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In this paper, we extend the study of fixed point properties of semitopological semigroups of continuous mappings in locally convex spaces to the setting of completely regular topological spaces. As applications, we establish a general fixed point theorem, a convergence theorem and an application to amenable locally compact groups.
Let
$X,Y$
be two Hilbert spaces, let E be a subset of
$X,$
and let
$G\colon E \to Y$
be a Lipschitz mapping. A famous theorem of Kirszbraun’s states that there exists
$\tilde {G} : X \to Y$
with
$\tilde {G}=G$
on E and
$ \operatorname {\mathrm {Lip}}(\tilde {G})= \operatorname {\mathrm {Lip}}(G).$
In this note we show that in fact the function
$\tilde {G}:=\nabla _Y( \operatorname {\mathrm {conv}} (g))( \cdot , 0)$
, where
defines such an extension. We apply this formula to get an extension result for strongly biLipschitz mappings. Related to the latter, we also consider extensions of
$C^{1,1}$
strongly convex functions.
We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $\unicode[STIX]{x1D6FF}$-dense (the orbit meets every ball of radius $\unicode[STIX]{x1D6FF}$), weakly dense and such that $\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.
The existence and nonexistence of semi-trivial or coexistence steady-state solutions of one-dimensional competition models in an unstirred chemostat are studied by establishing new results on systems of Hammerstein integral equations via the classical fixed point index theory. We provide three ranges for the two parameters involved in the competition models under which the models have no semi-trivial and coexistence steady-state solutions or have semi-trivial steady-state solutions but no coexistence steady-state solutions or have semi-trivial or coexistence steady-state solutions. It remains open to find the largest range for the two parameters under which the models have only coexistence steady-state solutions. We apply the new results on systems of Hammerstein integral equations to obtain results on steady-state solutions of systems of reaction-diffusion equations with general separated boundary conditions. Such type of results have not been studied in the literature. However, these results are very useful for studying the competition models in an unstirred chemostat. Our results on Hammerstein integral equations and differential equations generalize and improve some previous results.
For
$p\geq 2$
, let
$E$
be a 2-uniformly smooth and
$p$
-uniformly convex real Banach space and let
$A:E\rightarrow E^{\ast }$
be a Lipschitz and strongly monotone mapping such that
$A^{-1}(0)\neq \emptyset$
. For given
$x_{1}\in E$
, let
$\{x_{n}\}$
be generated by the algorithm
$x_{n+1}=J^{-1}(Jx_{n}-\unicode[STIX]{x1D706}Ax_{n})$
,
$n\geq 1$
, where
$J$
is the normalized duality mapping from
$E$
into
$E^{\ast }$
and
$\unicode[STIX]{x1D706}$
is a positive real number in
$(0,1)$
satisfying suitable conditions. Then it is proved that
$\{x_{n}\}$
converges strongly to the unique point
$x^{\ast }\in A^{-1}(0)$
. Furthermore, our theorems provide an affirmative answer to the Chidume et al. open problem [‘Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces’, SpringerPlus4 (2015), 297]. Finally, applications to convex minimization problems are given.
A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein–Avidan and Slomka to infinite-dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.
In this paper, we initiate the study of fixed point properties of amenable or reversible semitopological semigroups in modular spaces. Takahashi’s fixed point theorem for amenable semigroups of nonexpansive mappings, and T. Mitchell’s fixed point theorem for reversible semigroups of nonexpansive mappings in Banach spaces are extended to the setting of modular spaces. Among other things, we also generalize another classical result due to Mitchell characterizing the left amenability property of the space of left uniformly continuous functions on semitopological semigroups by introducing the notion of a semi-modular space as a generalization of the concept of a locally convex space.
We show how some Ulam stability issues can be approached for functions taking values in 2-Banach spaces. We use the example of the well-known Cauchy equation
$f(x+y)=f(x)+f(y)$
, but we believe that this method can be applied for many other equations. In particular we provide an extension of an earlier stability result that has been motivated by a problem of Th. M. Rassias. The main tool is a recent fixed point theorem in some spaces of functions with values in 2-Banach spaces.
We introduce a notion of modulated topological vector spaces, that generalises, among others, Banach and modular function spaces. As applications, we prove some results which extend Kirk’s and Browder’s fixed point theorems. The theory of modulated topological vector spaces provides a very minimalist framework, where powerful fixed point theorems are valid under a bare minimum of assumptions.
We consider a nonlinear Robin problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter δ. The relative size of each periodic perforation is determined by a positive parameter ε. Under suitable assumptions, such a problem admits a family of solutions which depends on ε and δ. We analyse the behaviour the energy integral of such a family as (ε, δ) tends to (0, 0) by an approach that represents an alternative to asymptotic expansions and classical homogenization theory.
