Wei et al. Journal of Inequalities and Applications (2017) 2017:227 DOI 10.1186/s13660-017-1506-9
RESEARCH
Open Access
Modified forward-backward splitting midpoint method with superposition perturbations for the sum of two kinds of infinite accretive mappings and its applications Li Wei1*† , Liling Duan1† , Ravi P Agarwal2† , Rui Chen1† and Yaqin Zheng3† *
Correspondence:
[email protected] 1 School of Mathematics and Statistics, Hebei University of Economics and Business, 47 Xuefu Road, Shijiazhuang, China Full list of author information is available at the end of the article † Equal contributors
Abstract In a real uniformly convex and p-uniformly smooth Banach space, a modified forward-backward splitting iterative algorithm is presented, where the computational errors and the superposition of perturbed operators are considered. The iterative sequence is proved to be convergent strongly to zero point of the sum of infinite m-accretive mappings and infinite θi -inversely strongly accretive mappings, which is also the unique solution of one kind variational inequalities. Some new proof techniques can be found, especially, a new inequality is employed compared to some of the recent work. Moreover, the applications of the newly obtained iterative algorithm to integro-differential systems and convex minimization problems are exemplified. Keywords: p-uniformly smooth Banach space; θi -inversely strongly accretive mapping; γi -strongly accretive mapping; μi -strictly pseudo-contractive mapping; perturbed operator
1 Introduction and preliminaries Let X be a real Banach space with norm · and X ∗ be its dual space. ‘→’ denotes strong convergence and x, f is the value of f ∈ X ∗ at x ∈ X. The function ρX : [, +∞) → [, +∞) is called the modulus of smoothness of X if it is defined as follows: x + y + x – y – : x, y ∈ X, x = , y ≤ t . ρX (t) = sup A Banach space X is said to be uniformly smooth if ρXt(t) → , as t → . Let p > be a real number, a Banach space X is said to be p-uniformly smooth with constant Kp if Kp > such that ρX (t) ≤ Kp t p for t > . It is well known that every p-uniformly smooth ∗ Banach space is uniformly smooth. For p > , the generalized duality mapping Jp : X → X © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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is defined by Jp x := f ∈ X ∗ : x, f = xp , f = xp– ,
x ∈ X.
In particular, J := J is called the normalized duality mapping. For a mapping T : D(T) X → X, we use F(T) and N(T) to denote its fixed point set and zero point set, respectively; that is, F(T) := {x ∈ D(T) : Tx = x} and N(T) = {x ∈ D(T) : Tx = }. The mapping T : D(T) X → X is said to be () non-expansive if Tx – Ty ≤ x – y
for ∀x, y ∈ D(T);
() contraction with coefficient k ∈ (, ) if Tx – Ty ≤ kx – y
for ∀x, y ∈ D(T);
() accretive [, ] if for all x, y ∈ D(T), Tx – Ty, j(x – y) ≥ , where j(x – y) ∈ J(x – y); m-accretive if T is accretive and R(I + λT) = X for ∀λ > ; () θ -inversely strongly accretive [] if for θ > , ∀x, y ∈ D(T), there exists jp (x – y) ∈ Jp (x – y) such that Tx – Ty, jp (x – y) ≥ θ Tx – Typ
for ∀x, y ∈ X;
() γ -strongly accretive [, ] if for each x, y ∈ D(T), there exists j(x – y) ∈ J(x – y) such that Tx – Ty, j(x – y) ≥ γ x – y for some γ ∈ (, ); () μ-strictly pseudo-contractive [] if for each x, y ∈ X, there exists j(x – y) ∈ J(x – y) such that
Tx – Ty, j(x – y) ≤ x – y – μ x – y – (Tx – Ty)
for some μ ∈ (, ). If T is accretive, then for each r > , the non-expansive single-valued mapping JrT : R(I + rT) → D(T) defined by JrT := (I + rT)– is called the resolvent of T []. Moreover, N(T) = F(JrT ). Let D be a nonempty closed convex subset of X and Q be a mapping of X onto D. Then Q is said to be sunny [] if Q(Q(x) + t(x – Q(x))) = Q(x) for all x ∈ X and t ≥ . A mapping Q of X into X is said to be a retraction [] if Q = Q. If a mapping Q is a retraction, then Q(z) = z for every z ∈ R(Q), where R(Q) is the range of Q. A subset D of X is said to be a sunny non-expansive retract of X [] if there exists a sunny non-expansive retraction of X onto D and it is called a non-expansive retract of X if there exists a non-expansive retraction of X onto D.
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It is a hot topic in applied mathematics to find zero points of the sum of two accretive mappings, namely, a solution of the following inclusion problem: ∈ (A + B)x.
(.)
For example, a stationary solution to the initial value problem of the evolution equation ∂u + (A + B)u, ∂t
u() = u
(.)
can be recast as (.). A forward-backward splitting iterative method for (.) means each iteration involves only A as the forward step and B as the backward step, not the sum A + B. The classical forward-backward splitting algorithm is given in the following way: xn+ = (I + rn B)– (I – rn A)xn ,
n ∈ N.
(.)
Some of the related work can be seen in [–] and the references therein. In , Wei et al. [] extended the related work of (.) from a Hilbert space to the real smooth and uniformly convex Banach space and from two accretive mappings to two finite families of accretive mappings: ⎧ ⎪ x ∈ D, ⎪ ⎪ ⎪ ⎪ ⎨y = Q [( – α )(x + e )], n D n n n ⎪zn = ( – βn )xn + βn [a yn + N ai JrAi (yn – rn,i Bi yn )], ⎪ n,i ⎪ i= ⎪ ⎪ ⎩ xn+ = γn ηf (xn ) + (I – γn T)zn , n ∈ N ∪ {},
(.)
where D is a nonempty, closed and convex sunny non-expansive retract of X, QD is the sunny non-expansive retraction of E onto D, {en } is the error, Ai and Bi are m-accretive mappings and θ -inversely strongly accretive mappings, respectively, where i = , , . . . , N . T : X → X is a strongly positive linear bounded operator with coefficient γ and f : X → X is a contraction. N am = , < am < . The iterative sequence {xn } is proved to converge m= strongly to p ∈ N N(A i + Bi ), which solves the variational inequality i= (T – ηf )p , J(p – z) ≤
(.)
for ∀z ∈ N i= N(Ai + Bi ) under some conditions. The implicit midpoint rule is one of the powerful numerical methods for solving ordinary differential equations, and it has been extensively studied by Alghamdi et al. They presented the following implicit midpoint rule for approximating the fixed point of a nonexpansive mapping in a Hilbert space H in []: x ∈ H,
xn+ = ( – αn )xn + αn T
xn + xn+ ,
n ∈ N,
(.)
where T is non-expansive from H to H. If F(T) = ∅, then they proved that {xn } converges weakly to p ∈ F(T) under some conditions.
