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In this paper, we consider the existence of positive solutions to a singular semipositone boundary value problem of nonlinear fractional differential equations. By applying the fixed point index theorem, some new results for the existence of positive solutions are obtained. In addition, an example is presented to demonstrate the application of our main results.

We use the fixed point index theory of condensing mapping in cones discuss the existence of positive solutions for the following boundary value problem of fractional differential equations in a Banach space E $$\begin{aligned} \left\{ \begin{array}{ll} -D^{\,\beta }_{0^{+}}u(t)=f(t,u(t)),\quad t\in J, \\ u(0)=u^{\prime }(0)=\theta ,\quad u(1)=\rho \int _{0}^{1}u(t)dt,\\ \end{array} \right. \end{aligned}$$ where both $$2

We provide new results on the existence of nonzero positive weak solutions for a class of second order elliptic systems. Our approach relies on a combined use of iterative techniques and classical fixed point index. Some examples are presented to illustrate the theoretical results., Comment: 13 pages

We study the homogeneous Dirichlet boundary value problem of a singular $$(p_1,p_2)$$ -Laplacian system with multiparameters. The global structure of multiparameters for existence, nonexistence and multiplicity of positive solutions can be obtained. Proofs are mainly based on the upper and lower solution method, the fixed point index theory in a cone, generalized Picone identity, global continuum theorem and degree arguments. As an application, we can establish the global result of the positive radial and decaying solutions for a class of quasilinear elliptic systems in exterior domains.

In this paper we prove the existence of continua of nonradial solutions for the Lane–Emden equation in the annulus. In a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones. Next, using the fixed point index in cone, we produce nonradial solutions with a new type of symmetry. This result also applies to solutions with fixed signed, showing that the set of solutions to the Lane–Emden problem has a very rich and complex structure.

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.

Motivated by boundary value problems we give new results for a class of nonlinear Hammerstein integral operators acting in a cone to have a fixed point index equal to one. The idea is to allow the nonlinearity to be large on one part of its domains provided it is sufficiently small on a second part. Stronger results are obtained when the nonlinearity is decreasing on the second part of its domain. This allows new classes of nonlinearities to be treated and existence of a positive solution is established under weaker conditions than in previous works. The results are flexible and are not tied to any particular boundary conditions but can be applied to very many problems. We give several examples including applications to problems arising in chemical reactor theory.

In this paper, we give improved results on the existence of positive solutions for the following one-dimensional p-Laplacian equation with nonlinear boundary conditions: $$ \textstyle\begin{cases} (\phi_{p} ( y'' )) ' + b ( t ) g ( t, y ( t ) ) = 0, \quad 0 < t < 1, \\ \lambda_{1}\phi_{p} ( y ( 0 ) ) - \beta_{1} \phi_{p} ( y' ( 0 ) ) = 0, \\ \lambda_{2}\phi_{p} ( y ( 1 ) ) + \beta_{2} \phi_{p} ( y' ( 1 ) ) = 0,\qquad y'' ( 0 ) = 0, \end{cases} $$ where $\phi_{p} ( s ) = | s | ^{ p-2 } s$ , $p >1 $ . Constructing an available integral operator and combining fixed point index theory, we establish some optimal criteria for the existence of bounded positive solutions. The interesting point of the results is that the term $b ( t ) $ may be singular at $t=0$ and/or $t=1$ . Moreover, the nonlinear term $g(t, y)$ is also allowed to have singularity at $y=0$ . In particular, our results extend and unify some known results.

This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient termwhere , , is continuous. Under the conditions that the nonlinearity may be of superlinear or sublinear growth in u and , existence results of positive radial solutions are obtained. For the superlinear case, the growth of f in is restricted to quadratic growth. The superlinear and the sublinear growth of the nonlinearity of f are described by inequality conditions instead of the usual upper and lower limits conditions. Our discussion is based on the fixed point index theory in cones.

In this paper we study the existence of positive solutions for the fractional $p$-Laplacian boundary value problem \[\left\{\aligned & D_{0+}^\beta (\phi_p (D_{0+}^\alpha u(t)))=f(t,u(t)), t\in (0,1),\\ & u(0)=u'(0)=0, u'(1)=au'(\xi), D_{0+}^\alpha u(0)=0, D_{0+}^\alpha u(1)= b D_{0+}^\alpha u(\eta), \endaligned \right.\] where $2 1$, $\phi_p^{-1}=\phi_q$, $1/p+1/q=1$, $0

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.

