Inverse problems and nonlinear integral transforms

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We establish a theory of a nonlinear integral transform and obtain a global theorem for an inverse problem in bifurcation theory. Based upon the result we consider a problem of determining a nonlinearity of an autonomous differential equation of a period function, namely, a relation between periods and amplitudes to prove a global existence of nonlinear terms realizing a prescribed, Lipschitz continuous period function and characterize the nonlinear terms. This gives a complete answer to a classical inverse problem in a nonlinear autonomous oscillation.