This is apreprint of a paperwhose final anddefiniteform will bepublished in Journal of Computational and Applied Mathematics, ISSN: 0377-0427. Paper Submitted 04/Jul/2015; Revised 14/Dec/2015 and 03/Jan/2016; Accepted for publication 08/Jan/2016. EXISTENCE OF SOLUTION TO A LOCAL FRACTIONAL NONLINEAR 6 DIFFERENTIAL EQUATION 1 0 2 BENAOUMEURBAYOURANDDELFIMF.M.TORRES n a Abstract. We prove existence of solution to a local fractional nonlinear differential equation J withinitialcondition. Forthatweintroducethenotionoftubesolution. 9 ] A C 1. Introduction . h Fractional calculus is a branch of mathematical analysis that studies the possibility of taking t nonintegerorderpowersofthedifferentiationand/orintegrationoperators. Eventhoughtheterm a m “fractional” is a misnomer, it has been widely accepted for a long time: the term was coined by the famous mathematician Leibniz in 1695 in a letter to L’Hopital [26]. In the paper What [ is a fractional derivative? [23], Ortigueira and Machado distinguish between local and nonlocal 1 fractional derivatives. Here we are concerned with local operators only. Such local approach to v the fractionalcalculus dates backatleastto 1974,to the use ofthe fractionalincrementalratioin 6 [9]. For an overview and recent developments of the localapproachto fractional calculus we refer 2 1 the reader to [22, 24, 27, 28] and references therein. 2 Recently, Khalil et al. introduced a new well-behaved definition of local fractional (noninteger 0 order) derivative, called the conformable fractional derivative [21]. The new calculus is very . 1 interesting and is getting an increasing of interest – see [7, 11] and references therein. In [1], 0 Abdeljawadproveschainrules,exponentialfunctions,Gronwall’sinequality,fractionalintegration 6 by parts, Taylor power series expansions and Laplace transforms for the conformable fractional 1 calculus. Furthermore, linear differential systems are discussed [1]. In [5], Batarfi et al. obtain : v the Green function for a conformable fractional linear problem and then introduce the study of i X nonlinear conformable fractional differential equations. See also [3] where, using the conformable fractionalderivative,asecond-orderconjugateboundaryvalueproblemisinvestigatedandutilizing r a the corresponding positive fractional Green’s function and an appropriate fixed point theorem, existence ofapositive solutionis proved. ForabstractCauchyproblemsofconformablefractional systems see [2]. Here we are concerned with the following problem: x(α)(t)=f(t,x(t)), t∈[a,b], a>0, (1) (x(a)=x0, where f : [a,b]× R → R is a continuous function, x(α)(t) denotes the conformable fractional derivative of x at t of order α, α ∈ (0,1). For the first time in the literature of conformable fractional calculus, we introduce the notion of tube solution. Such idea of tube solution has been used with success to investigate existence of solutions for ordinary differentiable equations [12, 13], delta and nabla differential equations on time scales [6, 14, 16], and dynamic inclusions [15]. Roughly speaking, the tube solution method generalizes the method of lower and upper solution [8, 10, 17, 25]. 