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Positive Solutions of a Fractional Boundary Value Problem with Changing Sign Nonlinearity
Yongqing Wang,Lishan Liu,Yonghong Wu
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/149849
Abstract: We discuss the existence of positive solutions of a boundary value problem of nonlinear fractional differential equation with changing sign nonlinearity. We first derive some properties of the associated Green function and then obtain some results on the existence of positive solutions by means of the Krasnoselskii's fixed point theorem in a cone.
Multiple positive solutions for nonlinear second-order m-point boundary-value problems with sign changing nonlinearities
Fuyi xu,Zhenbo Chen,Feng Xu
Electronic Journal of Differential Equations , 2008,
Abstract: In this paper, we study the nonlinear second-order m-point boundary value problem $$displaylines{ u''(t)+f(t,u)=0,quad 0leq t leq 1, cr eta u(0)-gamma u'(0)=0,quad u(1)=sum _{i=1}^{m-2}alpha_{i} u(xi_{i}), }$$ where the nonlinear term $f$ is allowed to change sign. We impose growth conditions on $f$ which yield the existence of at least two positive solutions by using a fixed-point theorem in double cones. Moreover, the associated Green's function for the above problem is given.
Multiple positive solutions for second-order three-point boundary-value problems with sign changing nonlinearities
Jian Liu,Zengqin Zhao
Electronic Journal of Differential Equations , 2012,
Abstract: In this article, we study the second-order three-point boundary-value problem $$displaylines{ u''(t)+a(t)u'(t)+f(t,u)=0,quad 0 leq t leq 1, cr u'(0)=0,quad u(1)=alpha u(eta), }$$ where $0
Positive Solutions to a Generalized Second-Order Difference Equation with Summation Boundary Value Problem
Thanin Sitthiwirattham,Jessada Tariboon
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/569313
Abstract: By using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem Δ2(?1)
Positive solutions of a second-order difference equation with summation boundary value problem  [PDF]
T. Sitthiwirattham,S. Ratanapun
International Journal of Mathematical Analysis , 2013,
Abstract: In this paper, we study the existence of positive solutions tothe summation boundary value problem$$Delta^2u(t-1)+a(t)f(u)=0,~~~~~~tin {1,2,...,T},$$$$u(0)=0,~~~~~u(T+1)=alpha displaystyle sum_{s=1}^eta u(s),$$where $f$ is continuous, $Tgeq 3$ is a fixed positive integer, $etain {1,2,...,T-1}$, $0 Keywords Positive solution --- Boundary value problem --- Fixed point theorem --- Cone
Positive solutions of a second-order difference equation with summation boundary value problem II  [PDF]
T. Sitthiwirattham,S. Ratanapun
International Journal of Mathematical Analysis , 2013,
Abstract: In this paper, we study the existence of positive solutions to thethree-point summation boundary value problem$$Delta^2y(t-1)+a(t)f(y(t))=0,~~~~~~tin {1,2,...,T},$$$$y(0)=0,~~~y(T+1)=alphadisplaystyle sum_{s=1}^eta y(s),$$where $f$ is continuous, $Tgeq 3$ is a fixed positive integer, $etain {1,2,...,T-1}$, $0 Keywords Positive solution --- Boundary value problem --- Fixed point theorem --- Cone
Positive solutions for second-order singular three-point boundary-value problems with sign-changing nonlinearities  [cached]
Caisheng Ji,Baoqiang Yan
Electronic Journal of Differential Equations , 2010,
Abstract: In this article, we study the existence and uniqueness of the positive solution for a second-order singular three-point boundary-value problem with sign-changing nonlinearities. Our main tool is a fixed-point theorem.
