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This paper deals with the existence and multiplicity of weak solutions for the following Kirchhoff‐type problem M∬ℝ2N1p(x,y)|u(x)−u(y)|p(x,y)|x−y|N+s(x,y)p(x,y)dxdy(−Δ)p(·)s(·)u(x)=f(x,u)inΩ,u(x)=0inℝN\Ω,$$ \left\{\begin{array}{ll}M\left({\iint}_{{\mathrm{\mathbb{R}}}^{2N}}\frac{1}{p\left(x,y\right)}\frac{{\left|u(x)-u(y)\right|}^{p\left(x,y\right)}}{{\left|x-y\right|}^{N+s\left(x,y\right)p\left(x,y\right)}}\kern0.1em dx\kern0.1em dy\right){\left(-\Delta \right)}_{p\left(\cdotp \right)}^{s\left(\cdotp \right)}u(x)=f\left(x,u\right)& \kern0.1em \mathrm{in}\kern0.4em \Omega, \\ {}u(x)=0& \kern0.1em \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}^N\backslash \Omega, \end{array}\right. $$where M$$ M $$ models a Kirchhoff coefficient and (−Δ)p(·)s(·)$$ {\left(-\Delta \right)}_{p\left(\cdotp \right)}^{s\left(\cdotp \right)} $$ is a variable s(·)$$ s\left(\cdotp \right) $$‐order p(·)$$ p\left(\cdotp \right) $$‐fractional Laplace operator, with s(·):ℝ2N→(0,1)$$ s\left(\cdotp \right):{\mathrm{\mathbb{R}}}^{2N}\to \left(0,1\right) $$ and p(·):ℝ2N→(1,∞)$$ p\left(\cdotp \right):{\mathrm{\mathbb{R}}}^{2N}\to \left(1,\infty \right) $$. Here, Ω⊂ℝN$$ \Omega \subset {\mathrm{\mathbb{R}}}^N $$ is a bounded smooth domain with N>s(x,y)p(x,y)$$ N>s\left(x,y\right)p\left(x,y\right) $$ for any (x,y)∈Ω‾×Ω‾$$ \left(x,y\right)\in \overline{\Omega}\times \overline{\Omega} $$. By using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle in two cases when the Carathéodory function f(x,u)$$ f\left(x,u\right) $$ having special structure, we establish conditions ensuring the multiplicity results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions for a fourth order boundary value problem coupled to functional perturbed clamped beam boundary conditions. Our main ingredient is the classical fixed point index. The problem investigated is an extension of other problems studied in previous papers by covering very general nonlocal perturbed conditions on the boundary. [ABSTRACT FROM AUTHOR]

We analyze a general class of coupled systems of stationary Navier–Stokes type equations with variable coefficients and non-homogeneous terms of reaction type in the incompressible case. Existence of solutions satisfying the homogeneous Dirichlet condition in a bounded domain in ℝ N , N ≤ 3 , and localization results for the corresponding kinetic energy and enstrophy are obtained by using a variational approach and the fixed point index theory. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵ 2 u = f (u , ▵ u) on an annular domain Ω in R N with the Navier boundary conditions u | ∂ Ω = 0 and ▵ u | ∂ Ω = 0 , where f : R + × R − → R + is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f (ξ , η) to have superlinear or sublinear growth on ξ , η as | (ξ , η) | → 0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones. [ABSTRACT FROM AUTHOR]

The paper deals with existence, localization and multiplicity of radial positive solutions in the annulus or the ball, for the Neumann problem involving a general ϕ -Laplace operator. Our results apply in particular to the classical Laplacian and to the mean curvature operators in the Euclidean and Minkowski spaces. Numerical experiments with the MATLAB object-oriented package Chebfun are performed to obtain numerical solutions for some concrete equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions to a Hadamard-type fractional integral boundary value problem using fixed point index. We construct a new linear operator and obtain our main results under some conditions concerning the spectral radius of this linear operator. Our method improves and generalizes some results in the literature. [ABSTRACT FROM AUTHOR]

