)) + Df (PC (x), x) ≤ Df (y, x), ∀x ∈ E, y ∈ C.
Let f : E → (–∞, +∞] be a convex and Gâteaux diﬀerentiable function. The modulus of
the total convexity of f at x ∈ int dom f is the function vf (x, ·) : [, +∞) → [, +∞] deﬁned
by
vf (x, t) := inf Df (y, x) : y ∈ dom f , y – x = t .
The function f is called totally convex at x if vf (x, t) >  whenever t > . The function f is
called totally convex if it is totally convex at any point x ∈ int dom f and is said to be totally
convex on bounded sets if vf (B, t) >  for any nonempty bounded subset B of E and t > ,
where the modulus of the total convexity of the function f on the set B is the function
vf : int dom f × [, +∞) → [, +∞] deﬁned by
vf (B, t) := inf vf (x, t) : x ∈ B ∩ dom f .
Lemma . [] If x ∈ dom f , then the following statements are equivalent:
(i) the function f is totally convex at x;
(ii) for any sequence {yn } ⊂ dom f ,
lim Df (yn , x) = 
n→+∞
⇒
lim yn – x = .
n→+∞
Recall that the function f called sequentially consistent [] if for any two sequences {xn }
and {yn } in E such that the ﬁrst one is bounded
lim Df (yn , xn ) = 
n→+∞
⇒
lim yn – xn = .
n→+∞
Lemma . [] The function f is totally convex on bounded sets if and only if the function
f is sequentially consistent.
Lemma . [] Let f : E → R be a Gâteaux diﬀerentiable and totally convex function. If
x ∈ E and the sequence {Df (xn , x )} is bounded, then the sequence {xn } is bounded too.
Lemma . [] Let f : E → R be a Gâteaux diﬀerentiable and totally convex function,
x ∈ E and let C be a nonempty, closed, and convex subset of E. Suppose that the sequence
{xn } is bounded and any weak subsequential limit of {xn } belongs to C. If Df (xn , x ) ≤
f
f
Df (PC (x ), x ) for any n ∈ R, then {xn } converges strongly to PC (x ).
A mapping T is said to be Bregman ﬁrmly nonexpansive [], if for all x, y ∈ C,
∇f (Tx) – ∇f (Ty), Tx – Ty ≤ ∇f (x) – ∇f (y), Tx – Ty
or, equivalently,
Df (Tx, Ty) + Df (Ty, Tx) + Df (Tx, x) + Df (Ty, y) ≤ Df (Tx, y) + Df (Ty, x).
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A point p ∈ C is said to be asymptotic ﬁxed point of a map T, if there exists a sequence
ˆ
{xn } in C which converges weakly to p such that limn→∞ xn – Txn = . We denote by F(T)
the set of asymptotic ﬁxed points of T. A point p ∈ C is said to be strong asymptotic ﬁxed
point of a map T, if there exists a sequence {xn } in C which converges strongly to p such
˜
that limn→∞ xn – Txn = . We denote by F(T)
the set of strong asymptotic ﬁxed points
of T. Let f : E → R, a mapping T : C → C is said to be Bregman relatively nonexpansive
ˆ
[] if F(T) = ∅, F(T)
= F(T) and Df (p, T(x)) ≤ Df (p, x) for all x ∈ C and p ∈ F(T). The map
˜
T : C → C is said to be Bregman weak relatively nonexpansive if F(T) = ∅, F(T)
= F(T)
and Df (p, T(x)) ≤ Df (p, x) for all x ∈ C and p ∈ F(T). T is said to be quasi-Bregman relatively nonexpansive if F(T) = ∅, and Df (p, T(x)) ≤ Df (p, x) for all x ∈ C and p ∈ F(T). In
[] quasi-Bregman relatively nonexpansive is called left quasi-Bregman relatively nonexpansive. A map T : C → C is called right quasi-Bregman relatively nonexpansive []
if F(T) = ∅, and Df (T(x), p) ≤ Df (x, p) for all x ∈ C and p ∈ F(T). T is said to be quasiBregman strictly pseudocontractive if there exist a constant k ∈ [, ) and F(T) = ∅ such
that Df (p, Tx) ≤ Df (p, x) + kDf (x, Tx) for all x ∈ C and p ∈ F(T). In particular, T is said
to be quasi-Bregman relatively nonexpansive if k =  and T is said to be quasi-Bregman
pseudocontractive if k = .
