Affiliation:
1. Department of Mathematics, Luoyang Normal University , Luoyang , Henan , China
Abstract
Abstract
In this article, a functional boundary value problem involving mixed fractional derivatives with
p
(
x
)
p\left(x)
-Laplacian operator is investigated. Based on the fixed point theorems and Mawhin’s coincidence theory’s extension theory, some existence theorems are obtained in the case of non-resonance and the case of resonance. Some examples are supplied to verify our main results.
Reference37 articles.
1. E. Acerbi and G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Rational Mech. Anal. 156 (2001), 121–140, DOI: https://doi.org/10.1007/s002050100117.
2. L. Diening, Theoretical and Numerical Results for Electrorheological Fluids, Ph.D. Thesis, University ofFreiburg, Germany, 2002.
3. M. Ružička, Electrorheological fluids: Modeling and mathematical theory, in: Lecture Notes in Mathematics, vol. 1784, Springer-Verlag, Berlin, 2000.
4. Y. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406, DOI: https://doi.org/10.1137/050624522.
5. V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 29 (1987), 33–36, DOI: https://doi.org/10.1070/IM1987v029n01ABEH000958.