This paper mainly deals with the following k-Hessian system with the nonlinear gradients S k (λ (D 2 u i)) + | ∇ u i | k = φ i (| v | , - u 1 , - u 2 , ... , - u n) , i n Ω , u i = 0 , i = 1 , 2 , ... , n , o n ∂ Ω , where 1 ≤ k ≤ N , n ≥ 2 , Ω is the open unit ball in R N (N ≥ 2) and S k (λ (D 2 u)) is the k-Hessian operator of u. Some results about the existence of radial solutions are obtained by making some appropriate assumptions about R + n -monotone matrices for φ i . Based on the Jensen integral inequality and fixed point theorem, we have overcome the computational difficulties of k-Hessian system with gradients, and obtained the conclusion that the system has at least one radial solution and at least two radial solutions. [ABSTRACT FROM AUTHOR]

Let M be a closed 1-connected smooth 4-manifolds, and let r be a non-negative integer. We study the problem of finding minimal number of r-periodic points in the smooth homotopy class of a given map f : M → M . This task is related to determining a topological invariant D r 4 [ f ] , defined in Graff and Jezierski (Forum Math 21(3):491–509, 2009), expressed in terms of Lefschetz numbers of iterations and local fixed point indices of iterations. Previously, the invariant was computed for self-maps of some 3-manifolds. In this paper, we compute the invariants D r 4 [ f ] for the self-maps of closed 1-connected smooth 4-manifolds with definite intersection forms (i.e., connected sums of complex projective planes). We also present some efficient algorithmic approach to investigate that problem [ABSTRACT FROM AUTHOR]

In this work, we investigate a Riemann-Liouville-type impulsive fractional integral boundary value problem. Using the fixed point index, we obtain two existence theorems on positive solutions under some conditions concerning the spectral radius of the relevant linear operator. Our method improves and generalizes some results in the literature.

By means of a recent Birkhoff-Kellogg type theorem, we discuss the solvability of a fairly general class of parameter-dependent fourth order retarded differential equations subject to functional boundary conditions. We seek solutions within a translate cone of nonnegative functions. We provide an example to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we study a class of diffusive predator-prey models with mutual interference among the predators while searching for food. Under Dirichlet boundary condition, by the fixed point index theory, we provide the necessary and sufficient conditions for the existence of coexistence states (i.e., stationary pattern). This is a strong contrast to the corresponding Neumann boundary problem, which exhibits bistability and admits no patterns. [ABSTRACT FROM AUTHOR]

In this paper, the existence of positive solutions of the following third-order three-point boundary value problem with p-Laplacian {(Φp(u"(t)))' + f(t, u(t)) = 0, t (0, 1), u(0) = au(η), u(1) = au(η), u"(0) = 0, is studied, where Φp(s) = |s|p-2s, p > 1. By using the fixed point index method, we establish sufficient conditions for the existence of at least one or at least two positive solutions for the above boundary value problem. The main result is demonstrated by providing an example as an application. [ABSTRACT FROM AUTHOR]

We consider the nonlinear conjugate multi-point boundary value problem with feedback control that finds functions $ u $ and $ q $ satisfying$ \begin{align*} & \mathcal Lu(t) = q(t)F[t,u(t),u'(t),..., u^{(m-1)}(t)],\,\, q(t) \in \Phi[t,u(t)],\,\,\, t\in I, \hfill \\ & u^{(j)}(t_i) = 0, \,\,\,\,\, 1\leq i \leq l,\,\,\,\, 0\leq j\leq k_i-1, \hfill \end{align*} $where $ \mathcal L $ is a linear differential operator of order $ m $, $ I = [0,1] $, $ 0 = t_1<...

