Nonexistence of subnormal solutions for linear complex di¤erential equations

Abstract

Throughout this work, we have been talking about the possibility of generalization of some results related to second-order complex di¤erential equations to higher-order complex di¤erential equations in analogous manner or di¤erent manner, and extension for other results. For example, we generalized the results of Li and Yang : Theorem 2.2.2 and Theorem 2.2.3 to Theorem 3.2.1 and Theorem 3.2.2, and these last two theorems are extensions for results of Liu and Yang, Chen and Shon : Theorem 3.1.2 and Theorem 3.1.3. Theorem 2.2.3 is generalized to Theorem 3.2.5 and we considered at that case, the equation f(k)+

Pk􀀀1 (ez) + Qk􀀀1 􀀀 e􀀀z f(k􀀀1)+ +

P1 (ez) + Q1 􀀀 e􀀀z f0+

P0 (ez) + Q0 􀀀 e􀀀z f = 0 with deg P1 = deg P0 and degQ1 = degQ0: From that, we hope to solve the next problem : What can be said about the subnormal solutions of the equation above if we suppose that deg Pj = deg P0 and degQj = degQ0 ; 8j = 0; : : : ; k 􀀀 1 ? or, What are the hypotheses that guarantee that the equation above doesn t have subnor- mal solution ? We hope also, study the existence or nonexistence of subnormal solutions of the equa- tion of the general form f00 + P(eA)f0 + Q(eB)f = 0 where P(z);Q(z) are polynomials in z, with deg P = degQ and ;A(z) and B(z) are polynomials in z with degA = degB or A(z) and B(z) are transcendental entire functions with p(A) = p(B); here, p denote the p-iterated order. See [23].Another problem is about the existence or nonexistence of subnomral solutions of the equation f00 +

P1(eA) + Q1(e􀀀A)

f0 +

P1(eB) + Q2(e􀀀B)

f = 0 where P(z);Q(z);A(z) and B(z) are polynomials in z. Other questions are raised about di¤erential polynomials generated by the nontrivial solutions and especially nontrivial subnormal solutions of all forms of di¤erential equations mentioned in this thesis. The problems related to the di¤erential polynomials are about estimate the growth of order and if possible estimate the e-type order, oscillation theory, etc. University