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[EN] The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction obtained in \cite{DjebaMeb, Svet-Meb}, to the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of positive solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as a class of partial differential equations (PDEs for short)., Direction Générale de la Recherche Scientifique et du Développement Technologique DGRSDT. MESRS Algeria. Projet PRFU: C00L03UN060120180009

The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: $$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$ { u ( 4 ) ( t ) + β 1 u ″ ( t ) − α 1 u ( t ) = f 1 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; v ( 4 ) ( t ) + β 2 v ″ ( t ) − α 2 v ( t ) = f 2 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 ; v ( 0 ) = v ( 1 ) = v ″ ( 0 ) = v ″ ( 1 ) = 0 , where $f_{1},f_{2}\in C[(0,1)\times \mathbb{R}^{+}_{0}\times \mathbb{R}, \mathbb{R}]$ f 1 , f 2 ∈ C [ ( 0 , 1 ) × R 0 + × R , R ] , $\mathbb{R}_{0}^{+}=(0,+\infty )$ R 0 + = ( 0 , + ∞ ) . By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.

Aims/ Objectives: In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the n−dimensional elliptic system systems have been widely studied, but there is relatively little research on n-dimensional elliptic systems. We are very interested in this subject and want to study it. We give new conclusions on the existence, nonexistence and multiplicity of positive solutions for the n-dimensional elliptic system. Study Design: Study on the existence, nonexistence and multiplicity of positive solutions. Place and Duration of Study: School of Applied Science, Beijing Information Science & Technology University, September 2019 to present. Methodology: We prove the existence, nonexistence and multiplicity of positive solutions by the results of fixed point index. Results: We give new conclusions of existence, nonexistence and multiplicity of positive solutionsfor the system. Conclusion: We prove the existence, nonexistence and multiplicity of positive solutions to the n-dimensional elliptic system and give new conclusions.

In this survey we describe how the so-called Dold congruence arises in topology, and how it relates to periodic point counting in dynamical systems., 38 pages; survey

We define a topological characteristic, the fixed point index with respect to a convex closed subset of a Banach space for a class of completely fundamentally restrictible multivalued maps which can be represented as a composition of maps with aspheric values. This class includes, in particular, maps which are condensing with respect to a monotone nonsingular measure of noncompactness. Maps of this type naturally arise in the study of nonlinear systems with impulse effects. Applications of the index to some fixed point theorems are considered.

In this paper, we prove the existence, unbounded continuity of positive set for a multivalued equation containing a parameter of the form $x \in A \circ F(\lambda,x)$ and give applications in the control problem with multi-point boundary conditions and second order derivative operator \begin{equation} \left\{ \begin{array}{l} u^{\prime \prime }(t) +g(\lambda,t) f(u(t)) =0,\text{ }t\in (0,1) , \\ g(\lambda,t) \in F(\lambda,u(t)) \text{ a.e. on } J \\ u(0) =0, u( 1) =\sum_{i=1}^{m}\alpha_{i} u( \eta _{i}) \end{array} \right. \label{Eq3.1} \end{equation}

This work investigates the existence of multiple positive solutions for a system of two nonlinear higher-order fractional differential equations with ϕ-Laplacian operators and nonlocal conditions. A degenerate nonlinearity which obeys some general growth conditions is considered. The singularities are dealt with by approximating the fixed point operator. New existence results are presented by using the fixed point index theory. Examples of applications illustrate the theoretical results.

In this paper, we study the positive radial solutions for elliptic systems to the nonlinear BVP: , where Δu = div (∇u) and Δv = div (∇v) are the Laplacian of u, λ is a positive parameter, Ω = {x ∈ Rn : N > 2, |x| > r0, r0 > 0}, let i = [1,2] then Ki :[r0,∞] → (0,∞) is a continuous function such that limr→∞ ki(r) = 0 and is The external natural derivative, and : [0, ∞) → (0, ∞) is a continuous function. We discuss existence and multiplicity results for classes of f with a) fi > 0, b) fi fi = 0. We base our presence and multiple outcomes via the Sub-solutions method. We also discuss some unique findings.

