http://jipam.vu.edu.au/

Volume 7, Issue 2, Article 66, 2006

**GENERALIZED CO-COMPLEMENTARITY PROBLEMS IN**p-UNIFORMLY
**SMOOTH BANACH SPACES**

M. FIRDOSH KHAN AND SALAHUDDIN DEPARTMENT OFMATHEMATICS

ALIGARHMUSLIMUNIVERSITY

ALIGARH-202002, INDIA

khan_mfk@yahoo.com

*Received 14 October, 2005; accepted 30 December, 2005*
*Communicated by R.U. Verma*

ABSTRACT. The objective of this paper is to study the iterative solutions of a class of generalized co-complementarity problems inp-uniformly smooth Banach spaces, with the devotion of sunny retraction mapping,p-strongly accretive,p-relaxed accretive and Lipschitzian (or more generally uniformly continuous) mappings. Our results are new and represents a significant improvement of previously known results. Some special cases are also discussed.

*Key words and phrases: Generalized co-complementarity problems, Iterative algorithm, Sunny retraction, Sunny nonexpan-*
sive mapping,p-strongly accretive,p-relaxed accretive mapping, Lipschitzian mapping, Hausdorff
metric,p-uniformly smooth Banach spaces.

*2000 Mathematics Subject Classification. 49J40, 90C33, 47H10.*

**1. I****NTRODUCTION**

The theory of complementarity problems initiated by Lemke [19] and Cottle and Dantzing [10] in the early sixties and later developed by other mathematicians see for example [6, 9, 11, 14, 17, 22] plays an important role and is fundamental in the study of a wide class of problems arising in optimization, game theory, economics and engineering sciences [3, 6, 8, 11, 15].

On the other hand, the accretive operators are of interest because several physically resolvent problems can be modeled by nonlinear evolution systems involving operators of the accretive type. Very closely related to the accretive operators is the class of dissipative operators, where an operatorT is said to be dissipative if and only if(−T)is accretive. The concepts of strictly strongly andm-(or sometimes hyper-) dissipativity are similarly defined.

These classes of operators have attracted a lot of interest because of their involvement in evolution systems modeling several real life problems. Consequently several authors have stud- ied the existence, uniqueness and iterative approximations of solutions of nonlinear equations involving such operators, see [5, 12, 18] and the references cited therein.

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

011-06

It is our purpose in this article to establish the strong convergence of the iterative algorithm to a solution of the generalized co-complimentarity problems in p−uniformly smooth Banach spaces when the operators are accretive, strictly accretive, strongly accretive, relaxed accretive and Lipschitzian. Our iteration processes are simple and independent of the geometry of E and iteration parameters can be chosen at the start of the iteration process. Consequently, most important results known in this connection will be special cases of our problem.

**2. B****ACKGROUND OF****P****ROBLEM****F****ORMULATIONS**

Throughout this article, we assume thatE is a real Banach space whose norm is denoted by
k · k, E^{?} its topological dual space. CB(E) denotes the family of all nonempty closed and
bounded subsets ofE. D(·,·)is the Hausdorff metric onCB(E)defined by

max

sup

x∈A

d(x, B), sup

y∈B

d(A, y)

=D(A, B), where

d(x, B) = inf

y∈Bd(x, y) and d(A, y) = inf

x∈Ad(x, y),

dis the metric onE induced by the normk · k. As usual,h·,·iis the generalized duality pairing
betweenE andE^{?}. For1< p <∞, the mappingJ_{p} :E →2^{E}^{?} defined by

J_{p}(x) = {f^{?} ∈E^{?} :hx, f^{?}i=kfk · kxkandkfk=kxk^{p−1}} for allx∈E,

is called the duality mapping with gauge function φ(t) = t^{p−1}. In particular for p = 2, the
duality mappingJ_{2}with gauge functionφ(t) =tis called the normalized duality mapping. It is
known thatJ_{p}(x) =kxk^{p−2}J_{2}(x)for allx 6= 0andJ_{p} is single valued ifE^{?} is strictly convex.

