# Solvability of a boundary value problem at resonance

- A. Guezane-Lakoud
^{1}, - R. Khaldi
^{1}and - A. Kılıçman
^{2}Email author

**Received: **14 May 2016

**Accepted: **30 August 2016

**Published: **7 September 2016

## Abstract

This paper concerns the solvability of a nonlinear fractional boundary value problem at resonance. By using fixed point theorems we prove that the perturbed problem has a solution, then by some ideas from analysis we show that the original problem is solvable. An example is given to illustrate the obatined results.

### Keywords

Fractional boundary value problem at resonance Existence of solution Schaefer fixed point theorem Banach contraction principal### Mathematics Subject Classification

34A08 34B15## Background

In a recent study Mawhin (1972), Nieto investigated a resonance BVP by an other approach, that we will apply to a fractional boundary value problem to prove the existence of solutions.

*f*depends also on the derivative and the differential Eq. (1) is of fractional type.

Fractional boundary value problems at resonance have been investigated in many works such in Bai (2011), Han (2007), Infante and Zima (2008), where the authors applied Mawhin coincidence degree theory. Further for the existence of unbounded positive solutions of a fractional boundary value problem on the half line, see Guezane-Lakoud and Kılıçman (2014).

The organization of this work is as follows. In Sect. 2, we introduce some notations, definitions and lemmas that will be used later. Section 3 treats the existence and uniqueness of solution for the perturbed problem by using respectively Schaefer fixed point theorem and Banach contraction principal. Then by some analysis ideas, we prove that the problem (P) is solvable. Finally, we illustrate the obtained results by an example.

## Preliminaries

In this section, we present some Lemmas and Definitions from fractional calculus theory that can be found in Nieto (2013), Podlubny (1999).

###
**Definition 1**

###
**Definition 2**

*g*defined by

exists almost everywhere on [*a*, *b*] \(([\alpha ]\) is the integer part of \(\alpha ).\)

###
**Lemma 3**

*For*\(\alpha >0,\) \(g\in C(\left[ 0,1\right] ,{\mathbb {R}} ),\)

*the homogenous fractional differential equation*

*has a solution*

*where*, \(c_{i}\in {\mathbb {R}} ,\) \(i=0,{\ldots},n-1,\)

*here*

*n*

*is the smallest integer greater than or equal to*\(\alpha .\)

###
**Lemma 4**

*Let*
\(p,q\ge 0,\)
\(f\in L_{1}\left[ a,b\right] .\)
*Then*
\(I_{0^{+}}^{p}I_{0^{+}}^{q}f\left( t\right) =I_{0^{+}}^{p+q}f\left( t\right) =I_{0^{+}}^{q}I_{0^{+}}^{p}f\left( t\right)\)
*and*
\(^{c}D_{0^{+}}^{q}I_{0^{+}}^{q}f\left( t\right) =f\left( t\right) ,\)
*for all*
\(t\in \left[ a,b\right]\).

Now we start by solving an auxiliary problem.

###
**Lemma 5**

*Let*\(2<q<3\)

*and*\(y\in C\left[ 0,1\right] .\)

*The linear fractional boundary value problem*

*has a solution if and only if*\(I_{0^{+}}^{q}y(1)=0,\)

*in this case the solution can be written as*

*where*

###
*Proof*

*u*. We have

*c*by its value in (9) we obtain

*H*(

*t*,

*s*) is continuous according to both variables

*s*,

*t*on \(\left[ 0,1\right] \times \left[ 0,1\right]\) and is positive. \(\square\)

*u*we have to find

*v*and

*u*(1). Note \(v_{c}(t)=u(t)-t^{2}c,\) we try to solve for every \(v_{c}\) the problem

*u*is a solution of (1).

## Existence and uniqueness results

*E*be the Banach space of all functions \(u\in C^{1}\left[ 0,1\right]\) into \({\mathbb {R}} ,\) equipped with the norm \(\left\| u\right\| =\max \left( \left\| u\right\| _{\infty },\left\| u^{\prime }\right\| _{\infty }\right)\) where \(\left\| u\right\| _{\infty }=\max _{t\in \left[ 0,1\right] }\left| u\left( t\right) \right|\). Denote by \(L^{1}\left( \left[ 0,1\right] ,{\mathbb {R}} \right)\) the Banach space of Lebesgue integrable functions from \(\left[ 0,1\right]\) into \({\mathbb {R}}\) with the norm \(\left\| y\right\| _{L^{1}}=\int _{0}^{1}\left| y\left( t\right) \right| dt.\) Define the integral operator \(T:E\rightarrow E\) by

###
**Theorem 1**

*Assume that there exist nonnegative functions*

*g*,

*h*,

*k*\(\in\) \(L^{1}\left( \left[ 0,1\right] ,{\mathbb {R}} _{+}^{*}\right)\)

*and*\(0\le \alpha <1\)

*such that*

*Then the map*\(T_{c}\)

*has at least one fixed point*\(v^{*}\in E.\)

We apply Schaefer fixed point theorem to prove Theorem 1.

