Topological Fixed Point Theory Research Articles

An application of Sobolev's inequality to one-dimensional Kirchhoff equations

We consider nonlocal differential equations with convolution coefficients of the form−M((a⁎(g∘|u′|))(1))u″(t)=λf(t,u(t)), t∈(0,1), in which the coefficient function M imparts a nonlocal structure to the problem. The function g satisfies p-q growth. A model case occurs when g(t):=tp, where p>1, and a(t)≡1 – i.e., the problem−M(‖u′‖Lpp)u″(t)=λf(t,u(t)), t∈(0,1). By an application of the one-dimensional Sobolev inequality, together with a specially constructed order cone, we are able to demonstrate existence of at least one positive solution to this equation when subjected to Dirichlet boundary conditions. Our methodology utilises a more recently developed topological fixed point theory, which allows for the use of sets that are unbounded in the ambient norm.

The Reidemeister trace of an n-valued map

In topological fixed point theory, the Reidemeister trace is an invariant associated to a selfmap of a polyhedron which combines information from the Lefschetz and Nielsen numbers. In this paper we define the Reidemeister trace in the context of n-valued selfmaps of compact polyhedra. We prove several properties of the Reidemeister trace which generalize properties from the single-valued theory, and prove an averaging formula.

Convolution equations with variable time nonlocal coefficients

We consider a one-dimensional nonlocal equation of the form −M((a∗(g∘u))(t))u′′(t)=λf(t,u(t)),t∈(0,1)subject to boundary data. Using classical topological fixed point theory together with a nonstandard order cone, existence of at least one positive solution is established. The novelty lies in the fact that the nonlocal element, (a∗(g∘u))(t), is time varying.

Nonlocal differential equations with (p,q)$ {(p,q)}$ growth

AbstractThe author considers a class of nonlocal convolution‐type ordinary differential equations of the form where is a continuous function satisfying growth — that is, there exist constants , and such that where . Existence of at least one positive solution to this equation when subjected to given boundary data is studied by means of topological fixed point theory.

The Reidemeister spectrum of 2-step nilpotent groups determined by graphs

ABSTRACT In this paper we study the Reidemeister spectrum of finitely generated torsion-free 2-step nilpotent groups associated to graphs. We develop three methods, based on the structure of the graph, that can be used to determine the Reidemeister spectrum of the associated group in terms of the Reidemeister spectra of groups associated to smaller graphs. We illustrate our methods for several families of graphs, including all the groups associated to a graph with at most four vertices. We also apply our results in the context of topological fixed point theory for nilmanifolds.

Nonlocal Differential Equations with Convolution Coefficients and Applications to Fractional Calculus

Abstract The existence of at least one positive solution to a large class of both integer- and fractional-order nonlocal differential equations, of which one model case is - A ⁢ ( ( b * u q ) ⁢ ( 1 ) ) ⁢ u ′′ ⁢ ( t ) = λ ⁢ f ⁢ ( t , u ⁢ ( t ) ) , t ∈ ( 0 , 1 ) , q ≥ 1 , -A((b*u^{q})(1))u^{\prime\prime}(t)=\lambda f(t,u(t)),\quad t\in(0,1),\,q\geq 1, is considered. Due to the coefficient A ⁢ ( ( b * u q ) ⁢ ( 1 ) ) {A((b*u^{q})(1))} appearing in the differential equation, the equation has a coefficient containing a convolution term. By choosing the kernel b in various ways, specific nonlocal coefficients can be recovered such as nonlocal coefficients equivalent to a fractional integral of Riemann–Liouville type. The results rely on the use of a nonstandard order cone together with topological fixed point theory. Applications to fractional differential equations are given, including a problem related to the ( n - 1 , 1 ) {(n-1,1)} -conjugate problem.

