Bachar et al. Boundary Value Problems (2016) 2016:79 DOI 10.1186/s13661-016-0586-7

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Fractional Navier boundary value problems Imed Bachar1 , Habib Mâagli2 and Vicen¸tiu D. R˘adulescu3,4* *

Correspondence: [email protected] 3 Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, P.O. Box 1-764, Bucharest, 014700, Romania 4 Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, Craiova, 200585, Romania Full list of author information is available at the end of the article

Abstract We study the following fractional Navier boundary value problem: ⎧ α β ⎪ ⎨D (D u)(x) + u(x)f (x, u(x)) = 0, 0 < x < 1, limx→0+ Dα –1 (Dβ u)(x) = ξ , limx→0+ Dβ –1 u(x) = 0, ⎪ ⎩ u(1) = 0, Dβ u(1) = –ζ , where α , β ∈ (1, 2], Dα and Dβ stand for the standard Riemann-Liouville fractional derivatives, and ξ , ζ ≥ 0 are such that ξ + ζ > 0. Our purpose is to prove the existence, uniqueness, and global asymptotic behavior of a positive continuous solution, where f : (0, 1) × [0, ∞) → [0, ∞) is continuous and dominated by a function p satisfying appropriate integrability condition. MSC: 34A08; 34B15; 34B18; 34B27 Keywords: fractional Navier differential equations; positive solutions; Green’s function; perturbation arguments

1 Introduction The existence, uniqueness, and global asymptotic behavior of positive continuous solutions related to fractional differential equations have been studied by many researchers. Many fractional differential equations subject to various boundary conditions have been addressed; see, for instance, [–] and the references therein. It is known that fractional differential equations serve as a good tool to model many phenomena in various fields of science and engineering (see [–] and references therein for discussions of various applications). In [], the authors proved the existence and uniqueness of a positive solution to the following fractional boundary value problem: ⎧ ⎨Dα u(x) = u(x)ϕ(x, u(x)), ⎩limx→+ Dα– u(x) = –ξ ,

 < x < ,

(.)

u() = ζ ,

where  < α ≤ , ξ , ζ ≥  are such that ξ + ζ > , and ϕ(x, s) ∈ C + ((, ) × [, ∞)) satisfies appropriate conditions. Inspired by the above-mentioned paper, we aim at studying similar problem in the case of fractional Navier boundary value problem. More precisely, we are © 2016 Bachar et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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concerned with the problem ⎧ α β ⎪ ⎪ ⎨D (D u)(x) + u(x)f (x, u(x)) = ,  < x < , limx→+ Dβ– u(x) = , limx→+ Dα– (Dβ u)(x) = ξ , ⎪ ⎪ ⎩ u() = , Dβ u() = –ζ ,

(.)

where α, β ∈ (, ], and ξ , ζ ≥  are such that ξ + ζ > . The nonlinear term f (x, s) is required to be a nonnegative continuous function in (, ) × [, ∞) dominated by a function p belonging to the class Jα,β defined as follows. Definition . Let α, β ∈ (, ]. A nonnegative measurable function p on (, ) belongs to the class Jα,β iff 



t β– ( – t)α p(t) dt < ∞.

(.)



Next, we introduce the following notation. (i) B+ ((, )) is the set of nonnegative measurable functions in (, ). (ii) Let X be a metric space, we denote by C(X) (resp. C + (X)) the set of continuous (resp. nonnegative continuous) functions in X. (iii) For γ ∈ (, ], C–γ ([, ]) = {w ∈ C((, ]) : x → x–γ w(x) ∈ C([, ])}. (iv) For γ ∈ (, ], Gγ (x, s) is the Green function of the operator u → –Dγ u, with boundary data limx→+ Dγ – u(x) = u() = . From [], Lemma , we have Gγ (x, s) =

γ –     γ – , x ( – s)γ – – (x – s)+ (γ )

(.)

