The Study Of Existence Of Positive Solutions For The Fourth-order Four Point Boundary Value Problems With P-laplacian Operator

Our work in this paper is mainly on the positive solutions of fourth-order four point boundaryvalue problems with p-laplacian operator. On the hypothesis of different conditions for thenonlinear function, by employing different methods, we respectively obtain the necessary andsufficient condition for the existence and uniqueness of pseudo-C³[0,1] positive solution, theexistence and multiplicity results of C²[0,1] positive solutions.In chapter 1, we assume f∈C((0,1)×[0,+∞)×(-∞,0],[0,+∞)); for any fixed t∈(0,1),v∈(-∞,0],f(t,u,v) is non-decreasing in u; for any fixed t∈(0,1),u∈[0,+∞),f(t,u,v) is non-increasing in v; there exist 0＜α₁,α₂＜1 withα₁+α₂＜1, such thatf(t,cu,v)≥c^α₁f,(t,u,v),f(t,u,cv)≥c^α₂f(t,u,v),c∈(0,1]A necessary and sufficient condition for the existence of pseudo-C³[0,1] positive solution is given by means of the monotoneiterative technique. In chapter 2, let f∈C((0,1)×(0,+∞)×(-∞,0),[0,+∞)); for any fixedt∈(0,1),v∈(-∞,0),f(t,u,v) is non-increasing in u; for any fixed t∈(0,1),u∈(0,+∞),f(t,u,v),is non-decreasing in v.By using the upper and lower solutions method, the existence ofpositive solutions theorem for the boundary value problem is established. Later, we go aheadwith making further investigation of the fourth-order singular BVP, here, we construct specialcone, and make use of the cone expansion and compression theory to abtain at least two positivesolutions for the fourth-order singular boundary value problem.Worth mentioning, the existence results we gain in our paper is under p＞1 and p≥2. Whenp=2,the problem can be precisely turned into the more simple linear problem. Consequently,we can see our research with more generality.