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# On some boundary value problems with non-local and periodic conditions

## Abstract

In this work, we concern non-local and periodic boundary value problems. We will prove the existence of at least one solution of these problems such that the functions satisfy the growth condition. Hence, we will study the existence of at least one solution for a boundary value problem with periodic and integrable conditions.

## Introduction

Differential equations with non-local conditions were considered in many works (see [1], [2], [3], and [4]). Also, anti-periodic problems can be found in [5] and [6].

Here, we study the existence of at least one solution for the boundary value problem with non-local and periodic conditions:

$$\left\{ \begin{array}{c}x^{\prime\prime}(t)~=~f(t,~x(t),~x^{\prime}(t))~~~\text{a.e.}~t~\in~(0,2\pi),\\ x(0)~=~x(2\pi)~~\text{and}~~\sum_{k=1}^{m}~a_{k}~x(\tau_{k})~=~x_{0} \end{array}\right.$$
(1)

where x0âˆˆR, 0<Ï„1<Ï„2<â‹¯<Ï„m<2Ï€ and akâ‰ 0 for all k=1,2,â‹¯,m.

Also, the boundary value problem with integral and periodic conditions:

$$\left\{ \begin{array}{c}x^{\prime\prime}(t)~=~f(t,~x(t),~x^{\prime}(t))~~~\text{a.e.}~~t~\in~(0,2\pi),\\ x(0)~=~x(2\pi)~~\text{and}~~\int_{0}^{2\pi}~x(t)~dt~=~x_{0} \end{array}\right.$$
(2)

will be considered.

Problem (2) was studied in [7], but the author has not shown the equivalence between the differential problem (2) and the integral equation equivalent with it.

Here, we prove, by using nonlinear alternative of Leray-Schauder type, the existence of at least one solution for problem (1) such that the function f:IÃ—RÃ—Râ†’R, I=[0,2Ï€] satisfies the growth conditions.

## Preliminaries

### Theorem 1

(Nonlinear alternative of Leray-Schauder type) [8] Let E be a Banach space and Î© be a bounded open subset of E, 0âˆˆÎ© and $$T:\bar {\Omega }\rightarrow E$$ be a completely continuous operator. Then, either there exists xâˆˆâˆ‚Î©,Î»>1 such that T(x)=Î»x, or there exists a fixed point $$x^{\ast } \in \bar {\Omega }$$.

Denote by C(I) the space of all continuous functions defined on the interval I with norm

$$||u||_{C}~=~\sup_{t \in I}~|u(t)|$$

and by L1(I) the space of all Lebesgue integrable functions on the interval I with norm

$$||u||_{L_{1}}~=~\int_{I}~|u(t)|~dt.$$

The growth condition on the function f means that

$$|f(t,u)|~\leq ~a(t)~+~b~|u|,$$

where a(t)âˆˆL1, b is a nonnegative constant.

## Main results

Let the function f:IÃ—RÃ—Râ†’R satisfy the following assumptions:

1. (1)

f:IÃ—RÃ—Râ†’R is measurable in tâˆˆI for any (u1,u2)âˆˆRÃ—R

2. (2)

f:IÃ—RÃ—Râ†’R is continuous in (u1,u2)âˆˆRÃ—R for any tâˆˆI

3. (3)

There exist two positive constants b1,b2 and a function c(t)âˆˆL1(I) such that

$$|f(t,~u_{1},~u_{2})|~\leq~c(t)~+~b_{1}~|u_{1}|~+~b_{2}~|u_{2}|.$$

### Lemma 1

Let the assumptions (1)â€“(3) be satisfied. If the solution of the boundary value problem (1) exists, then it can be represented by

$$\begin{array}{@{}rcl@{}} x(t)&=&A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\\ &+&\left(t~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)~+~\int_{0}^{t}~(t~-~s)~y(s)~ds,\\ \end{array}$$

where

$$\begin{array}{@{}rcl@{}} y(t)&=&f\left(t,y_{1}(t),y_{2}(t)\right),\\ y_{1}(t)&=&A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\\ &+&\left(t~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)\\ &+&\int_{0}^{t}~(t~-~s)~y(s)~ds\\ \text{and}~~~y_{2}(t)&=&\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds~+~\int_{0}^{t} y(s)~ds,~~t\in I. \end{array}$$
(3)

