Positive Solutions Of Boundary Value Problem On Half Line

Analysis is the most important part of mathematics, and Differential Equations is the heart of Analysis.During the research on such problems. Topological play a role in it.For example,TopologicaI degree and the Variatimal Method that root in Topological become much effect theoretical tools. At the same time.the improvement of modern Differential Equations also improve the science fields such as Mathematics. Physics. Chemistry. Biology. Medicine, Economics. Engineering. Cybernetics and so on.The method of Topological degree is used in this issue to study boundary problems on half line.There are many authors interested in such subject.and get many wonderfull results([6]-[10]).It is realistic to study boundary problems on half line. This issue discuss such problems more generally on the basis of above references.Chapter 1 investigates the differential system on half line with parameterλ.Whereα,β,p≥0,q>0,α²+β²>0. In [11]-[15], the positive solutions on limited intervals were disussed.But there are few paper study differential system on half line. By using the fixed point index method, when the differential system satisfythe following conditions:(H₁) f_i:[0,+∞)×[0,+∞)×[0,+∞)→[0,+∞) continuous;for all R>0. wehave integral from n=0 to +∞,n_i(s)f_iR(s)ds<+∞.where f_iR(s)=(?){f(s,x₁,x₂)},n_i(s)=max{G_i(s, s).e^-r₁s},(?)∈[0, +∞),i=1,2;(H₂) (?),b,(?)∈[a,b](?)[0,+∞), satisfy f_i(s,x₁(s),x₂(s))≥g_i(s,x_i(s)),whereg_i:[0,+∞)×[0, +∞)→[0,+∞)increase on x, g(s,0)>0, (?)g_i(s,u)/u=+∞,i=1,2;then we can find 0<λ₁≤λ₂ such that:(i) There are at least 1 positive solutions while 0<λ≤λ₂;(ii) There are at least 2 positive solutions while 0<λ<λ₁;(iii) There are no positive solutions whileλ>λ₂.Chapter 2 study the follow boundary problem:andWhere integral from n=0 to +∞1/p(s) ds=+∞.We construct a special space and cone theorem is used toovercome the difficulty.Under such conditions separately:and We all get at least one positive solution.In Chapter 3 We also use fixed point method to study following problems. The first one is:where integral from n=0 to +∞1/p(s)ds<+∞. we get positive solutions under conditons (H₁)(H₂)(H₄) or (H₁)(H₃)(H₄).Where(H₁) p(t)>0,t∈(0, +∞), p(t)∈C(0,+∞),integral from n=0 to +∞1/p(s)dt<+∞,f∈C[(0, +∞)×[0,+∞)→[0.+∞)];(H₂) there exists continuous function g:(0, +∞)→(0, +∞).satisfy g(r)f(t,u)r,(?)g(r)/r=+∞,while t∈(0,+∞);(H₃) there exists continuous function l:(0, +∞)→(0, +∞),satisfy l(r)f(t, u)>f(t, ru), l(r)4) 01) f∈C(R⁺×R⁺×R⁺, (0, +∞)),f(t,(1+u(t))x,y)≤k(t)[h(x)+w(x)][g(y)+ r(y)],h increase in C(R⁺, (0,+∞)),w/h decrease in C(R^- ,R^-),g increase inC(R⁺, (0, +∞)),r/g decrease in C(R⁺,R⁺), and (?)k₀>0, s.t integral from n=0 to +∞p(s)k(s)g(k₀ 1+u(s)/u(s))ds<+∞;...