Abstract
In this article, using variational methods, we study the existence of solutions for the Kirchhoff-type problem involving tempered fractional derivatives
M (integral(R)vertical bar D(+)(alpha,lambda)u(t)vertical bar(2)dtD(-)(alpha,lambda)(D(+)(alpha,lambda)u(t)) = f(t, u(t)), t is an element of R,
u is an element of W-lambda(alpha,2)(R),
where D(+/-)(alpha,lambda)u(t) are the left and right tempered fractional derivatives of order alpha is an element of (1/2, 1], lambda > 0, W-lambda(alpha,2)(R) represent the fractional Sobolev space, f is an element of C(R X R, R) and M is an element of C(R+, R+).