### Proof

Let y=xâ€²â€²(t)=f(t,x,xâ€²). â–¡

Integrating both sides, we obtain

$$x^{\prime}(t)~-~x^{\prime}(0)~=~\int_{0}^{t}~y(s)~ds.$$

Integrating again, we get

$$\begin{array}{@{}rcl@{}} x(t)~=~x(0)&+&t~x^{\prime}(0)~+~\int_{0}^{t}~\int_{0}^{s}~y(\theta)~d\theta~ds\\ ~=~x(0)&+&t~x^{\prime}(0)~+~\int_{0}^{t}~(t~-~s)~y(s)~ds. \end{array}$$

From the boundary condition, we obtain

$$x^{\prime}(0)~=~\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds,$$

then

$$\begin{array}{@{}rcl@{}} x^{\prime}(t)&=&x^{\prime}(0)~+~\int_{0}^{t}~y(s)~ds\\ &=&\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds~+~\int_{0}^{t}~y(s)~ds. \end{array}$$
(4)

Now,

$$\begin{array}{@{}rcl@{}} x(t)&=&x(0)~+~t~x^{\prime}(0)~+~\int_{0}^{t}~(t~-~s)~y(s)~ds,\\ \text{then}~~~~x(\tau_{k})&=&x(0)~+~\tau_{k}~x^{\prime}(0)~+~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\\ \text{and}~~~\sum_{k=1}^{m}~a_{k}~x(\tau_{k})&=&x(0)\sum_{k=1}^{m}~a_{k}~~+~\sum_{k=1}^{m}~a_{k}~\tau_{k}~x^{\prime}(0)~+~\sum_{k=1}^{m}~a_{k}~ \int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds. \end{array}$$

Take $$A=(\sum _{k=1}^{m}~a_{k})^{-1}$$, then

$$x(0)~=~A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\tau_{k}~x^{\prime}(0)~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right).$$

Substituting the values of xâ€²(0) and x(0) in x(t), we get

$$\begin{array}{@{}rcl@{}} x(t)&=&A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\\ &+&\left(t~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)~+~\int_{0}^{t}~(t~-~s)~y(s)~ds.\\ \end{array}$$
(5)

Inserting (4) and (5) in xâ€²â€²(t) = f(t, x(t), xâ€²(t)), we get

$$\begin{array}{@{}rcl@{}} y(t)&=&f\left(t,y_{1}(t),y_{2}(t)\right)\\ &=&f\left(t,~A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\right.\\ &+&\left(t~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)\\ &+&\left.\int_{0}^{t}~(t~-~s)~y(s)~ds,~\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds~+~\int_{0}^{t} y(s)~ds\right),~~t\in[0,2\pi]. \end{array}$$

### Existence of solution

Define the operator T by

$$\begin{array}{@{}rcl@{}} T~y(t)&=&f\left(t,~y_{1}(t),~y_{2}(t)\right),~~t\in I \end{array}$$

where

$$\begin{array}{@{}rcl@{}} y_{1}(t)&=&A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\\ &+&\left(t~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)\\ &+&\int_{0}^{t}~(t~-~s)~y(s)~ds \end{array}$$

and

$$\begin{array}{@{}rcl@{}} y_{2}(t)&=&\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds~+~\int_{0}^{t} y(s)~ds. \end{array}$$

Firstly, we prove that the functional Eq. (3) has at least one solution yâˆˆL1(I); in order to do that, we will show that the operator T has a fixed point yâˆˆL1(I).

### Theorem 2

Let the function f:IÃ—RÃ—Râ†’R satisfy the assumptions (1)â€“(3) and the following assumption:

• Every solution y(.)âˆˆL1(I) to the equation

$$\begin{array}{@{}rcl@{}} y(t)&=&\gamma~f\left(t,~y_{1}(t),~y_{2}(t)\right)~~ \text{a.e. on}~I,~ \gamma ~\in ~(0,1) \end{array}$$

satisfies $$||y||_{L_{1}}\not =r$$ (r is arbitrary but fixed).

Then the operator T has a fixed point yâˆˆL1(I), which is a solution to Eq. (3).

