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Two Solutions for a Class of Fractional Boundary Value Problems with Mixed Nonlinearities

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Abstract

In this paper we investigate the existence of nontrivial solutions for the following fractional boundary value problem:

$$\begin{aligned} \left\{ \begin{array}{ll} \dfrac{d}{dt}\Bigl (\dfrac{1}{2} {_0D_t^{-\beta }(u'(t))}+\dfrac{1}{2} {_tD_T^{-\beta }(u'(t))}\Bigr )+\nabla F(t,u)=0,\quad a.e.\,\, t\in [0,T], \\ u(0)=u(T)=0, \end{array} \right. \end{aligned}$$

where \(_0D_t^{-\beta }\) and \(_tD_T^{-\beta }\) are the left and right Riemann-Liouville fractional integrals of order \(0\le \beta <1\), respectively, and \(\nabla F(t,u)\) is the gradient of F(t, u) at u. The novelty of this paper is that, when the nonlinearity F(t, u) involves a combination of superquadratic and subquadratic terms, we present some reasonable assumptions and establish one new criterion to guarantee the existence of at least two nontrivial solutions. Recent results in the literature are generalized and significantly improved.

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Authors and Affiliations

  1. Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China

    Ziheng Zhang

  2. Department of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Rong Yuan

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  1. Ziheng Zhang
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  2. Rong Yuan
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Correspondence to Ziheng Zhang.

Additional information

Communicated by Shangjiang Guo.

Project supported by the National Natural Science Foundation of China (Grant No. 11101304).

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Zhang, Z., Yuan, R. Two Solutions for a Class of Fractional Boundary Value Problems with Mixed Nonlinearities. Bull. Malays. Math. Sci. Soc. 41, 1233–1247 (2018). https://doi.org/10.1007/s40840-016-0386-3

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  • DOI: https://doi.org/10.1007/s40840-016-0386-3

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