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We present a fixed point theorem on topological cylinders in normed linear spaces for maps satisfying a property of stretching a space along paths. This result is a generalization of a similar theorem obtained by D. Papini and F. Zanolin. In view of the main result, we discuss the existence of fixed points for maps defined on different types of domains and we propose alternative proofs for classical fixed point theorems, as Brouwer, Schauder and Krasnosel'skiĭ ones. [ABSTRACT FROM AUTHOR]

In this paper, by virtue of fixed point index on cones, we obtain two existence theorems of positive solutions for a system of nonlinear semipositone fractional q-difference equations with q-integral boundary conditions. Concave functions and nonnegative matrices are used to characterize the coupling behavior of our nonlinearities. [ABSTRACT FROM AUTHOR]

We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the T-periodic solutions of the considered equations. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of at least two positive solutions for the following high-order fractional semipositone boundary value problem (SBVP, for short) with coupled integral boundary value conditions:.. where n - 1 < α ≤ n, n ≥ 3, 0 < η1, η2 ≤ 1, λ, λ1, λ2 are parameters and satisfy λ1λ2(η1η2) α < Γ²(α + 1), Dα0+ is the standard Riemann-Liouville derivative, and f, g are continuous and semipositone. By using the nonlinear alternative of Leray-Schauder type, Krasnoselskii's fixed point theorems, and the theory of fixed point index on cone, we establish some existence results of multiple positive solutions to the considered fractional SBVP. As applications, two examples are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

We prove new results regarding the existence of positive solutions for a nonlinear periodic boundary value problem related to the Liebau phenomenon. As a consequence we obtain new sufficient conditions for the existence of a pump in a simple model. Our methodology relies on the use of classical fixed point index. Some examples are provided to illustrate our theory. We improve and complement previous results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the existence of positive solutions for a class of fractional differential equations with Riemann-Stieltjes integral boundary conditions are investigated under some weak conditions concerning the spectral analysis of the relevant linear operator and Gelfand's formula by means of the fixed point index theorem in cones. The nonlinearity permits singularities not only at $$t=0,1,$$ but also at $$x_{i}=0\ (i=1,2,\ldots ,n-1)$$ . The results obtained herein generalize and improve some known results including singular and non-singular cases. [ABSTRACT FROM AUTHOR]

Using the theorem and properties of the fixed point index in a Banach space and applying a new method to dispose of the impulsive term, we prove that there exists a solvable interval of positive parameter λ in which the second order impulsive singular equation has two infinite families of positive solutions. Moreover, we also establish the new expression of Green's function for the above equation. Noticing that $\lambda>0$ and $c_{k}\neq0$ ( $k=1,2,\ldots,n$ ), our main results improve many previous results. This is probably the first time that the existence of two infinite families of positive solutions for second order impulsive singular parametric equations has been studied. [ABSTRACT FROM AUTHOR]

In this paper, a predator-prey model with predator-stage structured and diffusion is concerned. We deal with the system by endowing it with the homogeneous Neumann boundary conditions. We first give the a priori estimates of positive solutions for the reduced reaction diffusion system. Secondly, we discuss the effects of predator cannibalism and prey growth on the stability of nonnegative constant steady states of the model in detail. Thirdly, we investigate the nonexistence and existence of nonconstant positive solutions. Finally, we discuss the Hopf bifurcation created by diffusion. [ABSTRACT FROM AUTHOR]

In this paper, a kind of αth $(3<\alpha\leq4)$ order differential equation with two-point boundary conditions is considered. The existence result of a sign-changing solution is given by the topological degree theory and the fixed point index theory. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the competitive Leslie/Gower model (map) It is shown that T unconditionally admits a globally attracting 1-codimensional invariant hypersurface $$\varSigma $$ , called carrying simplex, such that every nontrivial orbit is asymptotic to one in $$\varSigma $$ . More general and easily checked conditions to guarantee the existence of carrying simplex for competitive maps are provided. An equivalence relation is defined relative to local stability of fixed points on $$\partial \varSigma $$ (the boundary of $$\varSigma $$ ) on the space of all three-dimensional Leslie/Gower models. Using a formula on the sum of the indices of all fixed points on the carrying simplex for three-dimensional maps, we list the 33 stable equivalence classes in terms of simple inequalities on the parameters $$c_i$$ and $$b_{ij}$$ and draw their orbits on $$\varSigma $$ . In classes 1-18, every nontrivial orbit tends to a fixed point on $$\partial \varSigma $$ . In classes 19-25, each map possesses a unique positive fixed point which is a saddle on $$\varSigma $$ , and hence Neimark-Sacker bifurcations do not occur. Neimark-Sacker bifurcation does occur within each of classes 26-31, while it does not occur in class 32. Each map from class 27 admits a heteroclinic cycle, which forms the boundary of $$\varSigma $$ . The criteria on the stability of heteroclinic cycles are also given. This classification makes it possible to further investigate various dynamical properties in respective class. [ABSTRACT FROM AUTHOR]

