The Solutions Of Several Classes Of Fourth-Order Differential Equations And Their Applications

In this paper,by using cone theory;the monotone iterative method and the fixed point index theory, we study the existence of unique and multiple solutions for sev-eral classes of the fourth-order differential equation boundary value problems.The new results arc obtained and the previous results are improved.This paper includes four chapters.In the first chapter,we introduce the background of the discussed problems and state the main results of this thesis.In the second chapter,we consider the fourth-order differential equation nonlinear boundary value problem where f:[0,1]×R×R→R, g:R→R. By using a new fixed point theorem,the sufficient condition for the problem to have unique positive solution is obtained.In particular,we do not demand continuity and compactness conditions.The previous result is improved.An example illustrating the main results is given.In the third chapter,we study the fourth-order nonlinear differential equation two point boundary value problem where f∈C([0,1]×R×R,R). This problem describes the deformations of an elastic beam,whose both ends are simply supported.We show that the problem has the criteri-ons of the existence for multiple positive solutions which involve the fixed point index theory and generalization Leggett-Williams'three solution theorem.As application we give two examples.In the fourth chapter,the fourth-order impulsive differential equation two point boundary value problem is discussed,where J=[0,1],f∈C(J×R×R×R,R),IOK∈C(R,R).IJk∈C(R,R),I2k∈C(R×R,R).By transforming this problem into the second-order im-pulsive differential equation boundary value problem and using the monotone iterative method and the fixed point index theory, the sufficient conditions for the problem to have minimal solution,maximal solution and multiple positive solutions are obtained in the order interval. An example is presented to illustrate the application of the obtained results.