Solutions And Applications Of Several Nonlinear Differential Equations

In later years,all sorts of nonlinear problems have resulted from mathemat-ics,physics,chemistry,biology,medicine,economics,engineering,cybernetics and so on.During the development of solving such problems,nonlinear func-tional analysis has been bing one of the most important research fields in modern mathematics.It mainly includes partial ordering method,topological degree method and the variational method.Also it provides a much effect theoretical tool for solving many nonlinear problems in the fields of the science and tech-nology.And what is more,it is an important approach for studying nonlinear differential equations arising from many applied mathematics.L.E.J.Brouwer had established the conception of topological degree for finite dimensional space in 1912.J.Leray and J.Schauder had extend the conception to completely con-tinuous field of Banach space in 1934,afterward E.Rothe,M.A.Krasnosel'skii, P.H.Rabinowitz,H.Amann,K.Deimling had carried on embedded research on topological degree and cone theory.Many well known mathematicians in China, say Zhang Gongqing,Guo Dajun,Chen Wenyuan,Ding Guanggui,Sun Jingxian, Liu Lishan and Zhao Zengqin etc.,had proud works in various fields of nonlinear functional analysis.(See[1-12]).The present paper mainly investigates existence of solutions,multiplicity for several classes of integral equations and differential equations singular boundary value problem by using topological degree,cone theory and monotone iterative technique.It is made up of six chapters and the main contents are as follows:Chapterâ… gives serval lemmas on existence of fixed point which will be used in next chapters.Chapterâ…¡investigates consider the solutions for the following initial value problems(IVP)of nonlinear second-order integro-differential equations of mixed type in ordered Banach spaces E, (?) where J=[0,a](a>0),x₀,x₁∈E,f∈C[J×E×E×E,E],and (Tu)(t)=integral from n=0 to t(k(t,s)u(s)ds),(Su)(t)=integral from n=0 to a(h(t,s)u(s)ds). here k(t,s)∈C[D,R⁺],h(t,s)∈C[D₀,R⁺],and D={(t,s)∈R×R|0≤s≤t≤a),D₀={(t,s)∈R×R|(t,s)∈J×J},R⁺=[0,+∞).We investigate the existence of the minimal and maximal solutions and the unique solutions of the (IVP)in sectionâ… and also obtained its global solutions and unique solutions as well as its iterative approximation in sectionâ…¡.In Chapterâ…¢,we study the positive solutions for a class of differential equations involving p-Laplacian operator.In sectionâ… ,by means of calculation of the fixed point index in cone,we research the existence of positive solutions and many positive solutions for the following nonlinear singular boundary value problem (?) whereφ_p(s)is p-Laplacian operator,i.e.φ_p(s)=|s|^p-2s,p>1,φ_q=(φ_p)^-1,1/p+ 1/q=l,α>0,β≥0,γ>0,δ≥0,ζ,η∈(0,1)is prescribed andζ<η, a:(0,1)→[0,∞).In sectionâ…¡,by using the method of defining operator by the reverse func-tion of Green function,we research the existence of positive solutions and many positive solutions for the following nonlinear n-order m-point singular boundary value problem with p-Laplacian operator (?) whereφ_p(s)is p-Laplacian operator,i.e.φ_p(s)=｜s｜^p-2s,p>1,φ_q=(φ_p)^-1,1/p+ 1/q=1,a(t):(0,1)→[0,∞),0<η₁<η₂<…<η_m-2<1,α_i>0 with sum from i=1 to m-2(α_iη_i^n-2)<1.In sectionâ…¢,we study the existence of countable many positive solutions of nonlinear singular boundary value systems with p-Laplacian operator: (?) (?) whereφ_pi(s)is p-Laplacian operator,i.e.,φ_pi(s)=|s|^pi-2s,pi>1,φ_qi= (φ_pi)^-1,1/pi+1/qi=1,α_i>0,β_i≥0,γ_i>0,δ_i≥0,α_i:[0,1]→[0,∞), and have countable many singularities on(0,1/2),i=1,2.In Chapterâ…£,we study the positive solutions of semipositone boundary value problems.In sectionâ… ,we study the following third-order two-point semipositone bound-ary value problems(SBVP): (?) where f(t,u):(0,1)×[0,+∞)→(-∞,+∞).And obtained the existence of positive solutions for semipositone boundary value problem(SBVP)under the conditions that the nonlinear term f(t,u):[0,1]×(0,+∞)→(-∞,+∞)satis-fies Caratheodory Condition and we also allow that the nonlinear term f is both semipositone and lower unbounded,i.e.,we delete the restrict about continuous-ness and lower bounded of the nonlinear term.In sectionâ…¡,we study the following semipositone(k,n-k)conjugate eigen-value problems(SCEP): (?) where n≥2,10 is positive parameter.We delete the restriction on lower bounded and on upper control-function of the nonlinear term. Without making any monotone-type assumption,by using the fixed-point index theory,we derive an explicit interval ofλsuch that for anyλin this interval.the existence of at least one positive solution to the semipositone conjugate eigenvalue problem(SCEP)under the conditions that the nonlinear term f(t,u):[0,1]×(0,+∞)→(-∞,+∞)is continuous,i.e.,we allow that the nonlinear term f is both semipositone and lower unbounded is guaranteed,and the existence of at least two solutions forλin an appropriate interval is also discussed. By using the lower and upper solutions argument and fixed index theorem in the frame of the ODE technique,Chapterâ…¤establishes study the existence and nonexistence of positive solutions for the following fourth-order nonlinear singular Sturm-Liouville boundary value problem(BVP) (?) whereλ>0 is positive real parameters,α_i,β_i,δ_i,γ_i≥0(i=1,2)are constants, and p∈C¹((0,1),(0,+∞)).Moreover g,p may be singular at t=0 and/or 1.Chapter VI investigates study the following nonlinear n-order and m-order multi-point singular boundary value system (?) with the following boundary conditions (?) where 1/2≤ξ₁<ξ₂<…<ξ_p<1,1/2≤η₁<η₂<…<η_q<1,and n,m≥3,p,q,n,m∈N.