Uniform convexity in \(\ell_{p(\cdot)}\)

In this work, we investigate the variable exponent sequence space \(\ell_{p(\cdot)}\). In particular, we prove a geometric property similar to uniform convexity without the assumption \(\limsup_{n \to \infty} p(n) lt; \infty\). This property allows us to prove the analogue to Kirk s fixed point theorem in the modular vector space \(\ell_{p(\cdot)}\) under Nakano s formulation.