Existence and approximation of solutions of nonlinear boundary value problems
In chapter two, we establish new results for periodic solutions of some second order non-linear boundary value problems. We develop the upper and lower solutions method to show existence of solutions in the closed set defined by the well ordered lower and upper solutions. We develop the method of quasilinearization to approximate our problem by a sequence of solutions of linear problems that converges to the solution of the original problem quadratically. Finally, to show the applicability of our technique, we apply the theoretical results to a medical problem namely, a biomathematical model of blood flow in an intracranial aneurysm. In chapter three we study some nonlinear boundary value problems with nonlinear nonlocal three-point boundary conditions. We develop the method of upper and lower solutions to establish existence results. We show that our results hold for a wide range of nonlinear problems. We develop the method of quasilinearization and show that there exist monotone sequences of solutions of linear problems that converges to the unique solution of the nonlinear problems. We show that the sequences converge quadratically to the solutions of the problem in the C1 norm. We generalize the technique by introducing an auxiliary function to allow weaker hypotheses on the nonlinearity involved in the differential equations. In chapter four, we extend the results of chapter three to nonlinear problems with linear four point boundary conditions. We generalize previously existence results studied with constant lower and upper solutions. We show by an example that our results are more general. We develop the method of quasilinearization and its generalization for the four point problems which to the best of our knowledge is the first time the method has been applied to such problems. In chapter five, we extend the results to second order problems with nonlinear integral boundary conditions in two separate cases. In the first case we study the upper and lower solutions method and the generalized method of quasilinearization for the Integral boundary value problem with the nonlinearity independent of the derivative. While in the second case we show the nonlinearity to depend also on the first derivative. Finally, in chapter six, we study multiplicity results for three point nonlinear boundary value problems. We use the method of upper and lower solutions and degree arguments to show the existence of at least two solutions for certain range of a parameter r and no solution for other range of the parameter. We show by an example that our results are more general than the results studied previously. We also study existence of at least three solutions in the pressure of two lower and two upper solutions for some three-point boundary value problems. In one problem, we employ a condition weaker than the well known Nagumo condition which allows the nonlinearity f(t, x, x’) to grow faster than quadratically with respect to x’ in some cases.