Entire Solutions of Two Certa in Types of Non-linear Differential-Difference Equations

摘要

In this paper, we describe entire solutions for two certain types of non-linear differential-difference equations of the form fn(z)+ωfn-1(z)f′(z)+q(z)eQ(z)f(z+c)=u(z)ev(z),documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} f^n(z)+omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c)=u(z)e^{v(z)}, end{aligned}$$end{document}and fn(z)+ωfn-1(z)f′(z)+q(z)eQ(z)f(z+c)=p1eλz+p2e-λz,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} f^n(z)+omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c)=p_1e^{lambda z}+p_2e^{-lambda z}, end{aligned}$$end{document}where q, Q, u, v are non-constant polynomials, c,λ,p1,p2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$c,lambda ,p_1,p_2$$end{document} are non-zero constants, and ωdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$omega $$end{document} is a constant. Our results improve and generalize some previous results.