where x+ = max(x, ). Proposition . (see []) Let  < γ ≤  and ϕ ∈ B+ ((, )). Then we have (i) For (x, s) ∈ (, ] × [, ],  (γ – ) H(x, s) ≤ Gγ (x, s) ≤ H(x, s), (γ ) (γ )

(.)

where H(x, s) := xγ – ( – s)γ – ( – max(x, s)).  (ii) The function x → Gγ ϕ(x) :=  Gγ (x, s)ϕ(s) ds belongs to C–γ ([, ]) if and only if  γ – ϕ(s) ds < ∞.  ( – s) (iii) If the map s → ( – s)γ – ϕ(s) ∈ C((, )) ∩ L ((, )), then Gγ ϕ belongs to C–γ ([, ]), and it is the unique solution of the problem ⎧ ⎨Dγ u(x) = –ϕ(x),

 < x < , ⎩limx→+ Dγ – u(x) = u() = . Throughout this paper, for α, β ∈ (, ], let G(x, s) be the Green function of the operator u → Dα (Dβ u) with Navier boundary conditions limx→+ Dβ– u(x) = limx→+ Dα– (Dβ u)(x) =

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u() = Dβ u() = . Then we have 



Gβ (x, t)Gα (t, s) dt.

G(x, s) =

(.)



For a given function p in B+ ((, )), we put 



κp := sup x,s∈(,) 

G(x, t)G(t, s) p(t) dt, G(x, s)

(.)

and we will prove that κp < ∞ if and only if p ∈ Jα,β . From here on, let ξ , ζ be two nonnegative constants such that ξ + ζ > , and θ (x) be the unique solution of the problem ⎧ α β ⎪ ⎪ ⎨D (D u)(x) = , ⎪ ⎪ ⎩

limx→+ D

β–

 < x < , limx→+ Dα– (Dβ u)(x) = ξ ,

u(x) = ,

(.)

β

u() = ,

D u() = –ζ .

We can easily verify that, for x ∈ (, ], θ (x) = ξ h (x) + ζ h (x), where 



h (x) =

Gβ (x, t)Gα (t, ) dt 

=

      xβ–  – xα + xβ– xα+ –  (α – )(α + β – ) (α + β)

(.)

and 



Gβ (x, t)t α– dt =

h (x) = 

 (α – ) β–   – xα . x (α + β – )

(.)

Note that from (.), (.), and (.) it follows that there exists a constant c >  such that, for each x ∈ (, ],  β– x ( – x) ≤ θ (x) ≤ cxβ– ( – x). c

(.)

To state our existence results, a combination of the following hypotheses are required. (A ) f is in C + ((, ) × [, ∞)). (A ) There exists p ∈ Jα,β ∩ C + ((, )) with κp ≤  such that, for each x ∈ (, ), the map s → s(p(x) – f (x, sθ (x))) is nondecreasing on [, ]. (A ) For each x ∈ (, ), the function s → sf (x, s) is nondecreasing on [, ∞). Our main results are the following. Theorem . Under conditions (A )-(A ), problem (.) admits a solution u ∈ C–β ([, ]) such that c θ (x) ≤ u(x) ≤ θ (x), where c ∈ (, ).

 < x ≤ ,

(.)

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Moreover, this solution is unique if hypothesis (A ) is also satisfied. Corollary . Let α, β ∈ (, ], and h be a nonnegative function in C  ([, ∞)) such that the map s → (s) = sh(s) is nondecreasing on [, ∞). Let q ∈ C + ((, )) and assume that the function q˜ (x) := q(x) max≤t≤θ(x)  (t) belongs to Jα,β . Then for λ ∈ [, κ ), the problem q˜

⎧ α β ⎪ ⎪ ⎨D (D u)(x) + λq(x)u(x)h(u(x)) = ,  < x < , limx→+ Dβ– u(x) = , limx→+ Dα– (Dβ u)(x) = ξ , ⎪ ⎪ ⎩ u() = , Dβ u() = –ζ ,

(.)

admits a unique solution u ∈ C–β ([, ]) such that ( – λκq˜ )θ (x) ≤ u(x) ≤ θ (x),

 < x ≤ .