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Combining the ideas of forward-backward method and midpoint method, Wei et al. extended the study of two finite families of accretive mappings to two infinite families of accretive mappings [] in a real q-uniformly smooth and uniformly convex Banach space: ⎧ ⎪ ⎪ ⎪x ∈ D, ⎪ ⎪ ⎨y = Q [( – α )(x + e )], n D n n n ∞ Ai yn +zn n ⎪ ⎪ zn = δn yn + βn i= ai Jrn,i [ – rn,i Bi ( yn +z )] + ζn en , ⎪ ⎪ ⎪ ⎩ xn+ = γn ηf (xn ) + (I – γn T)zn + e n ∈ N ∪ {}, n,
(.)
where {en }, {en } and {e n } are three error sequences, Ai : D → X and Bi : D → X are m-accretive mappings and θi -inversely strongly accretive mappings, respectively, where i ∈ N . T : X → X is a strongly positive linear bounded operator with coefficient γ , f : X → X is a contraction, ∞ n= an = , < an < , δn + βn + ζn ≡ for n ∈ N ∪ {}. The iterative sequence {xn } is proved to converge strongly to p ∈ ∞ i= N(Ai + Bi ), which solves the following variational inequality: (T – ηf )p , J(p – z) ≤ ,
z∈
∞
N(Ai + Bi ).
(.)
i=
In , Ceng et al. [] presented the following iterative algorithm to approximate zero point of an m-accretive mapping: ⎧ ⎪ ⎪ ⎨x ∈ X, ⎪ ⎪ ⎩
yn = αn xn + ( – αn )JrAn xn ,
(.)
xn+ = βn f (xn ) + ( – βn )[JrAn yn
– λn μn F(JrAn yn )],
n ∈ N ∪ {},
where T : X → X is a γ -strongly accretive and μ-strictly pseudo-contractive mapping, with γ + μ > , f : E → E is a contraction and A : X → X is m-accretive. Under some assumptions, {xn } is proved to be convergent strongly to the unique element p ∈ N(A), which solves the following variational inequality: p – f (p ), J(p – u) ≤ ,
∀u ∈ N(A).
(.)
The mapping F in (.) is called a perturbed operator which only plays a role in the construction of the iterative algorithm for selecting a particular zero of A, and it is not involved in the variational inequality (.). Inspired by the work mentioned above, in Section , we shall construct a new modified forward-backward splitting midpoint iterative algorithm to approximate the zero points of the sum of infinite m-accretive mappings and infinite θi -inversely strongly accretive mappings. New proof techniques can be found, the superposition of perturbed operators is considered and some restrictions on the parameters are mild compared to the existing similar works. In Section , we shall discuss the applications of the newly obtained iterative algorithms to integro-differential systems and the convex minimization problems. We need the following preliminaries in our paper.
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Lemma . ([]) Let X be a real uniformly convex and p-uniformly smooth Banach space with constant Kp for some p ∈ (, ]. Let D be a nonempty closed convex subset of X. Let A : D → X be an m-accretive mapping and B : D → X be a θ -inversely strongly accretive mapping. Then, given s > , there exists a continuous, strictly increasing and convex function ϕp : R+ → R+ with ϕp () = such that for all x, y ∈ D with x ≤ s and y ≤ s,
A
J (I – rB)x – J A (I – rB)y p r r ≤ x – yp – r pθ – Kp rp– Bx – Byp
– ϕp I – JrA (I – rB)x – I – JrA (I – rB)y .
In particular, if < r ≤ ( Kpθp ) p– , then JrA (I – rB) is non-expansive. Lemma . ([]) Let X be a real smooth Banach space and B : X → X be a μ-strictly pseudo-contractive mapping and also be a γ -strongly accretive mapping with μ +γ > . ). Then, for any fixed number δ ∈ (, ), I – δB is a contraction with coefficient – δ( – –γ μ Lemma . ([]) Let X be a real Banach space and D be a nonempty closed and convex subset of X. Let f : D → D be a contraction. Then f has a unique fixed point. Lemma . ([]) Let X be a real strictly convex Banach space, and let D be a nonempty closed and convex subset of X. Let Tm : D → D be a non-expansive mapping for each m ∈ N . Let {am } be a real number sequence in (,) such that ∞ m= am = . Suppose that ∞ ∞ ∞ m= F(Tm ) = ∅. Then the mapping m= am Tm is non-expansive and F( m= am Tm ) = ∞ m= F(Tm ). Lemma . ([]) In a real Banach space X, for p > , the following inequality holds: x + yp ≤ xp + p y, jp (x + y) ,
∀x, y ∈ X, jp (x + y) ∈ Jp (x + y).
Lemma . ([]) Let X be a real Banach space, and let D be a nonempty closed and convex subset of X. Suppose A : D → X is a single-valued mapping and B : X → X is m-accretive. Then F (I + rB)– (I – rA) = N(A + B) for ∀r > . Lemma . ([]) Let {an } be a real sequence that does not decrease at infinity, in the sense that there exists a subsequence {ank } so that ank ≤ ank + for all k ∈ N ∪ {}. For every n > n , define an integer sequence {τ (n)} as τ (n) = max{n ≤ k ≤ n : ak < ak+ }. Then τ (n) → ∞ as n → ∞ and for all n > n , max{aτ (n) , an } ≤ aτ (n)+ .
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Lemma . ([]) For p > , the following inequality holds: p p – p– b , ab ≤ ap + p p
for any positive real numbers a and b. Lemma . ([]) The Banach space X is uniformly smooth if and only if the duality mapping Jp is single-valued and norm-to-norm uniformly continuous on bounded subsets of X.
2 Strong convergence theorems Theorem . Let X be a real uniformly convex and p-uniformly smooth Banach space with constant Kp where p ∈ (, ] and D be a nonempty closed and convex sunny nonexpansive retract of X. Let QD be the sunny non-expansive retraction of X onto D. Let f : X → X be a contraction with coefficient k ∈ (, ), Ai : D → X be m-accretive mappings, Ci : D → X be θi -inversely strongly accretive mappings, Wi : X → X be μi -strictly pseudocontractive mappings and γi -strongly accretive mappings with μi + γi > for i ∈ N . Suppose i p– ) for {ωi() } and {ωi() } are real number sequences in (, ) for i ∈ N . Suppose < rn,i ≤ ( pθ Kp ∞ () ∞ () () i ∈ N and n ∈ N , κt ∈ (, ) for t ∈ (, ), ∞ i= ωi Wi < +∞, i= ωi = i= ωi = and ∞ n N(A + C ) = ∅. If, for each t ∈ (, ), we define Z : X → X by i i t i= Ztn u = tf (u) + ( – t)
I – κt
∞
ωi() Wi
i=
∞
ωi() JrAn,ii (I
– rn,i Ci )QD u ,
i=
then Ztn has a fixed point unt . Moreover, if κtt → , then unt converges strongly to the unique solution q of the following variational inequality, as t → : q – f (q ), J(q – u) ≤ ,
∀u ∈
∞
N(Ai + Ci ).