In this paper, we establish the results on the existence, nonexistence and multiplicity of positive solutions to singular boundary value problems involving $\varphi$-Laplacian. Our approach is based on the fixed point index theory. The interesting point is that a result for the existence of three positive solutions is given., Comment: This is a revised version of the manuscript which was submitted to a journal

In this paper we use the fixed point index and nonnegative matrices to study the existence of positive solutions for a class of fractional difference systems with coupled boundary conditions.

This article is to study a three-point boundary value problem of Hadamard fractional p-Laplacian differential equation. When our nonlinearity grows ( p &minus, 1 ) -superlinearly and ( p &minus, 1 ) -sublinearly, the existence of positive solutions is obtained via fixed point index. Moreover, using an increasing operator fixed-point theorem, the uniqueness of positive solutions and uniform convergence sequences are also established.

In this article, we consider a class of singular fractional differential equations with nonlocal boundary value conditions. The existence and multiplicity of positive solutions are derived by the fixed point index theory, and the nonlinearity $f(t,x)$ may be singular at $t=0,1$ and $x=0$ . The interesting point is that the existence results are closely associated with the relationship between 1 and the spectral radii corresponding to the relevant linear operators. An example is also given to demonstrate the validity of the main results.

We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of \begin{document}$ 33 $\end{document} stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes \begin{document}$ 1-25 $\end{document} and \begin{document}$ 33 $\end{document} ; there is always a heteroclinic cycle in class \begin{document}$ 27 $\end{document} ; Neimark-Sacker bifurcations may occur in classes \begin{document}$ 26-31 $\end{document} but cannot occur in class \begin{document}$ 32 $\end{document} . Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes \begin{document}$ 29, 31, 33 $\end{document} and those in class \begin{document}$ 27 $\end{document} with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.

In this article we study applications of the Schrödinger-type identity for obtaining transmutations via the fixed point index for nonlinear integral equations. It is possible to derive a wide range of transmutation operators by this method. Classical Riesz transforms are involved in the Schrödinger-type identity method as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace, and some generalized transforms. In this paper, we present a modified Schrödinger-type identity for solutions of a class of linear Schrödinger equations with mixed boundary conditions. The techniques used in our proofs are quite different, and most remarkably some of the proofs become simpler and more straightforward. As an application, we obtain the existence and uniqueness of a solution to the stationary Schrödinger equation in the sense of the Weyl law, which advances the recent results obtained in several articles even in a more general setting.

In this article, we investigate the existence of positive solutions to fourth-order problems with dependence on all derivatives in nonlinearities subject to the Stieltjes integral boundary conditions $$\begin{aligned}& \textstyle\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u'(0)+\beta _{1}[u]=0, \qquad u''(0)+\beta _{2}[u]=0, \qquad u(1)=\beta _{3}[u], \qquad u'''(1)=0, \end{cases}\displaystyle \end{aligned}$$ and $$\begin{aligned}& \textstyle\begin{cases} -u^{(4)}(t)=g(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u(0)=\alpha _{1}[u], \qquad u'(0)=\alpha _{2}[u], \qquad u''(0)=\alpha _{3}[u], \qquad u'''(1)=0, \end{cases}\displaystyle \end{aligned}$$ where $f: [0,1]\times \mathbb{R}_{+}\times \mathbb{R}_{-}^{3}\to \mathbb{R}_{+}$ , $g: [0,1]\times \mathbb{R}^{3}_{+}\to \mathbb{R}_{+}$ are continuous and $\beta _{i}[u]$ , $\alpha _{i}[u]$ ( $i=1,2,3$ ) are linear functionals involving Stieltjes integrals of signed measures. Some growth conditions are posed on nonlinearities f, g, meanwhile the spectral radii of corresponding linear operators are restricted, which means the superlinear or sublinear conditions. On the cones in $C^{3}[0,1]$ we apply the theory of fixed point index, the existence of positive solutions is obtained. We also give some examples under mixed multi-point and integral boundary conditions with sign-changing coefficients.