IfE =H is a Hilbert space, thenJ_{2}becomes the identity mapping onH.

* Proposition 2.1 ([7]). Let*E

*be a real Banach space. For*1 < p < ∞, the duality mapping J

_{p}:E →2

^{E}

^{?}

*has the following basic properties:*

(1) J_{p}(x)6=∅*for all*x∈E*and*D(J_{p})*(the domain of*J_{p}*)*=E,
(2) J_{p}(x) =kxk^{p−2}J_{2}(x)*for all*x∈E,(x6= 0),

(3) Jp(αx) = α^{p−1}Jp(x)*for all*α∈[0,∞),
(4) Jp(−x) =−Jp(x),

(5) J_{p} *is bounded i.e., for any bounded subset*A⊂E,J_{p}(A)*is a bounded subset in*E^{?}*,*
(6) J_{p} *can be equivalently defined as the subdifferential of the functional*ϕ(x) = p^{−1}kxk^{p}

*(Asplund [2]), i.e.,*

J_{p}(x) =∂ϕ(x) ={f ∈E^{?} : ϕ(y)−ϕ(x)≥(f, y−x), *for all*y∈E}

(7) E *is a uniformly smooth Banach space (equivalently,*E^{?} *is a uniformly convex Banach*
*space) if and only if*Jp*is single valued and uniformly continuous on any bounded subset*
*of*E*(Xu and Roach [24]).*

**Definition 2.1. Let**Ebe a real Banach space andK a nonempty subset ofE. LetT :K →2^{E}
be a multivalued mapping

(1) T is said to be accretive if for any x, y ∈ K, u ∈ T(x) and v ∈ T(y) there exists
j_{2} ∈J_{2}(x−y)such that

hu−v, j_{2}i ≥0,

or equivalently, there existsj_{p} ∈J_{p}(x−y),1< p <∞, such that

(2) T is said to be strongly accretive if for anyx, y ∈ K, u ∈ T(x) and v ∈ T(y)there
existsj_{2} ∈J_{2}(x−y)such that

hu−v, j_{2}i ≥kkx−yk^{2},

or equivalently, there existsj_{p} ∈J_{p}(x−y),1< p <∞such that
hu−v, j_{p}i ≥kkx−yk^{p},

for some constantk > 0.

The concept of a single-valued accretive mapping was introduced independently by Browder [5] and Kato [18] in 1967. An early fundamental result in the theory of accretive mappings which is due to Browder states that the following initial value problem,

du(t)

dt +T u(t) = 0, u(0) =u0, is solvable ifT is locally Lipschitzian and accretive onE.

More precisely, letN : E×E → E andm, g : E →E be the single-valued mappings and F, G, T : E → CB(E)the multivalued mappings. LetX be a fixed closed convex cone ofE.

DefineK :E →2^{E} by

K(z) = m(z) +X for allx∈E, z∈T(x).

**We shall study the following generalized co-complementarity problem (GCCP):**

Findx∈E,u∈F(x),v ∈G(x),z ∈T(x)such thatg(x)∈K(z)and

(2.1) N(u, v)∈(J(K(z)−g(x)))^{?},

where(J(K(z)−g(x)))^{?} is the dual cone of the setJ(K(z)−g(x)).

**2.1. Special Cases.**

(i) IfEis a Hilbert space,F, T are identity mappings andN(u, v) = Bx+Av, whereB, A are single-valued mappings, then Problem (2.1) reduces to a problem of findingx∈E, v ∈G(x)such thatg(x)∈K(z)and

(2.2) B(x) +A(v)∈(K(x)−g(x))^{?}

considered by Jou and Yao [16].