(.)
i=
Proof We split the proof into five steps. Step . Ztn : X → X is a contraction for t ∈ (, ), κt ∈ (, ) and n ∈ N . In fact, for ∀x, y ∈ X, using Lemmas . and ., we have
n
Z x – Zn y
t t
∞ ∞
()
() Ai
≤ t f (x) – f (y) + ( – t) ×
ωi (I – κt Wi ) ωi Jrn,i (I – rn,i Ci )QD x
i= i= ∞
∞
()
() Ai – ωi (I – κt Wi ) ωi Jrn,i (I – rn,i Ci )QD y
i= i= ∞ – γi ≤ tkx – y + ( – t) ωi() – κt – x – y μi i=
≤ – ( – k)t x – y,
which implies that Ztn is a contraction. By Lemma ., there exists unt such that Ztn unt = unt . ∞ () Ai () n That is, unt = tf (unt ) + ( – t)(I – κt ∞ i= ωi Wi )( i= ωi Jrn,i (I – rn,i Ci )QD ut ).
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Step . If limt→ κtt = , then {unt } is bounded for n ∈ N , < t ≤ a, where a is a sufficiently small positive number and unt is the same as that in Step . For ∀u ∈ ∞ i= N(Ai + Ci ), using Lemmas ., . and ., we know that ∞
n
u – u ≤ tk un – u + t f (u) – u + ( – t)κt ωi() Wi u t t i=
∞ ∞
() ()
+ ( – t) ωi (I – κt Wi ) ωi JrAn,ii (I – rn,i Ci )QD unt
i= i=
∞
()
– ωi JrAn,ii (I – rn,i Ci )QD u
i=
∞
ωi() Wi u ≤ tk unt – u + t f (u) – u + ( – t)κt i=
+ ( – t)
∞ i=
– γi
()
un – u
ωi – κt – t μi
∞
≤ t f (u) – u + ( – t + tk) unt – u + ( – t)κt ωi() Wi u. i=
Then
n
f (u) – u +
u – u ≤ t
κt t
∞
() i= ωi Wi u
–k
.
Since limt→ κtt = , then there exists a sufficiently small positive number a such that < κt < for < t ≤ a. Thus {unt } is bounded for n ∈ N and < t ≤ a. t () Ai n Step . If limt→ κtt = , then unt – ∞ i= ωi Jrn,i (I – rn,i Ci )QD ut → , as t → , for n ∈ N . Noticing Step , we have
∞
n () Ai n
ωi Jrn,i (I – rn,i Ci )QD ut
ut –
i=
∞
ωi() JrAn,ii (I – rn,i Ci )QD unt
≤ t f unt + t i=
+ ( – t)κt
∞ i=
ωi() Wi
∞ i=
ωi() JrAn,ii (I
– rn,i Ci )QD unt
→ , as t → . Step . If the variational inequality (.) has solutions, the solution must be unique. ∞ Suppose u ∈ ∞ i= N(Ai + Ci ) and v ∈ i= N(Ai + Ci ) are two solutions of (.), then u – f (u ), J(u – v ) ≤ ,
(.)
v – f (v ), J(v – u ) ≤ .
(.)
and
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Adding up (.) and (.), we get u – f (u ) – v + f (v ), J(u – v ) ≤ .
(.)
Since u – f (u ) – v + f (v ), J(u – v ) = u – v – f (u ) – f (v ), J(u – v ) ≥ u – v – ku – v = ( – k)u – v , then (.) implies that u = v . Step . If limt→ κtt = , then unt → q ∈ ∞ i= N(Ai + Ci ), as t → , which solves the variational inequality (.). Assume tm → . Set unm := untm and define μ : X → R by
μ(u) = LIM unm – u ,
u ∈ X,
where LIM is the Banach limit on l∞ . Let
K = x ∈ X : μ(x) = min LIM unm – x . x∈X
It is easily seen that K is a nonempty closed convex bounded subset of X. Since unm – ∞ () Ai n i= ωi Jrn,i (I – rn,i Ci )QD um → from Step , then for u ∈ K , μ
∞
∞
n () Ai
– rn,i Ci )QD u = LIM um – ωi Jrn,i (I – rn,i Ci )QD u
ωi() JrAn,ii (I
i=
i=
≤ LIM unm – u = μ(u),
∞ () Ai it follows that i= ωi Jrn,i (I – rn,i Ci )QD (K) ⊂ K ; that is, K is invariant under ∞ () Ai ω J (I – r C n,i i )QD . Since a uniformly smooth Banach space has the fixed point i= i rn,i () Ai property for non-expansive mappings, ∞ i= ωi Jrn,i (I – rn,i Ci )QD has a fixed point, say q , ∞ () Ai in K . That is, i= ωi Jrn,i (I – rn,i Ci )QD q = q ∈ D, which ensures from Lemmas . and . that q ∈ ∞ i= N(Ai + Ci ). Since q is also a minimizer of μ over X, it follows that, for t ∈ (, ), ≤
μ(q + tf (q ) – tq ) – μ(q ) t
= LIM
unm – q – tf (q ) + tq – unm – q t
unm – q – tf (q ) + tq , J(unm – q – tf (q ) + tq ) – unm – q t n n = LIM um – q , J um – q – tf (q ) + tq
+ t q – f (q ), J unm – q – tf (q ) + tq – unm – q /t.
= LIM
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Since X is uniformly smooth, then by letting t → , we find the two limits above can be interchanged and obtain LIM f (q ) – q , J unm – q ≤ .
(.)
Since unm – q = tm (f (unm ) – q ) + ( – tm )[(I – κtm rn,i Ci )QD unm ) – q ],
∞ () Ai () i= ωi Wi )( i= ωi Jrn,i (I
∞
–
then
n
u – q = un – q , J un – q m m m ≤ tm f unm – f (q ), J unm – q + tm f (q ) – q , J unm – q
∞
() Ai n + ( – tm )
ωi Jrn,i (I – rn,i Ci )QD um – q
unm – q
i=
∞ ∞
()
() Ai n
n ωi Wi ωi Jrn,i (I – rn,i Ci )QD um um – q
+ ( – tm )κtm
i=
i=
i=
i=
≤ ( – tm + tm k) unm – q + tm f (q ) – q , J unm – q
∞ ∞
()
() + ( – tm )κtm
ωi Wi ωi JrAn,ii (I – rn,i Ci )QD unm
unm – q .
