Nontrivial Limit Cycles in a Kind of Piecewise Smooth Generalized Abel Equation

In this paper, we focus on the number of nontrivial limit cycles in a kind of piecewise smooth generalized Abel equation dx/dt = {x(p)/p -1 + sigma(2)(i=1) epsilon(i)(A(i)(-)(t)x(p) + B-i(-)(t) x(2p-1)), -1 <= t <= 0, x is an element of R,-x(p)/p-1 + sigma(2)(i=1) epsilon(i)(A(i)(+)(t)x(p) + B-i(+)(t) x(2p-1)), 0 <= t <= 1, x is an element of R,where p is an element of Z(+)\ {1}, A(1)(+) = a(10)(+/-) + a(11)(+/-)t, B-+/-(1) = b(10)(+/-) + b(11)(+/-)t, A(2)(+/-) = a(20)(+/- )+ a(21)(+/-)t, B-2(+/-) = b(20)(+/-) + b(21)(+/-)t and |epsilon| << 1. Under the condition b(11)(+) &NOTEQUexpressionL; 0, employing Melnikov functions of any order and using properties of Chebyshev systems, we prove that if p is odd, then the maximum number of nontrivial limit cycles bifurcating from the periodic annulus of the unperturbed system is 6 and it is attainable, and if p is even, then the maximum number is 3, and it can be attained too.