(ii) IfGandg are identity mappings, then (2.2) reduces to findingx∈K(x)such that

(2.3) B(x) +A(y)∈(K(x)−x)^{?}

which is called a strongly nonlinear quasi complementarity problem, studied by Noor [22].

(iii) Ifmis a zero mapping, then (2.3) is equivalent to findingx∈E such that

(2.4) Bx+Ax∈E^{?} and hBx+Ax, xi= 0,

which is known as the mildly nonlinear complementarity problem, studied by Noor [21].

(iv) IfA is zero mapping, then (2.4) is equivalent to a problem of findingx ∈ E such that
Bx∈E^{?} and

(2.5) hBx, xi= 0,

considered by Habetler [14] and Karamardian [17].

**3. T****HE****C****HARACTERIZATION OF****P****ROBLEM AND****S****OLUTIONS**

In this section, we briefly consider some basic concepts and results, which will be used
throughout the paper. The real Banach spaceEis said to be uniformly smooth if its modulus of
smoothnessρ_{E}(τ)defined by

ρ_{E}(τ) = sup

kx+yk+kx−yk

2 −1 : kxk= 1,kyk ≤τ

satisfies ^{ρ}^{E}_{τ}^{(τ)} → 0 as τ → 0. It follows that E is uniformly smooth if and only if J_{p} is
single-valued and uniformly continuous on any bounded subset ofEand there exists a complete
duality between uniform convexity and uniform smoothness. E is uniformly convex (smooth)
if and only if E^{?} is uniformly smooth (convex). Recall that E is said to have the modulus of
smoothness of power type p > 1 (and E is said to bep-uniformly smooth) if there exists a
constantc >0such that

ρ_{E}(τ)≤cτ^{p} for0< τ <∞.

**Remark 3.1. It is known that all Hilbert spaces and Banach spaces,**L_{p}, l_{p}andW_{m}^{p} (1< p < ∞)
are uniformly smooth and

ρ_{E}(τ)<

1

p τ^{p}, 1< p <2

(E =L_{p}, l_{p}orW_{m}^{p})
p−1

2 τ^{2}, p≤2

thereforeE is ap-uniformly smooth Banach space with modulus of smoothness of power type
p <1andJ_{p}will always represent the single-valued duality mapping.

**Definition 3.1 ([4, 13]). Let** E be a p-uniformly smooth Banach space and let Ωa nonempty
closed convex subset ofE. A mappingQ_{Ω} :E →Ωis said to be

(i) retraction onΩifQ^{p}_{Ω} =Q_{Ω};

(ii) nonexpansive retraction if it satisfies the inequality

kQ_{Ω}(x)−Q_{Ω}(y)k ≤ kx−yk, for allx, y ∈E;
(iii) sunny retraction if for allx∈Eand for all−∞< t <∞

Q_{Ω}(Q_{Ω}(x) +t(x−Q_{Ω}(x)) =Q_{Ω}(x).

The following characterization of a sunny nonexpansive retraction mapping can be found in [4, 13].

**Lemma 3.2 ([4, 13]).** Q_{Ω}*is sunny nonexpansive retraction if and only if for all*x, y ∈E,
hx−Q^{?}_{Ω}, J(QΩ x−y)i ≥0.

* Lemma 3.3. Let*E

*be a real Banach space and*J

_{p}:E →2

^{E}

^{?}

*,*1< p < ∞

*a duality mapping.*

*Then, for any given*x, y ∈E, we have

kx+yk^{p} ≤ kxk^{p}+phy, j_{p}i, *for all*j_{p} ∈J_{p}(x+y).

*Proof. From Proposition 2.1, it follows that* J_{p}(x) = ∂ϕ(x) (subdifferential of ψ), where
ψ(x) =p^{−1}kxk^{p}. Also, it follows from the definition of the subdifferential ofψ that

ψ(x)−ψ(x+y)≥ hx−(x+y), j_{p}i,
for allj_{p} ∈J_{p}(x+y). Substitutingψ(x)byp^{−1}kxk^{p}, we have

kx+yk^{p} ≤ kxk^{p}+phy, jpi, for alljp ∈Jp(x+y).

