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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.

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

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.

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.

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 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.

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 $$x_{1}, x_{2}$$ .

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.

In this paper, we devote ourselves to investigating the existence of positive solution for a class of nonlocal elliptic equations. Our approach is based on the fixed point index theory.

The purpose of this paper is to study the existence of steady state in a linear cross-diffusion predator–prey model with Beddington–DeAngelis functional response. The proofs mainly rely on Fixed point index theory and analytical techniques.

For every \begin{document}$ n ∈ {\mathbb N}$\end{document} we construct orientation preserving planar homeomorphisms \begin{document}$ g_n$\end{document} such that \begin{document}$ Fix(g_n)=\{0\}$\end{document} , the fixed point index of \begin{document}$ g_n$\end{document} at \begin{document}$ 0$\end{document} , \begin{document}$ i_{{\mathbb R}^2}(g_n,0)$\end{document} , is equal to \begin{document}$ -n$\end{document} and the stable (respectively unstable) sets of \begin{document}$ g_n$\end{document} at \begin{document}$ 0$\end{document} decompose into exactly \begin{document}$ n+1$\end{document} connected branches \begin{document}$ \{S_{j}\}_{j ∈ \{1,2, \dots, n+1\}}$\end{document} (resp. \begin{document}$ \{U_{j}\}_{j ∈ \{1,2, \dots, n+1\}}$\end{document} ) such that: a) \begin{document}$ S_i \cap S_j= \{0\} = U_i \cap U_j$\end{document} for any \begin{document}$ i, j ∈ \{1,2, \dotsn+1\}$\end{document} with \begin{document}$ i\ne j$\end{document} . b) \begin{document}$ S_i \cap U_j= \{0\}$\end{document} for any \begin{document}$ i, j ∈ \{1,2, \dots n+1\}$\end{document} . c) For every \begin{document}$ j ∈ \{1,2, \dots n+1\}$\end{document} , \begin{document}$ S_j \setminus\{0\}$\end{document} and \begin{document}$ U_j \setminus \{0\}$\end{document} admit translation pseudo-arcs. This means that there exist pseudo-arcs \begin{document}$ K_j\subset S_j $\end{document} and points \begin{document}$ p_{j\star} , g_n(p_{j\star}) ∈ K_j$\end{document} , such that \begin{document}$ g_n(K_j)\cap K_j=\{ g_n(p_{j\star} )\} $\end{document} and \begin{document}$S_j\setminus \{ 0\}=\bigcup\limits_{m=-∞}^{∞} g_n^m (K_j)$ \end{document} and analogously for \begin{document}$ U_j$\end{document} . We also study the closure of the class of above homeomorphisms in the (complete) metric space of planar orientation preserving homeomorphisms.

In this paper, we obtain conditions of existence and uniqueness of periodic solutions for a nonlinear fourth-order differential equation utilizing explicit Green’s function and fixed point index theorem combining with operator spectral theorem. We discuss an iteration method for constant coefficient nonlinear differential equations and establish the theorem on the existence of positive solutions for a fourth-order boundary value problem with a variable parameter. Finally, we give an example to illustrate our results.

This paper is concerned with the second-order nonlinear Robin problem involving the first-order derivative: $$ \textstyle\begin{cases} u''+f(t,u,u^{\prime })=0, \\ u(0)=u'(1)-\alpha u(1)=0,\end{cases} $$ { u ″ + f ( t , u , u ′ ) = 0 , u ( 0 ) = u ′ ( 1 ) − α u ( 1 ) = 0 , where $f\in C([0,1]\times \mathbb{R}^{2}_{+},\mathbb{R}_{+})$ f ∈ C ( [ 0 , 1 ] × R + 2 , R + ) and $\alpha \in ]0,1[$ α ∈ ] 0 , 1 [ . Based on a priori estimates, we use fixed point index theory to establish some results on existence, multiplicity and uniqueness of positive solutions thereof, with the unique positive solution being the limit of of an iterative sequence. The results presented here generalize and extend the corresponding ones for nonlinearities independent of the first-order derivative.

Competition is a fundamental force shaping population size and structure as a result of limited availability of resources. In biomathematics, the biological models with competitive interactions exist widely. Furthermore, the nonlinear-diffusion (including self- and cross-diffusions) terms are incorporated to the biological models to better simulate the actual movement of species. Therefore, better compatibility with reality can be achieved by introducing nonlinear-diffusion into biological models with competitive interactions. As a result, a competition system with nonlinear-diffusion and nonlinear functional response is proposed and analyzed in this paper. We first briefly discuss the stability of trivial and semi-trivial solutions by spectrum analysis. Then the boundedness and the non-existence of steady states are studied. Based on the boundedness of the solutions, the existence of the steady states is also investigated by the fixed point index theory in a positive cone. The result shows that the two species can coexist when their diffusion and inter-specific competition pressures are controlled in a certain range.

We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on fixed point index based on parameters that are related to the definitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator.

