Positive solutions of first order boundary value problems with nonlinear nonlocal boundary conditions

Authors: SMITA PATI, SESHADEV PADHI

Abstract: 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.

Keywords: Positive solutions, Leray-Schauder fixed point theorem, nonlinear boundary conditions

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