Fixed point theory for cyclic generalized {(\varphi-\phi)}-contraction mappings

Fixed point results are presented for cyclic generalized ${(\phi-\varphi)}$ -contraction mappings on complete metric spaces (X, d). Our results extend previous results given by ?iri?, Moradi and Khojasteh, and Karap?nar.     

Nonsmooth analysis of vector-valued mappings with contributions to nondifferentiable programming

We define order Lipschitz mappings from a Banach space to an order complete vector lattice and present a nonsmooth analysis for such functions. In particular, we establish properties of a generalized directional derivative and gradient and derive results concerning a calculus of generalized gradients (i.e., calculation of the generalized gradient of f when f = f₁ + f₂, f = f · ₂, etc.). We show the relevance of the above analysis to nondifferentiaile programming by deriving optimality conditions for problems of the form min f(x) subject to x [euro] S. For S arbitrary we state the results in terms of cones of displacement of the feasible region at the optimal point; when S ={x ? A|g(x) ? B}, we obtain Kuhn-Tucker type results.     

On the convergence of the proximal algorithm for nonsmooth functions involving analytic features

We study the convergence of the proximal algorithm applied to nonsmooth functions that satisfy the ?jasiewicz inequality around their generalized critical points. Typical examples of functions complying with these conditions are continuous semialgebraic or subanalytic functions. Following ?jasiewicz’s original idea, we prove that any bounded sequence generated by the proximal algorithm converges to some generalized critical point. We also obtain convergence rate results which are related to the flatness of the function by means of ?jasiewicz exponents. Apart from the sharp and elliptic cases which yield finite or geometric convergence, the decay estimates that are derived are of the type O(k ^?s ), where s ∈ (0, + ∞) depends on the flatness of the function.     

Fixed Points of Sequence of Ćirić Generalized Contractions of Perov Type

Perov used the concept of vector valued metric space and obtained a Banach type fixed point theorem on such a complete generalized metric space. In this article, we study fixed point results for the new extensions of sequence of ?iri? generalized contractions on cone metric space, and we give some generalized versions of the fixed point theorem of Perov. The theory is illustrated with some examples. It is worth mentioning that the main result in this paper could not be derived from ?iri?’s result by the scalarization method, and hence indeed improves many recent results in cone metric spaces.     

Ishikawa iterative process with errors for nonlinear equations of generalized monotone type in Banach spaces

The concept of the operators of generalized monotone type is introduced and iterative approximation methods for a fixed point of such operators by the Ishikawa and Mann iteration schemes {xn} and {yn} with errors is studied. Let X be a real Banach space and T : D ? X → 2^D be a multi‐valued operator of generalized monotone type with fixed points. A new general lemma on the convergence of real sequences is proved and used to show that {xn} converges strongly to a unique fixed point of T in D. This result is applied to the iterative approximation method for solutions of nonlinear equations with generalized strongly accretive operators. Our results generalize many of know results. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A unification of the existence theorems of the nonlinear complementarity problem

The nonlinear complementarity problem is the problem of finding a point x in the n-dimensional Euclidean space,R ⁿ , such that x ? 0, f(x) ? 0 and 〈x,f(x)～ = 0, where f is a nonlinear continuous function fromR ⁿ into itself. Many existence theorems for the problem have been established in various ways. The aim of the present paper is to treat them in a unified manner. Eaves's basic theorem of complementarity is generalized, and the generalized theorem is used as a unified framework for several typical existence theorems.     

Pseudo-differential Operators,Cubature and Equidistribution on the 3D ball: An Approach Based on Orthonormal Basis Systems

In this paper, the distribution of points on a unit ball in ?³ is investigated. The ansatz is motivated by an approach for point grids on the unit sphere by Cui and Freeden. A formula for a generalized discrepancy is developed, which is then used to check the uniformity of point grids on a ball. The generalized discrepancy originates from an error bound for a quadrature (cubature) rule on the ball with uniform weights. In particular, we discuss the integration of functions from particular Sobolev spaces based on known orthonormal systems on the ball. This includes the introduction of a concept of pseudo-differential operators on the ball. Finally, different point grids are constructed on the ball and are compared by the discrepancy. Furthermore, numerical and graphical comparisons of the grids are presented.     

Subquadrangle m‐regular systems on generalized quadrangles

We introduce the notion of subquadrangle regular system of a generalized quadrangle. A subquadrangle regular system of order m on a generalized quadrangle of order (s, t) is a set ? of embedded subquadrangles with the property that every point lies on exactly m subquadrangles of ?. If m is one half of the total number of subquadrangles on a point, we call ? a subquadrangle hemisystem. We construct two infinite families of symplectic subquadrangle hemisystems of the Hermitian surface ??(3, q²), q odd, and two infinite families of symplectic subquadrangle hemisystems of ??₃(q²), q even. Some sporadic examples of symplectic subquadrangle regular systems of ??(3, q²) are also presented. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:28‐41, 2010     

On generalized weak subdifferentials and some properties

In this article, some characterizations for gw-subdifferentiability of functions from ? ⁿ to ? ^m are stated. Some criteria for gw-subdifferentiability of generalized lower locally Lipschitz functions and positively homogeneous functions are given. Furthermore, it is proved that every Lipschitz function is gw-subdifferentiable at any point in its domain. Finally, the relationship between directional derivative and gw-subdifferential is given and a convexity criteria for Fréchet differentiable function is given by using gw-subdifferential.     