Recent advances in the Lefschetz fixed point theory for multivalued mappings

In 1923 S. Lefschetz proved the famous fixed point theorem which is now known as the Lefschetz fixed point theorem (comp. [5], [9], [20], [21]. The multivalued case was considered for the first time in 1946 by S. Eilenberg and D. Montgomery ([10]). They proved the Lefschetz fixed point theorem for acyclic mappings of compact ANR-spaces (absolute neighborhood retracts (see [4] or [13]) using Vietoris mapping theorem (see [4], [13], [16]) as the main tool. In 1970 Eilenberg, Montgomery's result was generalized for acyclic mappings of complete ANR-s (see [17]). Next, a class of admissible multivalued mappings was introduced ([13] or [16]).
 Note that the class of admissible mappings is quite large and contains as a special case not only acyclic mappings but also infinite compositions of acyclic mappings. For this class of multivalued mappings several versions of the Lefschetz fixed point theorem were proved (comp. [11], [13]-15], [18], [19], [27]). In 1982 G. Skordev and W. Siegberg ([26]) introduced the class of multivalued mappings so-called now (1 − n)-acyclic mappings. Note that the class (1−n)-acyclic mappings contain as a special case n-valued mappings considered in [6], [12], [28]. We recommend [8] for the most important results connected with (1 − n)-acyclic mappings. Finally, the Lefschetz fixed point theorem was considered for spheric mappings (comp. [3], [2], [7], [23]) and for random multivalued mappings (comp. [1], [2], [13]). Let us remark that the main classes of spaces for which the Lefschetz fixed point theorem was formulated are the class of ANR-spaces ([4]) and MANR-spaces (multi absolute neighborhood retracts (see [27]). The aim of this paper is to recall the most important results concerning the Lefschetz fixed point theorem for multivalued mappings and to prove new versions of this theorem mainly for AANR-spaces (approximative absolute neighborhood retracts (see [4] or [13]) and for MANR-s. We believe that this article will be useful for analysts applying topological fixed point theory for multivalued mappings in nonlinear analysis, especially in differential inclusions.

Nonlocal differential equations with concave coefficients of convolution type

For a continuous function A:R→R the existence of at least one positive solution of the nonlocal differential equation −A∫01(g∘u)(s)dsu′′(t)=λf(t,u(t)),t∈(0,1)is considered in the specific case in which g is a concave, strictly increasing continuous function. As a model case we consider −A∫01|u(s)|qdsu′′(t)=λf(t,u(t)),t∈(0,1),where 0<q<1. The results rely on the use of a nonstandard order cone together with an application of topological fixed point theory. Due to this the coefficient function A is neither required to be strictly positive nor monotone nor satisfy any particular growth condition. As a consequence of the theory developed for the preceding problem we also provide results for the nonlocal equation −A((b∗(g∘u))(1))u′′(t)=λf(t,u(t)),t∈(0,1)where t↦(b∗(g∘u))(t) is a finite convolution.

A topological approach to a class of one-dimensional Kirchhoff equations

We consider, for q &gt; 1 q&gt;1 , the one-dimensional Kirchhoff-type problem − A ( ∫ 0 1 ( u ′ ( s ) ) q d s ) u ( t ) a m p ; = λ f ( u ( t ) ) , t ∈ ( 0 , 1 ) u ( 0 ) a m p ; = 0 u ′ ( 1 ) a m p ; = 0 \begin{equation} \begin {split} -A\left (\int _0^1\big (u’(s)\big )^q\ ds\right )u(t)&amp;=\lambda f\big (u(t)\big )\text {, }t\in (0,1)\\ u(0)&amp;=0\\ u’(1)&amp;=0\notag \end{split} \end{equation} and demonstrate the existence of at least one positive solution to this problem. The main contribution is to show that by using a nonstandard order cone together with topological fixed point theory much weaker conditions than usual can be imposed on the coefficient function A A .

Discrete Kirchhoff equations with sign-changing coefficients

The second-order nonlocal difference equation is considered subject to the right-focal boundary conditions The problem is a discrete version of a one-dimensional Kirchhoff-type equation. Existence of at least one positive solution is considered by means of a topological fixed-point theory. Weaker-than-usual conditions are proposed for the coefficient function A.

Topological analysis of doubly nonlocal boundary value problems

We consider both Hammerstein integral equations and nonlocal boundary value problems in possession of two different nonlocal elements. The first occurs in the differential equation itself and takes the form $$\Vert u\Vert _q^q$$ . The second occurs in the boundary condition and takes the form of a Stieltjes integral. Because the nonlocal elements are not necessarily related, a careful analysis is required to control each nonlocal element simultaneously. Topological fixed point theory is used to deduce existence of at least one positive solution to the boundary value problem. And we illustrate the application of the results with an example.

Complex Solutions and Stationary Scattering for the Nonlinear Helmholtz Equation

We study a stationary scattering problem related to the nonlinear Helmholtz equation $-\Delta u - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For a given incident free wave $\varphi \in L^\infty(\mathbb{R}^N)$, we prove the existence of complex-valued solutions of the form $u=\varphi+u_{\text{sc}}$, where $u_{\text{sc}}$ satisfies the Sommerfeld outgoing radiation condition. Since neither a variational framework nor maximum principles are available for this problem, we use topological fixed point theory and global bifurcation theory to solve an associated integral equation involving the Helmholtz resolvent operator. The key step of this approach is the proof of suitable a priori bounds.