### Proof

Let y be an arbitrary element in the open set $$B_{r}= \{y:||y||_{L_{1}}< r, r=\frac {||c||_{L_{1}}+2\pi b_{1} |A| |x_{0}|}{1-\left (8 \pi ^{2} b_{1}+2\pi ~b_{1}~|A|~|\sum _{k=1}^{m}~a_{k}~\tau _{k}|+4~\pi ~b_{2}\right)}>0\}$$. Then from the assumptions (1)â€“(3), we have

{\begin{aligned} ||Ty||_{L_{1}}&=\int_{0}^{2\pi}~|Ty(t)|~dt\\[-1pt] &=\int_{0}^{2\pi}\left|f\left(t,y_{1}(t),y_{2}(t)\right)\right|~dt\\[-1pt] &\leq\int_{0}^{2\pi}\left[|c(t)|~+~b_{1}~|y_{1}(t)|~+~b_{2}~|y_{2}(t)|\right]~dt\\[-1pt] &\leq||c||_{L_{1}}~+~b_{1}~\int_{0}^{2\pi}|y_{1}(t)|~dt~+~b_{2}~\int_{0}^{2\pi}|y_{2}(t)|~dt\\[-1pt] &\leq||c||_{L_{1}}~+~b_{1}~\int_{0}^{2\pi}~\left|A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\right.\\[-1pt] &+\left(t~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)\\[-1pt] &+\int_{0}^{t}~(t~-~s)~y(s)~ds\left|~dt~+~b_{2}~\int_{0}^{2\pi}~\left|\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds~+~\int_{0}^{t} y(s)~ds\right|~dt\right.\\[-1pt] &\leq||c||_{L_{1}}~+~b_{1}~\int_{0}^{2\pi}~\left|A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\right|~dt\\[-1pt] &+b_{1}\int_{0}^{2\pi}\left|\left(t~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right) \left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)\right|~dt\\[-1pt] &+b_{1}\int_{0}^{2\pi}\left|\int_{0}^{t}~(t~-~s)~y(s)~ds\left|~dt~+~b_{2}~\int_{0}^{2\pi} \right|\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right|~dt\\[-1pt] &+b_{2}~\int_{0}^{2\pi}~\int_{0}^{t}~|y(s)|~ds~dt\\[-1pt] &\leq||c||_{L_{1}}~+~b_{1}~\int_{0}^{2\pi}~|A~x_{0}|~dt~+~b_{1}~\int_{0}^{2\pi}~\left|A~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right|~dt\\[-1pt] &+b_{1}~\int_{0}^{2\pi}~t~\left|\frac{1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right|~dt\\[-1pt] &+b_{1}~\int_{0}^{2\pi}~\left|A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right|~\left|\frac{1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right|~dt\\[-1pt] &+b_{1}\int_{0}^{2\pi}\int_{s}^{2\pi}~(t~-~s)~|y(s)|~dt~ds~+~b_{2}~\int_{0}^{2\pi}~\int_{0}^{2\pi}~(1-\frac{s}{2\pi})~|y(s)|~ds~dt\\[-1pt] &+b_{2}~\int_{0}^{2\pi}~\int_{s}^{2\pi}~|y(s)|~dt~ds\\[-1pt] &\leq||c||_{L_{1}}~+~2\pi~b_{1}~|A|~|x_{0}|~+~b_{1}~\int_{0}^{2\pi}~\left|A~\sum_{k=1}^{m}~a_{k}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right|~dt\\[-1pt] &+b_{1}~\int_{0}^{2\pi}~t~\int_{0}^{2\pi}~(1-\frac{s}{2\pi})~|y(s)|~ds~dt+b_{1}~\int_{0}^{2\pi}~\left|A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right|~ \int_{0}^{2\pi}~(1-\frac{s}{2\pi})~|y(s)|~ds~dt\\[-1pt] &+b_{1}\int_{0}^{2\pi}\frac{(t~-~s)^{2}}{2}|_{s}^{2\pi}~|y(s)|~ds~+~b_{2}~\int_{0}^{2\pi}~\int_{0}^{2\pi}~|y(s)|~ds~dt\\[-1pt] &+b_{2}~\int_{0}^{2\pi}~(2\pi-s)~|y(s)|~ds\\[-1pt] &\leq||c||_{L_{1}}~+~2\pi~b_{1}~|A|~|x_{0}|~+~b_{1}~\int_{0}^{2\pi}~\left|A~\sum_{k=1}^{m}~a_{k}\left|~\int_{0}^{2\pi}~(2\pi)~|y(s)|~ds~dt\right.\right.\\[-1pt] &+b_{1}~\int_{0}^{2\pi}~t~\int_{0}^{2\pi}~|y(s)|~ds~dt+b_{1}~\int_{0}^{2\pi}~\left|A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right|~ \int_{0}^{2\pi}~|y(s)|~ds~dt\\[-1pt] &+b_{1}\int_{0}^{2\pi}\frac{(2\pi~-~s)^{2}}{2}~|y(s)|~ds~+~b_{2}~||y||_{L_{1}}~\int_{0}^{2\pi}~dt+2\pi~b_{2}~\int_{0}^{2\pi}~|y(s)|~ds\\[-1pt] &\leq||c||_{L_{1}}~+~2\pi~b_{1}~|A|~|x_{0}|~+~(2\pi)^{2}~b_{1}~||y||_{L_{1}}~+~b_{1}~||y||_{L_{1}}~\frac{(2\pi)^{2}}{2}\\[-1pt] &+2\pi~b_{1}~|A|~|\sum_{k=1}^{m}~a_{k}~\tau_{k}|~||y||_{L_{1}}~+~\frac{(2\pi)^{2}}{2}~b_{1}~||y||_{L_{1}}~+~2\pi~b_{2}~||y||_{L_{1}}~+~2\pi~b_{2}~||y||_{L_{1}}\\[-1pt] &\leq||c||_{L_{1}}~+~2\pi~b_{1}~|A|~|x_{0}|~+~4~\pi^{2}~b_{1}~||y||_{L_{1}}~+~2\pi^{2}~b_{1}~||y||_{L_{1}}\\[-1pt] &+2\pi~b_{1}~|A|~|\sum_{k=1}^{m}~a_{k}~\tau_{k}|~||y||_{L_{1}}~+~2\pi^{2}~b_{1}~||y||_{L_{1}}~+~4~\pi~b_{2}~~||y||_{L_{1}}\\[-1pt] &=||c||_{L_{1}}~+~2\pi~b_{1}~|A|~|x_{0}|~+~8~\pi^{2}~b_{1}~||y||_{L_{1}}+2\pi~b_{1}~|A|~|\sum_{k=1}^{m}~a_{k}~\tau_{k}|~||y||_{L_{1}}~+~4~\pi~b_{2}~~||y||_{L_{1}}. \end{aligned}}