Our main objective of this paper is to study the singular $ p $-Monge-Ampère problems: equations and systems of equations. New multiplicity results of nontrivial $ p $-convex radial solutions to a single equation involving $ p $-Monge-Ampère operator are first analyzed. Then, some new criteria of existence, nonexistence and multiplicity for nontrivial $ p $-convex radial solutions for a singular system of $ p $-Monge-Ampère equation are also established.

We prove new results on the existence of multiple solutions for elliptic inclusions with nonlinear boundary conditions of Neumann type. Our approach is topological and relies on the fixed point index for multivalued map. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity of convex radial solutions for the Monge-Amp$ \grave{\text e} $re equation involving the gradient $ \nabla u $: $ \begin{cases} \det (D^2u) = f(|x|, -u, |\nabla u|), x\in B, \\ u|_{\partial B} = 0, \end{cases} $ where $ B: = \{x\in \mathbb R^N: |x| < 1\} $. The fixed point index theory is employed in the proofs of the main results.

In this paper, we construct the fixed point index for a class of contractive mapping defined by a simulation mapping and a measure of noncompact-ness noted by Zµ− contraction maps. Then we establish some fixed point theorem for this mapping of the Krasnoselskii type. An Application to the integral equation is presented to support the results.

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵2u=f(u,▵u) on an annular domain Ω in RN with the Navier boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:R+×R−→R+ is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f(ξ,η) to have superlinear or sublinear growth on ξ,η as |(ξ,η)|→0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones.

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Cancer caused by malignant tumor is a very common and serious disease which can lead to people towards death. However, sometimes, cancer may be controlled if the cancer cells can be infected and split by particular virus with weak pathogenicity. To give a better understanding on tumor dynamics, in this paper, we deal with a reaction–diffusion tumor model, which comes from the process of tumor therapy with oncolytic virus. Our main objective is to analyze the biological compatibility of uninfected and infected tumor cells. In addition, we are concerned with the long‐time behavior, stability, and a priori estimates for the model. The compatibility conditions show that under certain range of parameters control, the infected tumor cells can coexist with normal tumor cells or be eliminated by oncolytic virus as time goes on; in the elimination case, it implies that the organism's complete recovery is highly possible as a result of the attack of oncolytic virus. An indirect idea in calculating fixed point index is proposed; correspondingly, a new condition on coexistence is presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the positive solutions for Hilfer frac-tional differential equation boundary value problems at resonance. First, we give the expression of the solution with Mittag-Leer function. Next, we ob-tain the existence of the positive solutions by using fixed point index theorem. Finally, we give relevant examples to prove our main results. [ABSTRACT FROM AUTHOR]

The purpose of this work is to establish an extension of a Leggett-Williams type expansion-compression fixed point theorem for a sum of two operators. As illustration, our approach is applied to prove the existence of non trivial nonnegative solutions for two-point BVPs and three-point BVPs. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of positive solutions for one dimensional Minkowski curvature problem with either singular weight function or singular nonlinear term. By virtue of fixed point arguments and perturbation technique, we establish the new existence results of positive solutions under different assumptions on the nonlinear term. Moreover, some examples are also given as applications. [ABSTRACT FROM AUTHOR]

This article discusses the existence of positive solutions for the system of second-order ordinary differential equation boundary value problems −u″(t)=f(t,u(t),v(t),u′(t)),t∈[0,1],−v″(t)=g(t,u(t),v(t),v′(t)),t∈[0,1],u(0)=u(1)=0,v(0)=v(1)=0,\left\{\begin{array}{l}-{u}^{^{\prime\prime} }\left(t)=f\left(t,u\left(t),v\left(t),u^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ -{v}^{^{\prime\prime} }\left(t)=g\left(t,u\left(t),v\left(t),v^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ u\left(0)=u\left(1)=0,\hspace{1em}v\left(0)=v\left(1)=0,\end{array}\right. where f,g:[0,1]×R+×R+×R→R+f,g:\left[0,1]\times {{\mathbb{R}}}^{+}\times {{\mathbb{R}}}^{+}\times {\mathbb{R}}\to {{\mathbb{R}}}^{+} are continuous. Under the related conditions that the nonlinear terms f(t,x,y,p)f\left(t,x,y,p) and g(t,x,y,q)g\left(t,x,y,q) may be super-linear growth or sub-linear growth on x,y,px,y,p, and qq, we obtain the existence results of positive solutions. For the super-linear growth case, the Nagumo condition (F3)\left(F3) is presented to restrict the growth of f(t,x,y,p)f\left(t,x,y,p) and g(t,x,y,q)g\left(t,x,y,q) on pp and qq. The super-linear growth or sub-linear growth of the nonlinear terms ff and gg is described by related inequality conditions instead of the usual independent inequality conditions about ff and gg. The discussion is based on the fixed point index theory in cones.

