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