In this paper we study some properties of isolated invariant continua for arbitrary homeomorphisms of $${{\mathbb {R}}}^n$$ . We study the existence of special isolating blocks for them which allow us to compute the fixed point indices of the iterates of arbitrary homeomorphisms at arbitrary isolated continua in dimension two. Among the consequences we would highlight the following

We study the existence, multiplicity, and uniqueness results of positive solutions for a fractional thermostat model. Our approach depends on the fixed point index theory, iterative method, and nonsymmetry property of the Green function. The properties of positive solutions depending on a parameter are also discussed.

We establish existence, localization and multiplicity results of positive solutions for general operator systems in ordered Banach spaces. Our main tool is the fixed point index in cones which we compute in suitable relatively open sets. In this context, each component of the fixed point operator can satisfy either the expansion condition or the compression condition. If some component of the operator is expansive, then we obtain multiplicity results. As an application, new results concerning systems of Hammerstein equations and systems of $$\phi $$ -Laplace equations are deduced.

In the present paper, we prove the existence of solutions $(\lambda_1,\lambda_2,u,v)\in\mathbb{R}^2\times H^1(\mathbb{R}^3,\mathbb{R}^2)$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1u=\mu_1 u^3+\beta uv^2\quad &\hbox{in}\;\mathbb{R}^3\\ -\Delta v+\lambda_2v=\mu_2 v^3+\beta u^2v\quad&\hbox{in}\;\mathbb{R}^3\\ u,v>0&\hbox{in}\;\mathbb{R}^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\mathbb{R}^3}u^2=a^2\quad\hbox{and}\;\int_{\mathbb{R}^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $\mu_1,\mu_2,\beta>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the fixed frequency case. And when the masses are prescribed, the standard approach to this problem is variational with $\lambda_1,\lambda_2$ appearing as Lagrange multipliers. Here we present a new approach based on bifurcation theory and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $\beta$ in a large range. We also give a result about the nonexistence of positive solutions. From which one can see that our existence theorem is almost the best. Especially, if $\mu_1=\mu_2$ we prove that normalized solutions exist for all $\beta>0$ and all $a,b>0$., Comment: 27 pages, 1 figure

In this paper we consider wildly ramified power series, \emph{i.e.}, power series defined over a field of positive characteristic, fixing the origin, where it is tangent to the identity. In this setting we introduce a new invariant under change of coordinates called the \emph{second residue fixed point index}, and provide a closed formula for it. As the name suggests this invariant is closely related to the residue fixed point index, and they coincide in the case that the power series have small multiplicity. Finally, we characterize power series with large multiplicity having the smallest possible multiplicity at the origin under iteration, in terms of this new invariant., Comment: 18 pages. arXiv admin note: text overlap with arXiv:1904.04494

In this paper, we study the existence of positive solutions for the system of second-order difference equations involving Neumann boundary conditions: -Δ2u1(t-1)=f1(t,u1(t),u2(t)), t∈[1,T]Z, -Δ2u2(t-1)=f2(t,u1(t),u2(t)), t∈[1,T]Z, Δui(0)=Δui(T)=0, i=1,2, where T>1 is a given positive integer, Δu(t)=u(t+1)-u(t), and Δ2u(t)=Δ(Δu(t)). Under some appropriate conditions for our sign-changing nonlinearities, we use the fixed point index to establish our main results.

In this paper, we study power series having a fixed point of multiplier 1. First, we give a closed formula for the residue fixed point index, in terms of the first coefficients of the power series. Then, we use this formula to study wildly ramified power series in positive characteristic. Among power series having a multiple fixed point of small multiplicity, we characterize those having the smallest possible lower ramification numbers in terms of the residue fixed point index. Furthermore, we show that these power series form a generic set, and, in the case of convergent power series, we also give an optimal lower bound for the distance to other periodic points., Comment: 29 pages; minor revision from v2

We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index., 11 pages

We study existence and multiplicity of positive radial solutions for a semipositone coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the Neumann boundary conditions are nonlocal. We use a nonstandard cone to establish existence of multiple solutions by means of fixed point index theory.

Motivated by the study of systems of higher order boundary value problems with functional boundary conditions, we discuss, by topological methods, the solvability of a fairly general class of systems of perturbed Hammerstein integral equations, where the nonlinearities and the functionals involved depend on some derivatives. We improve and complement earlier results in the literature. We also provide some examples in order to illustrate the applicability of the theoretical results., Comment: 13 pages

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.

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.

In this paper, using the theory of fixed point index, we prove some results on the multiplicity of positive solutions for a class of nonlocal elliptic problem.