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

El-Sayed, Ahmed M. A.; Gaafar, Fatma M.

doi:https://doi.org/10.1155/2018/9726475

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Volume 2018 | Article ID 9726475 | https://doi.org/10.1155/2018/9726475

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

Ahmed M. A. El-Sayed¹and Fatma M. Gaafar²

Academic Editor: Adrian Petrusel

Received12 Jun 2018

Revised24 Aug 2018

Accepted03 Sept 2018

Published08 Oct 2018

Abstract

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, periodic-integral 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.

1. Introduction

In this paper, we study the existence of at least one continuous solution for the second-order differential equationwhere is a given function satisfying Carathéodory’s conditions (i.e., for a.e. the function is continuous on and for each , the function is measurable on ) and is allowed to be singular at and .

We consider the following nonlocal and multipoint boundary conditions

periodic-deviated multipoint boundary conditions of the form:periodic and integral boundary conditions:and periodic and nonlocal integral conditions:where is a deviated continuous given function, i.e., , with , and is an increasing function and the integral is meant in the Riemann-Stieltjes sense for .

For the function we can take(i), , where (ii), , , and

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:

We list the following assumptions which we use in this paper: , and , where . There exists with such that has a growth condition of the form where for every closed interval .

We point out that our results can not be extended to the cases in [1, 2]. On the one hand the cases and 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.

The investigation of multipoint boundary value problems was started by Il’in and Moiseev [3]. From that time, many researchers investigate wide-ranging nonlinear multipoint nonlocal boundary value problems (see Chen et al. [4], Guo et al. [5], Ma [6], Webb [7], Webb and Zima [8], and Zhang and Liu [9] and the references cited therein).

The singular ordinary differential equations emerge in the fields of the theory of boundary layer, Newtonian fluid mechanics, gas dynamics, and so on. The theory of singular boundary value problems has turned into an essential region of examination during the last few decades; one may refer to Agarwal and O’Regan [10], Agarwal et al. [11], Cheng [12], Cui and Zou [13], Du and Zhao [14], Erbe and Mathsen [15], Feng and Ge [16], Han and Wu [17], Jiang et al. [18], Kiguradze [19], Li et al. [20], Li and Yan [21], B. Liu et al. [22], L. S. Liu et al. [23], Liua [24], Ma and Thompson [2], Moshinsky [25], Pandey and Barnwal [26], Sun and Zhang [27], Suna [28], Timoshenko [29], Wei [30], Yan et al. [31], Yaoa and Lina [32], Zhao [33], Zhang and Sun [34], and Zhang and Feng [35], and the references therein.

The existence of solutions of ordinary differential equation with integral and periodic boundary conditions has been widely considered in recent years; they constitute very important and interesting problems because they have various applications in thermoelasticity, chemical engineering, population dynamics, and underground water flow and include nonlocal and multipoint boundary conditions (see, for example, Feng et al. [36], Feng and Cong [37], Hua et al. [38], Jiang et al. [18], Nanware and Dhaigude [39], Song et al. [40], Staněk [41], Webb and Infante [42], Yan et al. [31], Yang [43], Yao [44], Zhang and Ge [45], and Zhang and Xu [46], and the references therein).

2. Preliminary Lemmas

We shall use the classical Banach spaces , , and . We denoted by the space of absolute continuous functions on and denoted . Let be the space consisting of functions that are absolutely continuous on every closed interval .

Let be the Banach spacewith the normLet be the Banach spaceequipped with the normwhere and .

For convenience we set the following notation: .

Lemma 1 (see [47]). Let . Then(i) , and(ii),

Theorem 2 (Schauder’s fixed point theorem). Let be a closed convex subset of a Banach space . If is continuous and is relatively compact, then has a fixed point in .

Theorem 3 (Leray-Schauder continuation principle; see, e.g., [48]). Let be a Banach space and be a compact map. Assume that there exists so that if for , then . Then is solvable.

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 byCondition givesand we haveAlso condition givesand substituting it in (15) we get (12).
Conversely, (cf. (12)) . MoreoverIntegrating from 0 to and interchanging the order of integration to get hence we getSimilarly integrate from to 1 and interchange the order of integration to get Hence we haveThen from (19) and (21) we have ; also from (12) we have .
This proves that the solution of singular second-order ordinary differential equation with periodic and deviated multipoint boundary conditions (1)-(2) is equivalent to the functional integral equation (12).

