construction of common fixed points for monotone ...€¦ · construction of common xed points for...

Construction of common fixed pointsfor monotone nonexpansive semigroups

Wojciech M. (Walter) KozlowskiUniversity of New South Wales

[email protected]

RMIT, Melbourne, 8 September 2017

W.M. Kozlowski, UNSW Monotone nonexpansive semigroups 1 / 21

Abstract

Common Fixed Point for Monotone Nonexpansive Semigroups

We discuss the monotone nonexpansive semigroups of nonlinear mappingsacting in Banach spaces equipped with partial order. We provide analgorithm for the construction of a common fixed point for suchsemigroups and prove its weak convergence. We give examples andinterpret our results in the context of the theory of differential equationsand dynamical systems.


Fixed point problem - Generic Applications

Let X be a Banach space or metric space. Let C ⊂ X and T : C → C.We want to find x ∈ C such that T (x) = x.

Equations

Every equation f(x) = 0 can be re-written as the fixed point problem forthe mapping T (x) = f(x) + x.

Let (M,d) be a complete metric space and ϕ : M → R+ be l.s.c.

Optimisation and variational problems (example)

Ekeland Variational Principle (1974): There exists v∗ ∈M such thatϕ(v∗)− ϕ(v) + d(v∗, v) > 0 for every v ∈M .Caristi Fixed Point (1975): If T : M →M satisfiesd(x, T (x)) ≤ ϕ(x)− ϕ(T (x)) for every x ∈M then T has a fixed point.

Oettli and Thera proved in 1993 that both theorems are equivalent!


Three main streams of FPT

Metric FPT initiated by Banach Fixed Point Theorem (1922)

Let (M,d) be a complete metric space, and T : M →M be a contraction,i.e. there exists a constant α < 1 such that d(T (x), T (y)) ≤ αd(x, y) forall x, y ∈M . Then, there exists a unique z ∈M such that T (z) = z.Moreover, for any x ∈M , there holds d(Tn(x), z)→ 0, where Tn is then-th iterate of T .

Order FPT initiated by Knaster-Tarski Theorem (1928, 1955)

Let (E,�) be a complete lattice. Then every order-preserving (monotone)mappping T : E → E has a fixed point.

Topological FPT initiated by Brouwer Theorem (1912)

Let B be a closed ball in Rn. Then every continuous mapppingT : B → B has a fixed point.


Fixed points existence for nonexpansive mappings

Browder(1965)

Let C 6= ∅ be a closed, convex, bounded subset of a uniformly convexBanach space X, and T : C → C be nonexpansive, i.e.‖T (x)− T (y)‖ ≤ ‖x− y‖ for all x, y ∈ C. Then, there exists a z ∈ Csuch that T (z) = z. The set of all fixed points is closed and convex.

Browder (1965)

Let C 6= ∅ be a closed, convex, bounded subset of a uniformly convexBanach space X, and {Ts}s∈J a commuting family of nonexpansivemappings Ts : C → C. Then, there exists a z ∈ C such that Ts(z) = z forall s ∈ J . The set of all such common fixed points is closed and convex.


Monotone nonexpansive mappings

Let X be a Banach space endowed with a partial order ”�” such that allorder intervals are convex and closed.

Monotone nonexpansive mappings

Let C ⊂ X be convex, closed and bounded. We say that T : C → C is amonotone nonexpansive mapping if it is monotone, x � y ⇒ T (x) � T (y)and

‖T (x)− T (y)‖ ≤ ‖x− y‖ (1)

for any x, y ∈ C such that x and y are comparable in the sense of thepartial order ”�”.

Since (1) needs to hold only for comparable x and y, hence T does nothave to be nonexpansive or even continuous.


Monotone nonexpansive semigroups

Let J be a nontrivial subsemigroup of [0,∞) such that 0 ∈ J .

