Lyapunov-type Inequalities And Boundary Value Problems Of Second Order Ordinary Differential Equations Across Resonance

The existence and uniqueness of solutions of a specific differential equation is one of the fundamental questions in the field of differential equations.In 18 th century,many brilliant mathematicians(such as I.Newton,G.W.Leibniz,James and John Bernoulli,A.C.Clairaut,L.Euler,J.L.Lagrange,J.F.Riccati,J.d'Alembert,etc)attempted to derive the general solutions of differential equations.With further research,mathematicians realized that it is impossible to obtain the general solutions of all differential equations.The experts then switched their attention to solve the equations with the additional provision of certain boundary conditions.The theory of boundary value problems is a very large and important area of differential equations.Many applications were made to physics,celestial mechanics,chemistry,biology,engineering science,economics and other parts of mathematics.In 1900,D.Hilbert posed 23 problems at the Second International Congress of Mathematicians in Paris as a challenge for the 20 th century.The 20 th problem was called The general problem of boundary value.Second order differential equations and associated boundary value problems have been extensively studied in the literature since Picard's time,not only because of their intrinsic mathematical importance,but also because of the huge variety of phenomena in nature they may be used to model.We shall be mainly interested in second order differential equationy??+ u(x)y = h(x)(9)with Dirichlet conditionsy(0)= 0 = y(1)(10)or Neumann conditionsy?(0)= 0,y?(1)= 0.(11)It is well-known that the nonhomogeneous boundary value problems(9)-(10)(or(9)-(11))have a unique solution if and only if their corresponding homogeneous equationy??+ f(x,y)= 0(12)has only trivial solution satisfying the related boundary conditions.Hence,we can establish unique solvability criteria with respect to BVP(9)-(10)(or(9)-(11))by constructing Lyapunov-type inequalities.In 1907,Lyapunov [42] obtained the following remarkable inequality if Hill's equationy??+ u(x)y = 0(14)has a nontrivial solution y(x)with Dirichlet boundary conditionsy(a)= 0 = y(b),(15)where a,b ? R with a < b,u is a real-valued and continuous function.Inequality(13)has been shown to be sharp in the sense that the constant 4 cannot be replaced by a larger number.There are plenty mathematicians,including A.Wintner,P.Hartman,A.Wintner,P.Hartman,A.Beurling,R.Brown,D.Hinton,G.Borg,R.Dahiya,B.Singh,S.B.Eliason,R.Ferreira,A.Ca (?)ada,J.A.Montero,S.Villegas and etc.,who work and make important contributions to extend the classical Lyapunov inequality.And Lyapunov's inequality and its various generalizations have proved to be useful tools in the study of oscillation theory,stability and numerous other applications for the theories of differential and difference equations(cf.[8–17,20,21,23,24,27–30,33,51,58,59]).The classical Lyapunov inequality(13)has a very natural geometric interpretation: Suppose that a < b are two real consecutive zeros of(14)and BVP(14)-(15)has a nontrival solution y(x)on [a,b].For the sake of simplicity,we will restrict ourselves to the case when a = 0 and b = 1.Then the area of the region between u(x)and x-axis over [0,1] must exceed 4.Albeit theoretically powerful,published results for Lyapunov-type inequalities are in some sense limited.In fact,the restrictions on u(x)seem too loose especially when u(x)is centered away from x-axis.It is thus natural to ask whether(13)admits an extension in these cases.Although there is an extensive literature on unique solvability of boundary value problems for second order ordinary differential equations,to the best of our knowledge,the conditions of the theorems established in the papers are mostly carried out for non-resonance case(cf.[9,19,38,39,47,48]).The works on BVPs relative to second-order ODEs across-resonance,especially across multi-resonance,are limited.Our aim is to establish unique solvability criteria for BVP(9)-(10)(or(9)-(11)),where A ? u(x)? B and there are many resonance points lying inside the interval [A,B].The thesis is structured as follows.Chapter 1 begins with a brief historical overview of ordinary differential equations and associated boundary value problems.A review of existing literature on Lyapunov-type inequalities and boundary value problems of second order ordinary differential equations across resonance was performed.At the end of this chapter,we summarize our main results.In Chapter2,we provide some basic concepts and theorems which are required in the proof.We establish some new Lyapunov-type inequalities for second order differential equations across multi-resonance with Dirichlet conditions in Chapter 3 by applying Pontryagin's maximum principle and fixed point theorems.Besides,we provide three new optimal criteria which guarantee the unique solvability of BVP(9)-(10).Some relevant examples are given to illustrate our results and most of which are so far not covered by the existing results.Variants of Lyapunov-type inequalities for nonlinear BVPs are also discussed.Some new Lyapunov-type inequalities for second order differential equations across multi-resonance with Neumann conditions are presented in Chapter 4.As in Chapter 3,we provide some new optimal criteria which guarantee the unique solvability of BVP(9)-(11).Some relevant examples are given to illustrate our results.Variants of Lyapunov-type inequalities for nonlinear BVPs are discussed.Chapter 5 provides a succinct summary of our work.