Study Of Solutions To Some Differential Equations (Systems) With Integral Initial Or Boundary Conditions

Boundary value problems of ordinary di?erential equations are impor-tant fields in studying ordinary di?erential equations, which arise from dif-ferent areas of applied sciences such as physics, chemistry, medical science,astronomy and biological engineering. In the recent years, many researchershave devoted themselves to the study of problems with integral boundaryconditions coming from heat conduction, underground water ?ow, thermo-elasticity and plasma physics, and have made great progress.In this thesis, we applied Leray-Schauder degree, fixed point theorem onthe corn and upper-lower solution method to prove the solvability, the exis-tence and multiplicity of positive solutions to some integral initial boundaryvalue problems of nonlinear ordinary di?erential equations(systems). Thisthesis consists of three chapters and the main contents are as follows:In Chapter one, we discuss the existence of positive solutions to integralinitial boundary value problems of ODE involving p-Laplace operators. The In order to transform the problems mentioned above into an equivalentintegral equations, we applied the alternative theorem and topological de-gree theory to obtain the existence of solutions to the problem. The mainresults are:Theorem 1 Suppose thatIn Chapter two, we study the existence and multiplicity of positivesolutions to two systems of second order ordinary di?erential equations withintegral boundary conditions. We mainly discuss the following problems First we transform each problem of system of di?erential equations intoproblems of an equivalent integral equations, and then find a proper cornby using resolvent of the operator related to the integral system. We appliyKrasnoselskii fixed point theorem on the corn to prove the existence of fixedpoints of the operator, which implies the solvability of the correspondingproblem. For the sake of simplicity, we need the following notations andassumptions. then Problem (3) have at least one positive solution (Problem (4) have atleast one positive solution).Theorem 5 Suppose that (A0) and (A1) hold. If f and g satisfy where l := {(x,y)∈P×P, (x,y) < l}, then Problems (3) have at leasttwo positive solutions (Problems (4) have at least two positive solutions).Theorem 6 Suppose that (A0) and (A1) hold. If f and g satisfytwo positive solutions (Problems (4) have at least two positive solutions).Chapter three is devoted to studying the following fourth order di?er-ential equations with separated boundary conditions Since the character of the two problems is that the nonlinearities ofboth problems depend on the lower order derivative of the unknowns, itis important for studying the solvability of solutions to give the properdefinitions of upper and lower solutions to each problem. First we give thedefinitions of solutions to Problems (5) and (6)Definition 1 Letα,β∈C4((0,1))∩C3([0,1]) satisfy Our main results are:Theorem 7 Suppose thatβ(t),α(t) are a pair of upper and lowersolutions to Problem (5). Let f∈C([0, 1]×R4, R) satisfy the one-sidedNagumo condition on the set E? = {(t,x0,x1,x2,x3)∈[0, 1]×R4 :α(t)x0β(t)}. If (t,x2,x3)∈[0,1]×R2, (α(t),α(t)) (x0,x1) (β(t),β(t)),and f satisfies where (x0,x1) (y0,y1) i.e. x0 y0 and x1 y1; hi : R ?→R (i = 1, 2) iscontinuous, and hi(u) 0 (i = 1, 2).(B3)φis a strictly increasing continuous function,φ(0) = 0,φ(R) = R.Then Problem (6) has at least one solution u(t), and for every t∈[0, 1], thefollowing inequalities hold...