Existence of uncountably many bounded positive solutions for a third order nonlinear neutral delay difference equation

This paper deals with the solvability of the third order nonlinear neutral delay difference equation 12 (an1(xn + βnxn−τ )) + 12f (n, xn−b1n , xn−b2n , . . . , xn−bkn ) +1g(n, xn−c1n , xn−c2n , . . . , xn−ckn ) = h(n, xn−d1n , xn−d2n , . . . , xn−dkn ), n ≥ n0, relative to the sequence {βn}n∈Nn0 ⊂ R. Using the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem, a few sufficient conditions of the existence of uncountably many bounded positive solutions for the equation are presented. Seven examples are included to demonstrate the advantages and effectiveness of the results presented in this paper.