星型关联代数与星式M?bius反演

本文借助于非标准组合论中的星型有限结构,定义了星型关联代数,从而建立了局部星型有限集上的M?bius反演.由此在一个超结构扩大中,在非标准意义下,将M?bius反演推广到局部标准无限半序集上.文中几例显示,在非标准领域里,本文结果为探索离散数学与连续数学的某些反演的统一性提供了一种可能途径.     

On maximal element theorems,variants of Ekeland’s variational principle and their applications

In this paper, we establish several different versions of generalized Ekeland’s variational principle and maximal element theorem for τ$\tau$-functions in ?$?$ complete metric spaces. The equivalence relations between maximal element theorems, generalized Ekeland’s variational principle, generalized Caristi’s (common) fixed point theorems and nonconvex maximal element theorems for maps are also proved. Moreover, we obtain some applications to a nonconvex minimax theorem, nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces.     

A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties

We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z, except for three exceptional cases. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several explicit formulae for the global height of any generalized flag variety. Oblatum 17-VI-1999 & 10-IX-2001?Published online: 19 November 2001     

Covers of the Abelian Variety of Generalized MV-Algebras

Using the categorical equivalence of the class of generalized MV-algebras with the class of unital ?-groups, we describe all varieties of symmetric top abelian unital ?-groups that cover the variety  _u? of abelian unital ?-groups. Equivalently, we describe all cover varieties of the variety of MV-algebras, ?, within the variety of generalized MV-algebras admitting only one negation and each of whose maximal ideals is normal. In particular, there are continuum many representable varieties of generalized MV-algebras that cover ?.     

Triangle Contact Representations and Duality

A?contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. De Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual. A?primal?Cdual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal?Cdual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a corner of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.     

Inequalities for the generalized trigonometric and hyperbolic functions

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p$p$-Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinovi?–Adamovi?’s inequality, Lazarevi?’s inequality, Huygens-type inequalities, Wilker-type inequalities, and Cusa–Huygens-type inequalities, are generalized to the case of generalized functions.     

Generalized second-order directional derivatives and optimization with C1,1 functions

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C¹,¹ functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gãteaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C^1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.     

A note on Hilbert and Beltrami systems

Hilbert and Beltrami (line- ) systems were introduced by H. Mohrmann, Math. Ann. 85 (1922) p.177- 183. These systems give examples of non- desarguesian affine planes, in fact, the earliest known examples are of this type. We describe a construction for “generalized Beltrami systems”, and show that every such system defines a topological affine plane with point set ?². Since our construction uses only the topological structure of ?²- planes, it is possible to iterate this process. As an application, we obtain an embeddability theorem for a class of two- dimensional stable planes, including Strambach’s exceptional SL₂R- plane.     

Singular solutions with exponential growth to Protter’s problems

Four-dimensional boundary value problems which were formulated by Proter for the nonhomogeneous wave equation are studied. They can be considered as multidimensional versions of the Darboux problems in ?². Protter’s problem is not well posed in the frame of classical solvability. On the other hand, it is known that the unique generalized solution may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. Some known results suggest that the solution may have at most exponential growth. We construct an infinitely smooth right-hand side function such that the corresponding generalized solution to Protter’s problem has an exponential singularity.     

On primitive points of elliptic curves with complex multiplication

Let E be an elliptic curve defined over Q and P∈E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the imaginary quadratic field k. Denote by M_E,P the set of rational primes ? such that ? splits in k, E has good reduction at ?, and P is a primitive point modulo ?. Under the generalized Riemann hypothesis, we can determine the positivity of the density of the set M_E,P explicitly.     

On a conjecture of Brouwer involving the connectivity of strongly regular graphs

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components.We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called Δ-spaces are counterexamples to Brouwer?s Conjecture. Using J.I. Hall?s characterization of finite reduced copolar spaces, we find that the triangular graphs T(m), the symplectic graphs Sp(2r,q) over the field Fq (for any q prime power), and the strongly regular graphs constructed from the hyperbolic quadrics O⁺(2r,2) and from the elliptic quadrics O⁻(2r,2) over the field F₂, respectively, are counterexamples to Brouwer?s Conjecture. For each of these graphs, we determine precisely the minimum number of vertices whose removal disconnects the graph into non-singleton components. While we are not aware of an analogue of Hall?s characterization theorem for Δ-spaces, we show that complements of the point graphs of certain finite generalized quadrangles are point graphs of Δ-spaces and thus, yield other counterexamples to Brouwer?s Conjecture.We prove that Brouwer?s Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles GQ(q,q) graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue −2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases.We leave as an open problem determining the best general lower bound for the minimum size of a disconnecting set of vertices of a strongly regular graph, whose removal disconnects the graph into non-singleton components.