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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.
We consider relatively Meir–Keeler condensing operators to study the existence of best proximity points (pairs) by using the notion of measure of noncompactness, and extend a result of Aghajani et al. [‘Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness’, Acta Math. Sci. Ser. B35 (2015), 552–566]. As an application of our main result, we investigate the existence of an optimal solution for a system of integrodifferential equations.
We prove the existence of common fixed points for monotone contractive and monotone nonexpansive semigroups of nonlinear mappings acting in Banach spaces equipped with partial order. We also discuss some applications to differential equations and dynamical systems.
Extending recent results by Cascales et al. [‘Plasticity of the unit ball of a strictly convex Banach space’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.110(2) (2016), 723–727], we demonstrate that for every Banach space
$X$
and every collection
$Z_{i},i\in I$
, of strictly convex Banach spaces, every nonexpansive bijection from the unit ball of
$X$
to the unit ball of the sum of
$Z_{i}$
by
$\ell _{1}$
is an isometry.
Let
$X$
and
$Y$
be two normed spaces over fields
$\mathbb{F}$
and
$\mathbb{K}$
, respectively. We prove new generalised hyperstability results for the general linear equation of the form
$g(ax+by)=Ag(x)+Bg(y)$
, where
$g:X\rightarrow Y$
is a mapping and
$a,b\in \mathbb{F}$
,
$A,B\in \mathbb{K}\backslash \{0\}$
, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result.
We consider the split common null point problem in Hilbert space. We introduce and study a shrinking projection method for finding a solution using the resolvent of a maximal monotone operator and prove a strong convergence theorem for the algorithm.
We extend the results of Schu [‘Iterative construction of fixed points of asymptotically nonexpansive mappings’, J. Math. Anal. Appl.158 (1991), 407–413] to monotone asymptotically nonexpansive mappings by means of the Fibonacci–Mann iteration process
where
$T$
is a monotone asymptotically nonexpansive self-mapping defined on a closed bounded and nonempty convex subset of a uniformly convex Banach space and
$\{f(n)\}$
is the Fibonacci integer sequence. We obtain a weak convergence result in
$L_{p}([0,1])$
, with
$1<p<+\infty$
, using a property similar to the weak Opial condition satisfied by monotone sequences.
Through appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as an α-fractal operator on , the space of all real-valued continuous functions defined on a compact interval I. With an eye towards connecting fractal functions with other branches of mathematics, in this paper we continue to investigate the fractal operator in more general spaces such as the space of all bounded functions and the Lebesgue space , and in some standard spaces of smooth functions such as the space of k-times continuously differentiable functions, Hölder spaces and Sobolev spaces . Using properties of the α-fractal operator, the existence of Schauder bases consisting of self-referential functions for these function spaces is established.