Existence of Solutions for Singular Second-Order Ordinary Differential Equations with Periodic and Deviated Nonlocal Multipoint Boundary Conditions

This paper studies the existence of continuous solutions for a class of nonlinear singular second-order ordinary differential equations subject to one of the following boundary conditions: periodic-deviated multipoint boundary conditions, periodicintegral boundary conditions, and periodic-nonlocal integral conditions in the Riemann-Stieltjes sense. An existence result based on the Schauder fixed point theorem and Leray-Schauder continuation principle is used to obtain at least one continuous solution for the singular second-order ordinary differential problems. Two examples are given to show the application of our results.


Journal of Function Spaces
The considered problem is to determine sufficient conditions on  guaranteeing that the boundary value problem (1), (2) or ( 1), ( 3) or ( 1), (4) has a solution.
Note.One must notice that the following periodic and multipoint boundary conditions are special cases of our periodic and multipoint boundary conditions: (  ) = 0,   ∈ (0, 1) .
We point out that our results can not be extended to the cases in [1,2].On the one hand the cases  1 and  2 = 1 which are used in them are not included in our hypothesis and on the other hand periodic and deviated multipoint boundary conditions are different from boundary conditions of them.Hence we consider a new class of assumptions and periodic and deviated multipoint boundary conditions.
Let E be the Banach space with the norm Let  be the Banach space ⋅   () and lim
The following lemmas play a pivotal role to define the solutions for the given problems.Lemma 4. The singular second-order ordinary differential equation with periodic and deviated multipoint boundary conditions ( 1)-( 2) is equivalent to the functional integral equation: Proof.The solution of ( 1) is given by ( − )  (,  () ,   ()) .
Define a nonlinear operator  by From ( 1 ), we have and we conclude that  :  → E is well defined.
Proof.From Lemma 4, we have the fact that () is a solution of problem (1)-( 2) if and only if () satisfies the integral equation (12).We apply Theorem 3 to obtain the existence of at least one solution of (1)-( 2) in .
To do this, by Lemma 6, it suffices to verify that the set of all possible solutions of the family of equations   =  (,  () ,   ()) , a.e. ∈ (0, 1) , is a priori bounded in  by a constant independent of  ∈ (0, 1).

Problems with Periodic and Integral Boundary Conditions
Following exactly the same arguments used in Sections 2 and 3, we can prove analogous results for the problems (1), ( 3) and ( 1), ( 4), so we omit the proof of main results of this section.Firstly, we deal with the problem (1), (3).By analogous method as in Lemma 4, we have the following result.Proof.The proof is similar to that of Lemmas 5 and 6.
Using the operator (58) and the method of proof for the results obtained in Section 3, we can establish the following result for the problem (1), (3).Theorem 13.Let  : [0, 1] ×  2 →  be a function satisfying the Carathéodory conditions.Assume that the assumption ( 1 ) holds.Then there exists at least one solution  ∈  of the nonlinear singular ordinary differential equation with periodic and integral conditions (1), (3)

represented by (57).
The existence of solutions for the problem (1), (4) can be proved by a similar way.