Nontrivial Solutions for Time Fractional Nonlinear Schrödinger-Kirchhoff Type Equations

Abstract

We study the existence of solutions for time fractional Schrödinger-Kirchhoff type equation involving left and right Liouville-Weyl fractional derivatives via variational methods.

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

“…Some worthwhile studies of Kirchhoff equations include those by Graef et al [15], Hssini [16], Matallah et al [17], and references therein. It should be mentioned that the existence of solutions to the Kirchhoff-type fractional differential equations addressed by variational methods has also received much attention from scholars [18][19][20][21][22][23], as models based on fractional order are better suited to describing the memory and hereditary properties of many processes and materials. Here, by using the mountain pass theorem and genus properties in the critical point theory, Chen et al [20] were concerned with the following Kirchhoff-type fractional Dirichlet problem with p-Laplacian:…”

“…Some worthwhile studies of Kirchhoff equations include those by Graef et al [15], Hssini [16], Matallah et al [17], and references therein. It should be mentioned that the existence of solutions to the Kirchhoff‐type fractional differential equations addressed by variational methods has also received much attention from scholars [18‐23], as models based on fractional order are better suited to describing the memory and hereditary properties of many processes and materials. Here, by using the mountain pass theorem and genus properties in the critical point theory, Chen et al [20] were concerned with the following Kirchhoff‐type fractional Dirichlet problem with $$$ p $$$‐Laplacian: $$$$ \left\{\begin{array}{l}{{\left(a+b{\int}_0^T{\left|{}_0{D}_t^{\alpha }u(t)\right|}^p dt\right)}^{p-1}}_t{D}_T^{\alpha }{\phi}_p\left({}_0{D}_t^{\alpha }u(t)\right)=f\left(t,u(t)\right),\kern0.30em t\in \left(0,T\right),\\ {}u(0)=u(T)=0,\end{array}\right.$$…”

Multiple Solutions for a Kirchhoff-type Fractional Coupled Problem with p-Laplacian

“…Unlike the two types of fractional order differential equations listed above, the Kirchhoff-type fractional order differential equations discussed in this study have received less attention to my knowledge. (see [12][13][14][15][16] and references therein). In these articles, just certain asymptotic conditions for the non-linear terms on the right-hand side of the equation are given to derive the existence and multiplicity results for the equation's solutions, ignoring the geometric conditions for the non-linear terms, which are addressed in this paper.…”

“…In Section2, some important definitions and lemmas are given in this section. In Section 3, we conclude that there are at least two different weak solutions to the equation (1) by combining Lemma 5 and Lemma 6, rather than using the traditional mountain pass theorem as in [2,10,12,13,16], and then Theorem 1 is used to show that the problem has at least two non-trivial solutions based on geometric considerations. Finally, we bring the paper to a close in Section 4.…”

Analysis of fractional boundary value problem with non local flux multi-point conditions on a Caputo fractional differential equation

“…These equations capture non local relations in space and time with memory essentials. Due to extensive applications of FDEs in engineering and science, research in this area has grown significantly all around the world., for instance, see [18], [11], [15] and the references cited therein. Recently, much interest has been created in establishing the existence of solutions for various types of boundary value problem of fractional order with nonlocal multi-point boundary conditions.…”