A topological approach to nonlocal elliptic partial differential equations on an annulus

AbstractFor we consider the nonlocal ordinary differential equation subject to the Dirichlet boundary conditions . Due to the term appearing in the equation, this is a class of nonlocal differential equations. By using a novel order cone we are able to establish existence of a positive solution to this problem by means of topological fixed point theory. The preceding problem is really a special case of a more general problem that we consider – namely, the existence of a positive radially symmetric solution to the nonlocal elliptic partial differential equation subject to , for , where Ω is an annular region when .

Fixed poin sets in digital topology, 1

&lt;p&gt;In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.&lt;/p&gt;&lt;p&gt;We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C&lt;sub&gt;n&lt;/sub&gt;) where C&lt;sub&gt;n&lt;/sub&gt; is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C&lt;sub&gt;n&lt;/sub&gt;, in which F(X) does not equal {0, 1, . . . , #X}.&lt;/p&gt;&lt;p&gt;We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.&lt;/p&gt;

Topological fixed point theory for compact multifunctions via homotopy and essential maps

In this paper we present a very simple result which will enable us to establish the topological transversality theorem in a general setting.

Recent results on the topological fixed point theory of multivalued mappings: a survey

AbstractIn this survey, we present current results from the topological fixed point theory of multivalued mappings which were obtained by ourselves in the last five years (see Andres and Górniewicz in Fixed Point Theory 12(2):255-264, 2011; Topol. Methods Nonlinear Anal. 40:337-358, 2012; Libertas Mathematica 33(1):69-78, 2013; Int. J. Bifurc. Chaos 24(11):1450148, 2014; J. Nonlinear Convex Anal. 16(6):1013-1023, 2015; Int. J. Bifurc. Chaos, 2015, to appear). Some abstract theorems are applied to differential inclusions and multivalued fractals. A part of the deterministic theory is randomized, including the applications to random differential inclusions.

Topological fixed point theory and applications to variational inequalities

This is the first part of a work on generalized variational inequalities and their applications in optimization. It proposes a general theoretical framework for the solvability of variational inequalities with possibly non-convex constraints and objectives. The framework consists of a generic constrained nonlinear inequality ( $\exists\hat{u}\in\Psi(\hat {u})$ , $\exists \hat{y}\in\Phi(\hat{u})$ , with $\varphi(\hat{u},\hat{y},\hat{u})\leq \varphi(\hat{u},\hat{y},v)$ , $\forall v\in\Psi(\hat{u})$ ) derived from new topological fixed point theorems for set-valued maps in the absence of convexity. Simple homotopical and approximation methods are used to extend the Kakutani fixed point theorem to upper semicontinuous compact approachable set-valued maps defined on a large class of non-convex spaces having non-trivial Euler-Poincare characteristic and modeled on locally finite polyhedra. The constrained nonlinear inequality provides an umbrella unifying and extending a number of known results and approaches in the theory of generalized variational inequalities. Various applications to optimization problems will be presented in the second part to this work to be published ulteriorly.

70 Years of asymptotic fixed point theory

In this expository article we aim to illustrate asymptotic fixed point theory from the viewpoints of its history, the main motivation from applications, the two basic techniques for proofs and a survey on the major results. The bibliography should cover most aspects but is, for sure, not complete. We confine ourselves to results from topological fixed point theory, i.e., metric fixed point theory is not at all touched.

Solvability of Variational Inequalities on Hilbert Lattices

This paper provides a systematic solvability analysis for (generalized) variational inequalities on separable Hilbert lattices. By contrast to a large part of the existing literature, our approach is lattice-theoretic, and is not based on topological fixed point theory. This allows us to establish the solvability of certain types of (generalized) variational inequalities without requiring the involved (set-valued) maps be hemicontinuous or monotonic. Some of our results generalize those obtained in the context of nonlinear complementarity problems in earlier work, and appear to have scope for applications. This is illustrated by means of several applications to fixed point theory, optimization, and game theory.

Fractional semilinear differential inclusions

In this paper, we first present a version of Filippov’s Theorem for fractional semilinear differential inclusions of the form, {(cDαy−Ay)(t)∈F(t,y(t)), a.e. t∈[0,b],y(0)=y0∈X, where cDα is the Caputo fractional derivative and F is a set-valued map. After several existence results, the compactness of solutions sets is also investigated. Some results from topological fixed point theory together with notions of measure of noncompactness are used.