Coincidence Theorems Research Articles

Periodic and homoclinic solutions of discontinuous Cohen–Grossberg neural networks with time-varying delays

In this paper, we investigate a class of discontinuous Cohen–Grossberg neural networks (DCGNNs) with time-varying delays. Under the framework of Filippov solution, by means of differential inclusions theory and the set-valued version of the Mawhin coincidence theorem, we firstly establish the existence results of 2kT-periodic solutions for the proposed DCGNNs. Secondly, by applying an approximation technique, we prove that the limit of the 2kT-periodic solutions exists and the limit is exactly the homoclinic solution of the proposed DCGNNs. Thirdly, by using the non-smooth analysis theory with Lyapunov-like approach, some new testable algebraic criteria are derived for ensuring the global exponential stability of the solutions. It should be regarded as the first time to study the homoclinic solutions of Cohen–Grossberg neural networks with discontinuous activations based on Filippov solution. Finally, convincing numerical examples and remarks are provided to substantiate the superiority and efficiency of obtained results.

Function and colorful extensions of the KKM theorem

We deal with an aggregate version of the KKM theorem. Given $n$ KKM families of special type on an $( n-1)$-dimensional simplex, we show that it is possible to choose a single element from every KKM family to get a KKM family on that simplex. We also introduce and study function KKM families as surrogates of KKM families on simplexes. We show a function version of the KKM theorem. The Coincidence Theorem is our main result, Brouwer's fixed is a special cases of that theorem.

A minimum theorem for epsilon Nielsen coincidences

Let f,g:X→Y be a pair of maps of a compact, connected, oriented Riemannian manifold, with or without boundary. In this paper we prove the minimum theorem (Wecken theorem) for the epsilon Nielsen coincidence number Nε(f,g).

Zero Preservation for a Family of Multivalued Functionals, and Applications to the Theory of Fixed Points and Coincidences

A theorem on the zero existence preservation for a parametric family of multivalued (α, β)-search functionals on an open subset of a metric space is proved. Several corollaries on the existence preservation for preimages of a closed subspace, for coincidence points, and for common fixed points under the action of a parametric family (a number of families) of mappings are obtained. The notion of a Zamfirescu-type pair of mappings is introduced, and a coincidence theorem for such pairs of mappings is obtained. In addition, a theorem on the coincidence existence preservation for a parametric family of such pairs of mappings is obtained. The obtained results imply several well-known theorems.

Summing multilinear operators by blocks: The isotropic and anisotropic cases

Unifying several directions of the development of the study of summing multilinear operators between Banach spaces, we construct a general framework that studies, under one single definition, multilinear operators that are summing with respect to sums taken over any number of indices, iterated and non-iterated sums (the isotropic and the anisotropic cases), sums over arbitrary blocks and over several different sequence norms. A large number of special classes of multilinear operators and of methods of generating classes of multilinear operators are recovered as particular instances. Ideal properties and coincidence theorems for the general classes are proved.

Lipschitz p-summing multilinear operators

We apply the geometric approach provided by Σ-operators to develop a theory of p-summability for multilinear operators. In this way, we introduce the notion of Lipschitz p-summing multilinear operators and show that it is consistent with a general panorama of generalization: Namely, they satisfy Pietsch-type domination and factorization theorems and generalizations of the inclusion Theorem, Grothendieck's coincidence Theorems, the weak Dvoretsky-Rogers Theorem and a Lindenstrauss-Pełczyńsky Theorem. We also characterize this new class in tensorial terms by means of a Chevet-Saphar-type tensor norm. Moreover, we introduce the notion of Dunford-Pettis multilinear operators. With them, we characterize when a projective tensor product contains ℓ1. Relations between Lipschitz p-summing multilinear operators with Dunford-Pettis and Hilbert-Schmidt multilinear operators are given.

Factorization and coincidence theorems for generalized adjoints

ABSTRACT As a consequence of a factorization theorem, we establish conditions on the Banach spaces E and F so that the k-adjoint of every linear operator/homogeneous polynomial from E to F belongs a given operator ideal . Applications are provided for operators/polynomials on and for the ideal of weakly compact operators.

A Topological Coincidence Theory for Multifunctions via Homotopy

A new simple result is presented which immediately yields the topological transversality theorem for coincidences.

R-KKM THEOREMS ON L-CONVEX SPACES AND ITS APPLICATIONS

In this paper, we introduce the generalized R − KKM mapping and the new class RψN − KKM(X, Y ), and we get some fixed point theorems, matching theorems, coincidence theorems, and minimax inequalities on L-convex spaces.

Search for Zeros of Functionals, Fixed Points, and Mappings Coincidence in Quasi-Metric Spaces

The cascade search principle for zeros of (α, β)-search functionals and consequent fixed point and coincidence theorems are proved for collections of single-valued and set-valued mappings of (b1, b2)-quasimetric spaces. These results are extensions of previous author’s results for metric spaces. In particular, a generalization is obtained for the recent result on coincidences of a covering mapping and a Lipschitz mappings of (b1, b2)-quasimetric spaces.

