| This thesis discusses the conception, classification, application and development survey of differential difference equation, differential integral equation and functional differential equation as well as methods of solutions to functional differential equation with one deviation, two deviations and special deviation.In chapter one, firstly the conception and classification of differential difference equation, differential integral equation and functional differential equation are introduced.Consider the following equationx'(t)=f(t,x(t),x(t-r1),x(t-rn)), (1)Where f:R×Rn×…×Rn→Rn,ri(i=1…n)are constants.Definition 1.1 Equation (1) is called differential difference equation where ri is called deviation.Definition 1.2 If ri≥0(i=1,2,…n),then (1) is called retarded differential difference equation, ri is called retarded amount.Definition 1.3 If ri≤0(i=1,2,…n) then (1) is called advanced differential difference equation, ri is called advanced amountDefinition 1.4 Equationx'(t)=f(t,x(t),x(t-r1),…,x(t-rn),x'(t-τ1),…,x'(t-τn)) (2)is called neutral differential difference equation ,where ri andτi are con- stants besides ri≥0(i=1,2,…n),τi>0(i=1,2,…m).Definition 1.5 If someτi>0,while someτi<0, then (2) is called mixed differential difference equation.Definition 1.6 Equationis called linear differential integral equation and equationis called non-linear differential integral equation.RFDE can be divided into three classifications: 1) bounded retarded functional differential equation; 2) non-bounded retarded functional differential equation; 3) infinite retarded functional differential equation.1) Bounded retarded functional differential equation Definition 1.7 Let D(?)R×C,f:D→Rn.Then equationx'(t)=f(t,xt) (3)is called bounded retarded functional differential equation, where x'(t) means the right derivative of t.Definition 1.8 If there are t0∈R,A≥0,x∈C([t0-r,t0+A],Rn,(t,xt)∈D,such that x(t) satisfies (3) in [t0-r,t0+A).Then x(t) is called the solution of (3).2) Non -bounded retarded functional differential equation.Definition 1.9 Assume p(t,θ) is the real value continuous functionin [σ,∞)×[-r,0],whereσ,r>0 are constants.If p(t,θ) satisfies the following(1) For fixed t,p(t,θ) is monotonous increasing function ofθ;(2) p(t,0)=t,t∈[σ,∞);(3) p(t,0)-p(t-r)≥λ>0,t∈[σ,∞), whereλis some constant;(4) p(t,-r) is non -decreasing. then p(t,θ) is called p function.For t0∈[σ,∞),A≥0,x∈C([p(t0,-r),t0+A]Rn),define (?)t,as (?)t,(θ)= x(p(t,θ)),t∈[t0,t0+A],θ[-r,0].Definition 1.10 LetΩ(?)[σ,∞)×C,f:Ω→Rn.Then equationx'(t)=f(t,(?)t) (4)is called p- retarded functional differential equation, where x'(t) means the right derivative of t.Definition 1.11 If there are t-0∈[σ,∞),A>0, x∈C([p(t0 ,-r),t0+A], Rn).(t,(?)t)∈Ωsuch that x(t) satisfies(4) in [t0,t0+A),then x(t)is called the solution of (4).(3) Infinite retarded functional differential equation.Assume B is (-∞,0]→Rn a function space.For t0∈R,A>0,x∈B([-∞,t0+A],Rn),define xt as xt(θ)=x(t+θ) t∈[t0,t0+A],θ∈[-r,0].Definition 1.12 LetΩ(?)R×B,f:Ω→Rn.Then equationx'(t)=f(t,xt) (5)is called infinite retarded functional differential equation, where x'(t) means the right derivative of t.Consider neutral differential difference equationx'(t)=f(t,x(t),x(t-r1(t)),x(t-r2 (t)),x'(t),x'(t-r1(t)),x'(t-r2(t))),(6) where f:R×Rn×…×Rn→Rn,0≤ri(t)≤r,(i=1,2,r>0) and neutral differtialintegral equationx'(t)=∫g(θ,x(θ),x'(θ))dθ, (7)where g:R×Rn×Rn→Rn,r>0.According to classification of retarded functional differential equation, equation (6) and (7) can be transformed into functional differential equationx'(t)=f(t,xt,x't)>(8)where f:R×C×C→Rn,x't∈C are defined by x't(θ)=(?)x(t+θ)Definition 1.13 Equation (8) is called neutral functional differential equation.Secondly, it is mainly to state the application of functional differential equations because of its wide application in physics, automatic control theory, theory of the population, medical problems, biological problems and economic problems.Finally, it is to introduce the four stages of the development survey of FDE and at the same time it is to outline the development of China's FDE concisely. In chapter two,firstly consider following the equations x'(t)= f(t,x(t),x(t-τ1(t))), (9) x'(t)=g(t,x(t),x(t-1(t),x'(t-τ1(t))), (10) x'(t)=h(t,x(t),x(t-τ1(t)),x(t-τ2(t))), (11)where f:R×Rn×Rn→Rn,g,h:R×R×Rn×Rn→Rn,τ1,τ2:R→R are differentiable continuations. Theorem 2.1 Assume equation (9) can be transformed into dp(t,x(t))=q(t-τ(t),x(t-τ(t)))d(t-τ(t)), (12)where p:R×Rn→Rn,q:R×Rn→Rn,τ:R→R are differentiable continuations.If there is constantαand x=φ(t) satisfies equations andthen x=φ(t) is the solution of (12). Theorem 2.2 Assume equation (10) can be transformed into dp1(t, x(t))=q1(t-τ(t),x(t-τ(t)),x'(t-τ(t)))d(t-r(t), (13)where p1,q1:R×Rn→Rn,τ:R→R are differentiable continuations.If there is constantα1 and x=φ(t) satisfy equationsandthen x =φ(t) is the solution of (13). Theorem 2.3 Assume equation (11) can be transformed into d(p2(t, x(t))+p3(t,x(t))=q2(t-τ1(t),x(t-τ1(t)))d(t-τ1(t))+q3(t-τ2(t),x(t-τ2(t)))d(t-τ2(t)) (14)where p2,p3,q2,q3:R×Rn→Rn,τ1,τ2:Rare differentiable continuation. If there are constantsα2,α3 and x=φ(t) satisfy equationsandthen x(t) is the solution of (14).On the basis of the above discussion, find out equationy2(x)+y2(x)y'2(x)-y2(x+yy')=Fwhen F<0,the solution is y(x) =(?)Finally, use equation (15) to discuss geometric quality of conic Definition 2.1 Assume curveΓ:y=y(x) (for picture), the normal line of any point p(x,y) inΓ:y=y(x) across x-axis R(x,y), from R and vertically line X-axis across Q(x,y) inΓ:y=y(x),soSegment (?) is called the normal line segment of curveΓ,marked by H1;Segment (?) is called the vertical line segment of curveΓ,marked by H2;H12-H22 is called the geometric volume of curveΓ,marked by F. The Main conclusion are:Theorem 2.4 the geometric volume of curveΓ:y2=2px+c is F=-p2,and H12.Theorem 2.5 the geometric volume of curveΓ:(?) isF=(?),and H12.Theorem 2.6 the geometric volume of curveΓ:(?) is F=(?),and when a>b,H12;a=b H1=H2,a1>H2.
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