Definition

A one-parameter family S = {Ts : s ∈ J} of mappings from C into itself issaid to be a monotone nonexpansive semigroup on C if S satisfies thefollowing conditions:

(i) all Ts are monotone nonexpansive mappings;

(ii) T0(x) = x for x ∈ C;

(iii) Tt+s(x) = Tt(Ts(x)) for x ∈ C and t, s ∈ J ;

(iv) for each x ∈ C, the mapping s 7→ Ts(x) is norm continuous;

(v) x � Ts(x) for x ∈ C and s ∈ A,where A is generating for J

We say that S is equicontinuous if the family of mappings{t 7→ Tt(x) : x ∈ C} is equicontinuous at t = 0.A set A ⊂ J is called a generating set for the parameter semigroup J if forevery 0 < u ∈ J there exist m ∈ N, s ∈ A, t ∈ A such that u = ms+ t.


Fixed point existence for monotone nonexpansive mappings

Bin Dehaish and Khamsi (2016)

Assume that X is uniformly convex in every direction. Let C ⊂ X beconvex, closed and bounded. Let T : C → C be a monotone nonexpansivemapping. Assume that there exists x0 ∈ C such that x0 and T (x0) arecomparable. Then T has a fixed point.As observed very recently by Espinola and Wisnicki, by using theKnaster-Tarski theorem, this result can be generalized to monotonemappings.

WMK (2017)

Assume X is uniformly convex. Let S be a monotone nonexpansivesemigroup on C. Assume in addition that there exists x ∈ C such thatx � Ts(x) for all s ∈ J . Then S has a common fixed point z ∈ Fix(S)such that x � z. Moreover, if f1, f2 ∈ Fix(S) are comparable thenf = cf1 + (1− c)f2 ∈ Fix(S) for every c ∈ [0, 1].


Example of a monotone nonexpansive mapping

Let X = L2([0, 1],R) and B be a closed ball in X centered at zero.Define the operator T : X → X by

T (x)(t) = g(t) +

∫ 1

0F (t, s, x(s))ds,

where g ∈ X and F : [0, 1]× [0, 1]×X → R is measurable and satisfies

0 ≤ F (t, s, x(s))− F (t, s, y(s)) ≤ x(t)− y(t),

for t, s ∈ [0, 1] and x, y ∈ X such that y ≤ x a.e. Assume also that thereexists a non-negative function h(·, ·) ∈ L2([0, 1]× [0, 1]) and M < 1

2 suchthat |F (t, s, u)| ≤ h(t, s) +M |u|, where t, s, u ∈]0, 1]. Then T is amonotone nonexpansive mapping acting within B. Moreover, ifg(t) +

∫ 10 F (t, s, 0)ds ≥ 0 for almost every t ∈ [0, 1] then T (0) ≥ 0.

Hence, T has a fixed point z ∈ B and z ≥ 0.


Example of a monotone nonexpansive semigroup

Consider the following Initial Value Problem (IVP) for an unknownfunction u(x, ·) : [0,∞)→ X:{

u(x, 0) = x

u′(x, t) + (I −Ht)(u(x, t)) = 0,

where Ht : C → C are monotone nonexpansive mappings with respect tothe order �. The notation u′(x, t) denotes the derivative of the functiont 7→ u(x, t). We assume that x � Ht(x) for every t ∈ [0,∞). It can beproved that that there exists a solution u(x, ·) for the (IVP) such thatx � u(x, t) for every t ∈ [0,∞). Let us denote Tt(x) = u(x, t). It can beproved that T = {Tt}t≥0 is a monotone nonexpansive semigroup. Thecommon fixed points of T can be interpreted as the stationary points ofthe dynamical process with the evolution function (t, x)→ Tt(x).


Opial property

Definition

A Banach space X is said to have the Opial property if for each sequence{xn} of elements of X such that xn ⇀ x, and for any y ∈ X such thaty 6= x

lim infn→∞

‖xn − x‖ < lim infn→∞

‖xn − y‖.

