The aim of this paper is to describe a continuation method combining the Chebyshev’s method and the convex acceleration of Newton’s method to solve nonlinear equations in Banach spaces. The semilocal convergence analysis of the method is established using recurrence relations under the assumption that the first Fréchet derivative satisfies the Hölder continuity condition. This condition is milder than the usual Lipschitz condition. The computation of second Fréchet derivative is also avoided. Two real valued functions and a real sequence are used to establish a convergence criterion of R-order (1+p), where p∈(0,1] is the order of the Hölder condition. An existence and uniqueness theorem along with the closed form of error bounds is derived in terms of a real parameter α∈[0,1]. Two numerical examples are worked out to demonstrate the efficacy of our convergence analysis. For both the examples, the convergence conditions hold for the Chebyshev’s method (α=0). However, for the convex acceleration of Newton’s method (α=1), these convergence conditions hold for the first example but fail for the second example. For particular values of α, our method reduces to the Chebyshev’s method (α=0) and the convex acceleration of Newton’s method (α=1).