The existence and uniqueness of solutions to the boundary value problem of a class of nonlinear 3rd-order differential equations were studied. Firstly, the results of the research on the boundary values of 3rd-order differential equations at home and abroad in recent years were combed. Then the boundary value problem of nonlinear 3rd-order differential equations with nonlinear boundary value conditions was put forth, and the solution to the related linear problem was explored. Finally, the Banach fixed point theorem was used to prove that the proposed boundary value problem has a unique solution. An example illustrates the applicability of the main results.

In this paper, we introduce new sequential fractional differential equations with mixed-type boundary conditions CDq+kCDq−1ut=ft,ut,CDq−1ut,t∈0,1,α1u0+β1u1+γ1Iruη=ε1,η∈0,1,α2u′0+β2u′1+γ2Iru′η=ε2, where q∈1,2 is a real number, k,r>0,αi,βi,γi,εi∈ℝ,i=1,2,CDq is the Caputo fractional derivative, and the boundary conditions include antiperiodic and Riemann-Liouville fractional integral boundary value cases. Our approach to treat the above problem is based upon standard tools of fixed point theory and some new inequalities of norm form. Some existence results are obtained and well illustrated through the aid of examples.

对于数学教育特别是小学数学教育,教育者们越来越认识到数学思想方法的渗透在数学教学中的重要性.《新课程标准》也加强了渗透数学思想方法的要求.数学微格教学是高师数学教育专业师范生的必修课,所以对即将走进小学数学课堂的师范生来说,应该加强数学思想方法渗透技能的培养和训练.

The properties of positive solutions were investigated for a class of SturmLiouville boundary value problems with p-Laplace operators. Based on the properties of p-Laplace operators, and according to the L’H?pital’s rule and the extreme value theorem for continuous functions on closed intervals, the SturmLiouville boundary value problems with p-Laplace operators were studied. The 2 necessary conditions for the existence of positive solutions were obtained. In the last part, the application of the main findings was given. The work enriches the content in the field of boundary value problems, and provides a new channel of using computer and iterative techniques to find approximate solutions to boundary value problems, meanwhile extending some conclusions in previous literatures.

新形势下的高等职业教育辅导员工作任重道远，引导大学生认识和评价自我，帮助他们树立正确的人生观和价值观，指导他们适应新的学习和生活方式，树立起良好的学风，形成优良的班风，是本文探讨的主要目的。

This paper expounds the establishment of conditions for lim(f(x))x→+∞=0,generalizes it and expands its application.

By applying the fixed-point theorem of strict-set-contraction,this paper establishes the existence of one solution or one positive solution to the generalized Sturm-Liouville m-point boundary value problem in Banach spaces.

By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of Sturm–Liouville boundary value problems for a nonlinear singular differential system. Some well-known results in the literature are generalized and improved. An example is presented to illustrate the application of our main result.

This paper studies a class of nonlinear four-point boundary value problems with p-Laplacian operator.Several sufficient conditions for the existence of positive solutions is obtained by constructing a completely continuous operator and combing fixed-point theorem of cone expansion and compression of norm type.

This paper studies a class of singular Sturm-Liouville boundary value problems of fourthorder differential equations.A necessary and sufficient condition for the existence of C~3[0,1]positive solutions as well as C~2[0,1]positive solutions of the nonlinear differential equations is established by means of the fixed point theorems on cones.

By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of Sturm–Liouville boundary value problems for a nonlinear singular second order ordinary differential system with a parameter. Some well-known results in the literature are generalized and improved. An example is presented to illustrate the application of our main result.

This paper discusses the existence of positive solutions for fourth-order m-point boundary value problems with a one-dimensional p-Laplacian operator{(ϕp(u″(t)))″−g(t)f(u(t))=0,t∈(0,1),u″(0)=u″(1)=0,au(0)−bu′(0)=∑i=1m−2aiu(ξi),cu(1)+du′(1)=∑i=1m−2biu(ξi), where ϕp(s)=|s|p−2s,p>1,f is a lower semi-continuous function. By using Krasnoselskii’s fixed point theorems in a cone, the existence of one positive solution and multiple positive solutions for nonlinear singular boundary value problems is obtained.

This paper deals with the existence of symmetric positive solutions for a class of singular Sturm–Liouville-like boundary value problems with a one-dimensional p-Laplacian operator. By using the fixed theorem of cone expansion and compression of norm type in a cone, the existence of positive solutions is established though nonlinear term contains the first derivative of unknown function.

By applying the fixed-point theorem of cone expansion and compression of norm type the existence of positive solutions is investigated for a class of singular nonlinear four point boundary value problems with one-dimensional p-Laplacian operator though nonlinear term contains first derivative of unknown function.

By applying Schauder fixed point theorem,we study the solvability of one-dimensional p-Laplacian operator equation for the nonlinear two-point boundary value problems with one-dimensional p-Laplacian operator and get some existence theorems.The main results show that the class of equations has at least one solution or positive solution if the height of nonlinear term is appropriate on a bounded subset of its definitional domain.

In this paper, we study the existence of positive solutions of nth order m-point boundary value problemu(n)(t)+a(t)f(u(t))=0,t∈(0,1),u(0)=u′(0)=⋯=u(n-2)(0)=0,u(1)=∑i=1m-2αiu(ηi),where n⩾2, m>2, 0<η1<η2<⋯<ηm-2<1,αi>0(i=1,2,…,m-2) with ∑i=1m-2αiηin-1<1. We obtained the existence of positive solutions by using the fixed point theorems of cone expansion and compression of norm type. The results obtained in this paper essentially improve and generalize some well-known results.