Very recently, Zhou and Gao [] introduced this deﬁnition of a quasi-strict pseudocontraction related to the function φ and proved the convergence of a hybrid projection algorithm to a ﬁxed point of a closed and quasi-strict pseudocontraction in a smooth and
uniformly convex Banach space. They studied the strong convergence of the following
scheme:
⎧
⎪
x ∈ E,
⎪
⎪
⎪
⎪
⎪
C,
⎨ C = x = C (x ),
⎪
⎪
⎪
Cn+ = {z ∈ Cn : φ(xn , Txn ) ≤
⎪
⎪
⎪
⎩x = n+
Cn+ (x ),

x
–k n
– z, Jxn – JTxn },
where Cn+ is the generalized projection from E onto Cn+ . They proved that the sequence
{xn } converges strongly to F(T) (x ).
Recently, Zegeye and Shahzad [] proved a strong convergence theorem for the common ﬁxed point of a ﬁnite family of right Bregman strongly nonexpansive mappings in
a reﬂexive Banach space. Alghamdi et al. [] proved a strong convergence theorem for
the common ﬁxed point of a ﬁnite family of quasi-Bregman nonexpansive mappings. Pang
et al. [] proved weak convergence theorems for Bregman relatively nonexpansive mappings. Shahzad and Zegeye [] proved a strong convergence theorem for multivalued
Bregman relatively nonexpansive mappings, while Zegeye and Shahzad [] proved a
strong convergence theorem for a ﬁnite family of Bregman weak relatively nonexpansive
mappings.
Motivated and inspired by the above works, in this paper, we prove a new strong convergence theorem for a ﬁnite family of closed quasi-Bregman strictly pseudocontractive
mapping and a system of equilibrium problems in a real reﬂexive Banach space. These
results generalize and improve several recent results. We showed by an example that the
class of quasi-Bregman strictly pseudocontractive mappings is a proper generalization of
the class of quasi-φ-Bregman strictly pseudocontractive mappings.
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2 Preliminaries
The next lemma will be useful in the proof of our main results.
Lemma . Let f : E → R be a Legendre function which is uniformly Fréchet diﬀerentiable
and bounded on subsets of E, let C be a nonempty, closed, and convex subset of E and let
T : C → C be a quasi-Bregman strictly pseudocontractive mapping with respect to f . Then,
for any x ∈ C, p ∈ F(T) and k ∈ [, ) the following hold:
Df (x, Tx) ≤
 ∇f (x) – ∇f (Tx), x – p .
–k
(.)
Proof Let x ∈ C, p ∈ F(T) and k ∈ [, ), by deﬁnition of T, we have
Df (p, Tx) ≤ Df (p, x) + kDf (x, Tx)
and, from (.), we obtain
Df (p, x) + Df (x, Tx) + ∇f (x) – ∇f (Tx), p – x ≤ Df (p, x) + kDf (x, Tx),
which implies
Df (x, Tx) ≤
 ∇f (x) – ∇f (Tx), x – p .
–k
This completes the proof.
Lemma . [] Let E be a real reﬂexive Banach space, f : E → (–∞, +∞] be a proper
lower semi-continuous function, then f ∗ : E∗ → (–∞, +∞] is a proper weak∗ lower semicontinuous and convex function. Thus, for all z ∈ E, we have
Df z, ∇f
∗
N
i=
ti ∇f (xi )
≤
N
ti Df (z, xi ).