This completes the proof.

* Theorem 3.4 ([9]). Let* E

*be a Banach space,*Ω

*a nonempty closed convex subset of*E, and m:E →E. Then for allx, y ∈E, we have

Q_{Ω+m(z)}(x) =m(z) +Q_{Ω}(x−m(z)).

We mention the following characterization theorem for the solution of a generalized co- complementarity problem which can be easily proved by using Lemma 3.2 and the argument of [1, Theorem 8.1].

* Theorem 3.5. Let*E

*be a real*p-uniformly smooth Banach space andX

*a closed convex cone*

*in*E. Let F, G, T : E → CB(E)

*be the multivalued mappings,*m, g : E → E

*the two*

*single-valued mappings and*N : E×E → E

*the nonlinear mapping. Let*K : E → 2

^{E}

*and*K(z) =m(z) +X

*for*x∈E. Then the following statements are equivalent:

(i) x ∈ E, u ∈ F(x), v ∈ G(x) *and* z ∈ T(x)*are solutions of the Problem (2.1), i.e.,*
g(x)∈K(z)*and*

N(u, v)∈(J(K(z)−g(x)))^{?}.
(ii) x∈E,u∈F(x),v ∈G(x)*and*z ∈T(x)*and*τ >0

g(x) =Q_{K(z)}[g(x)−τ N(u, v)].

Combining Theorem 3.4 and 3.5, we have the following result.

* Theorem 3.6. Let*E

*be a*p-uniformly smooth Banach space andX

*a closed convex cone in*B.

*Let*m, g:E →E *be the two single-valued mappings,*F, G, T :E → CB(E)*the multivalued*
*mappings and* N : E × E → E *a nonlinear mapping. Then the following statements are*
*equivalent:*

(i) x∈E,u∈F(x),v ∈G(x)*and*z ∈T(x)*are solutions of the Problem (2.1),*
(ii) x=x−g(x) +m(z) +Q_{X}[g(x)−τ N(u, v)−m(z)],*for some*τ > 0.

The following inequality will be used in our main results.

* Lemma 3.7. Let*E

*be a real Banach space and*j

_{p}: E →2

^{E}

^{?}

*,*1< p < ∞

*a duality mapping.*

*Then, for any given*x, y ∈E, we have

hx−y, j_{p}(x)−j_{p}(y)i ≤2d^{p}ρ_{E}

4kx−yk d

,
*where*

d^{p} =

kxk^{2}+kyk^{2}
2

.

*Proof. The proof of the above inequalities are the generalized form of the proof of Theorem*

3.4, and hence will be omitted.

**4. I****TERATIVE****A****LGORITHMS AND** **P****ERTINENT****C****ONCEPTS**

We now propose the following iterative algorithm for computing the approximate solution of
**(GCCP).**

**Algorithm 4.1. Let**g, m:E →E be the two single-valued mappings,F, G, T :E →CB(E)
the multivalued mappings andN :E×E →Ea nonlinear mapping.

For any givenx_{0} ∈E,u_{0} ∈F(x_{0}),v_{0} ∈G(x_{0})andz_{0} ∈T(x_{0}), let

x_{1} =x_{0}−g(x_{0}) +m(z_{0}) +Q_{X}[g(x_{0})−τ N(u_{0}, v_{0})−m(z_{0})]

whereτ > 0is a constant.

Sinceu_{0} ∈F(x_{0})∈ CB(E),v_{0} ∈G(x_{0})∈CB(E)andz_{0} ∈T(x_{0})∈CB(E), by Nadler’s
Theorem [20], there existsu_{1} ∈F(x_{1}),v_{1} ∈G(x_{1})andz_{1} ∈T(x_{1})such that

ku_{0}−u_{1}k ≤(1 + 1)D(F(x_{0}), F(x_{1})),
kv_{0} −v_{1}k ≤(1 + 1)D(G(x_{0}), G(x_{1})),
kz_{0}−z_{1}k ≤(1 + 1)D(T(x_{0}), T(x_{1})),
whereDis a Hausdorff metric onCB(E).