The main goal of this paper is investigating the existence of nonconstant positive steady states of a linear prey–predator cross-diffusion system with Beddington–DeAngelis and Tanner functional response. An analytical method and fixed point index theory plays a significant role in our main proofs.

We deal with a coupled system of k-Hessian equations: Sk( μ(D2u1)) = (-u1)) ∝ in B, Sk( μ(D2u1)) = (-u2)) in B, u1 < 0, u2< 0 in B, u1 = u2= 0 on ∂B, where k = 1,2, ⋯ , N, B is a unit ball in RN, N ≥ 2, α and β are positive constants. By using the fixed-point index theory in cone, we obtain the existence, uniqueness and nonexistence of radial convex solutions for some suitable constants α and β. Furthermore, by using a generalized Krein-Rutman theorem, we also obtain a necessary and sufficient existence condition of the convex solutions to a nonlinear eigenvalue problem. &nbsp

The research by J. R. Graef was supported in part by a University of Tennessee at Chattanooga SimCenter - Center of Excellence in Applied Computational Science and Engineering (CEACSE) grant. In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results. University of Tennessee at Chattanooga SimCenter - Center of Excellence in Applied Computational Science and Engineering (CEACSE) grant

We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of nonnegative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.

In this paper, we investigate the existence of positive solutions for a nonlocal fractional boundary value problem involving Caputo fractional derivative and nonlocal Riemann–Stieltjes integral boundary condition. By using the spectral analysis of the relevant linear operator and Gelfand’s formula, we obtain an useful upper and lower bounds for the spectral radius. Our discussion is based on the properties of the Green’s function and the fixed point index theory in cones.

In this paper, we discuss the positive solutions of beam equations with the nonlinearities including the slope and bending moment under nonlocal boundary conditions involving Stieltjes integrals. We pose some inequality conditions on nonlinearities and the spectral radius conditions on associated linear operators. These conditions mean that the nonlinearities have superlinear or sublinear growth. The existence of positive solutions is obtained by fixed point index on cones in $C^{2}[0,1]$C2[0,1], and some examples are given for beam equations subject to mixed integral and multi-point boundary conditions with sign-changing coefficients.

We investigate positive solution for high-order fractional differential equations with resonant boundary value conditions. By using the spectral theory and the fixed point index theory, both the nonexistence and existence results of the resonant boundary value problems are given.

The existence and nonexistence of semi-trivial or coexistence steady-state solutions of one-dimensional competition models in an unstirred chemostat are studied by establishing new results on systems of Hammerstein integral equations via the classical fixed point index theory. We provide three ranges for the two parameters involved in the competition models under which the models have no semi-trivial and coexistence steady-state solutions or have semi-trivial steady-state solutions but no coexistence steady-state solutions or have semi-trivial or coexistence steady-state solutions. It remains open to find the largest range for the two parameters under which the models have only coexistence steady-state solutions. We apply the new results on systems of Hammerstein integral equations to obtain results on steady-state solutions of systems of reaction-diffusion equations with general separated boundary conditions. Such type of results have not been studied in the literature. However, these results are very useful for studying the competition models in an unstirred chemostat. Our results on Hammerstein integral equations and differential equations generalize and improve some previous results.

In the present work, we have studied a diffusive tritrophic food chain model in which the species at each trophic level interact in accordance with Crowley–Martin functional response under mixed boundary conditions. Using degree theory and fixed point index-based methods, we have proved the existence of the positive solutions of the proposed system. We have proved the permanence of the positive solutions and existence of global attractor. The conditions for diffusion-driven instability have been obtained analytically. Moreover, the pattern formation due to diffusion-driven instability has been investigated numerically. We have shown the existence of the positive solutions both analytically and numerically.

In this paper, we study the fourth-order problem with the second derivative in nonlinearity under nonlocal boundary value conditions involving Stieltjes integrals. Some inequality conditions on nonlinearity and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on special cone. The conditions allow that the nonlinearity has superlinear or sublinear growth. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.

