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In the present paper, we find out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition, the completeness and the basis property in the elliptic part of the third kind of a domain in . We also consider a new boundaries condition and analyze the orthogonal basis of the eigenfunctions depending on parameters of the problem.doi:10.11648/j.pamj.20150406.16 fatcat:727sk4nxwbd2fgsokie74j4yuu