2010 Mathematics Subject Classification. 26A33,34A12. Key words and phrases. Existence of solutions, fractional differential equations, local fractional derivatives, conformablefractionalderivatives,initialvalueproblems. 1 2 B.BAYOURANDD.F.M.TORRES The paper is organized as follows. In Section 2, we present the main concepts of the local conformable fractional calculus and we give some useful preliminary results. In Section 3, we prove existence of solution to problem (1) by using the notion of tube solution and Schauder’s fixed-point theorem (see Theorem 19). We end with Section 4, where an illustrative example is given. 2. Preliminaries We consider fractional derivatives in the conformable sense [21]. Definition 1 (Conformable fractional derivative [21]). Let α ∈ (0,1) and f : [0,∞) → R. The conformable fractional derivative of f of order α is defined by Tα(f)(t) := limǫ 0 f(t+ǫt1−ǫα)−f(t) for all t>0. Often, we write f(α) instead of T (f) to denote the conformable fra→ctional derivative α of f of order α. In addition, if the conformable fractional derivative of f of order α exists, then we simply say that f is α-differentiable. If f is α-differentiable in some t ∈ (0,a), a > 0, and lim f(α)(t) exists, then we define f(α)(0):=lim f(α)(t). t 0+ t 0+ → → Theorem 2 ([21]). Let α∈(0,1] and assume f,g to be α-differentiable. Then, (1) T (af +bg)=aT (f)+bT (g) for all a,b∈R; α α α (2) T (fg)=fT (g)+gT (f); α α α (3) T (f/g)=(gT (f)−fT (g))/g2. α α α If, in addition, f is differentiable at a point t>0, then T (f)(t)=t1 αdf(t). α − dt Remark 3. From Theorem 2 it follows that if f ∈C1, then one has lim T (f)(t)=f (t) α ′ α 1 → and lim T (f)(t)=tf (t). (2) α ′ α 0 → So T (f) is “conformable” in the sense it coincides with f in the case α→1 and satisfies similar α ′ properties totheinteger-ordercalculus. Notethatthepropertylim T (f)6=f isnotuncommon α 0 α in fractional calculus, both for local and nonlocal operators: see, e.g→., the local fractional derivative of [19, 20], for which property (2) also holds [4]; and the classical nonlocal Marchaud fractional derivative, which is zero when α→0 [26]. Note, however, that we only have T (f)(t)=t1 αf (t) α − ′ in case f is differentiable. If one considers a function that is not differentiable at a point t, then the conformable derivative is not t1 αf (t). For applications we refer the reader to [11]. − ′ Example 4. Let α ∈ (0,1]. Functions f(t)= tp, p ∈ R, g(t) ≡ λ, λ ∈ R, h(t) = ect, c ∈ R, and β(t)=eα1tα, are α-differentiable with conformable fractional derivatives of order α given by (1) T (f)(t)=ptp α; α − (2) T (g)(t)=0; α (3) T (h)(t)=ct1 αect; α − (4) Tα(β)(t)=eα1tα. Remark 5. Differentiability implies α-differentiability but the contrary is not true: a nondiffer- entiable function can be α-differentiable. For a discussion of this issue see [21]. Definition 6 (Conformable fractional integral [21]). Let