In this short note, we prove, by means of classical fixed point index, an affine version of a Birkhoff–Kellogg‐type theorem in cones. We apply our result to discuss the solvability of a class of boundary value problems for functional differential equations subject to functional boundary conditions. We illustrate our theoretical results in an example. [ABSTRACT FROM AUTHOR]

We introduce the notion of a strong pair of periodic segments over [0, T] and we show its applications in detecting chaotic dynamics. We prove a various number of fixed point index formulas concerning periodic points of the Poincaré map. [ABSTRACT FROM AUTHOR]

In this paper we use the fixed point index theory to study the existence of positive radial solutions for a system of boundary value problems with semipositone second order elliptic equations. Some appropriate concave and convex functions are utilized to characterize coupling behaviors of our nonlinearities.

In this paper, we study the existence of positive solutions for a class of multi points boundary value problems. We introduce a completely continuous operator such that, the fixed points of this operator are positive solutions of the problem. We establish some theorems to prove the existence ofsolutions for this system.

The paper deals with the existence and localization of positive radial solutions for stationary partial differential equations involving a general $\phi $-Laplace operator in the annulus. Three sets of boundary conditions are considered: Dirichlet–Neumann, Neumann–Dirichlet and Dirichlet–Dirichlet. The results are based on the homotopy version of Krasnosel'skii's fixed point theorem and Harnack type inequalities, first established for each one of the boundary conditions. As a consequence, the problem of multiple solutions is solved in a natural way. Numerical experiments confirming the theory, one for each of the three sets of boundary conditions, are performed by using the MATLAB object-oriented package Chebfun.

A class of dissipative orientation preserving homeomorphisms of the infinite annulus, pairs of pants, or generally any infinite surface homeomorphic to a punctured sphere is considered. We prove that in some isotopy classes the local behavior of such homeomorphisms at a fixed point, namely the existence of so-called inverse saddle, impacts the topology of the attractor — it cannot be arcwise connected. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

By using the theory of fixed point index and spectral theory of linear operators, we study the existence of positive solutions for Riemann-Liouville fractional differential equations at resonance. Our approach will provide some new ideas for the study of this kind of problem.

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving ϕ$$ \phi $$‐Laplacian operators in a ball. In particular, p$$ p $$‐Laplacian and Minkowski‐curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack‐type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions. [ABSTRACT FROM AUTHOR]

We give a unified approach to study the existence of multiple positive solutions of nonlinear differential inclusions of the form −u′′(t)∈F(t,u(t)),a.e.t∈(0,1),$$\begin{equation*}\hskip7pc -u^{\prime \prime }(t)\in F(t,u(t)),\; \text{a.e.} \; t \in (0,1), \end{equation*}$$subject to various nonlocal boundary conditions. We study these problems via a perturbed integral inclusion of the form u(t)∈Bu(t)+∫01k(t,s)F(s,u(s))ds$ u(t)\in Bu(t) +\int _{0}^{1}k(t,s)F(s,u(s))\,ds$. [ABSTRACT FROM AUTHOR]

We study the existence of nonzero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with nonlocal functional weights and functional boundary conditions. We present two examples to shed light on the type of functionals and growth conditions that can be considered with our approach. [ABSTRACT FROM AUTHOR]