Our paper is organized as follows. In Section , we establish some properties of G(x, s). In particular, we prove the existence of a constant c >  such that, for all x, t, s ∈ (, ), G(x, t)G(t, s)  β– t ( – t)α ≤ ≤ ct β– ( – t)α . c G(x, s) This implies that κp < ∞ if and only if p ∈ Jα,β . In Section , for a given function p ∈ Jα,β with κp ≤  , we construct the Green function H(x, s) of the operator u → Dα (Dβ u) + p(x)u with boundary conditions limx→+ Dβ– u(x) = limx→+ Dα– (Dβ u)(x) = u() = Dβ u() = , and we derive some of its properties including the following: ( – κp )G(x, s) ≤ H(x, s) ≤ G(x, s) for all (x, s) ∈ (, ] × [, ] and   W ϕ = Wp ϕ + Wp (pW ϕ) = Wp ϕ + W (pWp ϕ) for ϕ ∈ B + (, ) , where W and Wp are defined by 







G(x, s)ϕ(s) ds and Wp ϕ(x) :=

W ϕ(x) := 

H(x, s)ϕ(s) ds,

x ∈ (, ].



Exploiting these results, we prove our main results by means of a perturbation argument.

2 Estimates on the Green function We recall the definition of the Riemann-Liouville derivative. Definition . (see [, , ]) The Riemann-Liouville derivative of fractional order γ >  of a function g is defined as Dγ g(x) :=

n  x d  (x – s)n–γ – g(s) ds, (n – γ ) dx 

where n –  ≤ γ < n ∈ N.

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Next, we prove some properties of G(x, s). Proposition . Let α, β ∈ (, ]. Then there exist two constants m >  and M >  such that, for all (x, s) ∈ (, ] × [, ], we have mxβ– ( – x)( – s)α– ≤ G(x, s) ≤ Mxβ– ( – x)( – s)α– .

(.)

Proof Using (.) and (.), we have 



G(x, s) =

Gβ (x, t)Gα (t, s) dt 

 (β)

≤





xβ– ( – x)( – t)β– Gα (t, s) dt 

xβ– ( – x) ≤ (β)(α)





( – t)β– t α– ( – s)α– dt 

= Mxβ– ( – x)( – s)α– . On the other hand, using again (.), (.), and the inequality  – max(x, s) ≥ ( – x)( – s), we get 



Gβ (x, t)Gα (t, s) dt

G(x, s) = 

≥

(β – ) (α – ) (β) (α)





xβ– ( – x)( – t)β t β– ( – s)α– dt 



= mxβ– ( – x)( – s)α– . Using Proposition ., we deduce the following.

Corollary . Let α, β ∈ (, ]. Then there exists a constant c >  such that, for all x, t, s ∈ (, ), we have  β– G(x, t)G(t, s) t ( – t)α ≤ ≤ ct β– ( – t)α . c G(x, s)

(.)

Proposition . Let α, β ∈ (, ], and p be a function in B+ ((, )). (i) There exists a constant c >  such that  c







t

β–



( – t) p(t) dt ≤ κp ≤ c α



t β– ( – t)α p(t) dt,

(.)



where κp is given by (.). In particular, κp < ∞ if and only if

p ∈ Jα,β .

(.)

(ii) For x ∈ (, ], we have W (θ p)(x) ≤ κp θ (x),

(.)

where θ (x) := ξ h (x) + ζ h (x), and h and h are given respectively in (.) and (.).

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Proof Let p be a function in B+ ((, )). (i) Inequalities in (.) follow immediately from (.) and (.). (ii) Since θ (x) := ξ h (x) + ζ h (x), it suffices to prove (.) for h and h . To this end, observe that from (.) it follows that, for each x, s ∈ (, ), lim

r→

G(s, r) G(s, ) h (s) = = . G(x, r) G(x, ) h (x)

So by Fatou’s lemma and (.) we deduce that 



G(x, s) 

h (s) p(s) ds ≤ lim inf r→ h (x)





G(x, s) 

G(s, r) p(s) ds ≤ κp , G(x, r)

that is, W (h p)(x) ≤ κp h (x) for x ∈ (, ]. Similarly, we prove that W (h p)(x) ≤ κp h (x) by observing that lim

r→

G(s, r) h (s) = . G(x, r) h (x)

This ends the proof.



