Some results on the transcendental entire solutions of certain type of non-linear shift-differential equations

In this paper, we have studied the nature and form of solutions of three types of non-linear shift-differential equations under the aegis of the following compact form: fn-m(z)L~(z,f)+q(z)eQ(z)f(z+c)=p1eλ1z+p2eλ2z,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f^{n-m}(z)\widetilde{L}(z,f)+q(z)e^{Q(z)}f(z+c)=p_1e^{\lambda _1z}+p_2e^{\lambda _2z}, \end{aligned}$$\end{document}where L~(z,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{L}(z,f)$$\end{document} is a linear differential polynomial in f; n, m are non-negative integers; q, Q respectively are two non-zero and non-constant polynomials; c,p1,p2,λ1,λ2∈C\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c, p_1, p_2,\lambda _1, \lambda _2\in {\mathbb {C}}\backslash \{0\}$$\end{document}. Our results improve two recent results of Chen et al. (Comput Methods Funct Theory 19(1):17–36, 2019) and Chen et al. (Comput Methods Funct Theory 21(2):199-218, 2021). Most importantly, in one of our result we have succeeded to answer partially the conjecture of Chen et al. (Comput Methods Funct Theory 19(1):17–36, 2019). We have also illustrated a handful number of examples to clarify our results.