Let M be a smooth compact and simply-connected manifold with simply-connected boundary $$\partial M$$ , r be a fixed odd natural number. We consider f, a $$C^1$$ self-map of M, preserving $$\partial M$$ . Under the assumption that the dimension of M is at least 4, we define an invariant $$D_r(f;M,\partial M)$$ that is equal to the minimal number of r-periodic points for all maps preserving $$\partial M$$ and $$C^1$$ -homotopic to f. As an application, we give necessary and sufficient conditions for a reduction of a set of r-periodic points to one point in the $$C^1$$ -homotopy class. [ABSTRACT FROM AUTHOR]

By means of topological methods, we provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of perturbed Hammerstein integral equations. In order to illustrate our theoretical results, we study some problems that occur in applied mathematics, namely models of chemical reactors, beams and thermostats. We also apply our theory in order to prove the existence of nontrivial radial solutions of systems of elliptic boundary value problems subject to nonlocal, nonlinear boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlocal fractional differential equation: where $1 < \alpha< 2$ , $0 < \xi< 1$ , $\eta\xi^{\alpha-1}= 1$ , $D^{\alpha}_{0+}$ is the standard Riemann-Liouville derivative, $f:[0,1]\times[0,+\infty)\rightarrow\mathbb{R}$ is continuous. The existence and uniqueness of positive solutions are obtained by means of the fixed point index theory and iterative technique. [ABSTRACT FROM AUTHOR]

We study the covering dimension of the set of (positive) solutions to various classes of nonlinear equations involving condensing and A-proper maps. It is based on the nontriviality of the fixed point index of a certain condensing map or on oddness of a nonlinear map. Applications to nonlinear singular integral equations and to semilinear ordinary and elliptic partial differential equations are given with finite or infinite dimensional null space of the linear part. [ABSTRACT FROM AUTHOR]

We study completely continuous maps in ordered Banach spaces having an invariant cone. We give conditions on minorants and majorants which yield the existence of at least one non-zero fixed point. [ABSTRACT FROM AUTHOR]

We study the existence of positive solutions for some nonlinear second order boundary value problems with nonlocal boundary conditions. The key boundary condition considered is of the form u(1) = α[u'], where a is a linear functional on C[0,1], that is, is given by a Riemann-Stieltjes integral α[v] = ί01 v(s)dA(s) where A is a function of bounded variation. It is important in our case that dA is not a positive measure but can be sign changing. [ABSTRACT FROM AUTHOR]

Consider the equation − M ( x , ‖ u ‖ ) Δ p u = λ f ( x , u , ∇ u ) − g ( x , u , ∇ u ) in Ω, u = 0 on ∂Ω. The main aim of this paper is to prove existence results for both non-degenerate and degenerate cases of the function M . Our approach relies on the fixed point index and the cone theoretic argument. [ABSTRACT FROM AUTHOR]

We present fixed point results for admissibly compact maps on cones in Fréchet spaces. We first extend the Krasnosel'skĭ fixed point theorem with order type cone-compression and cone-expansion conditions. Then, we extend the monotone iterative method to this context. Finally, we present fixed point results under a combination of the assumptions of the previous results. More precisely, we combine a cone-compressing or cone-extending condition only on one side of the boundary of an annulus with an assumption on the existence of an upper fixed point. In addition, we show that the usual monotonicity condition can be weaken. [ABSTRACT FROM AUTHOR]

A characterization of dynamically defined zeta functions is presented. It comprises a list of axioms, natural extension of the one which characterizes topological degree, and a uniqueness theorem. Lefschetz zeta function is the main (and proved unique) example of such zeta functions. Another interpretation of this function arises from the notion of symmetric product from which some corollaries and applications are obtained. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of positive solutions to a nonlinear third-order three point boundary value problem. The main tools are Krasnosel'skii fixed point theorem on cones, and the fixed point index theory.