Corollary . Let α, β ∈ (, ] and ϕ ∈ B + ((, )). Then x → W ϕ(x) ∈ C–β ([, ]) if and  only if  ( – s)α– ϕ(s) ds < ∞. Proof The assertion follows from (.) and the dominated convergence theorem.



Proposition . Let α, β ∈ (, ] and ϕ ∈ B + ((, )) be such that s → ( – s)α– ϕ(s) ∈ C((, )) ∩ L ((, )). Then W ϕ is the unique nonnegative solution in C–β ([, ]) of ⎧ ⎨Dα (Dβ u)(x) = ϕ(x),

 < x < , ⎩limx→+ Dβ– u(x) = limx→+ Dα– (Dβ u)(x) = u() = Dβ u() = . Proof Let ϕ ∈ B + ((, )). From (.) and the Fubini-Tonelli theorem we obtain  W ϕ(x) =



Gβ (x, t)Gα ϕ(t) dt,

(.)



 where Gα ϕ(t) =  Gα (t, s)ϕ(s) ds. Since the function s → ( – s)α– ϕ(s) ∈ C((, )) ∩ L ((, )), we deduce by Proposition . that Gα ϕ is the unique solution in C–α ([, ]) of ⎧ ⎨Dα v(x) = –ϕ(x),

 < x < , ⎩limx→+ Dα– v(x) = v() = .

(.)

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On the other hand, by using (.) we deduce that 



( – t)

β–



 Gα ϕ(t) dt ≤ (α)







( – t)





β–

t



α–

( – s)

α–

ϕ(s) ds dt



(β) ≤ (α – )(α + β – )





( – s)α– ϕ(s) ds < ∞.



Hence, the function t → ( – t)β– Gα ϕ(t) ∈ C((, )) ∩ L ((, )). Therefore, using (.) and Proposition ., we deduce that W ϕ is the unique solution in C–β ([, ]) of ⎧ ⎨Dβ u(x) = –G ϕ(x),  < x < , α ⎩limx→+ Dβ– u(x) = u() = .

(.)



Combining (.) and (.), we obtain the required result.

3 Proofs of main results Let α, β ∈ (, ]. For (x, s) ∈ (, ] × [, ], put H (x, s) = G(x, s) and 



Hn (x, s) =

G(x, t)Hn– (t, s)p(t) dt,

n ≥ .

(.)



Now, let H : (, ] × [, ] → R be defined by

H(x, s) =

∞ (–)n Hn (x, s),

(.)

n=

provided that the series converges. Lemma . Let α, β ∈ (, ] and m, M >  be as in (.). Let p ∈ Jα,β with κp < . Then on (, ] × [, ], we have (i) Hn (x, s) ≤ κpn G(x, s) for each n ∈ N. So, H(x, s) is well defined in (, ] × [, ]. (ii) For each n ∈ N, ln xβ– ( – x)( – s)α– ≤ Hn (x, s) ≤ rn xβ– ( – x)( – s)α– ,

(.)

where

 ln = mn+

n



t β– ( – t)α p(t) dt





n



and rn = Mn+

t β– ( – t)α p(t) dt 

 (iii) Hn+ (x, s) =  Hn (x, t)G(t, s)p(t) dt for each n ∈ N.   (iv)  H(x, t)G(t, s)p(t) dt =  G(x, t)H(t, s)p(t) dt. Proof By simple induction we prove (i), (ii), and (iii). (iv) By Lemma .(i) we have  ≤ Hn (x, t)G(t, s)p(t) ≤ κpn G(x, t)G(t, s)p(t) for n ≥  and all x, t, s ∈ (, ].

