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Existence of nontrivial and multiple solutions for two types of non-local problems with sublinear or superlinear nonlinearities are investigated by linking theorems and index theory in critical point theory. Some results in the literature are extended. Keywords Non-local(Fractional) problems Á Nontrivial(Multiple) solutions Á Linking theorems Á Index theory Mathematics Subject Classification 35R11 Á 35A15 Á 47A10 This article is part of the topical collection dedicated to Prof. Dajun Guo for his 85th birthday, edited by Yihong Du, Zhaoli Liu, Xingbin Pan, and Zhitao Zhang.

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