Fixed Point Index Research Articles

A global result for discrete ϕ-Laplacian eigenvalue problems

Concerned is the existence, nonexistence and multiplicity of positive solutions for discrete ϕ-Laplacian eigenvalue problems. By using lower and upper solutions method and fixed point index theory, a global result with respect to parameter is established.

Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions

Using the theory of fixed point index, we discuss the existence and multiplicity of non-negative solutions of a wide class of boundary value problems with coupled nonlinear boundary conditions. Our approach is fairly general and covers a variety of situations. We illustrate our theory in an example all the constants that occur in our theory.

Existence of multiple positive solutions for p-Laplacian multipoint boundary value problems on time scales

In this paper, we consider p-Laplacian multipoint boundary value problems on time scales. By using a generalization of the Leggett-Williams fixed point theorem due to Bai and Ge, we prove that a boundary value problem has at least three positive solutions. Moreover, we study existence of positive solutions of a multipoint boundary value problem for an increasing homeomorphism and homomorphism on time scales. By using fixed point index theory, sufficient conditions for the existence of at least two positive solutions are provided. Examples are given to illustrate the results.MSC:34B15, 34B16, 34B18, 39A10.

The Symmetric Solutions for Boundary Value Problems of Second-Order Singular Differential Equation

In this paper, by constructing a special operator and using fixed point index theorem of cone, we get the sufficient conditions for symmetric positive solution of a class of nonlinear singular boundary value problems with p-Laplace operator, which improved and generalized the result of related paper. Keywords—Banach space, cone, fixed point index, singular differential equation, p-Laplace operator, symmetric solutions.

Multiple Positive Solutions to $$m$$ m -Point Boundary Value Problem of Nonlinear Fractional Differential Equation

In this paper, we consider the existence and multiplicity of positive solutions to \(m\)-point boundary value problem of nonlinear fractional differential equation. By using the fixed point index theory, the multiplicity result of positive solutions is obtained.

Symmetric positive solutions to singular system with multi-point coupled boundary conditions

In this paper, we study the existence and multiplicity of symmetric positive solutions for a nonlinear system with multi-point coupled boundary conditions. The arguments are based upon a specially constructed cone and the fixed point index theorem in cones. An example is then given to demonstrate the applicability of our results.

Positive solutions for a fourth order discrete p‐Laplacian boundary value problem

AbstractIn this paper, we study the existence and multiplicity of positive solutions for the following fourth order nonlinear discrete p‐Laplacian boundary value problem urn:x-wiley:01704214:media:mma2766:mma2766-math-0001 where φp(s) = | s | p − 2s, p > 1, is continuous, T is an integer with T ≥ 5 and . By virtue of Jensen's discrete inequalities, we use fixed point index theory to establish our main results. Copyright © 2013 John Wiley & Sons, Ltd.

Existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter

In this paper, we study the existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter. By using the properties of the Green’s function, u 0-positive function and the fixed point index theory, we obtain some existence results of positive solution under some conditions concerning the first eigenvalue with respect to the relevant linear operator. The method of this paper is a unified method for establishing the existence of multiple positive solutions for a large number of nonlinear differential equations of arbitrary order with any allowed number of non-local boundary conditions.

Fixed point homomorphisms for parameterized maps

Let X be an ANR (absolute neighborhood retract), \({\Lambda}\) a k-dimensional topological manifold with topological orientation \({\eta}\) , and \({f : D \rightarrow X}\) a locally compact map, where D is an open subset of \({X \times \Lambda}\) . We define Fix(f) as the set of points\({{(x, \lambda) \in D}}\) such that \({x = f(x, \lambda)}\) . For an open pair (U, V) in \({X \times \Lambda}\) such that \({{\rm Fix}(f) \cap U \backslash V}\) is compact we construct a homomorphism \({\Sigma_{(f,U,V)} : H^{k}(U, V ) \rightarrow R}\) in the singular cohomologies H* over a ring-with-unit R, in such a way that the properties of Solvability, Excision and Naturality, Homotopy Invariance, Additivity, Multiplicativity, Normalization, Orientation Invariance, Commutativity, Contraction, Topological Invariance, and Ring Naturality hold. In the case of a \({C^{\infty}}\) -manifold \({\Lambda}\) , these properties uniquely determine \({\Sigma}\) . By passing to the direct limit of \({\Sigma_{(f,U,V)}}\) with respect to the pairs (U, V) such that \({K = {\rm Fix}(f) \cap U \backslash V}\) , we define a homomorphism \({\sigma_{(f,K)} : {H}_{k}({\rm Fix}(f), Fix(f) \backslash K) \rightarrow R}\) in the Cech cohomologies. Properties of \({\Sigma}\) and \({\sigma}\) are equivalent each to the other. We indicate how the homomorphisms generalize the fixed point index.

