Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives

Tunhua Wu, Xinguang Zhang and Yinan Lu

Abstract and Applied Analysis, 2012, vol. 2012, 1-16

Abstract:

We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term − ð ’Ÿ ð ›¼ ð ‘¥ ( ð ‘¡ ) = ð ‘ ( ð ‘¡ ) ð ‘“ ( ð ‘¡ , ð ‘¥ ( ð ‘¡ ) , ð ’Ÿ 𠜇 1 ð ‘¥ ( ð ‘¡ ) , ð ’Ÿ 𠜇 2 ð ‘¥ ( ð ‘¡ ) , … , ð ’Ÿ 𠜇 ð ‘› − 1 ð ‘¥ ( ð ‘¡ ) ) , 0 < ð ‘¡ < 1 , ð ’Ÿ 𠜇 ð ‘– ð ‘¥ ( 0 ) = 0 , 1 ≤ ð ‘– ≤ ð ‘› − 1 , ð ’Ÿ 𠜇 ð ‘› − 1 + 1 ð ‘¥ ( 0 ) = 0 , ð ’Ÿ 𠜇 ð ‘› − 1 ∑ ð ‘¥ ( 1 ) = ð ‘ âˆ’ 2 ð ‘— = 1 ð ‘Ž ð ‘— ð ’Ÿ 𠜇 ð ‘› − 1 ð ‘¥ ( 𠜉 ð ‘— ) , where ð ‘› − 1 < ð ›¼ ≤ ð ‘› , ð ‘› ∈ â„• and ð ‘› ≥ 3 with 0 < 𠜇 1 < 𠜇 2 < ⋯ < 𠜇 ð ‘› − 2 < 𠜇 ð ‘› − 1 and ð ‘› − 3 < 𠜇 ð ‘› − 1 < ð ›¼ − 2 , ð ‘Ž ð ‘— ∈ â„ , 0 < 𠜉 1 < 𠜉 2 < ⋯ < 𠜉 ð ‘ âˆ’ 2 < 1 satisfying ∑ 0 < ð ‘ âˆ’ 2 ð ‘— = 1 ð ‘Ž ð ‘— 𠜉 ð ›¼ − 𠜇 ð ‘› − 1 ð ‘— − 1 < 1 , ð ’Ÿ ð ›¼ is the standard Riemann-Liouville derivative, ð ‘“ ∶ [ 0 , 1 ] × â„ ð ‘› → â„ is a sign-changing continuous function and may be unbounded from below with respect to ð ‘¥ ð ‘– , and ð ‘ âˆ¶ ( 0 , 1 ) → [ 0 , ∞ ) is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:797398

DOI: 10.1155/2012/797398

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