Some classes of permutation polynomials of the form $$b(x^q+ax+\delta )^{\frac{i(q^2-1)}{d}+1}+c(x^q+ax+\delta )^{\frac{j(q^2-1)}{d}+1}+L(x)$$ over $$ {{{\mathbb {F}}}}_{q^2}$$

Let $q be a prime power and {{{\mathbb {F}}}}_q be a finite field with q elements. In this paper, we employ the AGW criterion to investigate the permutation behavior of some polynomials of the form \begin{aligned} b(x^q+ax+\delta )^{1+\frac{i(q^2-1)}{d}}+c(x^q+ax+\delta )^{1+\frac{j(q^2-1)}{d}}+L(x) \end{aligned} over {{{\mathbb {F}}}}_{q^2} with a^{1+q}=1, q\equiv \pm 1\pmod {d} and L(x)=-ax or...