FIXED POINT INDEX THEORY FOR WEAKLY INWARD MAPPINGS

In this paper we define a fixed point index theory for locally Lip., completely continuous and weakly inward mappings defined on closed convex sets in general Banach spaces where no other artificial conditions are imposed. This makes ns to deal with these kinds of mappings more easily. As obvious applications, some results in [3],[5],[7],[9],[10] are deepened and extended.

Existence of Sign-changing Solution for Three-point Boundary Value Problems

In this paper, by using the fixed-point index theory, we study the existence of sign-changing solution of some three-point boundary value problems {y ''(t) + f(y) = 0, t ∈ [0, 1], y' (0) = 0, y(1) = αy(η), where 0 < α < 1, 0 < η < 1, f : R → R is continuous, strictly increasing and f(0) = 0.

THE RELATION BETWEEN SIGN-CHANGING SOLUTION AND POSITIVE-NEGATIVE SOLUTIONS FOR NONLINEAR OPERATOR EQUATIONS AND ITS APPLICATIONS

By fixed point index theory and a result obtained by Amann, existence of the solution for a class of nonlinear operator equations x = Ax is discussed. Under suitable conditions, a couple of positive and negative solutions are obtained. Finally, the abstract result is applied to nonlinear Sturm-Liouville boundary value problem, and at least four distinct solutions are obtained.

Index Iteration Theory for Brake Orbit Type Solutions and Applications

In this paper,we give a survey on the index iteration theory of an index theory for brake orbit type solutions and its applications in the study of brake orbit problems including the Seifert conjecture and the minimal period solution problems in brake orbit cases.

QUANTIZATION COMMUTES WITH REDUCTION,A SURVEY

We review the themes relating to the proposition that“quantization commutes with reduction”([Q,R]=0),from symplectic manifolds to Cauchy-Riemann manifolds.

P-symmetric Subharmonic Solutions for Nonlinear Hamiltonian Systems

In this paper,we prove that for each positive k≡1 mod m there exists a P-symmetric kmτ-periodic solution xk for asymptotically linear mτ-periodic Hamiltonian systems,which are nonautonomous and endowed with a P-symmetry.If the P-symmetric Hamiltonian function is semi-positive,one can prove,under a new iteration inequality of the Maslov-type P-index,that xk_(1) and xk_(2) are geometrically distinct for k_(1)/k_(2)≥(2n+1)m+1;and xk_(1),xk_(2) are geometrically distinct for k_(1)/k_(2)≥m+1 provided xk_(1) is non-degenerate.

An Optimal Expansion Planning of Electric Distribution Network Incorporating Health Index and Non-network Solutions

The health status of distribution equipment and networks is not considered directly in existing distribution network planning methods.In order to effectively consider the health status and deal with the risk associated with load and renewable generation uncertainties,this paper presents a new optimal expansion planning approach for distribution network(EPADN)incorporating equipment’s health index(HI)and non-network solutions(NNSs).HI and relevant risk are used to help develop the optimal equipment replacement strategy and temporary NNSs are considered as promising options for handling the uncertainties of load growth,reliability requirements of power supply and output of distributed energy resources(DERs)at a lower cost than network alternatives.An EPADN model using network solutions(NSs)and NNSs is proposed.The planning objectives of the proposed model are safety,reliability,economy,and‘greenness’that are also the meaning of distribution network HI.A method integrating an improved niche genetic algorithm(INGA)and a spanning tree algorithm(STA)is fitted to solve the model presented here for real sized networks with a manageable computational cost.Simulation results of an actual 22-node distribution network in China,illustrate the effectiveness of the proposed approach.

INFINITE PERIODIC SOLUTIONS TO A CLASS OF SECOND-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

By means of variational structure and Z2 group index theory, we obtain infinite periodic solutions to a class of second-order neutral differential equations.