###
**Theorem 2**

*Let*
*A*
*be a completely continuous mapping of a Banach space*
*X*
*into it self, such that the set*
\(\left\{ x\in X:x=\lambda Ax,0<\lambda <1\right\}\)
*is bounded, then*
*A*
*has a fixed point.*

### Proof of Theorem 1

By Arzela-Ascoli Theorem we can easly show that \(T_{c}\) is a completely continuous mapping.

*H*(

*t*,

*s*) is continuous according to both variables

*s*,

*t*on \(\left[ 0,1\right] \times \left[ 0,1\right]\), nonnegative and \(0\le H(t,s)\le 2\) then using assumptions (14) and (15), we get

*s*,

*t*on \(\left[ 0,1\right] \times \left[ 0,1\right]\), nonnegative and \(0\le H_{t}(t,s)\le q+1\). We have

*v*is bounded independently of \(\lambda\), then Schaefer fixed point theorem implies \(T_{c}\) has at least a fixed point. Hence equation

*E*. The proof is complete. \(\square\)

The uniqueness result is given by the following Theorem:

###
**Theorem 3**

*Assume there exist nonnegative functions*\(g,h\in L^{1}\left( \left[ 0,1\right] ,{\mathbb {R}} _{+}\right)\)

*such that for all*\(x,y,\overline{x},\overline{y}\in {\mathbb {R}} ,\) \(t\in \left[ 0,1\right]\)

*one has*

*Then*\(T_{c}\)

*has a unique fixed point*\(v_{c}^{*}\)

*in*

*E*.

### Proof

*v*and \(w\in E\), then by (20) we get

Let us remark that under the assumptions of Theorem 3, the map \(\Psi :{\mathbb {R}}\rightarrow E,\) \(\Psi \left( c\right) =v_{c}^{*}\) is continuous. Moreover the map \(\Lambda :{\mathbb {R}}\rightarrow {\mathbb {R}},\) \(\Lambda =\Phi \circ \Psi ,\Lambda \left( c\right) =v_{c}^{*}\left( 1\right)\) is also continuous, where \(\Phi :E\rightarrow {\mathbb {R}},\) \(\Phi \left( v\right) =v\left( 1\right)\) and \(v_{c}^{*}\) is the unique fixed point of \(T_{c}\).

Let us show that the problem (1–2) is solvable.

###
**Theorem 4**

*Under the assumptions of Theorems*1

*and*3

*and if*

*uniformly on*\(\left[ 0.1\right]\),

*then the problem*(1–2)

*has at least one solution in*

*E*. \((\left( u,v\right) \rightarrow +\infty ,\)

*ie*. \(u\rightarrow +\infty\)

*and*\(v\rightarrow +\infty ).\)

###
*Proof*

The condition \(\lim _{\left( u,v\right) \rightarrow \pm \infty }f\left( t,u,v\right) =\pm \infty\) is assumed to avoid the case \(f(t,u(t),u^{\prime }(t))=y\left( t\right)\) where the problem may have no solution (in the case \(I_{0^{+}}^{q}y(1)\ne 0).\) If we prove that \(\lim _{c\rightarrow \pm \infty }\Lambda \left( c\right) =\pm \infty ,\) then there exists \(c^{*}\in {\mathbb {R}}\) such that \(\Lambda \left( c^{*}\right) =0\) consequently \(c^{*}=u_{c^{*}}(1)\) hence \(u_{c^{*}}(t)=v_{c^{*}}^{*}(t)+t^{2}c^{*}\) is a solution of the nonlinear problem (1–2).

Now taking into account (18) we get \(\lim _{c\rightarrow +\infty }\frac{\left\| v_{c}^{*}\right\| }{c}=0.\) Since the norms of \(\left( v_{c}^{*}(s)+cs^{2}\right)\) and \(\left( v_{c}^{*\prime }(s)+2cs\right)\) growth asymptotically as *c*, \(H\left( t,s\right)\) is nonnegative and continuous and \(\lim _{\left( u,v\right) \rightarrow \pm \infty }f\left( t,u,v\right) =\pm \infty ,\) then from (19) it yields \(\lim _{c\rightarrow \pm \infty }\Lambda \left( c\right) =\pm \infty .\) The proof is complete. \(\square\)

###
*Example 5*

*E*.

### Proof

*E*. It is easy to see that

From the above discussion and Theorem 4 we conclude that the problem (24) is solvable in *E*. \(\square\)

## Conclusion

## Declarations

### Authors' contributions

All authors read and approved the final manuscript

### Acknowledgements

First of all the authors would like to thank the referees for giving useful suggestions to improve the manuscript. The first author would also like to thank the University Putra Malaysia for the kind hospitality during her visit in December 2015. The third author acknowledges that this research was partially supported by the University Putra Malaysia.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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