The Reaserch Of Several Problems In Complex And Real Analysis

Nevanlinna theory can be seen the most important achievements in the preceding century to understand the properties of meromorphic functions. This theory is composed of two main theorems, which are called Nevanlinna’s first and second main theorems that had been significant breakthroughs in the development of the classical function theory.Today,many mathematics scholars still pay great attentions to this field which would always be based on Nevanlinna theory.The existence of Solutions of differential equations is an important part in functional analysis, and the topological degree theory and cone theory are very important methods in nonlinear function analysis.The present thesis involves some results of the author under the guidance of my supervisor Hongxun Yi. It consists of four parts and the matters are explained as below.In Chapter1, We introduce the general background of Nevanlinna Theo-ry. Meanwhile, we give some preliminary definitions and properties of nonlin-ear functional analysis, and several lemmas on the existence of fixed point.In Chapter2, we investigate the relationship between Borel directions and uniqueness of meromorphic functions and obtain some results of mero-morphic functions sharing four distinct values IM and one set in an angular domain containing a Borel line. Our result is an improvement of recent the-orem which given by Long and Wu [35]. In fact, we obtained the following result: Theorem0.1. Let.f be a meromorphic function of infinite order p(r), g∈M(p(r)),argz=θ(0<θ<2π)be one Borel direction of p(r)order of meromorphic function f,we assume that f and g share four distinct values aj(j=1,2,3,4)IM in Ω(θ-ε,θ+ε)and E(S,Ω(θ-ε,θ+ε),f)(?) E(S,Ω(θ ε,θ+ε)E,g),,for anyε(0<ε<π),where S={b1,…,bm},m>1and b1,…,bm∈C\{a1,a2,a3,a4).Then f and g share all values CM,thus it follows that either f=g or f is a Mobius transformation of g.Furthermore, if the number of the values in S is odd,then f=9.In Chapter3,we investigate the distribution and uniqueness of a class of differential-difference polynomials.In fact,we obtained the following result:Theorem0.2.Let f be transcendental entire function of finite order,a(z)(≠0)be a small function with respect to f,cj(j=1,2…d)be distinct finite complex numbers,n,m,d,k,vj(j=1,2…d)are non-negative integers. If n>k+2,then the differential-difference polynomial(fn(fm(z)-1)(?) f(z+cj)vj)(k)一a(z)has infinitely many zeros.Theorem0.3.Let f be transcendental entire function of finite order,a(z)(≠0)be a small function with respect to,f,cj(j=1,2...d)be distinct finite complex numbers,n,m,d,k,vj(J=1,2...d)are non-negative integers.If one of the following conditions holds:(i)n>k+2when m <k+2,(ii)n>2k-m+3when M>k+1.Then the differential-difference polynomial(fn(f(z)一1)m(?)f(z+cj)vj)(k)-a(z)has infinitely many zeros.Theorem0.4.Let,and g be transcendental entire functions of finite order, a(z)(≠0)be a common small function with respect to,andg,cj(j=1,2...d) be distinct finite complex numbers,n,m,d,k,vj(j=1,2…d)are non-negative integers. If n>2k+m+σ+5,and the differential-difference polynomial a(z)CM,then f=tg,where tm=tn+σ=1. Theorem0.5.Let f and g be transcendental entire functions of finite order, a(Z)(≠0)be a common small function with respect to f and g,cj(j=1,2…d) be distinct finite complex numbers,n,m,d,k,vj(j=1,2...d)are non-negative integers. If n>4k-m+σ+9,and the differential-difference polynomialIn Chapter4,we devoted to study the existence and multiplicity of positive solutions for the fourth order p-Laplacian boundary value problem involving impulsive effects where J=[0,1],f∈C([0,1]×R+,R+),Ik∈C(R+,R+)(R+:=[0,∞)). Based on a priori estimates achieved by uti-lizing properties of concave func-tions and Jensen’s inequality,we adopt fixed point index theory to establish our main results.Let p*:=max{1,p},p.:=min{1,p},K1:=2p*-1,k2:=m(p*-1), K3:=2p*-1,K4:=2m(p*-1),K5:=2p/p*p-2,k6:=2(m+1)(p*-1). We now list our hypotheses.(H1)There is a p>0such that0<y<p and0<t<1implies f(t,y)<ηppp,Ik(y)<ηkp,whereη,ηk>0satisfy(H2)There exist0<ro<p and.a1>0,a2>0satisfying such that f(t, y)> a1yp, Ik(y)> a2y,(?)t∈[0,1],0<y<r0,(7)where σ:=min t€[t1,tm] t(1-t)>0.(H3) There exist c>0and a3>0, a4>0satisfyingsuch that f(t,y)>a3yp-c, Ik(y)> a4y-c,(?)t∈[0,1], y>0.(8)(H4) There is a p>0such that σp<y<p and0<t<1implies f (t,y)>ζPPP, Ik(y)>-ζkAwhereζ,ζk>0satisfy(H5) There exist0<r0<p and b1>0, b2>0satisfyingsuch that f(t, y)<b1yp, Ik(y)<b2y,(?)t∈[0,1],0<y<r0.(9)(H6) There exist c>0and b3>0, b4>0satisfyingsuch that f(t, y)<b3yp+c,Ik(y)<b4y+c,(?)t∈[0,1], y>0.(10) In fact, we obtained the following result:Theorem0.6.Suppose that (H1)-(H3) are satisfied, then (6) has at least two positive solutions.Theorem0.7. Suppose that (H4)-(H6) are satisfied, then (6) has at least two positive solutions.Combining functional analysis with complex analysis is a very important research topic, we can continue to study the following question:Using the topological degree theory (mainly Brouwer degrees) in Non-linear functional analysis, we can study the normality of function family, especially for a continuous function. By studying the case of zeros of con-tinuous functions, we will resolve the problem of meromorphic functions and its derivative sharing continuous Functions. Currently there are very few results in this area, in1999, Bargmann, Bonk, Hinkkanen, Martin focus on this problem [2]. At present, this area still has a lot of problems to be solved.