Variational formulation for nonlinear impulsive fractional differential equations with (p, q)‐Laplacian operator

Abstract

In this paper, we focus on a class of generalized (p, q)‐Laplacian‐type impulsive fractional differential equation for 1 < p≤q < ∞, with a nonlinearity f containing fractional derivatives 0Dtαu and 0Dtβu simultaneously and being imposed on mild assumptions contrasting with previous works. The underlying idea is based on the Mountain pass theorem and the iterative technique to obtain the existence of nontrivial solutions for our equation. Our work extends and supplements some existing results.

New Multiplicity Results for a Boundary Value Problem Involving a ψ-Caputo Fractional Derivative of a Function with Respect to Another Function

“…Combining critical point theory and properties of fractional calculus of thee ψ-Caputo fractional integral and derivative, new multiplicity results of infinitely many solutions are established for the problem (6). Recently, some achievements available in the references discussed the existence and multiplicity results for ψ-Caputotype fractional boundary value problems via fixed point theorems [12,13], while few results were based on variational methods, even though variational methods are effective ways for studying the existence of solutions for fractional differential equations [14][15][16][17][18]. Moreover, some simple algebraic conditions are applied in the paper instead of the conventional asymptotic conditions used in previous articles because most nonlinear functions can not adapted for these asymptotic conditions.…”

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“…This structure are common in all around us, such as water pipes, molecular structures in medicine and biology and so on [12][13][14][15][16][17][18]. The model described by differential equations on star graphs has applications in chemical engineering, biology, physics and other fields [19][20][21][22][23]. At present, a few researchers have studied the existence results of solutions of differential equations on star graphs [24][25][26][27][28].…”

Existence and multiplicity of solutions for (p,q)$$ \left(p,q\right) $$‐Laplacian Kirchhoff‐type fractional differential equations with impulses

“…Under the (AR)‐condition, at least one nontrivial weak solution and infinitely many nontrivial weak solutions were obtained for problem (). On the other hand, the existence of solutions for impulsive fractional $$$ \left(p,q\right) $$$‐Laplacian boundary value problems by using variational methods was taken into consideration [9, 24, 25]. Li et al [9] considered the existence of solutions for an impulsive fractional coupled system of $$$ \left(p,q\right) $$$‐Laplacian type: …”