Therefore,
n
u – q ≤ m
f (q ) – q , J unm – q –k
∞ ∞
()
κt m
() Ai n
n
ωi Wi ωi Jrn,i (I – rn,i Ci )QD um um – q . +
tm
i=
Since
κtm tm
(.)
i=
→ , then from (.), (.) and the result of Step , we have LIM unm – q ≤
, which implies that LIM unm – q = , and then there exists a subsequence which is still denoted by {unm } such that unm → q . Next, we shall show that q solves the variational inequality (.). ∞ () Ai () n Note that unm = tm f (unm ) + ( – tm )(I – κtm ∞ i= ωi Wi )( i= ωi Jrn,i (I – rn,i Ci )QD um ), then ∞ for ∀v ∈ i= N(Ai + Ci ), !
∞
ωi() JrAn,ii (I
i=
= tm
– rn,i Ci )QD unm
! I – κt m
∞ i=
" n n – f um , J um – v
ωi() Wi
∞ i=
ωi() JrAn,ii (I
– rn,i Ci )QD unm
,J
unm
–v
"
! ∞ " ∞ () n () n Ai n u – t m κt m ωi Wi ωi Jrn,i (I – rn,i Ci )QD um , J um – v – tm m i= i=
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!∞ ∞ () () Ai n = ω (I – κtm Wi ) ωi Jrn,i (I – rn,i Ci )QC um tm i= i i= ∞ " ∞ () n () Ai ωi (I – κtm Wi ) ωi Jrn,i (I – rn,i Ci )QD v , J um – v – i=
!
i=
" ∞
κtm n
() n
– u –v – ω Wi v, J um – v tm m tm i= i !∞ ∞ " () () n Ai n ωi Wi ωi Jrn,i (I – rn,i Ci )QD um , J um – v + κt m ≤–
tm
i=
i=
$ ∞
n – γi ()
u – v ωi – κtm – – m μ i i=
#
∞
κtm () ω Wi v unm – v
tm i= i
∞
∞
()
()
Ai n
n ωi Wi ωi Jrn,i (I – rn,i Ci )QD um um – v
+ κt m
+
i=
≤
κt m tm
∞
ωi() Wi v unm – v
i=
+ κt m
i=
∞ i=
ωi() Wi
∞ i=
ωi() JrAn,ii (I
– rn,i Ci )QD unm
n
um – v
→ , as tm → . Since xn → q and J is uniformly continuous on each bounded subset of X, then taking the limits on both sides of the above inequality, q – f (q ), J(q – v) ≤ , which implies that q satisfies the variational inequality (.). Next, to prove the net {unm } converges strongly to q , as t → , suppose that there is another subsequence {untk } of {unt } satisfying untk → v as tk → . Denote untk by unk . A Then the result of Step implies that = limtk → (unk – ∞ ω() Jr i (I – rn,i Ci )QD unk ) = ∞ () Ai ∞i= i n,i v – i= ωi Jrn,i (I – rn,i Ci )QD v , which ensures that v ∈ i= N(Ai + Ci ) in view of Lemmas . and .. Repeating the above proof, we can also know that v solves the variational inequality (.). Thus q = v by using the result of Step . Hence unt → q , as t → , which is the unique solution of the variational inequality (.). This completes the proof. Theorem . Let X be a real uniformly convex and p-uniformly smooth Banach space with constant Kp where p ∈ (, ] and D be a nonempty closed and convex sunny nonexpansive retract of X. Let QD be the sunny non-expansive retraction of X onto D. Let f : X → X be a contraction with coefficient k ∈ (, ), Ai : D → X be m-accretive mappings, Ci : D → X be θi -inversely strongly accretive mappings, and Wi : X → X be μi strictly pseudo-contractive mappings and γi -strongly accretive mappings with μi + γi > for i ∈ N . Suppose {ωi() }, {ωi() }, {αn }, {βn }, {ϑn }, {νn }, {ξn }, {δn } and {ζn } are real number sequences in (, ), {rn,i } ⊂ (, +∞), {an } ⊂ X and {bn } ⊂ D are error sequences, where n ∈ N and i ∈ N . Suppose ∞ i= N(Ai + Ci ) = ∅. Let {xn } be generated by the following iterative
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algorithm: ⎧ ⎪ x ∈ D, ⎪ ⎪ ⎪ ⎪ ⎨u = Q (α x + β a ), n D n n n n () Ai ∞ un +vn ⎪vn = ϑn un + νn ⎪ ⎪ i= ωi Jrn,i (I – rn,i Ci )( ) + ξn bn , ⎪ ⎪ ∞ () Ai ⎩ () un +vn xn+ = δn f (xn ) + ( – δn )(I – ζn ∞ i= ωi Wi ) i= ωi Jrn,i (I – rn,i Ci )( ),
(.) n ∈ N.
Under the following assumptions: (i) αn + βn ≤ , ϑn + νn + ξn ≡ for n ∈ N ; ∞ () ∞ () (ii) ωi = i= ωi = ; i= ∞ ∞ ∞ ∞ (iii) n= an < +∞, n= bn < +∞, n= ( – αn ) < +∞, n= ξn < +∞, ∞ limn→∞ i= rn,i = ; (iv) limn→∞ δn = , ∞ n= δn = +∞; (v) – αn + an = o(δn ), ξn = o(δn ), ζn = o(ξn ), νn , as n → ∞; ∞ () pθi p– (vi) for i ∈ N , n ∈ N , i= ωi Wi < +∞, < rn,i ≤ ( Kp ) the iterative sequence xn → q ∈ ∞ N(A + C i i ), which is the unique solution of the variai= tional inequality (.). Proof We split the proof into four steps. Step . {vn } is well defined and so is {xn }. For s, t ∈ (, ), define Hs,t : D → D by Hs,t x := su + tH( u+x ) + ( – s – t)v, where H : D → D is non-expansive for x ∈ D and u, v ∈ D. Then, for ∀x, y ∈ D,
u + x u + y t
≤ x – y.
– Hs,t x – Hs,t y ≤ t
Thus Hs,t is a contraction, which ensures from Lemma . that there exists xs,t ∈ D such that Hs,t xs,t = xs,t . That is, xs,t = su + tH( u+x s,t ) + ( – s – t)v. Ai () Since ∞ i= ωi = and Jrn,i (I – rn,i Ci ) is non-expansive for n ∈ N and i ∈ N , then {vn } is well defined, which implies that {xn } is well defined. Step . {xn } is bounded. For ∀p ∈ ∞ i= N(Ai + Ci ), we can easily know that un – p ≤ αn xn – p + βn an + ( – αn )p. And
un + vn
vn – p ≤ ϑn un – p + νn
– p
+ ξn bn – p νn νn un – p + vn – p + ξn bn – p. ≤ ϑn + Thus vn – p ≤
ϑn + νn ξn bn – p un – p + – νn – νn
≤ αn xn – p + βn an + ( – αn )p + bn +
ξn p. – νn
(.)
