Nonlinear Singular And Pulse Equations Related Research

Nonlinear functional analysis is a important branch of modern analysis mathematics. Its richtheory and advanced method have provided with some effective theory tools for dealing withpractical problems emerging in the mathematical models corresponding to nonlinear differentialequations, partial differential equations, nonlinear integral equations and so on. In China, manywell known mathematicians including Gongqing Zhang, Dajun Guo, Wenyuan Chen, JingxianSun etc., have gained brilliant achievements in various fields of nonlinear functional analysis(see[28]-[41]).Much more attention has been devoted to singular nonlinear problems, which is a difficult andinteresting part of nonlinear functional analysis in the past few decades. In partial order Banachspace, nonlinear singular differential equation and impulsive differential equation are an importantresearch topic, which maybe obtain fruitful achievement. In recent years, they have been receivedhighly attention by mathematician and natural scientists in the world, since they arise in variousfields, such as atmospheric convection, biology, medicine, chemistry, economics, Newtonian ?uidmechanics, nuclear physics, theory of boundary layers and nonlinear optics etc.By making use of many kinds of advanced analysis tools which developed in nonlinear anal-ysis for past few years, such as topological degree, cone and partial order, fixed point index the-ory, fixed point theorem together with lower and upper solution method, maximum principle, andcomparison theorem, the present thesis mainly studies the existence and uniqueness of solution, it-erative scheme, error estimate and constructing iterative algorithms of converging the solution forseveral classes of nonlinear singular differential equation boundary value problem as well as non-linear impulsive integro-differential equations. We obtained many new significant results. Mostresults of the thesis are published in important journal of both domestic and overseas, for example《J. Math. Anal. Appl.》,《J. Comput. Appl. Math.》,《Appl. Math. Comput.》,《Appl.Math. Lett.》,《Nonlinear Funct. Anal. Appl.》,《Acta Math. Hungar.》,《Acta Math. Appl.Sin.》and so on.There are seven chapters in the thesis. The main results are:In Chapter 1, we introduce background and main works of the thesis. Some nonlinear func-tional analysis of preliminaries and several lemmas are also given. They play an important role innext chapters.In Chapter 2, we consider the existence of the following nonlinear singular systems of second and fourth-order ordinary differential equationswhere f∈C((0, 1)×R+×R+, R+), g∈C((0, 1)×R+, R+), R+ = [0, +∞), f, g may besingular at t = 0 and/or t = 1, moreover f may be singular at x = 0.Sufficient conditions of the existence of positive solutions to boundary value problems havebeen widely studied by many researchers. To seek necessary and sufficient condition for the exis-tence of positive solution of singular differential system is interesting and important, but difficult.However, we gain some new results.By using fixed point theorem and monotone iterative technique, we establish the existence ofpositive solutions for a class of nonlinear singular elastic beam differential system under reason-able conditions. Moreover, some uniqueness results of positive solution, estimate of error togetherwith the convergent rate of the iterative sequence of the solution are obtained. It is a new descrip-tion of the solution sequence. We also note that the iterative process is explicit, which can beimplemented by suitable numerical computation.In Chapter 3, we study nonlinear singular boundary value problems.In Section 3.1, we investigate the existence of positive solutions for the following singularsecond order Neumann boundary value problem (NBVP)where 0 < k <π2 is a constant, and the nonlinear terms f(t), g(t,x) may be singular at t = 0,t =1 and x = 0. f∈C((0,1), (0,+∞)), g∈C((0,1)×(0,+∞),(0,+∞)).Recently, many authors have been interested in studying the existence of positive solutionsfor singular boundary value problems, and many papers have focused on second order boundaryvalue problems which arise in a variety of different areas of applied mathematics and physics.However results on the existence of positive solutions for the singular Neumann boundary valueproblems taking on important application of the background in the fields of biology and medicinehave been few.By applying some properties of the Green's function as well as fixed point theory, we establishsome new results of the existence of positive solutions for the Neumann boundary value problems.The nonlinear term g(t,x) in our results only needs to satisfy a local monotone condition. In Section 3.2, we study the existence of positive solutions for the following third-order sin-gular boundary value problems(BVP)where a∈C((0, 1), (0, +∞)), F∈C((0, 1)×(0, +∞), (0, +∞)), and the nonlinear terma(t),F(t,x) may be singular at t = 0,t = 1 and x = 0.By employing certain inequality related partial order and constructing a special cone togetherwith fixed point theorem, we present some new sufficient conditions of the third-order boundaryvalue problems under nonlinear term satisfying a local monotone condition.In Section 3.3, the new existence theorems are that nonlinear term not only may be singularat t = 0, but also the explicit interval aboutμof the existence of positive solutions for Sturm-Liouville boundary value problems on the half-line is obtained. In Section 3.4 we give someexamples to show that our new results cover a wide range of functions.In Chapter 4, we present the following more general second-order nonlinear three pointboundary value problemswhereμ> 0 is a parameter,β> 0, 0 <η< 1, 0 <αη< 1, ? := (1 ?