Let

x_{2} =x_{1}−g(x_{1}) +m(z_{1}) +Q_{X}[g(x_{1})−τ N(u_{1}, v_{1})−m(z_{1})].

Sinceu_{1} ∈ F(x_{1}) ∈ CB(E), v_{1} ∈ G(x_{1}) ∈ CB(E)andz_{1} ∈ T(x_{1}) ∈ CB(E), there exists
u_{2} ∈F(x_{2}),v_{2} ∈G(x_{2})andz_{2} ∈T(x_{2})such that

ku_{1} −u_{2}k ≤(1 + 2^{−1})D(F(x_{1}), F(x_{2})),
kv_{1}−v_{2}k ≤(1 + 2^{−1})D(G(x_{1}), G(x_{2})),
kz1−z2k ≤(1 + 2^{−1})D(T(x1), T(x2)).

By induction, we can obtain{x_{n}},{u_{n}},{v_{n}}and{z_{n}}as

(4.1) x_{n+1} =x_{n}−g(x_{n}) +m(z_{n}) +Q_{X}[g(x_{n})−τ N(u_{n}, v_{n})−m(z_{n})].

un ∈F(xn);ku_{n}−un+1k ≤(1 + (1 +n)^{−1})D(F(xn), F(xn+1)),
v_{n}∈G(x_{n});kv_{n}−v_{n+1}k ≤(1 + (1 +n)^{−1})D(G(x_{n}), G(x_{n+1})),
z_{n}∈T(x_{n});kz_{n}−z_{n+1}k ≤(1 + (1 +n)^{−1})D(T(x_{n}), T(x_{n+1})),
n≥0, whereτ > 0is a constant.

These iteration processes have been extensively investigated by various authors for approx- imating either the fixed point of nonlinear mappings or solutions of nonlinear equations in Banach spaces or variational inequalities, variational inclusions, or complementarity problems in Hilbert spaces.

**Definition 4.1. A single valued mapping**g :E →E is said to be

(i) p-strongly accretive if for allx, y ∈E there existsjp ∈Jp(x−y)such that
hg(x)−g(y), j_{p}(x−y)i ≥κkx−yk^{p}

for some real constantκ∈(0,1)and1< p <∞.

(ii) Lipschitz continuous if for anyx, y ∈E, there exists constantβ >0, such that kg(x)−g(y)k ≤βkx−yk.

**Definition 4.2. A multivalued mapping**F :E →CB(E)is said to beD-Lipschitz continuous
if for anyx, y ∈E,

D(F(x), F(y))≤µkx−yk forµ > 0andD(·,·)is Hausdorff metric defined onCB(E).

**Definition 4.3. Let** F : E → CB(E) be a multivalued mapping. A nonlinear mappingN :
E×E →Eis said to be relaxed accretive with respect to the first argument of mapF, if there
exists a constantα >0such that

hN(u_{n},·)−N(u_{n−1},·), j_{p}(x_{n}−x_{n−1})i ≥ −αkx_{n}−x_{n−1}k^{p};
andN is Lipschitz continuous with respect to the first argument if

kN(u,·)−N(y,·)k ≤σkx−yk, forx, y ∈E,

whereσ > 0is a constant.

Similarly, we define the Lipschitz continuity ofN with respect to second argument.

**5. M****AIN****R****ESULTS**

In this section, we show that if E is a p-uniformly smooth Banach space, then the iterative process converges strongly to the given problem (2.1).

* Theorem 5.1. Let*E

*be a*p-uniformly smooth real Banach space withρ

_{E}(τ) ≤ cτ

^{p}

*for some*c > 0, 0< τ < ∞

*and*1 < p < ∞. LetX

*be a closed convex cone of*E.