The above inequality means that the operator T maps Br into L1. â–¡

Also, from assumption (2), we deduce that T maps Br continuously into L1(I).

Now, we will use Kolmogorov compactness criterion (see [9]) to show that T is compact. So, let â„µ be a bounded subset of Br. Then T(â„µ) is bounded in L1(I). Now we show that (Ty)hâ†’Ty in L1(I) as hâ†’0, uniformly with respect to TyâˆˆT â„µ.

Indeed:

$$\begin{array}{@{}rcl@{}} ||(Ty)_{h}~-~Ty||_{L_{1}}&=&\int_{0}^{2\pi}~|~(Ty)_{h}(t)~-~(Ty)(t)~|~dt\\ &=&\int_{0}^{2\pi}~\left|\frac{1}{h}~\int_{t}^{t+h}~(Ty)(s)~ds~-~(Ty)(t)\right|~dt\\ &\leq&\int_{0}^{2\pi}~\left(~\frac{1}{h}~\int_{t}^{t+h}~|~(Ty)(s)~-(Ty)(t)~|~ds~\right)~dt\\ &\leq&\int_{0}^{2\pi}~\frac{1}{h}~\int_{t}^{t+h}~|~f(s,y_{1}(s),~y_{2}(s))~-~f(t,y_{1}(t),~y_{2}(t))~|~ds~dt. \end{array}$$