It is well known that many classical spaces including the Hilbert space andBanach spaces lp with 1 < p <∞ posses the Opial property while suchimportant uniformly convex spaces like Lp[0, 1] ( 1 < p <∞) fail thiscondition for p 6= 2. In the context of monotone mappings the followingconcept can provide the remedy.


Monotone Opial property

Definition

We say that (X, ‖ · ‖,�) have the monotone Opial property if for eachnondecreasing sequence {xn} of elements of X such that xn ⇀ x, and forany y ∈ X such that y 6= x and xn � y for every n ∈ N

lim infn→∞

‖xn − x‖ < lim infn→∞

‖xn − y‖.

The norm ‖ · ‖ is called monotone if u � v � w implies thatmax{‖v − u‖, ‖w − v‖} ≤ ‖w − u‖ for any u, v, w ∈ X.

Alfuraidan and Khamsi (2017)

Let (X, ‖ · ‖,�) be a uniformly convex, partially ordered Banach space forwhich order intervals are convex and closed. Assume that the norm ‖ · ‖ ismonotone. Then X has the monotone Opial property.

As an important example Lp[0, 1] have the monotone Opial property forany p > 1.


Krasnoselskii-Ishikawa iteration process

Definition

Given λ ∈ (0, 1) and a sequence of parameters tn ∈ J . We will considerthe following iteration process, often called in the literature theKrasnoselskii-Ishikawa iteration process (KIS)

xn+1 = (1− λ)xn + λTtn(xn),

where x0 ∈ C is the starting element chosen so that x0 � Tt(x0) for t ∈ J .It can be proved that {xn} is nondecreasing.Also, Cx0 = C ∩ [x0,→) is nonempty, weakly compact and Ts(Cx0) ⊂ Cx0

for s ∈ J .Set K =

⋂∞n=0C ∩ [xn,→). Observe that K 6= ∅ is weakly compact and

that Ts(K) ⊂ K for s ∈ J .


Weak convergence of Krasnoselskii-Ishikawa iterationprocess to a common fixed point

Theorem (WMK 2017)

Let X be uniformly convex with monotone norm (hence with monotoneOpial property as well). Let S = {Tt : t ∈ J} be an equicontinuousmonotone nonexpansive semigroup of mappings on C ⊂ X where C isclosed, convex and bounded. Choose {tk} ⊂ J so that for every s ∈ Jthere exists a strictly increasing sequence {jk} such that {jk+1 − jk} isbounded and tjk+1

= s+ tjk for every k ∈ N. Let x0 ∈ C be such thatx0 � Ts(x0) for every s ∈ E, E = A. Let {xk} be a sequence given by(KIS). Then xk ⇀ w, where w is a common fixed point for S. Moreover,x0 � w and w is a minimal common fixed point with this property.

Typical examples of Banach spaces for which our Theorem can be appliedare: lp and Lp[0, 1] for p > 1; Orlicz spaces Lϕ with the Luxemburg normif ϕ is strictly convex and satisfies the condition ∆2, modular functionspaces with the ∆2 and (UUC2).


Key results needed for the proof the Convergence Theorem

From now on X is assumed to be uniformly convex.

Lemma 1 (WMK 2017)

Let {xk} be (KIS) sequence. Assume that for every s ∈ E there exists astrictly increasing, quasi-periodic sequence {jk : k = 1, 2, . . . } such thattjk+1

= s+ tjk . Then limk→∞ ‖Ts(xk)− xk‖ = 0 for every s ∈ J .

Lemma 2 (WMK 2017)

Assume that the norm in X is monotone. Let {xk} be (KIS) sequence. Ifxnk

⇀ w and for every s ∈ J , ‖Ts(xnk)− xnk

‖ → 0 then w ∈ Fix(S).

