Existence of bounded nonoscillatory solutions of first-order nonlinear neutral delay differential equations

The aim of this paper is to study the following first-order nonlinear neutral delay differential equation d dt [x.t/ C c.t/x.t 􀀀 /] C h.t/f .x.t 􀀀 1/; x.t 􀀀 2/; : : : ; x.t 􀀀 k// D g.t/; t t0; where c; h; g 2 C.Tt0;C1/; R/, > 0; i 2 RC for i 2 Jk D f1; : : : ; kg, and f 2 C.Rk; R/. By using the Schauder and Krasnoselskii fixed point theorems, we establish the existence of uncountably many bounded nonoscillatory solutions for the above equation. To dwell upon the importance and advantages of our results, seven nontrivial examples are included.