*Let*m, g :E → E

*be the two single-valued mappings,*F, G, T : E → CB(E)

*the multivalued mappings. Let*K :E →2

^{E}

*such that*K(z) =m(z) +X

*for all*x∈E,z ∈T(x)

*and the following conditions*

*hold.*

(i) g *and*m*are Lipschitz continuous;*

(ii) g *is strongly accretive;*

(iii) F, G*and*T *are*D-Lipschitz continuous;

(iv) N *is Lipschitz continuous with respect to the first as well as the second argument;*

(v) N *is* p-relaxed accretive with respect to the first argument with mapping F : E →
CB(E);

(vi)

q+ (1 +pατ+pc2^{2p+1}τ^{p}σ^{p}η^{p})^{1/p}+τ δξ <1
*and*

(5.1) q = 2(1−pκ+c2^{2p+1}pβ^{p})^{1/p}+ 2µρ.

*Then for any*x_{0} ∈E,u_{0} ∈F(x_{0}),v_{0} ∈G(x_{0})*and*z_{0} ∈T(x_{0})*the sequences*x_{n}, u_{n}, v_{n}
*and* z_{n} *generated by Algorithm 4.1, converge strongly to some* x ∈ E, u ∈ F(x),
v ∈G(x)*and*z ∈T(x), which solve the problem (2.1).

*Proof. By the iterative schemes (4.1) and Definition 3.1, we have*
kx_{n+1}−x_{n}k

=kx_{n}−g(x_{n}) +m(z_{n}) +Q_{X}[g(x_{n})−τ N(u_{n}, v_{n})−m(z_{n})]

− x_{n−1}+g(x_{n−1})−m(z_{n−1})−Q_{X}[g(x_{n−1})−τ N(u_{n−1}, v_{n−1})−m(z_{n−1})]k

≤ kxn−xn−1−(g(xn)−g(xn−1))k+km(zn)−m(zn−1k
+kQ_{X}[g(x_{n})−τ N(u_{n}, v_{n})−m(z_{n})]

−Q_{X}[g(xn−1)−τ N(un−1, vn−1)−m(zn−1)]k

≤2kx_{n}−xn−1 −(g(x_{n})−g(xn−1))k+ 2km(z_{n})−m(zn−1)k
+kx_{n}−x_{n−1}−τ(N(u_{n}, v_{n})−N(u_{n−1}, v_{n−1}))k

≤2kxn−xn−1 −(g(xn)−g(xn−1))k+ 2km(zn)−m(zn−1)k
+kx_{n}−xn−1−τ(N(u_{n}, v_{n})−N(un−1, v_{n}))k

(5.2)

+τkN(un−1, v_{n})−N(un−1, vn−1)k.

By Lemmas 3.3, 3.7, p-strongly accretive, Lipschitz continuity ofg and j_{p} ∈ J_{p}(x+y), we
have

kx_{n}−x_{n−1}−(g(x_{n})−g(x_{n−1}))k^{p}

≤ kxn−xn−1k^{p}+ph−(g(xn)−g(xn−1)), jpi

≤ kx_{n}−xn−1k^{p}−phg(x_{n})−g(xn−1), j_{p}(x_{n}−xn−1−(g(x_{n})−g(xn−1))i

≤ kx_{n}−xn−1k^{p}−phg(x_{n})−g(xn−1), j_{p}(x_{n}−xn−1)i

−phg(x_{n})−g(x_{n−1}), j_{p}(x_{n}−x_{n−1}−(g(x_{n})−g(x_{n−1})))−j_{p}(x_{n}−x_{n−1})i