α ∈ (0,1) and f : [a,∞) → R. The conformable fractional integral of f of order α from a to t, denoted by Ia(f)(t), is defined by α t f(τ) Ia(f)(t):= dτ, α τ1 α Za − where the above integral is the usual improper Riemann integral. Theorem 7 ([21]). If f is a continuous function in the domain of Ia, then T (Ia(f))(t)=f(t) α α α for all t≥a. EXISTENCE OF SOLUTION TO A LOCAL FRACTIONAL DIFFERENTIAL EQUATION 3 Notation 8. Let 0 < a < b. We denote by Jb[f] the value of the integral b f(t)dt, that is, α a a t1−α Jb[f]:=Ia(f)(b). α a α R Proposition 9. Assume f ∈L1([a,b],R), 00, the set of all real-valued functions f : [a,b] → R that are α-differentiable and for which the α-derivative is continuous. We often abbreviate C(α)([a,b],R) by C(α)([a,b]). The next lemma is a consequence of the conformable mean value theorem proved in [21] by noting the discussionunder Definition2.1in [1]. Note thatr(b)−r(a)=Ia r(α) (b) followsfrom α Lemma 2.8 in [1]. (cid:0) (cid:1) Lemma 11. Let r ∈ C(α)([a,b]), 0 < a < b, such that r(α)(t) < 0 on {t∈[a,b]:r(t)>0}. If r(a)≤0, then r(t)≤0 for every t∈[a,b]. Proof. Supposethecontrary. Ifthereexistst∈[a,b]suchthatr(t)>0,thenthereexistst ∈[a,b] such that r(t )=max (r(t)) >0 because r ∈C(α)([a,b]) and r(t) >0. There are tw◦o cases. a t b ◦ ≤ ≤ (i) if t > a, then there exists an interval [t ,t ] included in [a,t ] such that r(t) > 0 for all 1 t ∈ [t ◦,t ]. It follows from the assumption r(α)(◦t) < 0 for all t ∈◦[t ,t ] and Lemma 2.8 of [1] 1 1 that Iαt1 ◦r(α) (t )=r(t )−r(t1)<0, which contradicts the fact that r(◦t ) is a maximum. (ii) If t =a, then r(t◦)>0 is◦impossible from hypothesis. ◦ (cid:3) ◦ (cid:0) (cid:1) ◦ Theorem 12. If g ∈L1([a,b]), then function x:[a,b]→R defined by x(t):=e−α1(at)α(cid:18)eα1x0+αJta(cid:20)e−gα1((sas))α(cid:21)(cid:19) (3) is solution to problem x(α)(t)+ 1 x(t)=g(t), t∈[a,b], a>0, aα (x(a)=x0. Proof. Let x:[a,b]→R be the function defined by (3). We know from Theorems 2 and 7 that α x(α)(t)=t1−α(cid:18)−α1 (cid:18)a1(cid:19) αtα−1(cid:19)e−α1(at)α(cid:18)eα1x0+αJta(cid:20)e−gα1((sas))α(cid:21)(cid:19)+e−α1(at)α(cid:18)e−gα1((ta)t)α(cid:19) α =−(cid:18)a1(cid:19) e−α1(at)α(cid:18)eα1x0+αJta(cid:20)e−gα1((sas))α(cid:21)(cid:19)+g(t) α 1 =− x(t)+g(t). a (cid:18) (cid:19) We just obtained that x(α)(t)+ 1 αx(t)=g(t). On the other hand, a x(a)=e−α1(aa)α (cid:0)eα1(cid:1)x0+αJaa g1((ss))α =e−α1 eα1x0+0 =x0 (cid:18) (cid:20)e−α a (cid:21)(cid:19) (cid:16) (cid:17) and the proof is complete. (cid:3) Theorem 12 is enough for our purposes. It should be mentioned, however, that it can be generalizedby benefiting from Lemma 2.8 in [1] with its higher-orderversion[1, Proposition2.9]. 