Multiple Positive Solutions of Nonlinear Boundary Value Problems
نویسندگان
چکیده
where w, p are positive and continuous on (0, 1), which allows singularities at the endpoints, and f(y) ≥ 0 and continuous for y ∈ R. For a given positive integer N , we provide conditions on the nonlinear function f which guarantee existence of N positive solutions. Our motivation comes from previous work in the case w(x) ≡ p(x) ≡ 1 of Henderson and Thompson [6], which uses a fixed point theorem of Leggett and Williams [8] and the relevant Green’s functions to describe conditions on f which guarantee the existence of at least three positive solutions, and our previous paper [2] using initial value methods and extending those results to any number of positive solutions. Henderson and Thompson [6, 7] have also used their methods to prove results on the existence of three positive solutions to certain higher order boundary value problems, and Graef, et al [5] have approached the same problem using the Krasnosel’skii fixed point theorem; these papers have depended crucially on properties of Green’s functions. Consider the problem (1), (2) in the case that w(x) and p(x) are symmetric about x = 1/2. Clearly any solution of (1) on [0, 1/2] with y(0) = y0(1/2) = 0 can be reflected across x = 1/2 to give a solution of (1), (2). Thus we shall focus on the boundary value problem
منابع مشابه
Higher order multi-point fractional boundary value problems with integral boundary conditions
In this paper, we concerned with positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions by using a result from the theory of fixed...
متن کاملOn Approximate Stationary Radial Solutions for a Class of Boundary Value Problems Arising in Epitaxial Growth Theory
In this paper, we consider a non-self-adjoint, singular, nonlinear fourth order boundary value problem which arises in the theory of epitaxial growth. It is possible to reduce the fourth order equation to a singular boundary value problem of second order given by w''-1/r w'=w^2/(2r^2 )+1/2 λ r^2. The problem depends on the parameter λ and admits multiple solutions. Therefore, it is difficult to...
متن کاملExistence of triple positive solutions for boundary value problem of nonlinear fractional differential equations
This article is devoted to the study of existence and multiplicity of positive solutions to a class of nonlinear fractional order multi-point boundary value problems of the type−Dq0+u(t) = f(t, u(t)), 1 < q ≤ 2, 0 < t < 1,u(0) = 0, u(1) =m−2∑ i=1δiu(ηi),where Dq0+ represents standard Riemann-Liouville fractional derivative, δi, ηi ∈ (0, 1) withm−2∑i=1δiηi q−1 < 1, and f : [0, 1] × [0, ∞) → [0, ...
متن کاملPositive solutions for nonlinear systems of third-order generalized sturm-liouville boundary value problems with $(p_1,p_2,ldots,p_n)$-laplacian
In this work, byemploying the Leggett-Williams fixed point theorem, we study theexistence of at least three positive solutions of boundary valueproblems for system of third-order ordinary differential equationswith $(p_1,p_2,ldots,p_n)$-Laplacianbegin{eqnarray*}left { begin{array}{ll} (phi_{p_i}(u_i''(t)))' + a_i(t) f_i(t,u_1(t), u_2(t), ldots, u_n(t)) =0 hspace{1cm} 0 leq t leq 1, alpha_i u...
متن کاملExistence of positive solutions for a boundary value problem of a nonlinear fractional differential equation
This paper presents conditions for the existence and multiplicity of positive solutions for a boundary value problem of a nonlinear fractional differential equation. We show that it has at least one or two positive solutions. The main tool is Krasnosel'skii fixed point theorem on cone and fixed point index theory.
متن کامل