Multiple Positive Solutions of a Second Order Nonlinear Semipositone m

In this paper, we study a general second-order m-point boundary value problem for nonlinear singular dynamic equation on time scales uΔ∇(t)+a(t)uΔ(t)+b(t)u(t)+λq(t)f(t,u(t))=0, t∈(0,1)𝕋, u(ρ(0))=0, u(σ(1))=∑i=1m-2αiu(ηi). This paper shows the existence of multiple positive solutions if f is semipositone and superlinear. The arguments are based upon fixed-point theorems in a cone

Existence of positive solutions for third-order semipositone boundary value problems on time scales

In this paper, we consider the existence of positive solutions for a semipositone third-order nonlinear ordinary differential equation on time scales. In suitable growth conditions, by considering the properties on time scales and establishing a special cone, some new results on the existence of positive solutions are established when the nonlinearity is semipositone

Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale

In this paper, we are concerned with the existence of positive solution of the following semipositone boundary value problem on time scales: \begin{align*} (\psi(t)y^\Delta (t))^\nabla + \lambda_1 g(t, \,y(t)) + \lambda_2 h(t,\,y(t)) = 0, \,t \in [\rho(c), \,\sigma(d)]_\mathbb{T}, \end{align*} with mixed boundary conditions \begin{align*} \alpha y(\rho(c))-\beta \psi(\rho(c)) y^\Delta(\rho(c))=0,\\ \gamma y(\sigma(d))+\delta \psi(d) y^\Delta(d)=0, \end{align*} where $\psi:C[\rho(c),\, \sigma(d)]_\mathbb{T}$, $\psi(t)>0$ for all $t \in [\rho(c),\,\sigma(d)]_\mathbb{T}$; both $g$ and $h : [\rho(c),\,\sigma(d)]_\mathbb{T} \times [0,\,\infty) \to \mathbb{R}$ are continuous and semipositone. We have established the existence of  at least one positive solution or multiple positive solutions of the above boundary value problem by using fixed point theorem on a cone in a Banach space, when $g$ and $h$ are both superlinear or sublinear or one is superlinear and the other is sublinear for $\lambda_i>0;\,i=1,\,2$ are sufficiently small

Existence and asymptotic analysis of positive solutions for a singular fractional differential equation with nonlocal boundary conditions

In this paper, we focus on the existence and asymptotic analysis of positive solutions for a class of singular fractional differential equations subject to nonlocal boundary conditions. By constructing suitable upper and lower solutions and employing Schauder’s fixed point theorem, the conditions for the existence of positive solutions are established and the asymptotic analysis for the obtained solution is carried out. In our work, the nonlinear function involved in the equation not only contains fractional derivatives of unknown functions but also has a stronger singularity at some points of the time and space variables

Positive Solutions of a Singular Positone and Semipositone Boundary Value Problems for Fourth-Order Difference Equations

This paper studies the boundary value problems for the fourth-order nonlinear singular difference equations Δ4u(i−2)=λα(i)f(i,u(i)), i∈[2,T+2], u(0)=u(1)=0, u(T+3)=u(T+4)=0. We show the existence of positive solutions for positone and semipositone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems in a cone

Complex boundary value problems of nonlinear differential equations: Theory, computational methods, and applications

Editorial to the theme Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and Application

Multiple Positive Solutions to Multipoint Boundary Value Problem for a System of Second-Order Nonlinear Semipositone Differential Equations on Time Scales

We study a system of second-order dynamic equations on time scales (p1u1∇)Δ(t)-q1(t)u1(t)+λf1(t,u1(t),u2(t))=0,t∈(t1,tn),(p2u2∇)Δ(t)-q2(t)u2(t)+λf2(t,u1(t), u2(t))=0, satisfying four kinds of different multipoint boundary value conditions, fi is continuous and semipositone. We derive an interval of λ such that any λ lying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone

Existence of Positive Solutions for Higher Order Boundary Value Problems on Time Scales

In this paper, we establish the existence of at least one positive solution for the higher order boundary value problems on time scales by using the Krasnoselskii fixed point theorem

Multiple positive solutions of nonlinear singular m-point boundary value problem for second-order dynamic equations with sign changing coefficients on time scales

Let $\mathbb{T}$ be a time scale. In this paper, we study the existence of multiple positive solutions for the following nonlinear singular $m$-point boundary value problem dynamic equations with sign changing coefficients on time scales $$\left\{\begin{array}{lll} u^{\triangle\nabla}(t)+ a(t)f(u(t))=0, (0,T)_{\mathbb{T}}, \cr\ u^{\triangle}(0)=\sum_{i=1}^{m-2}a_{i}u^{\triangle}(\xi_i), \cr u(T)=\sum_{i=1}^{k}b_{i}u(\xi_i)-\sum_{i=k+1}^{s}b_{i}u(\xi_i)-\sum_{i=s+1}^{m-2}b_{i}u^{\triangle}(\xi_i), \end{array}\right.$$ where $1\leq k\leq s\leq m-2, a_i, b_i\in(0,+\infty)$ with $0<\sum_{i=1}^{k}b_{i}-\sum_{i=k+1}^{s}b_{i}<1, 0<\sum_{i=1}^{m-2}a_{i}<1, 0<\xi_1<\xi_2<\cdots<\xi_{m-2}<\rho(T)$, $f\in C( [0,+\infty),[0,+\infty))$, $a(t)$ may be singular at $t=0$. We show that there exist two positive solutions by using two different fixed point theorems respectively. As an application, some examples are included to illustrate the main results. In particular, our criteria extend and improve some known results