(.)
i=
In order to solve the equilibrium problem, let us assume that a bifunction g : C × C → R
satisﬁes the following conditions []:
(A) g(x, x) = , ∀x ∈ C;
(A) g is monotone, i.e., g(x, y) + g(y, x) ≤ , ∀x, y ∈ C;
(A) lim supt↓ g(x + t(z – x), y) ≤ g(x, y), ∀x, z, y ∈ C;
(A) the function y → g(x, y) is convex and lower semi-continuous.
f
The resolvent of a bifunction g [] is the operator Resg : E → C deﬁned by
Resfg (x) = z ∈ C : g(z, y) + ∇f (z) – ∇f (x), y – z ≥ , ∀y ∈ C .
(.)
From Lemma , in [], if f : (–∞, +∞] → R is a strongly coercive and Gâteaux diﬀerenf
tiable function, and g satisﬁes conditions (A)-(A), then dom(Resg ) = E. The following
f
lemma gives some characterization of the resolvent Resg .
Lemma . [] Let E be a real reﬂexive Banach space and C be a nonempty closed convex
subset of E. Let f : E → (–∞, +∞] be a Legendre function. If the bifunction g : C × C → R
satisﬁes the conditions (A)-(A), then the following hold:
Ugwunnadi et al. Fixed Point Theory and Applications 2014, 2014:231
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(i)
(ii)
(iii)
(iv)
(v)
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f
Resg is single-valued;
f
Resg is a Bregman ﬁrmly nonexpansive operator;
f
F(Resg ) = EP(g);
EP(g) is closed and convex subset of C;
f
for all x ∈ E and for all q ∈ F(Resg ), we have
Df q, Resfg (x) + Df Resfg (x), x ≤ Df (q, x).
(.)
3 Main result
Lemma . Let f : E → R be a Legendre function which is uniformly Fréchet diﬀerentiable
on bounded subsets of E, let C be a nonempty, closed, and convex subset of E and let T : C →
C be a quasi-Bregman strictly pseudocontractive mapping with respect to f . Then F(T) is
closed and convex.
Proof Let F(T) be nonempty set. First we show that F(T) is closed. Let {xn } be a sequence
in F(T) such that xn → z as n → ∞, we need to show that z ∈ F(T). From Lemma ., we
obtain
Df (z, Tz) ≤
 ∇f (z) – ∇f (Tz), z – xn .
–k
(.)
From (.), we have Df (z, Tz) ≤ , and from [], Lemma ., it follows that Tz = z. Therefore F(T) is closed.
Next, we show that F(T) is convex. Let z , z ∈ F(T), for any t ∈ (, ); putting z = tz +
( – t)z , we need to show that z ∈ F(T). From Lemma ., we obtain, respectively,
Df (z, Tz) ≤
 ∇f (z) – ∇f (Tz), z – z
–k
(.)
Df (z, Tz) ≤
 ∇f (z) – ∇f (Tz), z – z .
–k
(.)
and
Multiplying (.) by t and (.) by ( – t) and adding the results, we obtain
Df (z, Tz) ≤
 ∇f (z) – ∇f (Tz), z – z ,
–k
(.)
which implies Df (z, Tz) ≤ , and from [], Lemma ., it follows that Tz = z. Therefore
F(T) is also convex. This completes the proof.
We now prove the following theorem.