.

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 Therefore, the series n≥  Hn (x, t)G(t, s)p(t) dt converges. So by applying the dominated convergence theorem we deduce that 



H(x, t)G(t, s)p(t) dt =



∞  n=

=

n= 

=

(–)n Hn (x, t)G(t, s)p(t) dt



∞ 







(–)n G(x, t)Hn (t, s)p(t) dt



G(x, t)H(t, s)p(t) dt.





Proposition . Let α, β ∈ (, ] and p ∈ Jα,β with κp < . Then the function (x, s) → x–β H(x, s) ∈ C([, ] × [, ]). Proof Clearly, the function (x, s) → x–β H (x, s) ∈ C([, ] × [, ]). Assume that the function (x, s) → x–β Hn– (x, s) ∈ C([, ] × [, ]). Using Lemma .(i) and (.), we have, for all (x, s, t) ∈ [, ] × [, ] × (, ], x–β G(x, t)Hn– (t, s)p(t) ≤ κpn– x–β G(x, t)G(t, s)p(t) ≤ M ( – x)( – t)α– t β– ( – t)( – s)α– p(t) ≤ M t β– ( – t)α p(t). So by (.) and the dominated convergence theorem we conclude that the function (x, s) → x–β Hn (x, s) ∈ C([, ] × [, ]). From Lemma .(i) and (.) we deduce that x–β Hn (x, s) ≤ κpn x–β G(x, s) ≤ Mκpn .

(.)

Therefore, the series n≥ (–)n x–β Hn (x, s) is uniformly convergent on [, ] × [, ], and so the function (x, s) → x–β H(x, s) ∈ C([, ] × [, ]).  Lemma . Let α, β ∈ (, ] and p ∈ Jα,β with κp ≤  . Then for (x, s) ∈ (, ] × [, ], we have ( – κp )G(x, s) ≤ H(x, s) ≤ G(x, s).

(.)

Proof Let p ∈ Jα,β with κp ≤  . By Lemma .(i) we deduce that ∞   H(x, s) ≤ (κp )n G(x, s) = n=

 G(x, s).  – κp

(.)

Now, from the expression of H we have

H(x, s) = G(x, s) –

∞ (–)n Hn+ (x, s). n=

(.)

Bachar et al. Boundary Value Problems (2016) 2016:79

Since the series that

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 n≥  G(x, t)Hn (t, s)p(t) dt

H(x, s) = G(x, s) –

∞







(–)n

G(x, t)Hn (t, s)p(t) dt 

n= 

= G(x, s) –

converges, we conclude by (.) and (.)

G(x, t) 

∞   (–)n Hn (t, s) p(t) dt, n=

namely,   H(x, s) = G(x, s) – W pH(·, s) (x).

(.)

On the other hand, since   W pH(·, s) (x) ≤ =

   W pG(·, s) (x)  – κp κp  H (x, s) ≤ G(x, s),  – κp  – κp

(.)

we deduce that

H(x, s) ≥ G(x, s) –

κp  – κp G(x, s) = G(x, s) ≥ .  – κp  – κp

Hence, H(x, s) ≤ G(x, s), and by (.) we have   H(x, s) ≥ G(x, s) – W pG(·, s) (x) ≥ ( – κp )G(x, s).



Corollary . Let α, β ∈ (, ] and p ∈ Jα,β with κp ≤  . Let ϕ ∈ B + ((, )). Then   Wp ϕ ∈ C–β [, ]

 if and only if



( – s)α– ϕ(s) ds < ∞.



Proof The assertion follows from Proposition ., (.), and (.).



Lemma . Let α, β ∈ (, ] and p ∈ Jα,β with κp ≤  . Let h ∈ B + ((, )). Then we have, for x ∈ (, ], Wh(x) = Wp h(x) + Wp (pWh)(x) = Wp h(x) + W (pWp h)(x).

(.)