Positive solutions of nonlinear equations via comparison with linear operators

We discuss positive solutions of problems that arise from nonlinear boundary value problems in the particular situation where the nonlinear term $f(t,u)$ depends explicitly on $t$ and this dependence is crucial. We give new fixed point index results using comparisons with linear operators. These prove new results on existence of positive solutions under some conditions which can be sharp.

A Continuation Result for Forced Oscillations of Constrained Motion Problems with Infinite Delay

Abstract We prove a global continuation result for T-periodic solutions of a T-periodic parametrized second order retarded functional differential equation on a boundaryless compact manifold with nonzero Euler-Poincaré characteristic. The approach is based on the fixed point index theory for locally compact maps on ANRs. As an application, we prove the existence of forced oscillations of retarded functional motion equations defined on topologically nontrivial compact constraints. This existence result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case.

On Fixed Point Index Formula and its Applications in Dynamics

Abstract The aim of this note is to present a generalization of the topological method for detecting chaotic dynamics in a periodic local processes generated by non-autonomous ordinary differential equations, based on the notion of periodic segments.

Nontrivial solutions of Hammerstein integral equations with reflections

Using the theory of fixed point index, we establish new results for the existence of nonzero solutions of Hammerstein integral equations with reflections. We apply our results to a first-order periodic boundary value problem with reflections. MSC: Primary 34K10; secondary 34B15; 34K13

Continuity of the cone spectral radius

This paper concerns the question whether the cone spectral radius of a continuous compact order-preserving homogenous map on a closed cone in Banach space depends continuously on the map. Using the fixed point index we show that if there exist points not in the cone spectrum arbitrarily close to the cone spectral radius, then the cone spectral radius is continuous. An example is presented showing that continuity may fail, if this condition does not hold. We also analyze the cone spectrum of continuous order-preserving homogeneous maps on finite dimensional closed cones. In particular, we prove that for each polyhedral cone with m faces, the cone spectrum contains at most m-1 elements, and this upper bound is sharp for each polyhedral cone. Moreover, for each non-polyhedral cone, there exist maps whose cone spectrum contains a countably infinite number of distinct points.

Multiple positive solutions of boundary value problems for fractional order integro-differential equations in a Banach space

In this paper, we obtain the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space by means of fixed point index theory of completely continuous operators. MSC:26A33, 34B15.

The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie–Gower predator–prey model with Holling-type II functional responses

In this paper, we revisit a diffusive Leslie–Gower predator–prey model with Holling-type II functional responses and Dirichlet boundary condition. It is shown that multiple positive steady state solutions exist under certain conditions on the parameters, while for another parameter region, the positive steady state solution is unique and locally asymptotically stable. Results are proved by using bifurcation theory, fixed point index theory, energy estimates and asymptotic behavior analysis.

Eigenvalue Approach to Even Order System Periodic Boundary Value Problems

AbstractWe study an even order systemboundary value problemwith periodic boundary conditions. By establishing the existence of a positive eigenvalue of an associated linear system Sturm-Liouville problem, we obtain new conditions for the boundary value problem to have a positive solution. Our major tools are the Krein-Rutman theorem for linear spectra and the fixed point index theory for compact operators

Positive Solutions for Nonlinear Lidstone Boundary Value Problems

In this paper, we consider the existence of positive solutions for nonlinear Lidstone boundary value problems. An new existence result is obtained by applying the fixed point index theorem.

Positive solutions of a diffusive Leslie–Gower predator–prey model with Bazykin functional response

In this paper, we consider a diffusive Leslie–Gower predator–prey model with Bazykin functional response and zero Dirichlet boundary condition. We show the existence, multiplicity and uniqueness of positive solutions when parameters are in different regions. Results are proved by using bifurcation theory, fixed point index theory, energy estimate and asymptotical behavior analysis.

An order-type existence theorem and applications to periodic problems

Based on the fixed point index and partial order method, one new order-type existence theorem concerning cone expansion and compression is established. As applications, we present sufficient existence conditions for the first- and second-order periodic problems. MSC: 34B15