Résumé:
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)+
Pk1 (ez) + Qk1
ez
f(k1)+ +
P1 (ez) + Q1
ez
f0+
P0 (ez) + Q0
ez
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(eA)
f0 +
P1(eB) + Q2(eB)
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