In particular, if W (ph) < ∞, then 

     I – Wp (p·) I + W (p·) h = I + W (p·) I – Wp (p·) h = h.

Proof Let (x, s) ∈ (, ] × [, ]. Then by (.) we have   G(x, s) = H(x, s) + W pH(·, s) (x).

(.)

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Let h ∈ B + ((, )). Using the Fubini theorem, we obtain 



   H(x, s) + W pH(·, s) (x) h(s) ds

Wh(x) = 

= Wp h(x) + W (pWp h)(x). Using Lemma .(iv) and again the Fubini theorem, we have   



H(x, t)G(t, s)p(t)h(s) dt ds =



  



G(x, t)H(t, s)p(t)h(s) dt ds,



that is, Wp (pWh)(x) = W (pWp h)(x). So Wh(x) = Wp h(x) + W (pWp h)(x) = Wp h(x) + Wp (pWh)(x).



Proposition . Let α, β ∈ (, ] and p ∈ Jα,β ∩ C((, )) with κp ≤  . Let ϕ ∈ B + ((, )) be such that s → ( – s)α– ϕ(s) ∈ C((, )) ∩ L ((, )). Then Wp ϕ ∈ C–β ([, ]), and it is the unique nonnegative solution of the problem ⎧ ⎨Dα (Dβ u)(x) + p(x)u(x) = ϕ(x),

 < x < ,

⎩limx→+ Dβ– u(x) = limx→+ Dα– (Dβ u)(x) = u() = Dβ u() = ,

(.)

satisfying ( – κp )W ϕ ≤ u ≤ W ϕ.

(.)

Proof By Corollary . the function x → p(x)Wp ϕ(x) ∈ C((, )). Using (.) and (.), we have that there exists c ≥  such that  Wp ϕ(x) ≤ W ϕ(x) ≤ M



xβ– ( – x)( – s)α– ϕ(s) ds = cxβ– ( – x).

(.)



Therefore,  







( – s)α– p(s)Wp ϕ(s) ds ≤ c

sβ– ( – s)α p(s) ds < ∞.



Hence, by Proposition . the function u = Wp ϕ = W ϕ – W (pWp ϕ) satisfies the equation ⎧ ⎨Dα (Dβ u)(x) = ϕ(x) – p(x)u(x),

 < x < , ⎩limx→+ Dβ– u(x) = limx→+ Dα– (Dβ u)(x) = u() = Dβ u() = . By integration of inequalities (.) we obtain (.).

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Let us prove the uniqueness. Let v ∈ C–β ([, ]) be another solution of problem (.) satisfying v ≤ W ϕ. Put v˜ := v + W (pv). Since the function s → ( – s)α– p(s)v(s) ∈ C((, )) ∩ L ((, )), then by Proposition . it follows that ⎧ ⎨Dα (Dβ v˜ )(x) = ϕ(x),

 < x < , ⎩limx→+ Dβ– v˜ (x) = limx→+ Dα– (Dβ v˜ )(x) = v˜ () = Dβ v˜ () = . From the uniqueness in Proposition . we conclude that v˜ := v + W (pv) = W ϕ. So 

     I + W (p·) (v – u)+ = I + W (p·) (v – u)– ,

where (v – u)+ = max(v – u, ) and (v – u)– = max(u – v, ). From (.), (.), (.), and (.), there exists a constant c˜ > , such that   W p|v – u| ≤ ˜cW (pθ ) ≤ ˜cκp θ < ∞. 

Therefore, u = v by Lemma ..

Proof of Theorem . Consider ξ ≥  and ζ ≥  with ξ + ζ > . Let α, β ∈ (, ] and p ∈ Jα,β ∩ C((, )) be such that (A ) is satisfied. Let     S := u ∈ B + (, ) : ( – κp )θ ≤ u ≤ θ , where θ (x) := ξ h (x) + ζ h (x), and h and h are defined respectively by (.) and (.). Define the operator F on S by    F u = θ – Wp (pθ ) + Wp p – f (·, u) u . By (.) and (.) we have Wp (pθ ) ≤ W (pθ ) ≤ κp θ ≤ θ .

