Abstract
In this paper, a research has been done about the existence of solutions to the Dirichlet boundary value problem for p-Laplacian fractional differential equations which include instantaneous and noninstantaneous impulses. Based on the critical point principle and variational method, we provide the equivalence between the classical and weak solutions of the problem, and the existence results of classical solution for our equations are established. Finally, an example is given to illustrate the major result.
Highlights
Fractional calculus involves arbitrary order derivatives and integration, so it plays a very important role in various fields such as physical engineering, medical image processing, mathematics, chemical engineering, and electricity
Many scholars did research on the theory of fractional differential equations continuously and have made enormous achievements; readers who are interested in these kinds of researches can refer to relevant literature
Motivated by the above-mentioned work, the paper focuses on the existence of solutions for the following pLaplacian fractional differential equations with instantaneous and noninstantaneous impulses, and if Dx Fiðt, uðtÞ − uðti + 1ÞÞ = f iðt, uðtÞÞ, the following problem reduces to (2): 8 >>>>>>>>>>>>
Summary
Fractional calculus involves arbitrary order derivatives and integration, so it plays a very important role in various fields such as physical engineering, medical image processing, mathematics, chemical engineering, and electricity. It is worth noting that by using the critical point theory and variational method, Zhao et al proved the existence and multiplicity of nontrivial solutions to the problem of nonlinear nontransient impulsive differential equations (see [21]). Journal of Function Spaces some scholars have obtained the existence results of solutions for noninstantaneous impulsive differential equations by variational methods (see [22,23,24]). Motivated by the above-mentioned work, the paper focuses on the existence of solutions for the following pLaplacian fractional differential equations with instantaneous and noninstantaneous impulses, and if Dx Fiðt, uðtÞ − uðti + 1ÞÞ = f iðt, uðtÞÞ, the following problem reduces to (2):.
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