≤ kx_{n}−xn−1k^{p}−p^{k}kx_{n}−xn−1k^{p}+ 2pd^{p}ρ_{E}

4kg(x_{n})−g(xn−1)k
d

≤ kxn−xn−1k^{p}−p^{k}kxn−xn−1k^{p}+ 2pd^{p}c4^{p}kg(x_{n})−g(x_{n−1})k^{p}
d^{p}

≤ kx_{n}−xn−1k^{p}−pkkx_{n}−xn−1k^{p}+ 2^{2p+1}cpβ^{p}kx_{n}−xn−1k^{p}

≤(1−pk+c2^{2p+1}pβ^{p})kxn−xn−1k^{p}.
(5.3)

By the Lipschitz continuity ofmandD-Lipschitz continuity ofT, we have
km(z_{n})−m(zn−1)k ≤µkz_{n}−zn−1k

≤µ(1 +n^{−1})D(T(x_{n}), T(xn−1))

≤µ(1 +n^{−1})ρkxn−xn−1k.

(5.4)

SinceF is η-Lipschitz continuous, Gis ξ-Lipschitz continuous and N is Lipschitz contin-
uous with respect to the first and second arguments with positive constants σ and δ respec-
tively. Using a similar argument to that of Xiaolin He [23], we have for everyun, u^{0}_{n} ∈F(xn),
N(un, v) = N(u^{0}_{n}, v). On the other handun−1 ∈F(xn−1), and from the definition of Hausdorff
metric and compactness ofF(x_{n}), there is au^{0}_{n} ∈F(x_{n})such that

ku^{0}_{n}−un−1k ≤D(F(xn), F(xn−1)).

Hence

kN(u_{n}, v_{n})−N(un−1, v_{n})k=kN(u^{0}_{n}, v_{n})−N(un−1, v_{n})k

≤σku^{0}_{n}−un−1k

≤σ(1 +n^{−1})D(F(xn), F(xn−1))

≤σ(1 +n^{−1})ηkx_{n}−x_{n−1}k.

(5.5)

Similarly, we get

(5.6) kN(un−1, v_{n})−N(un−1, vn−1)k ≤δξ(1 +n^{−1})kx_{n}−xn−1k.

By using Lemma 3.7, thep-relaxed accretive mapping and (5.5), we have
kx_{n}−xn−1−τ(N(u_{n}, v_{n})−N(un−1, v_{n}))k^{p}

≤ kx_{n}−xn−1k^{p}

−pτhN(u_{n}, v_{n})−N(u_{n−1}, v_{n}), j_{p}(x_{n}−x_{n−1}−τ(N(u_{n}, v_{n})−N(u_{n−1}, v_{n})))i

≤ kx_{n}−xn−1k^{p}−pτhN(u_{n}, v_{n})−N(un−1, v_{n}), j_{p}(x_{n}−xn−1)i

−phτ(N(u_{n}, v_{n})−N(un−1, v_{n})),

j_{p}(x_{n}−xn−1−τ(N(u_{n}, v_{n})−N(un−1, v_{n})))−j_{p}(x_{n}−xn−1)i

≤ kx_{n}−xn−1k^{p}+pτ αkx_{n}−xn−1k^{p}+ 2pd^{p}ρ_{E}

τ4kN(u_{n}, v_{n})−N(un−1, v_{n})k
d

≤ kx_{n}−xn−1k^{p}+pατkx_{n}−xn−1k^{p}+ 2pτ^{p}c4^{p}kN(u_{n}, v_{n})−N(un−1, v_{n})k^{p}

≤ kx_{n}−xn−1k^{p}+pατkx_{n}−xn−1k^{p}+pτ^{p}c2^{2p+1}(1 +n^{−1})^{p}σ^{p}η^{p}kx_{n}−xn−1k^{p}

≤(1 +pατ +pc2^{2p+1}(1 +n^{−1})^{p}τ^{p}σ^{p}η^{p})kx_{n}−xn−1k^{p}.
(5.7)