(.)

Using (A ), we get  ≤ f (·, u) ≤ p

for all u ∈ S .

Next, we prove that FS ⊆ S . Indeed, using (.) and (.), we have, for u ∈ S ,

F u ≤ θ – Wp (pθ ) + Wp (pu) ≤ θ

(.)

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and

F u ≥ θ – Wp (pθ ) ≥ ( – κp )θ . Observe that, by (A ), F becomes nondecreasing on S . Define the sequence {vn } by v = ( – κp )θ and vn+ = F vn for n ∈ N. Since FS ⊆ S , we have v = F v ≥ v , and by the monotonicity of F we deduce that ( – κp )θ = v ≤ v ≤ · · · ≤ vn ≤ vn+ ≤ θ . Using (A )-(A ) and the dominated convergence theorem, we deduce that the sequence {vn } converges to a function u ∈ S satisfying      u = I – Wp (p·) θ + Wp p – f (·, u) u , that is, 

     I – Wp (p·) u = I – Wp (p·) θ – Wp uf (·, u) ,

and by (.) we have W (pu) ≤ W (pθ ) ≤ θ < ∞. Therefore, by Lemma . we deduce that   u = θ – W uf (·, u) .

(.)

We claim that u is a solution. Indeed, from (.) and (.), there exists a constant c >  such that   ( – s)α– u(s)f s, u(s) ≤ ( – s)α– θ (s)p(s) ≤ csβ– ( – s)α p(s).

(.)

So, by Proposition . the function W (uf (·, u)) ∈ C–β ([, ]). This implies by (.) that u ∈ C–β ([, ]). Now, since the function s → ( – s)α– u(s)f (s, u(s)) ∈ C((, )) ∩ L ((, )), we deduce by Proposition . that u is a solution. It remains to prove the uniqueness. Let v be another solution in C–β ([, ]) to problem (.) satisfying (.). Since v ≤ θ , we deduce by (.) that    ≤ v(s)f s, v(s) ≤ θ (s)p(s) ≤ csβ– ( – s)p(s). This implies that s → ( – s)α– v(s)f (s, v(s)) ∈ C((, )) ∩ L ((, )). Let v˜ := v + W (vf (·, v)). By Proposition ., we have ⎧ α β ⎪ ⎪ ⎨D (D v˜ )(x) = , ⎪ ⎪ ⎩

limx→+ D v˜ () = ,

β–

 < x < ,

v˜ (x) = , Dβ v˜ () = –ζ .

limx→+ Dα– (Dβ v˜ )(x) = ξ ,

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Hence,   v = θ – W vf (·, v) .

(.)

Let ω : (, ) → R be defined by

ω(z) =

⎧ ⎨ v(z)f (z,v(z))–u(z)f (z,u(z))

if v(z) = u(z),

⎩

if v(z) = u(z).

v(z)–u(z)

By (A ), ω ∈ B + ((, )) and from (.) and (.) we deduce that 

     I + W (ω·) (v – u)+ = I + W (ω·) (v – u)– ,

where (v – u)+ = max(v – u, ) and (v – u)– = max(u – v, ). From (A ) we have ω ≤ p. So by using (.) and (.) we obtain   W ω|v – u| ≤ W (pθ ) ≤ κp θ < ∞. 

Hence, u = v by (.).

Proof of Corollary . The statement follows from Theorem . with f (x, t) = λq(x)h(t),  (t) = th(t) and p(x) := λq(x) max≤t≤θ(x)  (t). Example . Let σ ≥ , ν ≥ , and q ∈ C + ((, )) be such that 



t (β–)(+σ +ν) ( – t)α+σ +ν q(t) dt < ∞.