Existence of Positive Solutions for Fourth-Order Boundary Value Problems with Sign-Changing Nonlinear Terms  [PDF]
Xingfang Feng,Hanying Feng
ISRN Mathematical Analysis , 2013, DOI: 10.1155/2013/349624
Abstract: the existence of positive solutions for a fourth-order boundary value problem with a sign-changing nonlinear term is investigated. By using Krasnoselskii’s fixed point theorem, sufficient conditions that guarantee the existence of at least one positive solution are obtained. An example is presented to illustrate the application of our main results. 1. Introduction In this paper, we consider the existence of positive solutions to the following fourth-order boundary value problem (BVP): where is a positive parameter, is continuous and may be singular at , and is Lebesgue integrable and has finitely many singularities in . Boundary value problems for ordinary differential equations play a very important role in both theory and applications. They are used to describe a large number of physical, biological, and chemical phenomena. The work of Timoshenko [1] on elasticity, the monograph by Soedel [2] on deformation of structures, and the work of Dulcska [3] on the effects of soil settlement are rich sources of such applications. There has been a great deal of research work on BVPs for second and higher order differential equations, and we cite as recent contributions the papers of Anderson and Davis [4], Baxley and Haywood [5], and Hao et al. [6]. For surveys of known results and additional references, we refer the readers to the monographs by Agarwal et al. [7, 8]. Many authors have studied the existence of positive solutions for fourth-order boundary value problems where the nonlinearity takes nonnegative values, see [9–13]. However, for problems with sign-changing nonlinearities, only a few studies have been reported. Owing to the importance of high order differential equations in physics, the existence and multiplicity of the solutions to such problems have been studied by many authors, see [9, 12–17]. They obtained the existence of positive solutions provided is superlinear or sublinear in by employing the cone expansion-compression fixed point theorem. In [18], by using the strongly monotone operator principle and the critical point theory to discuss BVP the authors established some sufficient conditions for to guarantee that the problem has a unique solution, at least one nonzero solution, or infinitely many solutions. In [10], Feng and Ge considered the fourth-order singular differential equation subject to one of the following boundary conditions: where . By using a fixed point index theorem in cones and the upper and lower solutions method, the authors discussed the existence of positive solutions for the above BVP. However, most papers only focus
Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems  [PDF]
Guglielmo Feltrin,Fabio Zanolin
Mathematics , 2015,
Abstract: We prove the existence of positive periodic solutions for the second order nonlinear equation $u" + a(x) g(u) = 0$, where $g(u)$ has superlinear growth at zero and at infinity. The weight function $a(x)$ is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.
Sign-Changing Solutions for Nonlinear Operator Equations  [PDF]
Yanbin Sang
Journal of Operators , 2014, DOI: 10.1155/2014/379056
Abstract: The existence of six solutions for nonlinear operator equations is obtained by using the topological degree and fixed point index theory. These six solutions are all nonzero. Two of them are positive, the other two are negative, and the fifth and sixth ones are both sign-changing solutions. Furthermore, the theoretical results are applied to elliptic partial differential equations. 1. Introduction In recent years, motivated by some ecological problems, much attention has been attached to the existence of sign-changing solutions for nonlinear partial differential equations (see [1–4] and the references therein). We note that the proofs of main results in [1–4] depend upon critical point theory. However, some concrete nonlinear problems have no variational structures [5]. To overcome this difficulty, in [6], Zhang studied the existence of sign-changing solution for nonlinear operator equations by using the cone theory and combining uniformly positive condition. Xu [7] studied multiple sign-changing solutions to the following -point boundary value problems: where , , . We list some assumptions as follows.(A1)Suppose that the sequence of positive solutions to the equation ?is ;(A2) , is a continuous function, , and for all ;(A3)let and . There exist positive integers and such that (A4)there exists such that for all with . Theorem 1 (see [7]). Suppose that conditions are satisfied. Then the problem (1) has at least two sign-changing solutions. Moreover, the problem (1) also has at least two positive solutions and two negative solutions. Based on [7], many authors studied the sign-changing solutions of differential and difference equations. For example, Yang [8] considered the existence of multiple sign-changing solutions for the problem (1). Compared with Theorem 1, Yang employed the following assumption which is different from .(A′4)There exists such that Pang et al. [9] investigated multiple sign-changing solutions of fourth-order differential equation boundary value problems. Moreover, Wei and Pang [10] established the existence theorem of multiple sign-changing solutions for fourth-order boundary value problems. Y. Li and F. Li [11] studied two sign-changing solutions of a class of second-order integral boundary value problems by computing the eigenvalues and the algebraic multiplicities of the corresponding linear problems. He et al. [12] discussed the existence of sign-changing solutions for a class of discrete boundary value problems, and a concrete example was also given. Very recently, Yang [13] investigated the following discrete fourth Neumann
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