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Using Lemma . and (.), we have, for n ∈ N ,
xn+ – p ≤ δn f (xn ) – f (p) + δn f (p) – p
∞ ∞
() un + vn
() Ai + ( – δn ) I – ζn – p
ωi Wi ωi Jrn,i (I – rn,i Ci )
i=
i=
∞
ωi() Wi p ≤ δn kxn – p + δn f (p) – p + ( – δn )ζn i=
∞ ∞
un + vn
+ ( – δn )
ωi() (I – ζn Wi ) ωi() JrAn,ii (I – rn,i Ci )
i= i=
∞ ∞
ωi() (I – ζn Wi ) ωi() JrAn,ii (I – rn,i Ci )p
–
i=
i=
∞
ωi() Wi p ≤ δn kxn – p + δn f (p) – p + ( – δn )ζn
+ ( – δn ) – ζn –
∞
– γi μi
ωi()
i=
i=
× αn xn – p + βn an + ( – αn )p + bn + #
≤ ( – δn ) – ζn –
+ ( – δn ) – ζn –
∞
i= ∞
ωi()
i=
– γi μi
ωi()
– γi μi
× βn an + ( – αn )p + bn + + ( – δn )ζn
∞
$
ξn p – νn
+ δn k xn – p + δn f (p) – p
ξn p – νn
ωi() Wi p.
(.)
i=
By using the inductive method, we can easily get the following result from (.): ∞ () ωi Wi p f (p) – p , xn+ – p ≤ max x – p, i= () –γi –k – ∞ i= ωi μi n ∞ () – γi ( – δk ) – ζk – ωi + μi i= k= ξk × βk ak + ( – αk )p + bk + p . – νk Therefore, from assumptions (iii) and (vi), we know that {xn } is bounded. Step . There exists q ∈ ∞ i= N(Ai + Ci ), which solves the variational inequality (.). Using Theorem ., we know that there exists unt such that unt = tf (unt ) + ( – t)(I – ∞ () Ai () n κt ∞ i= ωi Wi )( i= ωi Jrn,i (I – rn,i Ci )QD ut ) for t ∈ (, ). Moreover, under the assump-
Wei et al. Journal of Inequalities and Applications (2017) 2017:227
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tion that κtt → , unt → q ∈ ∞ i= N(Ai + Ci ), as t → , which is the unique solution of the variational inequality (.). Step . xn → q , as n → ∞, where q is the same as that in Step . Set C := sup{αn xn + βn an – q p– , q αn xn + βn an – q p– : n ∈ N}, then from Step and assumption (iii), C is a positive constant. Using Lemma ., we have un – q p ≤ αn xn – q p – p( – αn ) q , Jp (αn xn + βn an – q ) + pβn an , Jp (αn xn + βn an – q ) ≤ αn xn – q p + C ( – αn ) + C an .
(.)
Using Lemma ., we know that vn – q ≤ ϑn un – q + νn p
p
∞
Ai ωi()
Jrn,i (I
i=
un + vn – rn,i Ci )
p
– q
+ ξn bn – q νn νn un – q p + vn – q p ≤ ϑn +
p ∞
un + vn p– ()
– Ci q
– νn ωi rn,i θi p – rn,i Kp Ci
i= p
– νn
∞ i=
– I
un + vn un + vn Ai – r I – J ωi() ϕp
C n,i i rn,i
– JrAn,ii
p (q – rn,i Ci q )
+ ξn bn – q .
Therefore, vn – q p ≤
ϑn + νn ξn un – q p + bn – q p – νn – νn
p
∞
νn () un + vn p–
– Ci q
– ω rn,i θi p – rn,i Kp Ci
– νn i= i
∞
un + vn un + vn νn () Ai
– rn,i Ci – ω ϕp I – Jrn,i – νn i= i
– I – JrAn,ii (q – rn,i Ci q )
.
Now, from (.)–(.) and Lemmas . and ., we know that for n ∈ N , xn+ – q p
∞
() un + vn
Ai = δn f (xn ) – q + ( – δn ) ωi Jrn,i (I – rn,i Ci ) – q
– ( – δn )ζn
∞ i=
ωi() Wi
k=
∞ i=
ωi() JrAn,ii (I
un + vn – rn,i Ci )
p
(.)
Wei et al. Journal of Inequalities and Applications (2017) 2017:227
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p
un + vn
– q
≤ ( – δn )
+ pδn f (xn ) – f (q ), Jp (xn+ – q ) + pδn f (q ) – q , Jp (xn+ – q ) " !∞ ∞ () () un + vn Ai – p( – δn )ζn ωi Wi ωi Jrn,i (I – rn,i Ci ) , Jp (xn+ – q ) i= i= un – p vn – p ≤ ( – δn ) + + pδn kxn – q xn+ – q p– + pδn f (q ) – q , Jp (xn+ – q ) " !∞ ∞ () () un + vn Ai – p( – δn )ζn ωi Wi ωi Jrn,i (I – rn,i Ci ) , Jp (xn+ – q ) i= i= ξn ≤ ( – δn )un – p + ( – δn ) bn – q p – νn
p ∞
νn () un + vn p–
– ωi rn,i θi p – rn,i Kp Ci – Ci q
– νn i=
∞
un + vn νn () un + vn Ai – r ωi ϕp
C I – J n,i i rn,i
– νn i=
– I – JrAn,ii (q – rn,i Ci q )
–
+ pkδn xn – q xn+ – q p– + pδn f (q ) – q , Jp (xn+ – q ) !∞ ∞ " () () + v u n n , Jp (xn+ – q ) – p( – δn )ζn ωi Wi ωi JrAn,ii (I – rn,i Ci ) i= i=
≤ ( – δn )xn – p p + C – αn + an + kδn xn – q p
ξn bn – q p – νn
∞
un + vn νn () un + vn Ai I – J – r – ( – δn ) ωi ϕp
C n,i i rn,i
– νn i=
– I – JrAn,ii (q – rn,i Ci q )
+ pδn f (q ) – q , Jp (xn+ – q ) + kδn xn+ – q p +
+ ζn
∞ i=
ωi() Wi
∞ i=
ωi() JrAn,ii (I
un + vn – rn,i Ci )
Jp (xn+ – q ) ,
which implies that xn+ – q p ≤
– δn ( – k) C ( – αn + an ) xn – q p + – δn k – δn k ξn bn – q p + pδn f (q ) – q , Jp (xn+ – q ) + – δn k – νn
Wei et al. Journal of Inequalities and Applications (2017) 2017:227
+ ζn
∞
ωi() Wi
i=
∞
Page 15 of 22
ωi() JrAn,ii (I
i=
un + vn – rn,i Ci )
Jp (xn+ – q )
∞
un + vn νn () un + vn Ai
ω ϕp I – Jrn,i – rn,i Ci – ( – δn ) – νn – δn k i= i
– I – JrAn,ii (q – rn,i Ci q )
.