Theorem . Let C be a nonempty, closed, and convex subset of a real reﬂexive Banach
space E and f : E → R a strongly coercive Legendre function which is bounded, uniformly
Fréchet diﬀerentiable and totally convex on bounded subset of E. For each k = , , . . . , m, let
N
} be a ﬁnite family of
gk be a bifunction from C × C to R satisfying (A)-(A) and let {Ti=
Li -Lipschitzian, i = , , , . . . , N , closed and quasi-Bregman strictly pseudocontractive self
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N
∞
mappings of C such that F := ( m
k= EP(gk )) ∩ ( i= F(Ti )) = ∅. Let {xn }n= be a sequence
generated by x = x ∈ C, C = C and
⎧
⎪
x ∈ C,
⎪
⎪
⎪
⎪
⎪
y = ∇f ∗ (αn ∇f (xn ) + ( – αn )∇f (Tn xn )),
⎪
⎪ n
⎪
f
⎪
⎪
j = , , , . . . , m,
⎨ uj,n = Resg
j yn ,
m
∗
wn = ∇f ( j= βj,n ∇f (uj,n )),
⎪
⎪

⎪
C
⎪
n+ = {w ∈ Cn : Df (xn , wn ) ≤ –k ∇f (xn )
⎪
⎪
⎪
⎪
– ∇f (Tn xn ), xn – w + ∇f (Tn xn ) – ∇f (wn ), xn – w},
⎪
⎪
⎪
⎩ x = Pf (x), n ∈ N,
n+
Cn+
(.)
where Tn = Tn(mod N) , and k ∈ [, ), for each i = , , . . . , N , Ti is uniformly continuous; sup∞
pose {αn }∞
n= and {βj,n }n= , j = , , . . . , m are sequences in (, ) such that (i) lim infn→∞ ( –
m
f
f
αn ) > , (ii) j= βj,n = , n ≥ . Then {xn }∞
n= converges strongly to PF (x), where PF is the
Bregman projection of E onto F.
Proof The proof is divided into six steps.
N
Step I. Show that F = ( m
j= EP(gj )) ∩ ( i= F(Ti )) is closed and convex. From Lemma .,
N
m
i= F(Ti ) is closed and convex and from (iv) of Lemma .,
j= EP(gj ) is closed and conm
N
vex. So, F = ( j= EP(gj )) ∩ ( i= F(Ti )) is closed and convex.
Step II. Show that Cn is closed and convex for all n ≥ . For n = , C = C is closed and
convex. Assume that Ch is closed and convex for some h > . For w ∈ Ch+ , one obtains
Df (xh , wh ) ≤
 ∇f (xh ) – ∇f (Th xh ), xh – w
–k
+ ∇f (Th xh ) – ∇f (wh ), xh – w ;
using the fact that ∇f (xh ) – ∇f (Th xh ), · and ∇f (Th xh ) – ∇f (wh ), · are continuous and
linear in E, for h ≥ , Ch+ is closed and convex.
Step III. Show that F ⊂ Cn for every n ≥ . Note that F ⊂ C = C. Suppose F ⊂ Ch , for
f
h ≥ , then for all w ∈ F ⊂ Ch , since uj,h = Resgj (yh ) for each j = , , . . . , m, from (.) and
Lemma ., we have
Df (w, wh ) = Df w, ∇f
∗
m
βj,n ∇f (uj,n )
j=
≤
m
βjh Df (w, ujh )
j=
≤
m
βjh Df (w, yh )
j=
= Df (w, yh );
also from (.) and (.), we obtain
Df (w, yh ) = Df w, ∇f ∗ αh ∇f (xh ) + ( – αh )∇f (Th xh )
≤ αh Df (w, xh ) + ( – αh )Df (w, Th xh )
(.)
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≤ αh Df (w, xh ) + ( – αh ) Df (w, xh ) + kDf (xh , Th xh )
≤ Df (w, xh ) + kDf (xh , Th xh )
≤ Df (w, xh ) +
k ∇f (xh ) – ∇f (Th xh ), xh – w .
–k
(.)
But, from (.),
Df (w, wh ) = Df (w, xh ) + Df (xh , wh ) + ∇f (xh ) – ∇f (wh ), w – xh .
(.)
From (.), (.), and (.), we obtain
Df (xh , wh ) ≤
k ∇f (xh ) – ∇f (Th xh ), xh – w
–k
+ ∇f (xh ) – ∇f (wh ), xh – w
k ∇f (xh ) – ∇f (Th xh ), xh – w
–k
+ ∇f (xh ) – ∇f (Th xh ), xh – w
+ ∇f (Th xh ) – ∇f (wh ), xh – w
k
=
+  ∇f (xh ) – ∇f (Th xh ), xh – w
–k
+ ∇f (Th xh ) – ∇f (wh ), xh – w
=
=
 ∇f (xh ) – ∇f (Th xh ), xh – w
–k
+ ∇f (Th xh ) – ∇f (wh ), xh – w .
