Positive Solutions Of Fourth-order Problem With Dependence On The Derivative In Nonlinearity Under Non-local Boundary Value Conditions

In this thesis,we investigate the fourth-order problem with dependence on the second derivative in nonlinearity under non-local boundary value conditions,#12 where f:[0,1]× R+× R-?R+ is continuous and ?[u]is linear functional involving Stieltjes integral.This equation models the stationary state of the deflection of elastic beam and the second derivative stands for the bending moment stiffness.Some inequality conditions on nonlinearity f and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on special cone in C2[0,1].The conditions allow that f(t,x1,x2)has superlinear or sublinear growth in x1,x2.Under the hypotheses(C1-C3)suppose that:(F1)there exist constants a2,b2?0,r>0,such that f(t,x1,x2)?a2x1-b2x2,the spectral radius r(L2)<1,for all(t,x1,x2)?[0,1]×[0,r]×[-r,0],(F2)there exist positive constants a1,b1,C0,satisfying min(?),such that f(t,x1,x2)?a1x1-b1x2-C0,for all(t,x1,x2)?[0,1]x R+x R-,then BVP has at least one positive solution.Under the hypotheses(C,-C3)suppose that:(F3)there exist constants a1,b1,C0?0 such that f(t,x1,x2)?a1x1-b1x2+C0,for all(t,x1,x2)?[0,1]x R+×R-,the spectral radius r(L1)<1,(F4)there exist constants a2,b2?0,r>0 such that f(t,x1,x2)?a2x1-b2x2,the spectral radius r(L2)?1,for all(t,x1,x2)?[0,1]×[0,r]×[-r,0],then BVP has at least one positive solution.Some examples are given to illustrate the theorems respectively under boundary value conditions of integral type and multi-point type with sign-changing coefficients.