Let f : M → M be a self-map of a compact manifold and n ∈ N. The least number of n-periodic points in the smooth homotopy class of f may be smaller than in the continuous homotopy class. We ask: for which self-maps f : M → M the two minima are the same, for each prescribed multiplicity? In the study of self-maps of tori and compact Lie groups a necessary condition appears. Here we give a criterion which helps to decide whether the necessary condition is also sufficient. We apply this result to show that for self-maps of the product of the lens space M = L(3) × L(3) the necessary condition is also sufficient. [ABSTRACT FROM AUTHOR]

We give a unified approach to study the existence of multiple positive solutions of nonlinear differential inclusions of the form −u′′(t)∈F(t,u(t)),a.e.t∈(0,1),$$\begin{equation*}\hskip7pc -u^{\prime \prime }(t)\in F(t,u(t)),\; \text{a.e.} \; t \in (0,1), \end{equation*}$$subject to various nonlocal boundary conditions. We study these problems via a perturbed integral inclusion of the form u(t)∈Bu(t)+∫01k(t,s)F(s,u(s))ds$ u(t)\in Bu(t) +\int _{0}^{1}k(t,s)F(s,u(s))\,ds$. [ABSTRACT FROM AUTHOR]

In this work a new functional expansion-compression fixed point theorem of Leggett-Williams type is developed for a class of mappings of the form T + F, where (I - T) is Lipschitz invertible map and F is a k -set contraction. The arguments are based upon recent fixed point index theory in cones of Banach spaces for this class of mappings. As application, our approach is applied to prove the existence of nontrivial nonnegative solutions for three-point boundary value problem. [ABSTRACT FROM AUTHOR]

In this paper, we present some results on fixed point index calculations for multivalued mappings and apply them to prove the existence of solutions to multivalued equations in ordered space, under flexible conditions for the positive eigenvalue.

In the article, we investigate three classes of fourth-order boundary value problems with dependence on all derivatives in nonlinearities under the boundary conditions involving Stieltjes integrals. A Gronwall-type inequality is employed to get an a priori bound on the third-order derivative term, and the theory of fixed-point index is used on suitable open sets to obtain the existence of positive solutions. The nonlinearities have quadratic growth in the third-order derivative term. Previous results in the literature are not applicable in our case, as shown by our examples.

In this paper, under some super- and sub-linear growth conditions, we study the existence of positive solutions for a high-order Riemann–Liouville type fractional integral boundary value problem involving fractional derivatives. Our analysis methods are based on the fixed point index and nonsymmetric property of the Green function. Additionally, we provide some valid examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of South China Normal University (Natural Science Edition) / Huanan Shifan Daxue Xuebao (Ziran Kexue Ban) is the property of Journal of South China Normal University (Natural Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem u (4) = f (t , u , u ′ , u ″ , u ‴) on [ 0 , 1 ] with the boundary condition u (0) = u (1) = u ″ (0) = u ″ (1) = 0 , which models a statically bending elastic beam whose two ends are simply supported, where f : [ 0 , 1 ] × R + × R × R − × R → R + is continuous. Some precise inequality conditions on f guaranteeing the existence of positive solutions are presented. The inequality conditions allow that f (t , u , v , w , z) may be asymptotically linear, superlinear, or sublinear on u , v , w , and z as | (u , v , w , z) | → 0 and | (u , v , w , z) | → ∞ . Our discussion is based on the fixed point index theory in cones. [ABSTRACT FROM AUTHOR]

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).

Abstract 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: { 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 , $$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$ where f 1 , f 2 ∈ C [ ( 0 , 1 ) × R 0 + × R , R ] $f_{1},f_{2}\in C[(0,1)\times \mathbb{R}^{+}_{0}\times \mathbb{R}, \mathbb{R}]$ , R 0 + = ( 0 , + ∞ ) $\mathbb{R}_{0}^{+}=(0,+\infty )$ . 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.