Opuscula Mathematica
Opuscula Math. 36, no. 1 (2016), 69-79

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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About the authors ▼
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
About the article ▼
Communicated by Alexander Domoshnitsky.
Received: 2014-06-17.
Revised: 2015-05-26.
Accepted: 2015-05-26.
Published online: 2015-09-19.
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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