Three positive solutions to initial-boundary value problems of nonlinear delay differential equations

In this paper, we consider the existence of triple positive solutions to the boundary value problem of nonlinear delay differential equation $$\left\{ \begin{array}{lll} (\phi(x'(t)))^{\prime} + a(t)f(t,x(t),x'(t),x_{t})=0, \ \ 0 < t<1,\\ x_{0}=0,\\ x(1)=0, \end{array}\right.$$ where $\phi: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing homeomorphism and positive homomorphism with $\phi(0)=0,$ and $x_t$ is a function in $C([-\tau,0],\mathbb{R})$ defined by $x_{t}(\sigma)=x(t+\sigma)$ for $-\tau \leq \sigma\leq 0.$ By using a fixed-point theorem in a cone introduced by Avery and Peterson, we provide sufficient conditions for the existence of triple positive solutions to the above boundary value problem. An example is also presented to demonstrate our result. The conclusions in this paper essentially extend and improve the known results

Triple positive solutions of BVP for second order ODE with one dimensional Laplacian on the half line

By applying Leggett-Williams fixed point theorem in a suitably constructed cone, we obtain the existence of at least three bounded positive solutions for a boundary value problem on the half line. Our result improves and complements some of the work in the literature

Positive solutions of complementary Lidstone boundary value problems

We consider the following complementary Lidstone boundary value problem (−1)my (2m+1)(t) = F(t, y(t), y′ (t)), t ∈ [0, 1] y(0) = 0, y(2k−1)(0) = y (2k−1)(1) = 0, 1 ≤ k ≤ m. The nonlinear term F depends on y ′ and this derivative dependence is seldom investigated in the literature. Using a variety of fixed point theorems, we establish the existence of one or more positive solutions for the boundary value problem. Examples are also included to illustrate the results obtained.Published versio