Existence And Uniqueness Of Positive Solutions Of Fourth Order Eigenvalue Problems Of Differential Equations

Existence and Uniqueness of Positive Solutions of fourth order Eigenvalue Problems of DifferentialEquationsSingular nonlinear boundary value problem (SBVP.for short) have resulted from nuclear physice,gas dynamics. Newtonian fluid machanics, the theorey of boundary layer, nonlinear optices and so on. Prom 1980s,such problems have received a great deal of attention by many research . Therefore they become a new study field, and there are many excellent results.In recent years,the fourth-order boundary value problems can describe the deformation of an elastic beam equilibrium state and have comprehensive application in elasticity mechanics and engineering phusics.Because either he function or the variable itself.which is mathematical model resulted from some important actual problems,may be singular at endpoints,the study of high order SBVP becomes very active.However,few papers have been reported on the same problems for multi-point SBVP.So the content that we studied has important theoretical and applicable value.On the basis of above discussions, the paper by using the fix point index number theory, the mix monotonous operator's fix point theorem and the Leggett-Williams fix point theorem,we study the warp of fourth order Boundary Value Problems of Differential Equations,and obtains some useful results. The main results are the following:In the second chapter,we investigates fourth order boundary value problem problems of two parameterWe make the following assumptions:(H₁):f : [0,1]×[0,∞)→[0,∞) is continuous:(H₂):α,β∈R such thatβ≤(?)(?),α≥-(?),(?),(?)+(?)≤1;(H₃): f(x,y) is nondecreasing in y for all x∈E [0,1];(H₄):f(x,0) > c> 0 for all x∈[0,1],where c is a number;(H₅):f_∞= lim_{y→∞}(?) =∞for all x∈[0,1].by using Lebesgue dominated convergence theorem and the property of fixed point index of complete continuous operators,we obain the following results:Theorem 1 If hypotheses (H₁)-(H₅) ,there exists someλ^*> 0,such that (i) ifλ>λ^*,the problems(2.1) has no positive solutions.(ii) ifλ=λ^*,the problems(2.1) has at least one positive solutions;(iii) if 0 <λ<λ^*,the problems(2.1)has at least two positives solutions:; in the third chapter,we consider the fourth order eigenvalue problem of two parameterWe make the following hypotheses:(H₁) : f : [0,1]×[0,∞)→[0,∞)are continuous;(H₂) :there existα,β∈R, such thatβ≤(?),α≥-(?),(?)+(?)≤1;(H₃) : f(x,y)=q(x)[g(y)+h{y)],g:[0,+∞)→[0.+∞) is continuous and nondecreasing ; h :[0, +∞)→[0,+∞) is continuous and non-increasing ; q∈((0,1), (0,∞)). Using the fixed point theory of mix monotone operators.we obtain the following results:Theorem 2 If hypotheses (H₁), (H₂),(H₃^'),we suppose that there existsα∈(0, l),such thatandFor allx∈(0,1),y>0,q∈((0,1), (0,∞)),ifis to be satisfied, the problem (3.1) has only one positive solutions y^*(x).In the forth Chapter,we consider the fourth order boundary value problem problems of two parameterWe make the following hypotheses:(H₁) :f : [0,1]×[0,∞)→[0,∞) is continuous;(H₂) :there existα,β∈R,such thatβ≤(?),α≥-(?),(?)+(?)≤1;(H₃^"):f(x,y) < N₁k,(x,y)∈(0,1)×[-k,k],and kis a nonnegative constant:(H₄^"):f(x,y)≥N₂h,(x,y)∈(0,1)×[h,(?)],0<δ<1.andh is a nonnegative constant;(H₅^"):f(x,y)≤N₁c,(x,y)∈(0,1)×[-c,c],and c is a nonnegative constant.where Using the Leggett-Williams' fixed point theory,we obtain the following results:Theorem 3 If (H₁), (H₂), (H₃^" ), (H₄^"), (H₅^") hold,we suppose there exist some nonnegative constant k,h,and c,such that0 < k < h <δc, we define a nonnegative continuous continuous functionφ: P→[0,∞),such thatThen, the problems(2.1) has at least three positives solutions: y₁.y₂å’Œy₃.such that...