In this paper, by employing the AGW criterion and determining the number of solutions to some equations over finite fields \(\mathbb {F}_{p^{2m}}\) with odd characteristic, we further investigate the permutation behavior of polynomials with given forms. Some of them are the permutation polynomials with the form \((x^{p^m}-x+\delta )^s+L(x)\) , others are the permutation polynomials with the form \((x^{p^m}-x+\delta )^{s_1}+(x^{p^m}-x+\delta )^{s_2}+L(x)\) , where \(L(x)=ax^{p^m}+ax\) , \(a\in \mathbb {F}_{p^m}^*\) and \(p\in \{3,5\}\) .