Let (t) = t σ + ln( + t ν ) and q˜ (t) := q(t) max≤s≤θ(t)  (s). Since q˜ ∈ Jα,β , then for ξ ≥ , ζ ≥  with ξ + ζ >  and λ ∈ [, κ ), the problem q˜

⎧ α β σ + ν ⎪ ⎪ ⎨D (D u)(x) + λq(x)u (x) ln( + u (x)) = , ⎪ ⎪ ⎩

limx→+ D u() = ,

β–

u(x) = ,

limx→+ D

α–

 < x < ,

β

(D u)(x) = ξ ,

β

D u() = –ζ ,

has a unique solution u ∈ C–β ([, ]) such that ( – λαq˜ )θ (x) ≤ u(x) ≤ θ (x) for x ∈ (, ]. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia. 2 College of Sciences and Arts, Department of Mathematics, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh, 21911, Saudi Arabia. 3 Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, P.O. Box 1-764, Bucharest, 014700, Romania. 4 Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, Craiova, 200585, Romania.

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Acknowledgements The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Research group NO (RG-1435-043). The authors would like to thank the anonymous referees for their careful reading of the paper. Received: 16 February 2016 Accepted: 5 April 2016 References 1. Agarwal, RP, O’Regan, D, Stanˇek, S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57-68 (2010) 2. Bachar, I, Mâagli, H: Positive solutions for superlinear fractional boundary value problems. Adv. Differ. Equ. 2014, 240 (2014) 3. Bai, Z, Lü, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005) 4. Giacomoni, J, Kumar Mishra, P, Sreenadh, K: Fractional elliptic equations with critical exponential nonlinearity. Adv. Nonlinear Anal. 5, 57-74 (2016) 5. Graef, JR, Kong, L, Kong, Q, Wang, M: Existence and uniqueness of solutions for a fractional boundary value problem with Dirichlet boundary condition. Electron. J. Qual. Theory Differ. Equ. 2013, 55 (2013). doi:10.14232/ejqtde.2013.1.55 6. Kaufmann, ER, Mboumi, E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 2008, 3 (2008) 7. Mâagli, H, Mhadhebi, N, Zeddini, N: Existence and estimates of positive solutions for some singular fractional boundary value problems. Abstr. Appl. Anal. 2014, Article ID 120781 (2014) 8. Mâagli, H, Mhadhebi, N, Zeddini, N: Existence and exact asymptotic behavior of positive solutions for a fractional boundary value problem. Abstr. Appl. Anal. 2013, Article ID 420514 (2013) 9. Metzler, R, Klafter, J: Boundary value problems for fractional diffusion equations. Physica A 278, 107-125 (2000) 10. Pucci, P, Xiang, M, Zhang, B: Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations. Adv. Nonlinear Anal. 5, 27-55 (2016) 11. Xu, X, Jiang, D, Yuan, C: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal. 71, 4676-4688 (2009) 12. Zhang, X, Liu, L, Wu, Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55, 1263-1274 (2012) 13. Diethelm, K, Freed, AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In: Keil, F, Mackens, W, Voss, H, Werther, J (eds.) Scientific Computing in Chemical Engineering. II. Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217-224. Springer, Heidelberg (1999) 14. Gaul, L, Klein, P, Kempfle, S: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81-88 (1991) 15. Glockle, WG, Nonnenmacher, TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46-53 (1995) 16. Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) 17. Kilbas, A, Srivastava, H, Trujillo, J: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) 18. Kilbas, AA, Trujillo, JJ: Differential equations of fractional order: methods, results and problems. I. Appl. Anal. 78, 153-192 (2001) 19. Kilbas, AA, Trujillo, JJ: Differential equations of fractional order: methods, results and problems. II. Appl. Anal. 81, 435-493 (2002) 20. Mainardi, F: Fractional calculus: some basic problems in continuum and statical mechanics. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Calculus Mechanics, pp. 291-348. Springer, Vienna (1997) 21. Miller, K, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) 22. Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999) 23. Samko, S, Kilbas, A, Marichev, O: Fractional Integrals and Derivative. Theory and Applications. Gordon and Breach, Yverdon (1993) 24. Tarasov, V: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, New York (2011)