This work concerns the existence and the multiplicity of solutions for singular boundary-value problems with a variable coefficient, posed on the positive half-line. When the nonlinearity is positive but may have a space singularity at the origin, the existence of single and twin positive solutions is obtained by means of the fixed point index theory. The singularity is treated by approximating the nonlinearity, which is assumed to satisfy general growth conditions. When the nonlinearity is not necessarily positive, the Schauder fixed point theorem is combined with the method of upper and lower solutions on unbounded domains to prove existence of solutions. Our results extend those in [18] and are illustrated with examples.

We discuss the nonexistence of positive solutions for nonlinear boundary value problems. In particular, we discuss necessary restrictions on parameters in nonlocal problems in order that (strictly) positive solutions exist. We consider cases that can be written in an equivalent integral equation form which covers a wide range of problems. In contrast to previous work, we do not use concavity arguments, instead we use positivity properties of an associated linear operator which uses ideas related to the $u_0$-positive operators of Krasnosel'skii.

In this article we study the existence and multiplicity of positive solutions for the system of second-order boundary value problems involving first order derivatives $$displaylines{ -u''=f(t, u, u', v, v'),cr -v''=g(t, u, u', v, v'),cr u(0)=u'(1)=0,quad v(0)=v'(1)=0. }$$ Here $f,gin C([0,1]imes mathbb{R}_+^{4}, mathbb{R}_+)(mathbb{R}_+:=[0,infty))$. We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing Jensen's integral inequality for concave functions and $mathbb{R}_+^2$-monotone matrices.

In this article we study the existence of positive solutions for the system of higher order boundary-value problems involving all derivatives of odd orders $$displaylines{ (-1)^mw^{(2m)} =f(t, w, w',-w''',dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',dots, (-1)^{n-1}z^{(2n-1)}), cr (-1)^nz^{(2n)} =g(t, w, w',-w''',dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',dots, (-1)^{n-1}z^{(2n-1)}), cr w^{(2i)}(0)=w^{(2i+1)}(1)=0quad (i=0,1,dots, m-1),cr z^{(2j)}(0)=z^{(2j+1)}(1)=0quad (j=0,1,dots, n-1). } $$ Here $f,gin C([0,1]imesmathbb{R}_+^{m+n+2},mathbb{R}_+)$ $(mathbb{R}_+:=[0,+infty))$. Our hypotheses imposed on the nonlinearities $f$ and $g$ are formulated in terms of two linear functions $h_1(x)$ and $h_2(y)$. We use fixed point index theory to establish our main results based on a priori estimates of positive solutions achieved by utilizing nonnegative matrices.

In this paper, we study the existence and multiplicity of positive solutions of a class of nonlinear fractional boundary value problems with Dirichlet boundary conditions. By applying the fixed point theory on cones we establish a series of criteria for the existence of one, two, any arbitrary finite number, and an infinite number of positive solutions. A criterion for the nonexistence of positive solutions is also derived. Several examples are given for demonstration.

This article concerns the existence and multiplicity of positive solutions for the singular Sturm-Liouville boundary value problem $$displaylines{ (p(t)u'(t))'+h(t)f(t,u(t))=0,quad 0

In the present paper we prove distributional chaos for the Poincaré map in the perturbed equation \[\dot{z}=\left(1 + e^{i\kappa t} |z|^2\right)\bar{z}^2 - N e^{-i\frac{\pi}{3}}.\] Heteroclinic and homoclinic connections between two periodic solutions bifurcating from the stationary solution \(0\) present in the system when \(N = 0\) are also discussed.

Using the theory of the fixed point index in a cone and the Leray-Schauder degree, this paper investigates the existence and multiplicity of nontrivial solutions for a class of fourth order m-point boundary-value problems.

By means of fixed point index, we establish sufficient conditions for the existence of positive solutions to p-Laplacian difference equations. In particular, the nonlinear term is allowed to change sign.

In this paper, we study the following semipositone dynamic system on time scales $$displaylines{ -x^{DeltaDelta}(t)=f(t,y)+p(t), quad tin(0,T)_{mathbb{T}},cr -y^{DeltaDelta}(t)=g(t,x), quad tin(0,T)_{mathbb{T}},cr x(0)=x(sigma^{2}(T))=0, cr alpha{y(0)}-eta{y^{Delta}{(0)}}= gamma{y(sigma(T))}+delta{y^{Delta}(sigma(T))}=0. }$$ Using fixed point index theory, we show the existence of at least one positive solution. The interesting point is the that nonlinear term is allowed to change sign and may tend to negative infinity.

