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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]

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 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 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]

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 existence of positive radial solutions for the following class of quasi‐linear elliptic systems −Δpu=f(|x|,u,v)inB,−Δqv=g(|x|,u,v)inB,(u,v)=(0,0)on∂B,\begin{equation*}\hspace*{13.5pc} \left\lbrace \def\eqcellsep{&}\begin{array}{rcll} -\Delta _p u &=& f(|x|,u,v) & \text{in}\quad B, \\ -\Delta _q v &=& g(|x|,u,v) & \text{in}\quad B, \\ (u,v) &=& (0,0) & \text{on}\quad \partial B, \end{array} \right. \end{equation*}where the nonlinearities f,g∈C(B×[0,+∞)2;[0,+∞))$ f, g \in C\big (B \times [0,+\infty)^2;\,[0,+\infty)\big)$ satisfy some local superlinear property at +∞$+\infty$. Here B is the unity ball in RN${\mathbb {R}}^N$ and 1

This paper deals with the existence, nonexistence, and multiplicity of positive solutions of the coupled system of Riemann‐Liouville fractional differential equations together with multipoint boundary conditions containing fractional derivatives at nonlocal points. We use the theory of fixed point index approach to prove our results exhibiting the existence and the nonexistence of positive solutions. [ABSTRACT FROM AUTHOR]

New coexistence fixed point theorems in product Banach spaces are established via the classical theory of fixed point index for compact maps defined on cones. Each component of a coexistence fixed point is nonzero. These coexistence fixed point theorems are applied to obtain results on coexistence solutions of systems of Hammerstein integral equations, second‐order ordinary differential equations with general separated boundary conditions, and one‐dimensional competition model of Ricker types. [ABSTRACT FROM AUTHOR]

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

Using the theory of fixed point index, we discuss the existence and multiplicity of nonnegative solutions of a wide class of boundary value problems with coupled nonlinear boundary conditions. Our approach is fairly general and covers a variety of situations. We illustrate in an example that all the constants that occur in our theory can be computed. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and multiplicity of positive solitary wave solutions for nonlinear Schrödinger equations with two parameters. The proof is based on the method of upper and lower solutions and the fixed point index. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of positive solutions for the following fourth order nonlinear discrete p-Laplacian boundary value problem [ABSTRACT FROM AUTHOR]