Lemma 3 (WMK 2017)

If w ∈ Fix(S) is such that x0 � w then w ∈ K and there exists r ∈ Rsuch that limk→∞ ‖xk − w‖ = r.


Sketch of the proof of the Convergence Theorem

Restrict to Cx0 . By Lemma 1, limk→∞ ‖Ts(xk)− xk‖ = 0 for every s ∈ J .Consider two weak cluster points of {xk}, y and z. There exist then twosubsequences {yk} and {zk} of {xk} such that yk ⇀ y and zk ⇀ z. ByLemma 2, y, z ∈ Fix(S). By Lemma 3 the following limits exist

r1 = limk→∞

‖xk − y‖, r2 = limk→∞

‖xk − z‖,

and y, z ∈ K. Claim: y = z. Assume to the contrary that y 6= z. Notexk � y and xk � z because y, z ∈ K. By monotone Opial property

r1 = lim infk→∞

‖yk − y‖ < lim infk→∞

‖yk − z‖ = r2

= lim infk→∞

‖zk − z‖ < lim infk→∞

‖zk − y‖ = r1.

The contradiction implies y = z, which means that {xk} has at most oneweak cluster point in weakly-compact Cx0 , hence xk ⇀ w. ApplyingLemma 2 again, we conclude that w ∈ Fix(S). Since w ∈ K, x0 � w.


Sketch of the proof of the Convergence Theorem, cont.

It remains to be proved that w is a minimal common fixed point greaterthan or equal to x0. To this end let us assume to the contrary that thereexists z ∈ Fix(S) such that x0 � z and z ≺ w. By Lemma 3, xk � z fork = 1, 2, . . . . Using the monotone Opial property we obtain that

lim infn→∞

‖xn − w‖ < lim infn→∞

‖xn − z‖

On the other hand, since the norm is monotone, we conclude fromxn � z ≺ w that

‖xn − z‖ ≤ ‖xn − w‖.

The contradiction completes the proof.


Example of a construction of a sequence {tk}

Let us order the elements from E so that 0 = α1 < α2 < α3 < . . ..We will consider segments, that is, finite sequences of the real numbers ofthe form B = {β1, β2, . . . , βm}. Given a real number r we definerB = {rβ1, rβ2, . . . , rβm}.Let us define three initial segments:B1 = {α1, α2},B2 = 2{α1, α2, α1, α3},B3 = 3{α1, α2, α1, α3, α1, α2, α1, α4}.Generally, given a segment Bn we define Bn+1 = (n+ 1)Dn+1, whereDn+1 is constructed as a concatenation of Bn to itself and replacing thelast element by αn+1. Concatenation of all segments B1, B2, . . . will give asequence {tk} with required properties.Note: that tn →∞.


Conclusion

Let X be uniformly convex with monotone norm (hence with monotoneOpial property as well). Let S = {Tt : t ∈ J} be an equicontinuousmonotone nonexpansive semigroup of mappings on C ⊂ X where C isclosed, convex and bounded. Choose {tk} ⊂ J so that for every s ∈ Jthere exists a strictly increasing sequence {jk} such that {jk+1 − jk} isbounded and tjk+1

= s+ tjk for every k ∈ N. Let x0 ∈ C be such thatx0 � Ts(x0) for every s ∈ E, E = A. Let {xk} be a sequence given by(KIS). Then xk ⇀ w, where w is a common fixed point for S. Moreover,x0 � w and w is a minimal common fixed point with this property.

Typical examples of Banach spaces for which our Theorem can be appliedare: lp and Lp[0, 1] for p > 1; Orlicz spaces Lϕ with the Luxemburg normif ϕ is strictly convex and satisfies the condition ∆2, modular functionspaces with the ∆2 and (UUC2).


Construction of common fixed pointsfor monotone nonexpansive semigroups

Wojciech M. (Walter) KozlowskiUniversity of New South Wales

[email protected]

RMIT, Melbourne, 8 September 2017


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