Using the theory of fixed point index, this paper discusses the existence of at least one positive solution and the existence of multiple positive solutions for the singular three-point boundary value problem: $$displaylines{ y''(t)+a(t)f(t,y(t),y'(t))=0,quad 0

Let X be a real infinite-dimensional Banach space and ψ a measure of noncompactness on X. Let Ω be a bounded open subset of X and A : Ω → X a ψ-condensing operator, which has no fixed points on ∂Ω.Then the fixed point index, ind(A,Ω), of A on Ω is defined (see, for example, ([1] and [18]). In particular, if A is a compact operator ind(A,Ω) agrees with the classical Leray-Schauder degree of I −A on Ω relative to the point 0, deg(I −A,Ω,0). The main aim of this note is to investigate boundary conditions, under which the fixed point index of strict- ψ-contractive or ψ-condensing operators A : Ω → X is equal to zero. Correspondingly, results on eigenvectors and nonzero fixed points of k-ψ-contractive and ψ-condensing operators are obtained. In particular we generalize the Birkhoff-Kellog theorem [4] and Guo’s domain compression and expansion theorem [17]. The note is based mainly on the results contained in [7] and [8].

This paper weals with the existence of positive solutions of the fully second-order boundary value problem [ABSTRACT FROM AUTHOR]

We consider elliptic Dirichlet problem − Δ p u = λ f ( x , u , ∇ u ) − g ( x , u ) in Ω , u = 0 on ∂ Ω . Assume that the nonlinearity f satisfies certain growth condition. Using the fixed point index, the Leggett–Williams theorem on fixed point and the arguments on monotone minorant, we prove the existence and multiplicity results for the equation. This extends some known results. [ABSTRACT FROM AUTHOR]

The first part of this paper is a general approach towards chaotic dynamics for a continuous map $$f:X\supset M\rightarrow X$$ which employs the fixed point index and continuation. The second part deals with the differential equation with state-dependent delay. For a suitable parameter $$\alpha $$ close to $$5\pi /2$$ we construct a delay functional $$d_{{\varDelta }}$$ , constant near the origin, so that the previous equation has a homoclinic solution, $$h(t)\rightarrow 0$$ as $$t\rightarrow \pm \infty $$ , with certain regularity properties of the linearization of the semiflow along the flowline $$t\mapsto h_t$$ . The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion. [ABSTRACT FROM AUTHOR]

We consider the perturbed Hammerstein equation where φ and ψ are linear functionals, which can be represented as Stieltjes integrals with signed measures, and is a parameter. We demonstrate that by imposing some mild conditions on and , together with an assumption that the functionals decompose in a particular way, it follows that this problem can have multiple positive solutions even if no conditions at all are imposed on f except for a positivity assumption and continuity. Applications are, in particular, given to boundary value problems with nonlocal boundary conditions, and one of these existence results allows to be affine and to be linear. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity of positive solutions for a system of nonlinear singular higher-order fractional differential equations with fractional multi-point boundary conditions. The main tool used in the proof is fixed point index theory. Some limit type conditions for ensuring the existence of positive solutions are given, and our conditions are suitable for more general functions. [ABSTRACT FROM AUTHOR]

A new definition of essential fixed points is introduced for a large class of multivalued maps. Two abstract existence theorems are presented for approximable maps on compact ANR-spaces in terms of a nontrivial fixed point index, or a nontrivial Lefschetz number and a zero topological dimension of the fixed point set. The second one is applied to the periodic dissipative Marchaud differential inclusions for obtaining the existence of a discretely essential subharmonic solution. Three simple illustrative examples are supplied. [ABSTRACT FROM AUTHOR]

A fixed point theorem for weakly inward A-proper maps defined on cones in Banach spaces is established using a fixed point index for such maps. The result generalizes a theorem in Deimling (Nonlinear Functional Analysis, ) for weakly inward maps defined on a cone in $\mathbb{R}^{n}$. We then apply the theorem to a Picard boundary value problem and obtain the existence of a positive solution. [ABSTRACT FROM AUTHOR]

In this paper using fixed point index theory we study the existence of positive solutions for a second order boundary value problem on time scales. Our existence theorems will be expressed under some conditions concerning the first eigenvalue of a relevant linear operator. [ABSTRACT FROM AUTHOR]