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Open Access Research

An inverse problem related to a half-linear eigenvalue problem

Wei-Chuan Wang1* and Yan-Hsiou Cheng2

Author Affiliations

1 Center for General Education, National Quemoy University, Kinmen, 892, Taiwan, ROC

2 Department of Mathematics and Information Education, National Taipei University of Education, Taipei, 106, Taiwan, ROC

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Boundary Value Problems 2014, 2014:65  doi:10.1186/1687-2770-2014-65

Published: 24 March 2014

Abstract

We study an inverse problem on the half-linear Dirichlet eigenvalue problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3">View MathML</a> and r is a positive function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4">View MathML</a>. Using eigenvalues and nodal data (the lengths of two consecutive zeros of solutions), we reconstruct <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M5">View MathML</a> and its derivatives. Our method is based on (Law and Yang in Inverse Probl. 14:299-312, 779-780, 1998; Shen and Tsai in Inverse Probl. 11:1113-1123, 1995), and our result extends the result in (Shen and Tsai in Inverse Probl. 11:1113-1123, 1995) for the linear case to the half-linear case.

MSC: 34A55, 34B24, 47A75.