Since

{\begin{aligned} ||f||_{L_{1}}&\leq&||c||_{L_{1}}+2\pi~b_{1}~|A|~|x_{0}|+8~\pi^{2}~b_{1}~||y||_{L_{1}}+2\pi~b_{1}~|A|~|\sum_{k=1}^{m}~a_{k}~\tau_{k}|~||y||_{L_{1}}+4~\pi~b_{2}~~||y||_{L_{1}}, \end{aligned}}

we have that f in L1(I). So, we have (see [10])

$$\frac{1}{h}~\int_{t}^{t+h}~|~f(s,y_{1}(s),~y_{2}(s))~-~f(t,y_{1}(t),~y_{2}(t))~|~ds~\rightarrow~0,$$

for a.e. tâˆˆI. So, T(â„µ) is relatively compact, that is, T is a compact operator.

Now from assumption (4) and Theorem 1, we get that T has a fixed point yâˆˆL1(I).

### Theorem 3

If the assumptions of Theorem 2 are satisfied, then the periodic and non-local boundary value problem (1) has at least one solution xâˆˆC1(I).

### Proof

Let x(t) be a solution of (5)

$$\begin{array}{@{}rcl@{}} x(t)&=&A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\\ &+&\left(t~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)~+~\int_{0}^{t}~(t~-~s)~y(s)~ds, \end{array}$$

by differentiation, we obtain

$$\begin{array}{@{}rcl@{}} x^{\prime}(t)&=&\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds~+~\int_{0}^{t}~y(s)~ds. \end{array}$$

Since Theorem 2 proved that yâˆˆL1(I), then by differentiating again, we get

$$x^{\prime\prime}(t)~=~y(t)~=~f(t,~x(t),~x^{\prime}(t)).$$

Substituting respectively by x=0 and x=2Ï€ in (5), we get

$$\begin{array}{@{}rcl@{}} x(0)&\,=\,&A\left(x_{0}\,-\,\sum_{k=1}^{m}a_{k}\int_{0}^{\tau_{k}}~(\tau_{k}-s)~y(s)~ds\right)\,+\,\left(\,-\,~A\sum_{k=1}^{m}a_{k}\tau_{k}\right)\left(\frac{-1}{2\pi} \int_{0}^{2\pi}(2\pi-s)~y(s)~ds\right)\\ \end{array}$$
(6)

and

\begin{aligned} x(2\pi)&=A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\\ &+\left(2\pi~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)~+~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\\ &=A\left(x_{0}-\sum_{k=1}^{m}a_{k}\int_{0}^{\tau_{k}}~(\tau_{k}-s)~y(s)~ds\right)+\left(-~A\sum_{k=1}^{m}a_{k}\tau_{k}\right)\left(\frac{-1}{2\pi} \int_{0}^{2\pi}(2\pi-s)~y(s)~ds\right).\\ \end{aligned}
(7)

From (6) and (7), we get x(0)=x(2Ï€). â–¡

Also,

{\begin{aligned} x(\tau_{k})&=A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\\ &+\left(\tau_{k}~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right) +\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds,\\ a_{k}~x(\tau_{k})&=a_{k}~A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\\ &+a_{k}~\left(\tau_{k}-A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)\left(\frac{-1}{2\pi}\int_{0}^{2\pi}(2\pi-s)y(s)~ds\right) +a_{k}~\int_{0}^{\tau_{k}}(\tau_{k}-s)y(s)~ds,\\ \sum_{k=1}^{m}~a_{k}~x(\tau_{k})&=\sum_{k=1}^{m}~a_{k}~A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)y(s)~ds\right)\\ &+\sum_{k=1}^{m}~a_{k}\left(\tau_{k}-A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)\left(\frac{-1}{2\pi}\int_{0}^{2\pi}(2\pi-s)y(s)~ds\right) +\sum_{k=1}^{m}~a_{k}\int_{0}^{\tau_{k}}(\tau_{k}-s)y(s)~ds\\ &=x_{0}~. \end{aligned}}

Then the periodic and non-local boundary value problem (1) is equivalent to the integral Eq. (5). Hence problem (1) has at least one solution xâˆˆC1(I).