(.)
This shows that w ∈ Ch+ , which implies F ⊂ Cn for every n ≥ .
f
Step IV. Show that limn→∞ Df (xn , x) exists. From (.), xn = PCn x, which from (a) of
Lemma . implies
∇f (x) – ∇f (xn ), y – xn ≤ ,
∀y ∈ Cn .
Since F ⊂ Cn , we have
∇f (x) – ∇f (xn ), w – xn ≤ ,
∀w ∈ F.
(.)
From (b) of Lemma . we have
f
f
Df (xn , x) = Df PCn x, x ≤ Df (w, x) – Df w, PCn x
≤ Df (w, x),
∀n ≥ , w ∈ F.
(.)
This implies that {Df (xn , x)} is bounded, from Lemma ., {xn } is bounded. By the conf
struction of Cn , we have xm ∈ Cm ⊂ Cn , and xn = PCn x, for any positive integer m ≥ n.
Then we obtain
f
f
Df (xm , xn ) = Df xm , PCn x ≤ Df (xm , x) – Df PCn x, x
= Df (xm , x) – Df (xn , x).
(.)
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In particular,
Df (xn+ , xn ) ≤ Df (xn+ , x) – Df (xn , x).
f
f
Since xn = PCn x and xn+ = PCn+ x ∈ Cn+ ⊂ Cn , we obtain Df (xn , x) ≤ Df (xn+ , x), ∀n ≥ .
This shows that {Df (xn , x)} is nondecreasing and hence the limit limn→∞ Df (xn , x) exists.
Thus from (.), taking the limit as m, n → ∞, we obtain limn→∞ Df (xm , xn ) = . Since f
is totally convex on bounded subsets of E, f is sequentially consistent (see []). It follows
that xm – xn →  as m, n → ∞. Hence {xn } is Cauchy sequence in C. As {xn } is Cauchy
in a complete space E, there exists p ∈ E such that xn → p as n → ∞. Clearly p ∈ C.
Since Df (xm , xn ) → , as m, n → ∞, we have in particular
lim Df (xn+ , xn ) = ,
n→∞
(.)
and this further implies that
lim xn+ – xn = .
n→∞
(.)
Step V. Next we show that xn → p ∈ F.
f
Since xn+ = PCn+ x ∈ Cn+ , we have from (.)
Df (xn , wn ) ≤
 ∇f (xn ) – ∇f (Tn xn ), xn – xn+
–k
+ ∇f (Tn xn ) – ∇f (wn ), xn – xn+ ,
(.)
(.)
which implies that limn→∞ Df (xn , wn ) = . Since f is totally convex on bounded subsets of
E, f is sequentially consistent (see []). It follows that
lim xn – wn = .
n→∞
(.)
From (.) and (.), we have
lim xn+ – wn = .
n→∞
(.)
Since f is uniformly Fréchet diﬀerentiable, it follows from Lemma . that ∇f is uniformly
continuous and f is uniformly continuous on bounded subsets of E (see [], Theorem .).
Hence
lim ∇f (xn+ ) – ∇f (wn ) = 
(.)
lim f (xn+ ) – f (wn ) = .
(.)
n→∞
and
n→∞
Since xn+ ∈ Cn+ , it follows from (.), (.) that
f (xn+ ) – f (wn ) – ∇f (wn ), xn+ – wn
= Df (xn+ , wn ) ≤ Df (xn+ , yn ) ≤ Df (xn+ , xn ) +
k ∇f (xn ) – ∇f (Tn xn ), xn – xn+ ,
–k
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which implies from (.), (.), (.), and (.) that
lim Df (xn+ , yn ) = .
n→∞
From the sequential consistency of f , we have
lim xn+ – yn = ;
(.)
n→∞
from (.) and (.), we obtain
lim xn – yn = ,
(.)
n→∞
which implies that yn → p ∈ C, since xn → p ∈ C. From the uniform continuity of ∇f , we
have
lim ∇f (xn ) – ∇f (yn ) = .
n→∞
(.)