From Step , if we set C = sup{
∞
∞ () Ai () i= ωi Wi ( i= ωi Jrn,i (I
– rn,i Ci )( un +vn )), xn –
q p– : n ∈ N}, then C is a positive constant. Let εn() =
ξn δn (–k) () , εn = δn (–k) [C ( – αn + an ) + –ν bn – q p + pδn f (q ) – q , Jp (xn+ – –δn k n A () () ∞ ν q ) + ζn C ] and εn = ( – δn ) –νn n –δn k i= ωi ϕp ((I – Jrn,ii )( un +vn – rn,i Ci ( un +vn )) – (I – A Jrn,ii )(q – rn,i Ci q )).
Then xn+ – q p ≤ – εn() xn – q p + εn() εn() – εn() .
(.)
Our next discussion will be divided into two cases. Case . {xn – q } is decreasing. If {xn – q } is decreasing, we know from (.) and assumptions (iv) and (v) that ≤ εn() ≤ εn() εn() – xn – q p + xn – q p – xn+ – q p → , ∞
Ai () un +vn i= ωi ϕp ((I – Jrn,i )(
A
– rn,i Ci ( un +vn )) – (I – Jrn,ii )(q – rn,i Ci q )) → Ai () un +vn – , as n → +∞. Then, from the property of ϕp , we know that ∞ i= ωi (I – Jrn,i )( which ensures that
A
rn,i Ci ( un +vn )) – (I – Jrn,ii )(q – rn,i Ci q ) → , as n → +∞. Note that limn→∞ ∞ i= rn,i = , then
∞
u + v un + vn
n n () Ai – ωi Jrn,i (I – rn,i Ci )
i= ≤
∞ i=
+
un + vn Ai Ai
(I – r (I – r – I – J ωi()
C ) C )q I – J n,i i n,i i
rn,i rn,i
∞ i=
∞
un + vn
C + ωi() rn,i
ωi() rn,i Ci q
i
i=
→ , as n → ∞. Now, our purpose is to show that limsupn→∞ εn() ≤ , which reduces to showing that limsupn→∞ f (q ) – q , Jp (xn+ – q ) ≤ .
Wei et al. Journal of Inequalities and Applications (2017) 2017:227
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Let unt be the same as that in Step . Since unt ≤ unt – q + q , then {unt } is bounded, as t → . Using Lemma . again, we have
n un + vn p
u –
t
∞
un + vn
n () Ai = ut – ωi Jrn,i (I – rn,i Ci )
i=
p un + vn
– +
i=
p ∞
un + vn
≤ unt – ωi() JrAn,ii (I – rn,i Ci )
i= !∞ " () u u + v + v + v u n n n n n n – , Jp unt – +p ωi JrAn,ii (I – rn,i Ci ) i=
∞ ∞
()
() n Ai n ωi Wi ωi Jrn,i (I – rn,i Ci )QD ut = tf ut + ( – t) I – κt
i= i= p ∞ un + vn
() Ai ωi Jrn,i (I – rn,i Ci ) –
i= !∞ " () un + vn un + vn un + vn Ai n – , Jp ut – +p ωi Jrn,i (I – rn,i Ci ) i= !
∞
n un + vn p
+ pt f un –
ωi() JrAn,ii (I – rn,i Ci )QD unt ≤ ut – t
i= ∞ ∞ () () κt Ai n – ( – t) ωi Wi ωi Jrn,i (I – rn,i Ci )QD ut , t i= i= " ∞ un + vn () Ai n Jp ut – ωi Jrn,i (I – rn,i Ci ) i= !∞ " () un + vn un + vn un + vn Ai n – , Jp ut – , ωi Jrn,i (I – rn,i Ci ) +p i= ∞
ωi() JrAn,ii (I
un + vn – rn,i Ci )
which implies that ! t
∞ i=
ωi() JrAn,ii (I – rn,i Ci )QD unt – f unt
∞ ∞ () κt () Ai n ωi Wi ωi Jrn,i (I – rn,i Ci )QD ut , + ( – t) t i= i= " ∞ un + vn () Ai n Jp ut – ωi Jrn,i (I – rn,i Ci ) i=
∞
p–
un + vn
un + vn un + vn
() Ai n
. – ωi Jrn,i (I – rn,i Ci ) – ≤
u
t
i=
Wei et al. Journal of Inequalities and Applications (2017) 2017:227
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() Ai n So, limt→ limsupn→+∞ ∞ – f (unt ) + κtt ( – t) × i= ωi Jrn,i (I – rn,i Ci )QD ut ∞ () Ai ∞ () Ai ∞ () un +vn n n i= ωi Wi ( i= ωi Jrn,i (I – rn,i Ci )QD ut ), Jp (ut – i= ωi Jrn,i (I – rn,i Ci )( )) ≤ . () Ai () Ai ∞ n Since unt → q , then ∞ i= ωi Jrn,i (I – rn,i Ci )QD ut → i= ωi Jrn,i (I – rn,i Ci )QD q = q , as t → . Noticing that !
q – f (q ), Jp q –
∞
ωi() JrAn,ii (I
i=
un + vn – rn,i Ci )
"
un + vn – rn,i Ci ) = q – f (q ), Jp q – i= " ∞ un + vn () Ai n ωi Jrn,i (I – rn,i Ci ) –Jp ut – i= ! " ∞ un + vn () Ai n ωi Jrn,i (I – rn,i Ci ) + q – f (q ), Jp ut – i= ! ∞ un + vn () Ai ωi Jrn,i (I – rn,i Ci ) = q – f (q ), Jp q – i= " ∞ un + vn () Ai n ωi Jrn,i (I – rn,i Ci ) –Jp ut – i= ! ∞ ωi() JrAn,ii (I – rn,i Ci )QD unt + f unt + q – f (q ) –
!
∞
ωi() JrAn,ii (I
i=
∞ () κt () Ai n – ( – t) ωi Wi ωi Jrn,i (I – rn,i Ci )QD ut , t i= i= " ∞ un + vn () Ai n ωi Jrn,i (I – rn,i Ci ) Jp ut – i= !∞ () ωi JrAn,ii (I – rn,i Ci )QD unt – f unt + ∞
i=
∞ ∞ () κt () Ai n ωi Wi ωi Jrn,i (I – rn,i Ci )QD ut , + ( – t) t i= i= " ∞ un + vn () Ai n J ut – , ωi Jrn,i (I – rn,i Ci ) i= () Ai un +vn we have limsupn→+∞ q – f (q ), Jp (q – ∞ i= ωi Jrn,i (I – rn,i Ci )( )) ≤ . () Ai From assumptions (iv) and (v) and Step , we know that xn+ – ∞ i= ωi Jrn,i (I – rn,i Ci )( un +vn ) → and then limsupn→+∞ q – f (q ), Jp (q – xn+ ) ≤ . Thus limsupn→∞ εn() ≤ . Employing (.) again, we have xn – q p ≤
xn – q p – xn+ – q p εn()
+ εn() .
