The Existence And Multiplicity Of Solutions For Several Boundary Value Problems

Nonlinear functional analysis is a research field of mathematics with pro-found theories and extensive applications. Its general theories and methods can deal with nonlinear problems which appeared in computational mathe-matics, cybernetics, optimized theory, dynamic system, etc. Most of all, its fixed point theorems are widely used to investigate solvability of many classes of nonlinear differential equations, nonlinear integral equations and some other types of equations. The boundary value problems of nonlinear differential e-quations are important subjects both in the theory of differential equations and in the applications of nonlinear functional analysis since they have extensive applications in both theory and reality.In the past few decades, integer order differential equations, difference equations, dynamic equations on time scales etc. many kinds of differential equations subject to a series of boundary value conditions, such as two points, multiple points, integral boundary value, nonlinear boundary value, are inves-tigated widely by many researchers. Most of all, owing to the impotence in the mathematic theory and specific physical explanation, the boundary value problems for integer order differential equations have been researched exten-sively and abundant research results have been obtained. Now, researchers focus on fractional differential equations since they have been widely used cybernetics, in diffusion and transport theory, viscoelastic mechanics, signal processing, non-newtonian fluid mechanics etc. The boundary value problems of fractional order equations, especially their numerical analysis and numerical solutions, have received highly attention of the domestic and foreign mathe-matics and natural science fields and have become one of the hottest issues in the international research fields.In the paper, some theories and methods of nonlinear functional analysis have been employed, such as cone theory, fixed point theory, Krasnosel’skii-Zabreiko fixed point theorem, to consider firstly the existence and multiplicity of positive solutions for four kinds of boundary value problems. In the final chapter, the existence and uniqueness of extremal solutions and numerical it-eration technique for nonlinear fractional differential equation have been inves-tigated via monotone iterative method and upper and lower solutions method. The paper comprises six chapters.In Chapter I, the background of nonlinear functional analysis, research situation of boundary value problems of nonlinear differential equations and some basic concepts and theorems have been introduced.In Chapter II, the existence and multiplicity results for positive solutions are derived to a kind of integer high-order impulsive differential equation with integral boundary value conditions where f01u(t)da(t) and fo1u(t)dβ(t) are Riemann-Stieltjes integrals. We first transform the problem into an equivalent integral equation so that we can con-struct a special cone. By constructing an integral operator for an appropriate linear problem without impulsive effects, its first eigenvalue and eigenfunc-tion are obtained, which are used to obtain the optimal growth conditions of nonlinear term f. Finally, employing the Krasnosel’skii-Zabreiko fixed point theorem, we establish the existence results for positive solutions of the bound-ary value problem.In Chapter Ⅲ, the existence and multiplicity of positive solutions for the following second-order p-Laplacian impulsive boundary value problem has been studied By constructing an integral operator for an appropriate linear Robin boundary value problem without impulsive effects, we obtain its first eigenvalue and eigenfunction. By the Krasnosel’skii-Zabreiko fixed point theorem and Jensen’s inequality, the existence and multiplicity results of positive solutions for the problem are established under the optimal conditions that derived from the first eigenvalue.In Chapter IV, the following discrete boundary value problems system where △u(k)= u(k+1) - u(k),△2u(k)= △(△u(k)) is considered under conditions of stronger coupling behaviors of f and g characterized by convex and concave functions. By the Krasnosel’skii-Zabreiko fixed point theorem, Jensen’s inequality and the first eigenvalue of a relative linear operator, the existence results of at least one or two positive solutions are obtained.In Chapter V, the existence and multiplicity results for positive solutions have been studied to the following fourth-order p-Laplacian dynamic equa-tions on time scale Under growth conditions on the nonlinearity f, some existence results of at least two and three positive solutions for the above problem are obtained by virtue of Guo-Krasnosel’skii fixed theorem and Leggett-Williams fixed point theorems on cone. In particular, our nonlinearity f may be both sublinear and superlinear.In Chapter VI, an extremal solutions and numerical iteration algorithm for the following nonlinear fractional boundary value problems with p-Laplacian are investigated where 0<α,β≤1,1<α+β≤2, g ∈C(R ×R,R). Well-defined monotone iterative sequences of upper and lower solutions are constructed and are proved that they converge uniformly to the actual solution of the problem under with-out assuming the usual monotonicity type conditions of the f. The existence and uniqueness of the extremal solution are discussed. A numerical iteration procedure is introduced and an accurate approximate solution is obtained for an example of fractional boundary value problem. Some relative errors and graphs are given for illustration.Throughout the paper, on the analytical hand, the main method to in-vestigate existence of solutions is as followings. ones first obtains the first eigenvalues and eigenfunctions of appropriate linear problems, which are used to obtain the optimal conditions on nonlinear terms by Jensen’s inequalities and p-Laplacian operator’s property. Under the conditions, employing the Krasnosel’skii-Zabreiko fixed point theorem establishes the existence and mul-tiplicity of positive solutions for the boundary value problems. On the nu-merical hand, the results of existence and uniqueness of extremal solution are proved without assuming the usual monotonicity type conditions, which is the novelty of this part. The theoretical analysis, numerical iterative algorithm as well as an example are presented to validate the proposed method that solve the nonlinear problem by solving iteratively a sequence of linear problem. The errors imply that the approximation of upper and lower solutions is quick and efficient. Thus, the results in the paper are new, enrich the applications of lower and upper solutions and the literature of analytical and numerical study of the fractional differential equations. Having studied thoroughly, some new interesting results under weaker conditions have been obtained and most of them have been published.