From (.), we have
∇f (Tn xn ) – ∇f (xn ) =
 ∇f (xn ) – ∇f (yn ),
 – αn
which implies from (.) that
lim ∇f (Tn xn ) – ∇f (xn ) = .
n→∞
(.)
Since f is strongly coercive and uniformly convex on bounded subsets of E, f ∗ is uniformly
Fréchet diﬀerentiable on bounded sets. Moreover, f ∗ is bounded on bounded sets, and
from (.) we obtain
lim Tn xn – xn = .
n→∞
(.)
On the other hand, we see that
xn – Tn+l xn ≤ xn – xn+l + xn+l – Tn+l xn+l + Tn+l xn+l – Tn+l xn ≤ ( + L)xn – xn+l + xn+l – Tn+l xn+l for all l ∈ {, , . . . , N}, where L := sup≤i≤N Li . It follows from (.) and (.) that
lim xn – Tn+l xn = 
n→∞
for all l ∈ {, , . . . , N}. Thus
lim xn – Tl xn = 
n→∞
(.)
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for all l ∈ {, , . . . , N}. Since xn → p as n → ∞, by the closedness of Tl for each l ∈
{, , . . . , N}, we obtain p ∈ N
l= F(Tl ).
Also, since yn → p as n → ∞, we have from Lemma ., for each j = , , . . . , m,
 ≤ Df (p, ujn ) = Df p, Resfgj yn ≤ Df (p, yn ) →  as n → ∞.
Then we have from Lemma . that limn→∞ p – ujn = , for each j = , , . . . , m. Consequently, we have
ujn – yn ≤ ujn – p + p – yn →  as n → ∞.
(.)
From the uniform continuity of ∇f , for each j = , , . . . , m we have
lim ∇f (ujn ) – ∇f (yn ) = .
(.)
n→∞
From (.), we have, for j = , , . . . , m,
gj (ujn , y) + ∇f (ujn ) – ∇f (yn ), y – ujn ≥ ,
∀y ∈ C.
Furthermore, using (A) in the last inequality, we obtain
∇f (ujn ) – ∇f (yn ), y – ujn ≥ gj (y, ujn ),
∀y ∈ C.
By (A), (.), and ujn → p as n → ∞, we have
gj (y, p) ≤ ,
∀y ∈ C.
(.)
Let zt := ty + ( – t)p for t ∈ (, ] and y ∈ C. This implies that zt ∈ C. This yields gj (zt , p) ≤ .
It follows from (A) and (A) that
 = gj (zt , zt ) ≤ tgj (zt , y) + ( – t)gj (zt , p)
≤ tgj (zt , y),
and hence
 ≤ gj (zt , y).
From condition (A), we obtain
gj (p, y) ≥ ,
∀y ∈ C and ∀j ∈ {, , , . . . , m}.
This implies that p ∈ EP(gj ), for each j = , , . . . , m. Thus, p ∈
m
p∈F =( N
i= F(Ti )) ∩ ( j= EP(gj )).
m
j= EP(gj ).
Hence we have
f
Step VI. Finally, we show that p = PF x. Setting n → ∞ in (.), we obtain
∇f (x) – ∇f (p), w – p ≤ ,
∀w ∈ F.
f
By (a) of Lemma ., we have p = PF x.
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Here we give an example of a quasi-Bregman strictly peudocontractive mapping which
is not quasi-φ strictly pseudocontractive mapping; this shows that the former class is a
generalization of the latter.
