Opuscula Math. 36, no. 1 (2016), 69-79
http://dx.doi.org/10.7494/OpMath.2016.36.1.69

Opuscula Mathematica

# Positive solutions of boundary value problems with nonlinear nonlocal boundary conditions

Smita Pati
D. K. Hota

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.

Mathematics Subject Classification: 34B08, 34B18, 34B15, 34B10.

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• Birla Institute of Technology, Department of Mathematics, Mesra, Ranchi - 835215, India
• Smita Pati
• Birla Institute of Technology, Department of Mathematics, Mesra, Ranchi - 835215, India
• D. K. Hota
• Indira Gandhi Institute of Technology, Department of Mechanical System Design, Sarang - 759146, India
• Communicated by Alexander Domoshnitsky.
• Revised: 2015-05-26.
• Accepted: 2015-05-26.
• Published online: 2015-09-19.