Now from (5.2) – (5.7), we get

kx_{n}−xn−1k ≤2(1−pk+c2^{2p+1}pβ^{p})^{1/p}kx_{n}−xn−1k+ 2µρ(1 +n^{−1})kx_{n}−xn−1k
+ (1 +pατ+pc2^{2p+1}(1 +n^{−1})^{p}τ^{p}σ^{p}η^{p})^{1/p}kxn−xn−1k

+τ δξ(1 +n^{−1})kx_{n}−x_{n−1}k

≤[2(1−pk+c2^{2p+1}pβ^{p})^{1/p}+ 2µρ(1 +n^{−1})

+ (1 +pατ+pc2^{2p+1}(1 +n^{−1})^{p}τ^{p}σ^{p}η^{p})^{1/p}+τ δξ(1 +n^{−1})]

≤[q_{n}+ (1 +pατ +pc2^{2p+1}(1 +n^{−1})^{p}τ^{p}σ^{p}η^{p})^{1/p}
+τ δξ(1 +n^{−1})]kx_{n}−xn−1k

≤θ_{n}kx_{n}−xn−1k,
(5.8)

where

θ_{n}=q_{n}+ (1 +pατ+pc2^{2p+1}(1 +n^{−1})^{p}τ^{p}σ^{p}η^{p})^{1/p}+τ δξ(1 +n^{−1})
and

(5.9) qn = 2(1−pk+c2^{2p+1}pβ^{p})^{1/p}+ 2µρ 1 +n^{−1}
.
Letting

(5.10) θ =q+ (1 +pατ +pc2^{2p+1}τ^{p}σ^{p}η^{p})^{1/p}+τ δξ
and

q= 2(1−pk+c2^{2p+1}pβ^{p})^{1/p}+ 2µρ.

We know thatθn→θasn→ ∞. From condition (5.1), it follows thatθ <1. Henceθn<1,
forn sufficiently large. Consequently{x_{n}}is a Cauchy sequence and this converges to some
x∈E. By Algorithm 4.1 and theD-Lipschitz continuity ofF, GandT, it follows that

ku_{n}−un−1k ≤(1 +n^{−1})D(F(x_{n}), F(xn−1))

≤(1 +n^{−1})ηkx_{n}−xn−1k,
kv_{n}−vn−1k ≤(1 +n^{−1})D(G(x_{n}), G(xn−1))

≤(1 +n^{−1})ξkx_{n}−xn−1k,
kz_{n}−zn−1k ≤(1 +n^{−1})D(T(x_{n}), T(xn−1))

≤(1 +n^{−1})ρkxn−xn−1k,

which means that{u_{n}}, {v_{n}} and{z_{n}}are all Cauchy sequences inE. Therefore there exist
u ∈ E, v ∈ E and z ∈ E such that u_{n} → u, v_{n} → v and z_{n} → z as n → ∞. Since
g, m, F, G, T, N andQ_{X} are all continuous, we have

x=x−g(x) +Q_{X}[g(x)−τ N(u, v)−m(z)].

Finally, we prove thatu∈F(x). In fact, sinceu_{n}∈F(x_{n})and
d(un, F(x))≤max

(

d(un, F(x)), sup

u∈F(x)

d(T(xn), u) )

≤max (

sup

y∈F(xn)

d(y, F(x)), sup

u∈F(x)

d(F(x_{n}), u)
)

=D(F(x_{n}), F(x)),

We have

d(u, F(x))≤ ku−unk+d(un, F(x))

≤ ku−u_{n}k+D(F(x_{n}), F(x))

≤ ku−u_{n}k+ηkx_{n}−xk →0 asn→ ∞,

which implies thatd(u, F(x)) = 0. SinceF(x)∈CB(E), it follows thatu∈F(x). Similarly, we can prove thatG(x)∈CB(E)i.e.,v ∈G(x)andT(x)∈CB(E)i.e.,z ∈T(x). Hence by

Theorem 3.6, we get the conclusion.

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