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Upper and Lower Solutions Method for a Class of Singular Fractional Boundary Value Problems with p-Laplacian Operator

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Abstract and Applied Analysis
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Abstract

The upper and lower solutions method is used to study the 𝑝 -Laplacian fractional boundary value problem 𝐷 𝛾 0 + ( 𝜙 𝑝 ( 𝐷 𝛼 0 + 𝑢 ( 𝑡 ) ) ) = 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) ) , 0 < 𝑡 < 1 , 𝑢 ( 0 ) = 0 , 𝑢 ( 1 ) = 𝑎 𝑢 ( 𝜉 ) , 𝐷 𝛼 0 + 𝑢 ( 0 ) = 0 , and 𝐷 𝛼 0 + 𝑢 ( 1 ) = 𝑏 𝐷 𝛼 0 + 𝑢 ( 𝜂 ) , where 1 < 𝛼 , 𝛾 ⩽ 2 , 0 ⩽ 𝑎 , 𝑏 ⩽ 1 , 0 < 𝜉 , 𝜂 < 1 . Some new results on the existence of at least one positive solution are obtained. It is valuable to point out that the nonlinearity 𝑓 can be singular at 𝑡 = 0 , 1 or 𝑢 = 0 .

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