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Assumption (iv) implies that liminfn→∞ lim xn – q p ≤ liminf
n→∞
xn –q p –xn+ –q p ()
εn
xn – q p – xn+ – q p εn()
n→∞
= . Then
+ limsup εn() ≤ . n→∞
Then the result that xn → q follows. Case . If {xn – q } is not eventually decreasing, then we can find a subsequence {xnk – q } so that xnk – q ≤ xnk+ – q for all k ≥ . From Lemma ., we can define a subsequence {xτ (n) – q } so that max{xτ (n) – q , xn – q } ≤ xτ (n)+ – q for all n > n . This enables us to deduce that (similar to Case ) ≤ ετ()(n) ≤ ετ()(n) ετ()(n) – xτ (n) – q p + xτ (n) – q p – xτ (n)+ – q p → , and then copying Case , we have limn→∞ xτ (n) – q = . Thus ≤ xn – q ≤ xτ (n)+ – q → , as n → ∞. This completes the proof. Remark . Theorem . is reasonable if we suppose X = D = (–∞, +∞), take f (x) = x , x , θi = i , ωi() = ωi() = i , αn = – n , βn = n , ϑn = δn = n , ξn = ζn = Ai x = Ci x = xi , Wi x = i+ a n = bn =
, n
γi =
i+
, μi =
i+ – + i+
i+ –
, rn,i =
n+i
for n ∈ N and i ∈ N .
Remark . Our differences from the main references are: (i) the normalized duality mapping J : E → E∗ is no longer required to be weakly sequentially continuous at zero as that in []; A (ii) the parameter {rn,i } in the resolvent Jrn,ii does not need satisfying the condition ∞ ‘ n= |rn+,i – rn,i | < +∞ and rn,i ≥ ε > for i ∈ N and some ε > ’ as that in [] or []; (iii) Lemma . plays an important role in the proof of strong convergence of the iterative sequence, which leads to different restrictions on the parameters and different proof techniques compared to the already existing similar works.
3 Applications 3.1 Integro-differential systems In Section ., we shall investigate the following nonlinear integro-differential systems involving the generalized pi -Laplacian, which have been studied in []: ⎧ (i) p – ∂u (x,t) (i) i (i) (i) r – (i) ⎪ ⎪ ∂t – div[(C(x, t) + |∇u | ) ∇u ] + ε|u | i u ⎪ ⎪ ⎪ ⎨ + g(x, u(i) , ∇u(i) ) + a ∂ % u(i) dx = f (x, t), (x, t) ∈ × (, T), ∂t
pi – ⎪ ⎪ –ϑ, (C(x, t) + |∇u(i) | ) ∇u(i) ∈ βx (u(i) ), ⎪ ⎪ ⎪ ⎩ (i) u (x, ) = u(i) (x, T), x ∈ , i ∈ N,
(x, t) ∈ × (, T),
(.)
where is a bounded conical domain of a Euclidean space RN (N ≥ ), is the boundary of with ∈ C and ϑ denotes the exterior normal derivative to . ·, · and | · | denote the Euclidean inner-product and the Euclidean norm in RN , respectively. T is a positive (i) ∂u(i) (i) , , . . . , ∂u ) and x = (x , x , . . . , xN ) ∈ . βx is the subdifferential of constant. ∇u(i) = ( ∂u ∂x ∂x ∂xN
Wei et al. Journal of Inequalities and Applications (2017) 2017:227
Page 19 of 22
ϕx , where ϕx = ϕ(x, ·) : R → R for x ∈ . a and ε are non-expansive constants, ≤ C(x, t) ∈ ∞ p ∞ max{pi ,p } ∞ ,pi i i (, T; Lmax{pi ,pi } ()) ()), f (x, t) ∈ ∞ i= Vi := i= L (, T; W i= Wi := i= L and g : × RN+ → R are given functions. Just like [], we need the following assumptions to discuss (.). N ∞ Assumption {pi }∞ i= is a real number sequence with N+ < pi < +∞, {θi }i= is any real N ∞ number sequence in (, ] and {ri }i= is a real number sequence satisfying N+ < ri ≤ min{pi , pi } < +∞. pi + p = and ri + r = for i ∈ N . i
i
Assumption Green’s formula is available. Assumption For each x ∈ , ϕx = ϕ(x, ·) : R → R is a proper, convex and lowersemicontinuous function and ϕx () = . Assumption ∈ βx () and for each t ∈ R, the function x ∈ → (I + λβx )– (t) ∈ R is measurable for λ > . Assumption Suppose that g : × RN+ → R satisfies the following conditions: (a) Carathéodory’s conditions; (b) Growth condition. & & & & &g(x, r , . . . , rN+ )&max{pi ,pi } ≤ &hi (x, t)&pi + bi |r |pi , where (r , r , . . . , rN+ ) ∈ RN+ , hi (x, t) ∈ Wi and bi is a positive constant for i ∈ N ; (c) Monotone condition. g is monotone in the following sense:
g(x, r , . . . , rN+ ) – g(x, t , . . . , tN+ ) ≥ (r – t )
for all x ∈ and (r , . . . , rN+ ), (t , . . . , tN+ ) ∈ RN+ . Assumption For i ∈ N , let Vi∗ denote the dual space of Vi . The norm in Vi , · Vi , is defined by
u(x, t) = V
'
i
T
u(x, t) pi ,p dt W i ()
p
i
,
u(x, t) ∈ Vi .
Definition . ([]) For i ∈ N , define the operator Bi : Vi → Vi∗ by '
T
w, Bi u =
'
pi – C(x, t) + |∇u| ∇u, ∇w dx dt + ε
'
T
' |u|ri – uw dx dt
for u, w ∈ Vi . Definition . ([]) For i ∈ N , define the function i : Vi → R by '
T
'
ϕx u| (x, t) d(x) dt
i (u) =
for u(x, t) ∈ Vi .
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Definition . ([]) For i ∈ N , define Si : D(Si ) = {u(x, t) ∈ Vi : u(x, T)} → Vi∗ by ∂u ∂ Si u = +a ∂t ∂t
∂u ∂t
∈ Vi∗ , u(x, ) =
' u dx.
Lemma . ([]) For i ∈ N , define a mapping Ai : Wi → Wi as follows: D(Ai ) = u ∈ Wi |there exists an f ∈ Wi such that f ∈ Bi u + ∂i (u) + Si u , where ∂i : Vi → Vi∗ is the subdifferential of i . For u ∈ D(Ai ), we set Ai u = {f ∈ Wi |f ∈ Bi u + ∂i (u) + Si u}. Then Ai : Wi → Wi is m-accretive, where i ∈ N .
