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

Seshadev Padhi
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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  • Seshadev Padhi
  • 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.
  • Received: 2014-06-17.
  • Revised: 2015-05-26.
  • Accepted: 2015-05-26.
  • Published online: 2015-09-19.
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Cite this article as:
Seshadev Padhi, Smita Pati, D. K. Hota, Positive solutions of boundary value problems with nonlinear nonlocal boundary conditions, Opuscula Math. 36, no. 1 (2016), 69-79, http://dx.doi.org/10.7494/OpMath.2016.36.1.69

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