The semilocal convergence of a third order iterative method used for solving nonlinear operator equations in Banach spaces is established by using recurrence relations under the assumption that the second Fr´echet derivative of the involved operator satisfies the ${omega}$-continuity condition given by $||F^{primeprime}(x)-F^{primeprime}(y)||{leq}{omega}(||x-y||)$, $x,y{in}{Omega}$, where, ${omega}(x)$ is a nondecreasing continuous real function for x > 0, such that ${omega}(0){geq}0$. This condition is milder than the usual Lipschitz/H$ddot{o}$lder continuity condition on $F^{primeprime}$. A family of recurrence relations based on two constants depending on the involved operator is derived. An existence-uniqueness theorem is established to show that the R-order convergence of the method is (2+$p$), where $p{in}(0,1]$. A priori error bounds for the method are also derived. Two numerical examples are worked out to demonstrate the efficacy of our approach and comparisons are elucidated with a known result.