Lemma . ([]) Define Ci : D(Ci ) = Lmax{pi ,pi } (, T; W ,max{pi ,pi } ()) ⊂ Wi → Wi by (Ci u)(x, t) = g(x, u, ∇u) – f (x, t) for ∀u(x, t) ∈ D(Ci ) and f (x, t) is the same as that in (.), where i ∈ N . Then Ci : D(Ci ) ⊂ Wi → Wi is continuous and strongly accretive. If we further assume that g(x, r , . . . , rN+ ) ≡ r , then Ci is θi -inversely strongly accretive, where i ∈ N . Lemma . ([]) For f (x, t) ∈ lution u(i) (x, t) ∈ Wi for i ∈ N .
∞
i= Wi ,
integro-differential systems (.) have a unique so-
Lemma . ([]) If ε ≡ , g(x, r , . . . , rN+ ) ≡ r and f (x, t) ≡ k, where k is a constant, then u(x, t) ≡ k is the unique solution of integro-differential systems (.). Moreover, {u(x, t) ∈ ∞ ∞ i= Wi |u(x, t) ≡ k satisfying (.)} = i= N(Ai + Ci ). Remark . ([]) Set p := infi∈N (min{pi , pi }) and q := supi∈N (max{pi , pi }). Let X := Lmin{p,p } (, T; Lmin{p,p } ()), where p + p = . Let D := Lmax{q,q } (, T; W ,max{q,q } ()), where q + q = . Then X = Lp (, T; Lp ()), D = Lq (, T; W ,q ()) and D ⊂ Wi ⊂ X, ∀i ∈ N . Theorem . Let D and X be the same as those in Remark .. Suppose Ai and Ci are the same as those in Lemmas . and ., respectively. Let f : X → X be a fixed contractive mapping with coefficient k ∈ (, ) and Wi : X → X be μi -strictly pseudo-contractive mappings and γi -strongly accretive mappings with μi + γi > for i ∈ N . Suppose that {ωi() }, {ωi() }, {αn }, {βn }, {ϑn }, {νn }, {ξn }, {δn }, {ζn }, {rn,i }, {an } ⊂ X and {bn } ⊂ D satisfy the same conditions as those in Theorem ., where n ∈ N and i ∈ N . Let {xn } be generated by the following iterative algorithm: ⎧ ⎪ ⎪x ∈ D, ⎪ ⎪ ⎪ ⎨u = Q (α x + β a ), n D n n n n ∞ () Ai ⎪ ⎪vn = ϑn un + νn i= ωi Jrn,i (I – rn,i Ci )( un +vn ) + ξn bn , ⎪ ⎪ ⎪ ∞ () Ai ⎩ () un +vn xn+ = δn f (xn ) + ( – δn )(I – ζn ∞ i= ωi Wi ) i= ωi Jrn,i (I – rn,i Ci )( ),
(.) n ∈ N.
If, in integro-differential systems (.), ε ≡ , g(x, r , . . . , rN+ ) ≡ r and f (x, t) ≡ k, then under the following assumptions in Theorem ., the iterative sequence xn → q ∈
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∞
i= N(Ai + Ci ), which is the unique solution of integro-differential systems (.) and which satisfies the following variational inequality: for ∀y ∈ ∞ i= N(Ai + Ci ),
(I – f )q (x, t), J q (x, t) – y ≤ .
3.2 Convex minimization problems Let H be a real Hilbert space. Suppose hi : H → (–∞, +∞) are proper convex, lowersemicontinuous and nonsmooth functions [], suppose gi : H → (–∞, +∞) are convex and smooth functions for i ∈ N . We use ∇gi to denote the gradient of gi and ∂hi the subdifferential of hi for i ∈ N . The convex minimization problems are to find x∗ ∈ H such that hi x∗ + gi x∗ ≤ hi (x) + gi (x),
i ∈ N,
(.)
for ∀x ∈ H. By Fermats’ rule, (.) is equivalent to finding x∗ ∈ H such that ∈ ∂hi x∗ + ∇gi x∗ ,
i ∈ N.
(.)
Theorem . Let H be a real Hilbert space and D be the nonempty closed convex sunny non-expansive retract of H. Let QD be the sunny non-expansive retraction of H onto D. Let f : H → H be a contraction with coefficient k ∈ (, ). Let hi : H → (–∞, +∞) be proper convex, lower-semicontinuous and nonsmooth functions and gi : H → (–∞, +∞) be convex and smooth functions for i ∈ N . Let Wi : H → H be μi -strictly pseudo-contractive mappings and γi -strongly accretive mappings with μi + γi > for i ∈ N . Suppose {ωi() }, {ωi() }, {αn }, {βn }, {ϑn }, {νn }, {ξn }, {δn }, {ζn }, {rn,i } ⊂ (, +∞), {an } ⊂ H and {bn } ⊂ D satisfy the same conditions as those in Theorem ., where n ∈ N and i ∈ N . Let {xn } be generated by the following iterative algorithm: ⎧ ⎪ ⎪ ⎪x ∈ D, ⎪ ⎪ ⎨u = Q (α x + β a ), n D n n n n ∞ () ∂hi ⎪ ⎪ vn = ϑn un + νn i= ωi Jrn,i (I – rn,i ∇gi )( un +vn ) + ξn bn , ⎪ ⎪ ⎪ ∞ () ∂hi ⎩ () un +vn xn+ = δn f (xn ) + ( – δn )(I – ζn ∞ i= ωi Wi ) i= ωi Jrn,i (I – rn,i ∇gi )( ),
(.) n ∈ N.
If, further, suppose ∇gi is θi -Lipschitz continuous and hi + gi attains a minimizer, then {xn } converges strongly to the minimizer of hi + gi for i ∈ N . Proof It follows from [] that ∂hi is m-accretive. From [], since ∇gi is θi -Lipschitz continuous, then ∇gi is θi -inversely strongly accretive. Thus Theorem . ensures the result. This completes the proof. Acknowledgements Supported by the National Natural Science Foundation of China (11071053), Natural Science Foundation of Hebei Province (A2014207010), Key Project of Science and Research of Hebei Educational Department (ZD2016024), Key Project of Science and Research of Hebei University of Economics and Business (2016KYZ07), Youth Project of Science and Research of Hebei Educational Department (QN2017328) and Science and Technology Foundation of Agricultural University of Hebei (LG201612).
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Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the manuscript. All authors read and approved the final manuscript. Author details 1 School of Mathematics and Statistics, Hebei University of Economics and Business, 47 Xuefu Road, Shijiazhuang, China. 2 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA. 3 College of Science, Agricultural University of Hebei, Baoding, China.
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