We study the second-order ordinary differential equation y(t) = −f(t, y(t), y(t)), 0 ≤ t ≤ 1, subject to the multi-point boundary conditions αy(0)± βy(0) = 0, y(1) = m−2 ∑

abstract.   the sturm-liouville boundary value problem of the multi-order fractional differential equation  is studied. results on the existence of solutions are established. the analysis relies on a weighted function space and a fixed point theorem. an example is given to illustrate the efficiency of the main theorems.

and Applied Analysis 3 Definition 2.2 see 18 . The standard Riemann-Liouville fractional derivative of order α > 0 of a continuous function y : a,∞ → R is given by D a y t 1 Γ n − α ( d dt )n ∫ t a t − s n−α−1y s ds, 2.2 where n α 1, provided that the integral on the right-hand side converges. Definition 2.3 see 18 . The Riemann-Liouville fractional integral of order α > 0 of a function y : a,∞...

We study sufficient conditions for the existence of positive solutions to the m-point integral boundary value problem u + a(t)f (u) = 0, t∈ (0, 1), u(0) = 0, u(1) = m−1 i=1 α i η i η i−1 u(s)ds, where 0 = η 0 < η 1 < η 2 < · · · < η m−2 < η m−1 = 1, 0 < m−1 i=1 α i (η 2 i − η 2 i−1) < 2, α i ≥ 0 for i ∈ {1,. .. , m − 3} ∪ {m − 1} and α m−2 > 0. a ∈ C([0, 1], [0, ∞)) and f ∈ C([0, ∞), [0, ∞)). W...

and Applied Analysis 3 For existence theorems for fractional differential equations and applications, we refer to the survey 14 . Concerning the definitions and basic properties, we refer the reader to 15 . 2. Preliminaries and Previous Results For the convenience of the reader, we present here some definitions, lemmas, and results that will be used in the proofs of our main results. Definition...

Under suitable conditions on f t u , ( ) , the nonlinear second-order m-point boundary value problem ′′ + ( ) = u t f t u t ( ) , ( ) 0 , 0 1 < < t u( ) 0 0 = , u a u i i i m ( ) ( ) 1 1 2 = = − ∑ x has at least one positive solution. In this paper, the authors examine the positive solutions of nonlinear second-order m-point boundary value problem. Ma, 2001; Sun, 2005) for some recent results o...

The purpose of this article is to establish the existence of multiple positive solutions of the dynamic equation on time scales φ uΔ t ∇ h t f t, u t , uΔ t 0, t ∈ 0, T T , subject to the multipoint boundary condition uΔ 0 0, u T ∑m−2 i 1 aiu ξi , where φ : R → R is an increasing homeomorphism and satisfies the relation φ xy φ x φ y for x, y ∈ R, which generalizes the usually p-Laplacian operat...

In this paper, we study second-order m-point boundary value problem { u′′(t) + a(t)u′(t) + f(t, u) = 0, 0 ≤ t ≤ 1, u′(0) = 0, u(1) = ∑k i=1 aiu(ξi)− ∑m−2 i=k+1 aiu(ξi), where ai > 0(i = 1, 2, · · ·m − 2), 0 < ∑k i=1 ai − ∑m−2 i=k+1 ai < 1, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, a ∈ C([0, 1], (−∞, 0)) and f is allowed to change sign. We show that there exist two positive solutions by using Leggett-Will...

We propose a fuzzy-based approach aiming at finding numerical solutions to some classical problems. We use the technique of F-transform to solve a second-order ordinary differential equation with boundary conditions. We reduce the problem to a system of linear equations and make experiments that demonstrate applicability of the proposed method. We estimate the order of accuracy of the proposed ...