<p>In this paper, we study the nonlinear Riemann boundary value problem with square roots that is represented by a Cauchy-type integral with kernel density in variable exponent Lebesgue spaces. We discuss the odd-order zero-points distribution of the solutions and separate the single valued analytic branch of the solutions with square roots, then convert the problem to a Riemann boundary value problem in variable exponent Lebesgue spaces and discuss the singularity of solutions at individual zeros belonging to curve. We consider two types of cases those where the coefficient is Hölder and those where it is piecewise Hölder. Then we solve the Hilbert boundary value problem with square roots in variable exponent Lebesgue spaces. By discussing the distribution of the odd-order zero-points for solutions and the method of symmetric extension, we convert the Hilbert problem to a Riemann boundary value problem. The equivalence of the transformation is discussed. Finally, we get the solvable conditions and the direct expressions of the solutions in variable exponent Lebesgue spaces.</p>