Example . Let E = R, C = [–, ] and deﬁne T, f : [–, ] → R by f (x) = x and Tx = x,
for all x ∈ [–, ]. We want to show that T is a quasi-Bregman strictly pseudocontractive
but not quasi-φ strictly pseudocontractive.
Proof From the deﬁnition it is clear that f is proper, lower semi-continuous, and convex,
and also F(T) = {}. By the deﬁnition of quasi-Bregman strict pseudocontractivity, we ﬁnd
k ∈ [, ) such that Df (p, Tx) ≤ Df (p, x) + kDf (x, Tx) for all x ∈ C and p ∈ F(T). Now,
D(, Tx) = f () – f (Tx) – ∇f (Tx),  – Tx
=  – x – ∇f (x),  – x
= –x – , –x
= –x + x = x,
D(, x) = f () – f (x) – ∇f (x),  – x
(.)
=  – x – , –x
= –x + x = 
(.)
and
D(x, Tx) = f (x) – f (Tx) – ∇f (Tx), x – Tx
= x – x – , x – x
= –x + x = x.
(.)
From (.), (.), and (.), we obtain
D(, Tx) = x ≤ x
≤  + kx,
∀x ∈ [–, ], k ∈ [, )
≤ D(, x) + kD(x, Tx),
∀x ∈ [–, ], k ∈ [, ).
D(, Tx) ≤ D(, x) + kD(x, Tx),
∀x ∈ [–, ], k ∈ [, ).
Therefore
Hence, T is a quasi-Bregman strictly pseudocontractive map.
Further,
φ(, Tx) = || –  , J(Tx) + |Tx|
=  –  , J(x) + |x|
= |x| ,
(.)
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φ(, x) = || –  , J(x) + |x|
=  –  , J(x) + |x|
= |x|
(.)
and
φ(x, Tx) = |x| –  x, J(Tx) + |Tx|
= |x| –  x, J(x) + |x|
= |x| –  x, J(x) + |x|
= |x| – |x| + |x|
= |x| .
(.)
Since |x| > |x| + k|x| , for all k ∈ [, ) and for all x ∈ [–, ],
φ(, Tx) ≤ φ(, x) + kφ(x, Tx),
∀x ∈ [–, ]
cannot hold for any k ∈ [, ). Hence, T is not a quasi-φ strictly pseudocontractive map.
4 Numerical example
In this section we discuss the direct application of Theorem . on a typical example on a
real line. Consider the following:
E = R,
g(z, y) = y + yz – z ,
C = [–, ],


Tx = –x,
∇f (x) = x,
f (x) = x ,


f ∗ x∗ = sup x∗ , x – f (x) : x ∈ E ,

f ∗ (z) = z ,

∇f ∗ (z) =

z,

αn ∇f (xn ) + ( – αn )∇f (Txn ) =
αn =
n+
,
n
–(n – )
xn ,
n
x = / ∈ C,
k = /,
then the scheme can be simpliﬁed as
yn = ∇f ∗ αn ∇f (xn ) + ( – αn )∇f (Txn ) ,
–(n – )
xn ,
n

un = Resfg yn = yn ,

wn = ∇f ∗ ∇f (un ) = un ,
Cn+ = w ∈ Cn : w ≤ xn –
∴ yn =
f
xn+ = PCn+ (x ) = xn –
( – k)(xn – wn )
,
[( + k)xn – ( – k)wn ]
( – k)(xn – wn )
.
[( + k)xn – ( – k)wn ]
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.
Author details
1
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria. 2 Department of
Mathematics, Ahmadu Bello University, Zaria, Nigeria. 3 Department of Mathematical Sciences, Bayero University Kano,
P.M.B. 3011, Kano, Nigeria.
Received: 18 March 2014 Accepted: 16 October 2014 Published: 17 Nov 2014
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Cite this article as: Ugwunnadi et al.: Strong convergence theorem for quasi-Bregman strictly pseudocontractive
mappings and equilibrium problems in Banach spaces. Fixed Point Theory and Applications 2014, 2014:231
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