EXISTENCE OF POSITIVE SOLUTIONS TO SUBLINEAR SECOND-ORDER PERIODIC BOUNDARY VALUE PROBLEM

In this paper, we consider a second-order periodic boundary value problem. By the topological degree theory and fixed point index theory, we prove the existence of positive solutions which gives the relationship between the first positive eigenvalue of the associated eigenvalue problem and the behavior of the nonlinear term of the system.

EXISTENCE OF POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, using the property of the corresponding Green’s function and fixed point index theory, some sufficient conditions for the multiplicity and nonexistence of positive solutions to a class of nonlinear fractional boundary value problem are obtained. Three examples are given to show the effectiveness of our results.

Homoclinic Orbits for First Order Hamiltonian Systems with Some Twist Conditions

In this paper, we study the nonperiodic first-order Hamiltonian system u = JL（t）u ＋ JH＇（t,u）, where HεCl（RxR2n）. With some assumptions on L, the corresponding Hamiltonianoperator has only discrete spectrum. By using the index theory for self-adjoint operator equation, we establish the existence of multiple homoclinic orbits for the asymptotically quadratic nonlinearty satisfying some twist conditions between infinity and origin.

Positive curvature, symmetry, and topology

We depict recent developments in the field of positive sectional curvature, mainly, but not exclusively, under the assumption of isometric torus actions. After an elaborate introduction to the field, we shall discuss various classification results, before we provide results on the computation of Euler characteristics. This will be the starting point for an examination of more involved invariants and further techniques. In particular, we shall discuss the Hopf conjectures, related decomposition results like the Wilhelm conjecture, results in differential topology and index theory as well as in rational homotopy theory, geometrically formal metrics in positive curvature and much more. The results we present will be discussed for arbitrary dimensions, but also specified to small dimensions. This survey article features mainly depictions of our own work interest in this area and cites results obtained in different collaborations; full statements and proofs can be found in the respective original research articles.

Metrics with Positive Scalar Curvature at Infinity and Localization Algebra

In this paper, the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.

ON THE EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR NEUMANN BOUNDARY VALUE PROBLEMS

By using fixed point index theory, we consider the existence of positive solutions for singular nonlinear Neumann boundary value problems. Our main results extend and improve many known results even for non-singular cases.

MULTI-PERIODIC SOLUTIONS TO A CLASS OF SECOND-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

By means of variational structure and Z 2 group index theory,we obtain infinite periodic solutions to a class of second-order neutral differential equations.

EXISTENCE OF POSITIVE SOLUTIONS TO SINGULAR SUBLINEAR SEMIPOSITONE NEUMANN BOUNDARY VALUE PROBLEM

The existence of positive solutions to a singular sublinear semipositone Neumann boundary value problem is considered. In this paper,the nonlinearity term is not necessary to be bounded from below and the function q(t) is allowed to be singular at t = 0 and t = 1.

MULTIPLICITY OF SOLUTIONS TO ASYMPTOTICALLY LINEAR SECOND-ORDER ORDINARY DIFFERENTIAL SYSTEM

In this paper,we consider an asymptotically linear second-order ordinary differential system with Dirchlet boundary value conditions. Under some conditions,we show the multiplicity of solutions to the system by the Morse theory and an index theory.

POSITIVE SOLUTIONS TO A CLASS OF SECOND-ORDER SINGULAR SEMIPOSITIVE NEUMANN BOUNDARY VALUE PROBLEM WITH GENERAL FORM

By constructing an explicit Green function and using the fixed point index theory on a cone, we present some existence results of positive solutions to a class of second-order singular semipositive Neumann boundary value problem, where the nonlinear term is allowed to be nonnegative and unbounded.

Symmetric Closed Characteristics on Symmetric Compact Convex Hypersurfaces in R^8

Let ∑be a C^3 compact symmetric convex hypersurface in R^8.We prove that when ∑ carries exactly four geometrically distinct closed characteristics,then all of them must be symmetric.Due to the example of weakly non-resonant ellipsoids,our result is sharp.