We study the existence of positive solutions on the half-line of a second order ordinary differential equation subject to functional boundary conditions. Our approach relies on a combination between the fixed point index for operators on compact intervals, a fixed point result for operators on noncompact sets, and some comparison results for principal and nonprincipal solutions of suitable auxiliary linear equations., 13 pages

We study a periodic Kolmogorov system describing two species nonlinear competition. We discuss coexistence and extinction of one or both species, and describe the domain of attraction of nontrivial periodic solutions in the axes, under conditions that generalise Gopalsamy conditions. Finally, we apply our results to a model of microbial growth and to a model of phytoplankton competition under the effect of toxins.

In this paper, we investigate the existence of positive solutions of a class of fractional three-point boundary value problem with an advanced argument by using fixed-point index theory. Our results improve and extend some known results in the literature. Two examples are given to demonstrate the effectiveness of our results.

In this article we combine the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. We define new spaces of functions on which study the equations, prove a version of Ascoli–Arzelà Theorem, develop the fixed point index results necessary to prove existence and multiplicity of solutions in these spaces and also illustrate the applicability of the theory with an example Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature SI

We investigate the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a positive parameter subject to nonlocal boundary conditions, which contain fractional derivatives and Riemann–Stieltjes integrals. The nonlinearity of the equation is nonnegative, and it may have singularities at its variables. In the proof of the main results, we use the fixed point index theory and the principal characteristic value of an associated linear operator. A related semipositone problem is also studied by using the Guo–Krasnosel’skii fixed point theorem.

This paper is concerned with multiple solutions for a class of nonlinear fourth-order boundary value problems with parameters. By constructing a special cone and applying fixed point index theory, the multiple solutions for the considered systems are obtained under some suitable assumptions. The main feature of obtained solutions (u(t),v(t)) is that the solution u(t) is positive, and the other solution v(t) may change sign. Finally, two examples with continuous function f1 being positive and f2 being semipositone are worked out to illustrate the main results.

We obtain a new fixed point index theory for a class of not necessarily continuous operators in cones. We employ this tool to prove a discontinuous version of Leggett–Williams’ fixed point theorem. Finally, we establish new existence and multiplicity criteria for a second-order three point BVP with discontinuous nonlinearities.

The purpose of this work is to establish an extension of a Bai-Ge type multiple fixed point theorem for a sum of two operators. The arguments are based upon recent fixed point index theory in cones of Banach spaces for a k-set contraction perturbed by an expansive operator. As illustration, our approach is applied to prove the existence of at least three nontrivial non-negative solutions for a class eigenvalue three-point BVPs for a class of fourth order ordinary differential equations (ODEs for short).

This paper deals with the existence of positive ω-periodic solutions for nth-order ordinary differential equation with delays in Banach space E of the form $$L_{n}u(t)=f\bigl(t,u(t-\tau_{1}),\ldots,u(t- \tau_{m})\bigr),\quad t\in\mathbb{R}, $$Lnu(t)=f(t,u(t−τ1),…,u(t−τm)),t∈R, where $L_{n}u(t)=u^{(n)}(t)+\sum_{i=0}^{n-1}a_{i} u^{(i)}(t)$Lnu(t)=u(n)(t)+∑i=0n−1aiu(i)(t) is the nth-order linear differential operator, $a_{i}\in\mathbb {R}$ai∈R ($i=0,1,\ldots,n-1$i=0,1,…,n−1) are constants, $f: \mathbb{R}\times E^{m}\rightarrow E$f:R×Em→E is a continuous function which is ω-periodic with respect to t, and $\tau_{i}>0$τi>0 ($i=1,2,\ldots,m$i=1,2,…,m) are constants which denote the time delays. We first prove the existence of ω-periodic solutions of the corresponding linear problem. Then the strong positivity estimation is established. Finally, two existence theorems of positive ω-periodic solutions are proved. Our discussion is based on the theory of fixed point index in cones.

In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables.

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving ϕ-Laplacian operator D0+αϕx,D0+βux+fx,ux=0,x∈0,1,D0+σu0=D0+βu0=0,u1=∫01 gtutdt+∑n=1n=+∞ αnuηn, where ϕ:0,1×R→R is a continuous function and D0+p is the Riemann-Liouville derivative for p∈α,β,σ. By using some properties of fixed point index, we obtain the existence results and give an example at last.

We study the existence and multiplicity of positive solutions for a system of nonlinear second-order difference equations subject to multi-point boundary conditions, under some assumptions on the nonlinearities of the system which contains concave functions. In the proofs of our main results we use some theorems from the fixed point index theory.

We establish a reaction–diffusion ecological–mathematical model with predator–prey dependent functional response and the Neumann boundary conditions. First, we study the bifurcation solution emitting from the unique positive constant solution by considering the local bifurcation. Finally, we mainly analyze the coexistence of the prey and predator with the help of the fixed point index theory, where we use a different approach in calculating one of the fixed point indices.

