On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order

In this paper, we study the existence of solutions of the operator equations p+λGfx=x in the Banach space C[I,E]. It is assumed the vector-valued function f is nonlinear Pettis-integrable. Some additional assumptions imposed on f are expressed in terms of a weak measure of noncompactness. To encompass the full scope of the paper, we investigate the existence of pseudo-solutions for the nonlinear boundary value problem of fractional type

under the Pettis integrability assumption imposed on f.     

The forward and inverse problems for a fractional boundary value problem

In this paper, the authors study the forward and inverse problems for a fractional boundary value problem with Dirichlet boundary conditions. The existence and uniqueness of solutions for the forward problem is first proved. Then an inverse source problem is considered.     

Second order three-point boundary value problems in abstract spaces

In this paper we investigate the existence of solutions of the nonhomogeneous three-point boundaryvalue problem We search for solutions of the above problem in the Banach space of continuous functions C（[O, 1], E） with the Pettis integrability assumptions imposed on $. Some classes of Pettis-integrable functions are described in the paper and exploited in the proofs of main results. We stress on a class of pseudo-solutions of considered problem. Our results extend previous results of the same type for both Bochner and Pettis integrability settings. Similar results are also proved for differential inclusions i.e. when f is a multivalued function.     

Positive solutions of a second-order integral boundary value problem

In this paper we study the existence of positive solutions of a second-order integral boundary value problems for ordinary differential equations. Our results presented here unify, generalize and substantially improve the existing results in the literature. Moreover, it is worthwhile to point out that our method will dispense with constructing a new Green function.     

Fractional boundary value problems with integral boundary conditions

In this article, we study a type of nonlinear fractional boundary value problem with integral boundary conditions. By constructing an associated Green's function, applying spectral theory and using fixed point theory on cones, we obtain criteria for the existence, multiplicity and nonexistence of positive solutions.     

一类四阶边值问题的正解

§ 1　IntroductionThe deformations of an elastic beam are described by a fourth-order two-pointbound-ary value problem[1 ] .The boundary conditions are given according to the controls at theends of the beam. For example,the nonlinear fourth order problemu(4) (x) =λa(x) f(u(x) ) ,u(0 ) =u′(0 ) =u′(1 ) =u (1 ) =0 (1 .1 ) λdescribes the deformations of an elastic beam whose one end fixed and the other slidingclamped.The existence of solutions of (1 .1 ) λhas been studied by Gupta[1 ] . But …     

On a fractional boundary value problem with a perturbation term

In this paper, the authors study a nonlinear fractional boundary value problem of order $\al$ with $2<\al<3$. The associated Green''s function is derived as a series of functions. Criteria for the existence and uniqueness of positive solutions are then established based on it.     

Existence and uniqueness of positive solution for nonlinear fractional q-difference equation with integral boundary conditions

This paper studies a class of nonlinear fractional $q$-difference equations with integral boundary conditions. By exploiting the properties of Green"s function and two fixed point theorems for a sum operator, the existence and uniqueness of positive solutions for the boundary value problem are established. Iterative schemes for approximating the solutions are also obtained. Explicit examples are given to illustrate main results.     

具超前变元的二阶微分方程三点边值问题的正解

§ 1　 IntroductionRecently,certain three-point boundary value problems for nonlinear ordinarydifferential equations have been studied by many authors[1— 6] .However,few papers havebeen published on the same problems for nonlinear functional differential equations.In thispaper,we are concerned with the following second order differential equation with anadvanced argumentu″(t) +λa(t) f(u(h(t) ) ) =0 ,t∈ (0 ,1 ) (1 .1 )with the three-point boundary conditionsu(0 ) =0 ,αu(η) =u(1 ) ,(1 .2 )…     

具有Caratheodory函数的四阶边值问题

利用上下解方法和Ｓｃｈａｕｄｅｒ不动点定理,讨论了一类具有Ｃａｒａｔｈｅｏｄｏｒｙ函数的四阶边值问题,给出了解存在的充要条件。     

奇异非线性二阶微分方程Neumann边值问题

研究了一个奇异非线性二阶微分方程的Neumann边值问题,通过摄动技巧和比较原理得到了所论问题解的存在唯一性。     

Positive solutions of singular Caputo fractional differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions of singular super-linear (or sub-linear) integral boundary value problems for fractional differential equation involving Caputo fractional derivative. Necessary and sufficient conditions for the existence of C³[0, 1] positive solutions are given by means of the fixed point theorems on cones. Our nonlinearity f(t, x) may be singular at t = 0 and/or t = 1.     

一类四阶差分方程多点边值问题的正解

为解决多点支撑弹性梁的正解的存在性问题,运用锥上不动点指数理论,研究一类含参四阶差分方程多点边值问题.获得了当参数在一定范围内取值时正解的存在性结果,得到了正解存在的充分条件.     

Periodic boundary value problem for the first order functional differential equations with impulses

This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is presented and the previous results are extended.     

具有脉冲的二阶三点边值问题存在性定理

In this paper, two existence theorems are given concerning the following 3-point boundary value problem of second order differential systems with impulses     

Upper and lower estimates for positive solutions of the higher order Lidstone boundary value problem

We consider the higher order Lidstone boundary value problem. New upper and lower estimates for positive solutions of the problem are obtained. A discussion of the sharpness of the estimates is included.