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

Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions

In this paper, we consider a four-point coupled boundary value problem for systems of the nonlinear semipositone fractional differential equation \begin{gather*}\left\{ \begin{array}{ll} \mathbf{D}_{0+}^\alpha u+\lambda f(t,u,v)=0,\quad 0<t<1, \lambda >0,\\ \mathbf{D}_{0+}^\alpha v+\lambda g(t,u,v)=0,\\ u^{(i)}(0)=v^{(i)}(0)=0, 0\leq i\leq n-2,\\ u(1)=av(\xi), v(1)=bu(\eta), \xi,\eta\in(0,1) \end{array}\right.\end{gather*} where $\lambda$ is a parameter, $a, b, \xi,\eta$ satisfy $\xi,\eta\in(0,1)$, $0<ab\xi\eta<1$, $\alpha \in(n-1, n]$ is a real number and $n\geq 3$, and $\mathbf{D}_{0+}^\alpha$ is the Riemann-Liouville's fractional derivative, and $f,g$ are continuous and semipositone. We derive an interval on $\lambda$ such that for any $\lambda$ lying in this interval, the semipositone boundary value problem has multiple positive solutions

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)&#x1D54B;, 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

Singular positone and semipositone boundary value problems of nonlinear fractional differential equations

We present some new existence results for singular positone and semipositone nonlinear fractional boundary value problem where μ &gt; 0, a, and f are continuous, α ∈ 3, 4 is a real number, and D α 0 is Riemann-Liouville fractional derivative. Throughout our nonlinearity may be singular in its dependent variable. Two examples are also given to illustrate the main results

Applications of Schauder’s Fixed Point Theorem to Semipositone Singular Differential Equations

We study the existence of positive periodic solutions of second-order singular differential equations. The proof relies on Schauder’s fixed point theorem. Our results generalized and extended those results contained in the studies by Chu and Torres (2007) and Torres (2007) . In some suitable weak singularities, the existence of periodic solutions may help