We investigate the existence and multiplicity of positive solutions for a nonlinear Riemann-Liouville fractional differential equation with a nonnegative singular nonlinearity, subject to Riemann-Stieltjes boundary conditions which contain fractional derivatives. In the proofs of our main results, we use an application of the Krein-Rutman theorem and some theorems from the fixed point index theory.

We are concerned with the existence and multiplicity of component-wise positive solutions for nonlinear system of Hammerstein integral equations with the weighted functions and the associated nonlinear eigenvalue problem. Our discussions are based on the product formula of fixed point index on product cones and the fixed point index theory. Moreover, we establish the existence and multiplicity of component-wise positive solutions for the associated nonlinear systems of second-order ordinary differential equations under the mixed boundary value conditions.

In this article, we study a class of fractional differential equations with resonant boundary value conditions. Some sufficient conditions for the existence of positive solutions are considered by means of the spectral theory of linear operator and the fixed point index theory.

In this paper we use the fixed point index to study the existence of positive solutions for a system of 2nth-order boundary value problems involving semipositone nonlinearities.

In this paper, we study the uniqueness and multiplicity of positive solutions of one-dimensional prescribed mean curvature equation $ \left\{ \begin{array}{l} - \left({\frac{{u'}}{{\sqrt {1 - u{'^2}} }}} \right)' = \lambda f\left(u \right), \\ u\left(x \right) \gt 0, - 1 \lt x \lt 1, \\ u\left({ - 1} \right) = u\left(1 \right) = 0, \end{array} \right. $ where $\lambda$ is a positive parameter. The main tool is the fixed point index in cones.

The main purpose of this paper is to establish the existence and multiplicity of positive solutions for a fourth-order boundary value problem with integral condition. By using a new technique of construct a positive cone, we apply the Krasnoselskii compression/expansion and Leggett-Williams fixed point theorems in cones to show our multiplicity results. Finally, a particular case is studied, and the existence of multiple solutions is proved for two different particular functions.

We discuss the existence and non-existence of non-negative, non-decreasing solutions of certain perturbed Hammerstein integral equations with derivative dependence. We present some applications to nonlinear, second order boundary value problems subject to fairly general functional boundary conditions. The approach relies on classical fixed point index theory., 11 pages

Some fourth-order two-point boundary value problems are considered in this paper. Firstly, the Green’s function is obtained by the use of the Laplace transform. Secondly, the first eigenvalue is given by using Ritz method. Then, by the use of the properties of self-adjoint operators and the fixed point index theory, the existence of positive solutions is obtained. Finally, an example is given to illustrate the main results.

We establish several criteria for the existence of positive periodic solutions of the multi-parameter differential systems $$\left \{ \textstyle\begin{array}{l} u'(t)+a_{1}(t)g_{1}(u(t))u(t)=\lambda b_{1}(t)f(u(t-\tau_{1}(t)),v(t-\zeta_{1}(t))), \\ v'(t)+a_{2}(t)g_{2}(v(t))v(t)=\mu b_{2}(t)g(u(t-\tau_{2}(t)),v(t-\zeta_{2}(t))), \end{array}\displaystyle \right . $${u′(t)+a1(t)g1(u(t))u(t)=λb1(t)f(u(t−τ1(t)),v(t−ζ1(t))),v′(t)+a2(t)g2(v(t))v(t)=μb2(t)g(u(t−τ2(t)),v(t−ζ2(t))), where the functions $g_{1}, g_{2}:[0,\infty)\to[0,\infty)$g1,g2:[0,∞)→[0,∞) are assumed to be unbounded. The analysis in the paper relies on the classical fixed point index theory. Our main findings improve and complement some existing results in the literature.