αη)+β(1 ?α) > 0, a∈C((0,1), (0,+∞)), a(t) may be singular at t = 0 and/or 1. f∈C([0,1]×(0,+∞),(0,+∞)),and f(t,x) may be singular at x = 0.Recently, under some stronger conditions imposed on nonlinear term, many authors havewidely studied for three point boundary value problems and obtained many nice results.Without any monotone or growth conditions imposed on the nonlinearity f(t,y), by con-structing an available integral operator and combing fixed point index theory with properties ofGreen's function under some conditions concerning the first eigenvalues corresponding to the rel-evant linear operator, we not only obtain the existence of positive solutions of the problem, butalso get the explicit interval about positive parameterμ. The interesting point of the results is thatthe term a(t) may be singular at t = 0 and/or t = 1. Moreover the nonlinearity f(t,y) is alsoallowed to have singularity at y = 0. The results obtained are novel.In Chapter 5, we study positive solutions of nonlinear singular differential equations on timesscales.In Section 5.1, we mainly introduce the developing history of dynamic equation on measurechain. We also present some basic concepts and preliminaries of measure chain analysis, which are necessary in the next sections.In Section 5.2, we are interested in the existence of positive solutions for the following dif-ferential equation on time scales of the form1u(t)(u(t)y?(t))? +λh(t)f(t,y(σ(t))) = 0, t∈[0,1],together with two point boundary conditionshereλ> 0 is a parameter, u(t) > 0 on (0,σ(1)) such that both the delta derivative of u(t) and theintegral 0σ(1)u?(ττ) exist and u,h∈C((0,σ(1)), (0, +∞)), f∈C([0,σ(1)]×[0, +∞), [0, +∞)),a,b,c,d≥0, such thatr := ub(c1) + u(σa(d1)) + ac 0σ(1)u?(ss) > 0.By applying fixed point index theorems for operators on a cone, existence criteria are devel-oped for positive solutions of Sturm-Liouville boundary value problem. Our results improve andextend many recent results.In Section 5.3, we consider the existence of positive solutions for the following second-orderdifferential equations nonlinear singular m?point boundary value problemswhere 0 <αi < T, i = 1, 2, 3,···,m ? 2, 0 <η1 <η2 <···<ηm?2 < T is a constant,αi < T, m≥3, f : (0,T)×(0,+∞) ?→[0,+∞) and g : (0,T) ?→[0,+∞) arecontinuous, and the nonlinear term f(t,x) may be singular at t = 0, t = T and x = 0.By making use of constructing exact lower and upper solution method as well as the maxi-mal principle, we obtain new sufficient conditions for the existence and uniqueness of Crd[0,T]positive solutions together with Crd[0,T] positive solutions for the nonlinear singular m?pointboundary value problems on time scales.In Chapter 6, we discuss the existence of positive periodic solutions for the following second-order differential equation where b(t) and g(t) are continuous w? periodic positive functions, f∈C(R×[0, +∞), [0, +∞)).We establish a new comparison theorem. By making use of constructing a special cone in Ba-nach space and applying Krein Rutmann theorem, the well-known fixed point index theory in thecone as well as employing a transforming technique, some new sufficient conditions of the exis-tence of positive periodic solutions for the second-order differential equations are presented undersome conditions concerning the first eigenvalues corresponding to the relevant linear operator.Some examples are presented to demonstrate the application of the results in this section.In Chapter 7, we study the existence of positive solution for the following second-order singu-lar nonlinear impulsive integro-differential equation of mixed type in a real Banach space (E,·)of the form: ?whereα,β,γ,δ≥0,ρ=βγ+αγ+αδ> 0, J = (0, 1), 0 < t1 < t2 <···< tm < 1,J =J \ {t1,t2,···,tm}, J = [0,1], J0 = (0,t1], J1 = (t1,t2], Jm = (tm,1], f∈C[J×P×P×P×P,P]. P is a positive cone in E, Ik∈C[P, P], Ik∈C[P,P],θis the zero element of E andin which K∈C[D,J], D = {(t,s)∈J×J : t≥s},H∈C[J×J,J], and K0 = max{K(t,s) :(t,s)∈D}, H0 = max{H(t,s) : (t,s)∈D}. ?y|t=tk and ?y |t=tk denote the jump of y(t) andwhere y(tk+ ), y (t+k ) and y(tk? ), y (tk? ) represent the right-hand limit and left-hand limit of y(t)and y (t) at t = tk respectively. h(t)∈C(J,R+) and may be singular at t = 0 and/or t = 1.By using fixed point index theory and fixed point theorem in a special cone for strict setcontraction operators, we show some new results of the existence of at least one positive solutionsfor the second-order singular nonlinear impulsive integro-differential equation boundary valueproblems. Our results improve and generalize recently related results.