As a continuation of previous work of the first author with Ranjbar [‘A variational inequality in complete CAT(0) spaces’, J. Fixed Point Theory Appl.17 (2015), 557–574] on a special form of variational inequalities in Hadamard spaces, in this paper we study equilibrium problems in Hadamard spaces, which extend variational inequalities and many other problems in nonlinear analysis. In this paper, first we study the existence of solutions of equilibrium problems associated with pseudo-monotone bifunctions with suitable conditions on the bifunctions in Hadamard spaces. Then, to approximate an equilibrium point, we consider the proximal point algorithm for pseudo-monotone bifunctions. We prove existence of the sequence generated by the algorithm in several cases in Hadamard spaces. Next, we introduce the resolvent of a bifunction in Hadamard spaces. We prove convergence of the resolvent to an equilibrium point. We also prove
$\triangle$
-convergence of the sequence generated by the proximal point algorithm to an equilibrium point of the pseudo-monotone bifunction and also the strong convergence under additional assumptions on the bifunction. Finally, we study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point without any additional assumption on the pseudo-monotone bifunction. Some examples in fixed point theory and convex minimization are also presented.
For a complex function
$F$
on
$\mathbb{C}$
, we study the associated composition operator
$T_{F}(f):=F\circ f=F(f)$
on Wiener amalgam
$W^{p,q}(\mathbb{R}^{d})\;(1\leqslant p<\infty ,1\leqslant q<2)$
. We have shown
$T_{F}$
maps
$W^{p,1}(\mathbb{R}^{d})$
to
$W^{p,q}(\mathbb{R}^{d})$
if and only if
$F$
is real analytic on
$\mathbb{R}^{2}$
and
$F(0)=0$
. Similar result is proved in the case of modulation spaces
$M^{p,q}(\mathbb{R}^{d})$
. In particular, this gives an affirmative answer to the open question proposed in Bhimani and Ratnakumar (J. Funct. Anal. 270(2) (2016), 621–648).
The implication
$(i)\Rightarrow (ii)$
of Theorem 2.1 in our article [1] is not true as it stands. We give here two correct statements which follow from the original proof.
In this work we are concerned with the existence of fixed points for multivalued maps defined on Banach spaces. Using the Banach spaces scale concept, we establish the existence of a fixed point of a multivalued map in a vector subspace where the map is only locally Lipschitz continuous. We apply our results to the existence of mild solutions and asymptotically almost periodic solutions of an abstract Cauchy problem governed by a first-order differential inclusion. Our results are obtained by using fixed point theory for the measure of noncompactness.
In the framework of fixed point theory, many generalizations of the classical results due to Krasnosel'skii are known. One of these extensions consists in relaxing the conditions imposed on the mapping, working with k-set contractions instead of continuous and compact maps. The aim of this work if to study in detail some fixed point results of this type, and obtain a certain generalization by using star convex sets.
Given two (real) normed (linear) spaces
$X$
and
$Y$
, let
$X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$
, where
$\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$
. It is known that
$X\otimes _{1}Y$
is
$2$
-UR if and only if both
$X$
and
$Y$
are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if
$X$
is
$m$
-dimensional and
$Y$
is
$k$
-UR, then
$X\otimes _{1}Y$
is
$(m+k)$
-UR. In the other direction, we observe that if
$X\otimes _{1}Y$
is
$k$
-UR, then both
$X$
and
$Y$
are
$(k-1)$
-UR. Given a monotone norm
$\Vert \cdot \Vert _{E}$
on
$\mathbb{R}^{2}$
, we let
$X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$
where
$\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$
. It is known that if
$X$
is uniformly rotund in every direction,
$Y$
has the weak fixed point property for nonexpansive maps (WFPP) and
$\Vert \cdot \Vert _{E}$
is strictly monotone, then
$X\otimes _{E}Y$
has WFPP. Using the notion of
$k$
-uniform rotundity relative to every
$k$
-dimensional subspace we show that this result holds with a weaker condition on
$X$
.
The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.
We study best proximity points in the framework of metric spaces with
$w$
-distances. The results extend, generalise and unify several well-known fixed point results in the literature.
In 1965, Browder proved the existence of a common fixed point for commuting families of nonexpansive mappings acting on nonempty bounded closed convex subsets of uniformly convex Banach spaces. The purpose of this paper is to extend this result to left amenable semigroups of nonexpansive mappings.
Let E be a uniformly convex and uniformly smooth real Banach space, and let E* be its dual. Let A : E → 2E* be a bounded maximal monotone map. Assume that A−1(0) ≠ Ø. A new iterative sequence is constructed which converges strongly to an element of A−1(0). The theorem proved complements results obtained on strong convergence of the proximal point algorithm for approximating an element of A−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.