In this paper, we are mainly concerned with existence of positive solutions for periodic boundary value problem of second-order impulsive differential equation with derivative in the nonlinearity $$\begin{aligned} \left\{ \begin{array}{ll} -u''+\rho ^{2} u=f(t, u, u'), &{} t \in J', \\ -\left. \Delta u'\right| _{t=t_{k}}=I_{k}(u(t_{k})), &{} k=1,2, \ldots m, \\ u(0)=u(2\pi ),\quad u^{\prime }(0)=u^{\prime }(2\pi ), &{} \end{array}\right. \end{aligned}$$ where $$f:[0,2 \pi ] \times {\mathbb {R}}^{+} \times {\mathbb {R}} \rightarrow {\mathbb {R}}^{+}$$ is continuous, $${\mathbb {R}}^{+}=[0,+\infty )$$ , $$J=[0,2 \pi ]$$ , $$ \rho >0$$ , $$J^{\prime }=J \backslash \left\{ t_{1}, t_{2}, \ldots t_{m}\right\} .$$ Some inequality conditions on nonlinearity f and the spectral radius condition of linear operator are presented that guarantee the existence of positive solution to the problem by the theory of fixed point index. The conditions allow that $$f\left( t, x_{1},x_{2}\right) $$ has superlinear or sublinear growth in $$x_{1}, x_{2}$$ .

We investigate the existence of positive solutions of a Riemann-Liouville fractional differential equation with sequential derivatives, a positive parameter and a nonnegative singular nonlinearity, supplemented with integral-multipoint boundary conditions which contain fractional derivatives of various orders and Riemann-Stieltjes integrals. Our general boundary conditions cover some symmetry cases for the unknown function. In the proof of our main existence result, we use an application of the Krein-Rutman theorem and two theorems from the fixed point index theory.

In this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions $$\begin{aligned}& \left \{ \textstyle\begin{array}{l}u^{(4)}(t)=h(t)f(t,u(t),u'(t),u''(t)),\quad t\in(0,1),\\ u(0)=u(1)=\beta_{1}[u],\qquad u''(0)+\beta_{2}[u]=0,\qquad u''(1)+\beta_{3}[u]=0, \end{array}\displaystyle \right . \end{aligned}$$ {u(4)(t)=h(t)f(t,u(t),u′(t),u″(t)),t∈(0,1),u(0)=u(1)=β1[u],u″(0)+β2[u]=0,u″(1)+β3[u]=0, where $f: [0,1]\times\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{-}\to \mathbb{R}_{+}$f:[0,1]×R+×R×R−→R+ is continuous, $h\in L^{1}(0,1)$h∈L1(0,1) and $\beta_{i}[u]$βi[u] is Stieltjes integral ($i=1,2,3$i=1,2,3). This equation describes the deflection of an elastic beam. Some inequality conditions on nonlinearity f are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in $C^{2}[0,1]$C2[0,1]. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.

In this article, we study the existence of positive solutions to a class of two-term fractional nonlocal boundary value problems. The existence and multiplicity of positive solutions are established by means of fixed point index theory. The nonlinearity f(t,x) $f(t,x)$ permits a singularity at t=0,1 $t = 0,1$ and x=0 $x=0$.

In this paper we use the fixed point index to study the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions. Here we use appropriate nonnegative matrices to depict the coupling behavior for our nonlinearities.

In this paper, we study existence and nonexistence of positive solutions for a class of Riemann–Stieltjes integral boundary value problems of fractional differential equations with parameters. By using the fixed point index theory, some new sufficient conditions for the existence of at least one, two and the nonexistence of positive solutions are obtained. The results we obtain show the influence of parameter λ and parameter a on the existence of positive solutions. Finally, some examples are given to illustrate our main results.

We investigate a class of semipositone fractional integral boundary value problem on the half-line. Using the fixed point index theorems in a cone, we obtain the existence result of positive solutions.

We generalize the main result on existence of nonzero nonnegative solutions of systems of second order elliptic boundary value problems obtained by Lan (2011). The motivation for the generalization is to propose and study the competition models of Ricker and Beverton–Holt types governed by such systems. To the best of our knowledge, there is little study on such continuous competition models although such difference equation and first order ordinary differential equation competition models have been widely studied. Our results enrich and develop the connections among the classical fixed point index theory, systems of second order elliptic boundary value problems and population dynamics.