### Theorem 4

If f:IÃ—RÃ—Râ†’R satisfies the assumptions of Theorem 2, then the boundary value problem (2) has at least one solution xâˆˆC1(I), and its solution is given by

$$\begin{array}{@{}rcl@{}} x(t)&=&\frac{1}{2\pi}~\left(x_{0}~-~\int_{0}^{2\pi}~\frac{(2\pi~-~s)^{2}}{2}~y(s)~ds\right)\\ &+&\left(t~-~\ \pi\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)~+~\int_{0}^{t}~(t~-~s)~y(s)~ds. \end{array}$$
(8)

Also,

$$\begin{array}{@{}rcl@{}} x^{\prime}(t)&=&x^{\prime}(0)~+~\int_{0}^{t}~y(s)~ds\\ &=&\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds~+~\int_{0}^{t}~y(s)~ds. \end{array}$$

### Proof

If we take ak=tkâˆ’tkâˆ’1,Ï„kâˆˆ(tkâˆ’1,tk) and 0<t1<t2<...<2Ï€, we get

$$\sum_{k=1}^{m}(t_{k} -t_{k-1})x(\tau_{k})~=~x_{0}.$$

By taking the limit as mâ†’âˆž, we get $$\int _{0}^{2\pi }x(t)dt =x_{0}$$. â–¡

Also, take the limit as mâ†’âˆž in (5):

$$\begin{array}{@{}rcl@{}} x(t)&=&A~\left(x_{0}~-~\sum_{k=1}^{m}~a_{k}~\int_{0}^{\tau_{k}}~(\tau_{k}~-~s)~y(s)~ds\right)\\ &+&\left(t~-~A~\sum_{k=1}^{m}~a_{k}~\tau_{k}\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)~+~\int_{0}^{t}~(t~-~s)~y(s)~ds,\\ \end{array}$$

we obtain (8):

$$\begin{array}{@{}rcl@{}} x(t)&=&\frac{1}{2\pi}~\left(x_{0}~-~\int_{0}^{2\pi}~\frac{(2\pi~-~s)^{2}}{2}~y(s)~ds\right)\\ &+&\left(t~-~\ \pi\right)~\left(\frac{-1}{2\pi}~\int_{0}^{2\pi}~(2\pi~-~s)~y(s)~ds\right)~+~\int_{0}^{t}~(t~-~s)~y(s)~ds. \end{array}$$

This completes the proof.

## Availability of data and materials

Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

## References

1. Boucherif, A.: First-order differential inclusions with nonlocal initial conditions. Appl. Math. Lett. 15, 409â€“414 (2002).

2. El-Sayed, A. M. A., Abd El-Salam, Sh. A.: On the stability of a fractional-order differential equation with nonlocal initial condition. EJQTDE. 29, 1â€“8 (2008).

3. El-Sayed, A. M. A., Abd El-Salam, Sh. A.: Nonlocal boundary value problem of a fractional-order functional differential equation. Inter. J. of Non. Sci. 7(4), 436â€“442 (2009).

4. Hamd-Allah, E. M. A.: On the existence of solutions of two differential equations with a nonlocal condition. JOEMS. 24, 367â€“372 (2016).

5. Ahmed, B., Nieto, J. J.: Anti-periodic fractional boundary value problems. Comp. Math. App. 62, 1150â€“1156 (2011).

6. Chen, Y., Nieto, J. J., Oâ€™Regan, D.: Anti-periodic solutions for fully nonlinear first-order differential equations. Math. Comp. Mod. 46, 1183â€“1190 (2007).

7. Feng, X., Cong, F.: Existence and uniqueness of solutions for the second order periodic-integrable boundary value problem. Bound. Value Probl., 1â€“13 (2017). https://doi.org/10.1186/s13661-017-0840-7.

8. Deimling, K.: Nonlinear Functional Analysis. Springer-Verlag (1985).

9. Dugundji, J., Granas, A.: Fixed point theory. Monografie Mathematyczne, PWN, Warsaw (1982).

10. Swartz, C.: Measure, Integration and function spaces. World Scientific, Singapore (1994).

## Acknowledgements

The author is very grateful to the referee for his valuable comments and suggestions which improved the original version of the paper.

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Abd El-Salam, S. On some boundary value problems with non-local and periodic conditions. J Egypt Math Soc 27, 38 (2019). https://doi.org/10.1186/s42787-019-0040-y