Periodic solution and its stability of a delayed Beddington‐DeAngelis type predator‐prey system with discontinuous control strategy

This paper investigates the periodic solution of a delayed Beddington‐DeAngelis (BD) type predator‐prey model with discontinuous control strategy. Firstly, the regularity and visibility analysis of the delayed predator‐prey model is carried out by using the principle of differential inclusion. Secondly, the positiveness and boundeness of the solution is discussed by employing the comparison theorem. Based on the boundary conditions of the model and the Mawhin‐like coincidence theorem, it is shown that the solution of the delayed BD system is asymptotically stable in finite time. Furthermore, it is found that there exists at least one periodic solution of the nonautonomous delayed predator‐prey model by using the principle of topological degree and set value mapping. Specially, when the nonautonomous delayed BD system degenerates into an autonomous system, some criteria are obtained to guarantee the convergence behavior of the harvesting solutions for the corresponding autonomous delayed BD system. Finally, numerical examples are given to demonstrate the applicability and effectiveness of main results. It is worthy to point out that the discontinuous control strategy is superior to the continuous harvesting policies adopted in existing literature.

Bounded lattice fuzzy coincidence theorems with applications

Bounded lattice fuzzy coincidence theorems with applications

Observations on relation-theoretic coincidence theorems under Boyd\u2013Wong type nonlinear contractions

In this article, we carry out some observations on existing metrical coincidence theorems of Karapinar et al. (Fixed Point Theory Appl. 2014:92, 2014) and Erhan et al. (J. Inequal. Appl. 2015:52, 2015) proved for Lakshmikantham–Ćirić-type nonlinear contractions involving (f,g)-closed transitive sets after proving some coincidence theorems satisfying Boyd–Wong-type nonlinear contractivity conditions employing the idea of (f,g)-closed locally f-transitive binary relation.

Periodic boundary value problems for two classes of nonlinear fractional differential equations

By using the coincidence degree theorem, we obtain a new result on the existence of solutions for a class of fractional differential equations with periodic boundary value conditions, where a certain nonlinear growth condition of the nonlinearity needs to be satisfied. Furthermore, we study another class of differential equations of fractional order with periodic boundary conditions at resonance. A new result on the existence of positive solutions is presented by use of a Leggett–Williams norm-type theorem for coincidences. Two examples are given to illustrate the main result at the end of this paper.

Coincidence Theorems in New Generalized Metric Spaces Under Locally g-transitive Binary Relation

<p>In this paper, we establish coincidence point theorems for contractive mappings, using locally g-transitivity of binary relation in new generalized metric spaces. In the present results, we use some relation theoretic analogues of standard metric notions such as continuity, completeness and regularity. In this way our results extend, modify and generalize some recent fixed point theorems, for instance, Karapinar et al [J. Fixed Point Theory Appl. 18(2016) 645-671], Alam and Imdad [Fixed Point Theory, in press].</p>

Coincedence theorem for reciprocally continuous systems of multi-valued and single-valued maps

In this paper we eliminate completely the requirement of continuity from the main results of Baillon- Singh [1], Gairola et al. [9] and Gairola-Jangwan [7] and prove a coincidence theorem for systems of single-valued and multi-valued maps on finite product of metric spaces using the concept of coordinatewise reciprocal continuity.

$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$

Let $X$ be a paracompact space, let $G$ be a finite group acting freely on $X$ and let $H$ a cyclic subgroup of $G$ of prime order $p$. Let $f:X\rightarrow M$ be a continuous map where $M$ is a connected $m$-manifold (orientable if $p>2$) and $f^* (V_k) = 0$, for $k\geq 1$, where $V_k$ are the $Wu$ classes of $M$. Suppose that $\ind X\geq n> (|G|-r)m$, where $r=\frac{|G|}{p}$. In this work, we estimate the cohomological dimension of the set $A(f,H,G)$ of $(H,G)$-coincidence points of $f$. Also, we estimate the index of a $(H, G)$-coincidence set in the case that $H$ is a $p$-torus subgroup of a particular group $G$ and as application we prove a topological Tverberg type theorem for any natural number $r$. Such result is a weak version of the famous topological Tverberg conjecture, which was proved recently fail for all $r$ that are not prime powers. Moreover, we obtain a generalized Van Kampen-Flores type theorem for any integer $r$.

Brondsted order in a metric space and generalizations of Caristi theorem

Fixed point and coincidence theorems for mappings of ordered sets and their metric counterparts generalizing the well-known Caristi fixed point theorem are presented.

Obstruction theory for coincidences of multiple maps

Let f1,...,fk:X→N be maps from a complex X to a compact manifold N, k≥2. In previous works [1,12], a Lefschetz type theorem was established so that the non-vanishing of a Lefschetz type coincidence class L(f1,...,fk) implies the existence of a coincidence x∈X such that f1(x)=...=fk(x). In this paper, we investigate the converse of the Lefschetz coincidence theorem for multiple maps. In particular, we study the obstruction to deforming the maps f1,...,fk to be coincidence free. We construct an example of two maps f1,f2:M→T from a sympletic 4-manifold M to the 2-torus T such that f1 and f2 cannot be homotopic to coincidence free maps but for anyf:M→T, the maps f1,f2,f are deformable to be coincidence free.

Coincidence point of L-fuzzy sets endowed with graph

In the present paper, coincidence theorems of a crisp mapping and a sequence of L-fuzzy mappings have been produced under graphic contractive conditions in connection with notions of \(D_{\alpha _{L}}\) and \(d_{L}^{\infty }\) distances on the class of L-fuzzy sets. Further, we obtain some fixed point theorems for L-fuzzy set-valued mappings and pull out a variety of current results on fixed points for fuzzy mappings and multivalued mappings in the literature. As applications, we acquire coincidence points of a sequence of multivalued mappings with a self mapping and prove the existence of solution for fuzzy integral equations.