Positive solutions of boundary value problems with nonlinear nonlocal boundary conditions

Abstract

We consider the existence of at least three positive solutions of a nonlinear first order problem with a nonlinear nonlocal boundary condition given by \[\begin{aligned} x^{\prime}(t)& = r(t)x(t) + \sum_{i=1}^{m} f_i(t,x(t)), \quad t \in [0,1],\\ \lambda x(0)& = x(1) + \sum_{j=1}^{n} \Lambda_j(\tau_j, x(\tau_j)),\quad \tau_j \in [0,1],\end{aligned}\] where \(r:[0,1] \rightarrow [0,\infty)\) is continuous; the nonlocal points satisfy \(0 \leq \tau_1 \lt \tau_2 \lt \ldots \lt \tau_n \leq 1\), the nonlinear function \(f_i\) and \(\tau_j\) are continuous mappings from \([0,1] \times [0,\infty) \rightarrow [0,\infty)\) for \(i = 1,2,\ldots ,m\) and \(j = 1,2,\ldots ,n\) respectively, and \(\lambda \gt 0\) is a positive parameter.

Keywords

• positive solutions
• Leggett-Williams fixed point theorem
• nonlinear boundary conditions