Define a nonlinear operator byFrom , we have and we conclude that is well defined.

For , we define an operator byFor and from and , we haveThen is well defined and bounded.

Lemma 5. Let be a function satisfying Carathéodory’s conditions and conditions and are fulfilled; then and

Proof. Firstly, to show , we have and by Lemma 1 and from (19), (21), and (24) we have and ; then ; moreover we haveNow sincethen we have . Now we have (27) and condition together with the fact that and for each ; we have andNow to prove that and exist, we prove , since Then which completes the proof of the lemma.

Lemma 6. is a completely continuous operator.

Proof. By the definition of and , let be a convergent sequence to in ; then . Since satisfies Carathéodory’s conditions, we haveFrom conditions and together with the fact that and for each , we obtain by applying Lebesgue dominated convergence theorem (cf. (24))Also we have (cf. (27)) Therefore we obtain ; that is, is a continuous operator.
Secondly, let be a bounded set. We will show that is a relatively compact subset of . Let ; let . From (27) and condition , we have Similarly we haveFrom (26) we obtain Then from (35) and by similar way as (34) we obtainSince and by using (i) of Lemma 1, thus from (34) and (37) we have the fact that is equicontinuous on . By Arzelá-Ascoli theorem, and are relatively compact and we have that fact that is a completely continuous operator.

3. Existence Results

Theorem 7. Let satisfy the Carathéodory conditions. Assume that assumptions and hold. Then problem (1)-(2) has at least one solution.

Proof. In order to apply Theorem 2, we define a ball , whereFirstly, we prove that . For , and from (25), we have Also from (27) Then Thus we have and Lemma 6 provides that is a completely continuous operator. Hence Theorem 2 implies the existence of a solution of problem (1)-(2) in .

Theorem 8. Let be a function satisfying the Carathéodory conditions. Assume that assumptions and hold. Then problem (1)-(2) has at least one solution in .

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 equationsis a priori bounded in by a constant independent of .
Let be a solution of (42)-(43) for some . Then, for , we have which implies thatSimilarly (cf. (17)), which implies thatNow from (42), (45), and (47) and condition , it follows that And we haveSince , from the above inequality, there exists a constant , such that which implies from (45) thatAlso from (47) we haveThereforeThen all conditions of Theorem 3 are satisfied and then problem (1)-(2) has at least one solution in .

Corollary 9. Under the assumptions of Theorem 8, the singular second-order ordinary differential equation with periodic and multipoint nonlocal boundary conditionshas at least one solution in represented by

Proof. Letting in problem (1)-(2), then from Theorem 8 we derive that singular second-order ordinary differential problem (53) has at least one solution given by (54).

Example 10. Consider the singular second-order ordinary differential problemwhere . Here, . Clearlywhere , , , and . We can see that and . Indeed we have , , , , and . Assumptions and hold; therefore, by Theorem 8, singular boundary value problem (55) has a continuous solution in .

4. 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.

Lemma 11. The singular second-order ordinary differential equation with periodic and integral boundary conditions (1), (3) is equivalent to the functional integral equation:We define an operator associated with the boundary value problem (1), (3) as follows: for From , we have Then is well defined and bounded.

Lemma 12. Let be a function satisfying Carathéodory’s conditions and is fulfilled; then is a completely continuous operator and

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 be a function satisfying the Carathéodory conditions. Assume that the assumption 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.

Theorem 14. Let be a function satisfying the Carathéodory conditions. Assume that the assumption holds. Then there exists at least one solution of the nonlinear singular ordinary differential equation with periodic and nonlocal integral conditionsrepresented by where is an increasing function and the integral is meant in the Riemann-Stieltjes sense for .

Example 15. Consider the singular second-order ordinary differential problemunder periodic and integral conditionsor under periodic and nonlocal integral conditionsHere, . Clearlywhere , , and , and we have and . Clearly, , , and . Assumption holds; therefore, by Theorems 13 and 14, singular boundary value problem (63) under the condition (64) or (65) has a continuous solution in .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Copyright

Copyright © 2018 Ahmed M. A. El-Sayed and Fatma M. Gaafar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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