creative_2022_31_1_101_108_001

Existence of positive solutions for 3n^th order boundary value problems involving p-Laplacian

 Sankar, R. R., Sreedhar, N. and Prasad, K. R.

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creative_2022_31_1_101_108

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This paper establishes the existence of positive solutions for 3n^{\text {th}} order differential equations with p-Laplacian operator

    \[(-1)^n[\phi_{p}(v^{(3n-3)}(t))]'''=g(t,v(t)), ~~t \in [0, 1],\]

satisfying the three-point boundary conditions

    \[\left. \begin{aligned} v^{(3i)}(0)=0&,~ v^{(3i+1)}(0)=0,~ v^{(3i+1)}(1)=\alpha_{ i+1}v^{(3i+1)}(\eta), \text{~for~} 0\leq i \leq n-2,\\ &[\phi_{p}(v^{(3n-3)}(t))]_{\text {at} ~ t=0}=0,~ [\phi_{p}(v^{(3n-3)}(t))]_{\text {at} ~ t=0}'= 0,\\ &~~[\phi_{p}(v^{(3n-3)}(t))]_{\text {at} ~ t=1}'=\alpha_{n}[\phi_{p}(v^{(3n-3)}(t))]_{\text {at} ~ t=\eta}', \end{aligned} \right\}\]

where n\geq 2,\ \eta\in(0,1), \alpha_{j}\in(0,\frac{1}{\eta}) is a constant for 1\leq j \leq n, by an application of Guo–Krasnosel’skii fixed point theorem.

 

 

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 Sankar, R. R., Prasad, K. R., Sreedhar, N.

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