We investigate a singular fractional differential equation with an infinite-point fractional boundary condition, where the nonlinearity $f(t,x)$ may be singular at $x = 0$ , and $g(t)$ may also have singularities at $t= 0$ or $t=1$ . We establish the existence of positive solutions using the fixed point index theory in cones.

This paper deals with the existence of positive ω-periodic solutions for third-order ordinary differential equation with delay $$u'''(t)+Mu(t)=f\bigl(t,u(t),u(t-\tau) \bigr),\quad t\in {\mathbb{R}}, $$ where $\omega>0$ and $M>0$ are constants, $f: {\mathbb{R}}^{3}\rightarrow {\mathbb{R}}$ is continuous, $f(t,x,y)$ is ω-periodic in t, and $\tau>0$ is a constant denoting the time delay. We show the existence of positive ω-periodic solutions when $0< M

In this paper we use the fixed point index to establish the existence of positive solutions for a system of first-order discrete fractional boundary value problems with semipositone nonlinearities. Concave and convex functions are adopted to characterize the coupling behavior of our nonlinearities.

In this paper we consider the existence of positive solutions of nth-order Sturm–Liouville boundary value problems with fully nonlinear terms, in which the nonlinear term f involves all of the derivatives $u',\ldots, u^{(n-1)}$ of the unknown function u. Such cases are seldom investigated in the literature. We present some inequality conditions guaranteeing the existence of positive solutions. Our inequality conditions allow that $f(t, x_{0}, x_{1},\ldots, x_{n-1})$ is superlinear or sublinear growth on $x_{0}, x_{1},\ldots, x_{n-1}$ . Our discussion is based on the fixed point index theory in cones.

This paper deals with the effect of parameters on properties of positive solutions and asymptotic behavior of an unstirred chemostat model with the Beddington–DeAngelis (denote by B–D) functional response under the Robin boundary condition. Firstly, we establish some a priori estimates and a sufficient condition for the existence of positive solutions (see (Feng et al. in J. Inequal. Appl. 2016(1):294, 2016)). Secondly, we study the effect of the small parameter $k_{1}$ and sufficiently large $k_{2}$ in B–D functional response, which shows that the model has at least two positive solutions. Thirdly, we investigate the case of sufficiently large $k_{1}$ . The results show that if $k_{1}$ is sufficiently large, then the positive solution of this model is determined by a limiting equation. Finally, we present an asymptotic behavior of solutions depending on time. The main methods used in this paper include the fixed point index theory, bifurcation theory, perturbation technique, comparison principle, and persistence theorem.

This paper investigates the existence of infinitely many positive solutions for the second-order n-dimensional impulsive singular Neumann system $$\begin{aligned}& -\mathbf{x}^{\prime\prime}(t)+ M\mathbf{x}(t)=\lambda {\mathbf{g}}(t)\mathbf{f} \bigl(t,\mathbf{x}(t) \bigr),\quad t\in J, t\neq t_{k}, \\& -\Delta {\mathbf{x}}^{\prime}|_{t=t_{k}}=\mu {\mathbf{I}}_{k} \bigl(t_{k},\mathbf{x}(t_{k}) \bigr),\quad k=1,2,\ldots ,m, \\& \mathbf{x}^{\prime}(0)=\mathbf{x}^{\prime}(1)=0. \end{aligned}$$ The vector-valued function x is defined by $$\begin{aligned}& \mathbf{x}=[x_{1},x_{2},\dots ,x_{n}]^{\top }, \qquad \mathbf{g}(t)=\operatorname{diag} \bigl[g_{1}(t), \ldots ,g_{i}(t), \ldots , g_{n}(t) \bigr], \end{aligned}$$ where $g_{i}\in L^{p}[0,1]$ for some $p\geq 1$ , $i=1,2,\ldots , n$ , and it has infinitely many singularities in $[0,\frac{1}{2})$ . Our methods employ the fixed point index theory and the inequality technique.

By topological arguments, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions of a class of perturbed nonlinear integral equations. These type of integral equations arise, for example, when dealing with boundary value problems where nonlocal terms occur in the differential equation and/or in the boundary conditions. Some examples are given to illustrate the theoretical results., 29 pages

We investigate the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations, subject to multipoint boundary conditions that contain fractional derivatives, by using some theorems from the fixed point index theory.

We consider a constrained semilinear evolution inclusion of parabolic type involving an \begin{document}$m$\end{document} -dissipative linear operator and a source term of multivalued type in a Banach space and topological properties of the solution map. We establish the \begin{document}$R_δ$\end{document} -description of the set of solutions surviving in the constraining area and show a relation between the fixed point index of the Krasnosel'skii-Poincare operator of translation along trajectories associated with the problem and the appropriately defined constrained degree of the right-hand side in the equation. This provides topological tools appropriate to obtain results on the existence of periodic solutions to studied differential problems.