In this paper, using variational methods, we study the existence of ground states solutions to the modified fractional Schr¨odinger equations with a generalized Choquard nonlinearity

In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative: HDαx(t)=Af(t,x(t))+∑i=1kCiHIβigi(t,x(t)),t∈(1,e), supplemented with fractional Hadamard boundary conditions: HDξx(1)=0,HDξx(e)=aHDα−ξ−12(HDξx(t))|t=δ,δ∈(1,e), where 1<α≤2, 0<ξ≤12, a∈(0,∞), 1<α−ξ<2, 0<βi<1, A,Ci, 1≤i≤k, are real constants, HDα is the Hadamard fractional derivative of order α and HIβi is the Hadamard fractional integral of order βi. By using some fixed point theorems, existence and uniqueness results are obtained. Finally, an example is given for demonstration.

In this paper, we study the number of limit cycles for discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are quadratic isochronous centers in a half-plane and the cubic isochronous centers in the other one. We deal with eight classes of discontinuous piecewise differential systems formed by these types of isochronous centers, and we provide an upper bound for the maximum number of limit cycles for these classes that can exhibit. Also, we show that the corresponding maximum number of limit cycles in four classes is reached.