Physical Sciences and Mathematics Commons^™

Positive Solutions Of First Order Boundary Value Problems With Nonlinear Nonlocal Boundary Conditions, Smita Pati, Seshadev Padhi

Turkish Journal of Mathematics

We consider the existence of positive solutions of the nonlinear first order problem with a nonlinear nonlocal boundary condition given by $x^{\prime}(t) = r(t)x(t) + \sum_{i=1}^{m} f_i(t,x(t)), t \in [0,1]$ $\lambda x(0) = x(1) + \sum_{j=1}^{n} \Lambda_j(\tau_j, x(\tau_j)),\tau_j \in [0,1],$ where $r:[0,1] \rightarrow [0,\infty)$ is continuous, the nonlocal points satisfy $0 \leq \tau_1 < \tau_2 < ... < \tau_n \leq 1$, the nonlinear functions $f_i$ and $\Lambda_j$ are continuous mappings from $[0,1] \times [0,\infty) \rightarrow [0,\infty)$ for $i = 1,2,...,m$ and $j = 1,2,...,n$ respectively, and $\lambda >1$ is a positive parameter. The Leray-Schauder theorem and Leggett--Williams fixed point theorem were used to prove our results.