4 B.BAYOURANDD.F.M.TORRES Theorem 13. If g ∈ L1([a,b]) and p(t) is continuous on [a,b], then the function x : [a,b] → R defined by 1 x(t)= (x(a)µ(a)+Ia(µg)(t)) (4) µ(t) α is a solution to the linear conformable equation x(α)(t)+p(t)x(t)=g(t), x(a)=x , a>0. (5) 0 Proof. Consider the integrating factor function µ(t) = eIαa(p)(t). Then, by means of item (3) of Example 4 and the Chain Rule [1, Theorem 2.11], one can see that µ(α)(t) = p(t)µ(t). Then, multiply (5) by function µ(t). By means ofthe productrule (item (2)of Theorem2), (5) turns to (x(t)µ(t))(α) =µ(t)g(t). (6) Apply Ia to (6) and use Lemma 2.8 in [1] to conclude that (4) holds. (cid:3) α Theorem 12 follows as a corollary from Theorem 13 by putting µ(t)=eα1(at)α and p(t)=eα1. Proposition 14. If x:(0,∞)→R is α-differentiable at t∈[a,b], then |x(t)|(α) = x(t)xα(t). x(t) | | Proof. From Definition 1 we have |x(t+ǫt1 α)|−|x(t)| |x(t)|(α) = lim − ǫ 0 ǫ → x t+ǫt1 α 2−x(t)2 − = lim ǫ→0ǫ(|x(cid:0)(t+ǫt1−α(cid:1))|+|x(t)|) x(t+ǫt1 α)2−x(t)2 1 − = lim · ǫ→0(cid:20) ǫ |x(t+ǫt1−α)|+|x(t)|(cid:21) = x(t)2 (α) 1 2|x(t)| (cid:2) (cid:3) 1 =2x(t)x(α)(t) , 2|x(t)| which proves the intended relation. (cid:3) 3. Main Result We begin by introducing the notion of tube solution to problem (1). Definition 15. Let (v,M) ∈ C(α)([a,b],R)×C(α)([a,b],[0,∞)). We say that (v,M) is a tube solution to problem (1) if (i) (y−v(t)) f(t,y)−v(α) ≤ M(t)M(α)(t) for every t ∈ [a,b] and every y ∈ R such that |y−v(t)|=M(t); (cid:0) (cid:1) (ii) v(α)(t)=f(t,v(t)) and M(α)(t)=0 for all t∈[a,b] such that M(t)=0; (iii) |x −v(a)|≤M(a). 0 Notation 16. We introduce the following notation: T(v,M):= x∈C(α)([a,b],R):|x(t)−v(t)|≤M(t), t∈[a,b] . n o Consider the following problem: x(α)+ 1 x(t)=f(t,x(t))+ 1 x(t), t∈[a,b], a>0, aα aα (7) (x(a)=x0, e e where M(t) (x(t)−v(t))+v(t) if |x−v(t)|>M(t), x(t):= x v(t) (8) | − | (x(t) otherwise. e EXISTENCE OF SOLUTION TO A LOCAL FRACTIONAL DIFFERENTIAL EQUATION 5 Let us define the operator N:C([a,b])→C([a,b]) by N(x)(t):=e−α1(at)α(cid:18)eα1x0+αJta(cid:20)f(s,xe(−s)α1)(+as)αa1αx(s)(cid:21)(cid:19). e e In the proof of Proposition 18 we use the concept of compact function. Definition 17 (See p. 112 of [18]). Let X, Y be topological spaces. A map f : X → Y is called compact if f(X) is contained in a compact subset of Y. Compact operators occur in many problems of classical analysis. Note that operator N is nonlinear because f is nonlinear. In the nonlinear case, the first comprehensive research on compact operators was due to Schauder [18, p. 137]. In this context, the Arzela`–Ascoli theorem asserts that a subset is relatively compact if and only if it is bounded and equicontinuous [18, p. 607]. Proposition 18. If (v,M) ∈ C(α)([a,b],R)×C(α)([a,b],[0,∞)) is a tube solution to (1), then N:C([a,b])→C([a,b]) is compact. Proof. Let {xn}n N be a sequence of C([a,b],R) convergingto x∈C([a,b],R). By Proposition9, ∈ |N(xn(t))−N(x(t))|=(cid:12)(cid:12)e−α1(as)α(cid:18)eα1x0+αJta(cid:20)f(s,xne(−s)α1)(+as)αa1αxn(s)(cid:21)(cid:19) (cid:12) e e −e−α1(as)(cid:12)(cid:12)α eα1x0+αJta f(s,x(s)1)(+s)aα1αx(s) (cid:18) (cid:20) e−α a (cid:21)(cid:19)(cid:12)(cid:12) K 1 e e (cid:12)1 ≤ CαJta f(s,xn(s))+ aαxn(s) − f(s,x(s))+ a(cid:12)(cid:12)αx(s) (cid:20)(cid:12)(cid:18) (cid:19) (cid:18) (cid:19)(cid:12)(cid:21) K (cid:12) 1 (cid:12) ≤ C αJta(cid:12)(cid:12)[|f(s,xen(s))−f(s,ex(s))|]+ aααJtae[|xn(s)−xe(s)|](cid:12)(cid:12) , (cid:18) (cid:19) where K := maxa s b{e−α1(as)α} andeC := minaes b{e−α1(as)α}. eWe needeto show that the sequence {gn}n N d≤efi≤ned by gn(s) := f(s,xn(s))+≤a1α≤xn(s) converges in C(α)([a,b]) to function g(s) = f(s,x(s)∈)+ a1αx(s). Since there is a constant R > 0 such that kxkC([a,b],R) < R, there exists an index N such that kxnkC([a,b],R) ≤eR for all ne>N. Thus, f is uniformly continuous on [a,b]×BR(0e). Thereforee, for ǫ>0 given, there is a δ >0 such that e e Cǫαaα |y−x|<δ < 2k(bα−aα) for all x,y ∈R; Cǫα |f(s,y)−f(s,x)|< 2k(bα−aα) for all s∈[a,b]. By assumption, one canfind an index Nˆ >N such thatkxn−xkC([a,b],R) <δ for n>Nˆ. In this case, e e K Cǫα 1 Cǫαaα |N(x )(t)−N(x)(t)|< Jb + Jb n C α a 2k(bα−aα) aαα a 2k(bα−aα) (cid:18) (cid:20) (cid:21) (cid:20) (cid:21)(cid:19) 2KCǫα = Jb[1] 2kC(bα−aα)α a ǫα bα−aα = bα−aα α =ǫ. This proves the continuity of N. We now show that the set N(C([a,b])) is relatively compact. Consider a sequence {yn}n N of N(C([a,b])) for all n ∈ N. It exists xn ∈ C([a,b]) such that ∈ 6 B.BAYOURANDD.F.M.TORRES y =N(x ). Observe that from Proposition 9 we have n n |N(xn)(t)|= e−α1(at)α eα1x0+αJta f(s,xn(s)1)(+s)aα1αxn(s) (cid:12) (cid:18) (cid:20) e−α a (cid:21)(cid:19)(cid:12) ≤K(cid:12)(cid:12)(cid:12) eα1|x0|+ C1αJba f(t,xn(es))+ a1αxn(es) (cid:12)(cid:12)(cid:12) (cid:18) (cid:20)(cid:12) (cid:12)(cid:21)(cid:19) 1 (cid:12) 1 (cid:12) ≤K eα1|x0|+ CαJba[|(cid:12)(cid:12)f(t,xen(s))|]+ CaeααJba(cid:12)(cid:12)[|xn(s)|] . (cid:18) (cid:19) Bydefinition,thereisanR>0suchthat|xn(s)|≤Reforalls∈[a,b]andealln∈N. The function f is compact on [a,b]×B (0) and we can deduce the existence of a constant A > 0 such that R |f(s,xn(s)| ≤ A for all s ∈ [a,b]. The seequence {yn}n N is uniformly bounded for all n ∈ N. ∈ Observe also that for t ,t ∈[a,b] we have 1 2 e |N(x )(t )−N(x )(t )| n 2 n 1 ≤B e−α1(ta1)α −e−α1(ta2)α + K(AC+R´) αJtt21[1] M(t)}. If t ∈ A, then by virtue of Proposition 14 we have (x(t)−v(t)) x(α)(t)−v(α)(t) (|x(t)−v(t)|−M(t))(α) = −M(α)(t). |x(t)−v(t)| (cid:0) (cid:1) Therefore, since (v,M) is a tube solution to problem (1), we have on {t∈A:M(t)>0} that (|x(t)−v(t)|−M(t))(α) (x(t)−v(t)) x(α)(t)−v(α)(t) = −M(α)(t) |x(t)−v(t)| (cid:0) (cid:1) (x(t)−v(t)) f(t,x(t))+ 1 x(t)− 1 x(t) −v(α)(t) = aα aα −M(α)(t) |x(t)−v(t)| (cid:0) (cid:0) (cid:1) (cid:1) (x(t)−v(t)) f(t,xe(t))−v(α)e(t) (x(t)−v(t))(x(t)−x(t)) = + −M(α)(t) M(t) aαM(t) (cid:0) (cid:1) (xe(t)−v(t)) f(t,xe(t))−v(α)(t) e M(t) e |x(t)−v(t)|2 = + −1 −M(α)(t) M(t) |x(t)−v(t)| aα|x(t)−v(t)| (cid:0) (cid:1) (cid:20) (cid:21) (xe(t)−v(t)) f(t,xe(t))−v(α)(t) M(t) |x(t)−v(t)| = + − −M(α)(t) M(t) aα aα (cid:0) (cid:1) (cid:20) (cid:21) Me(t)Mα(t) 1 e ≤ + [M(t)−|x(t)−v(t)|]−M(α)(t) M(t) aα <0. EXISTENCE OF SOLUTION TO A LOCAL FRACTIONAL DIFFERENTIAL EQUATION 7 On the other hand, we have on t∈{τ ∈A:M(τ)=0} that (x(t)−v(t)) f(t,x(t))+ 1 x(t)− 1 x(t) −v(α)(t) (|x(t)−v(t)|−M(t))(α) = aα aα −M(α)(t) |x(t)−v(t)| (cid:0) (cid:0) (cid:1) (cid:1) (x(t)−v(t)) f(t,xe(t))−v(α)e(t) 1 = − |x(t)−v(t)|−M(α)(t) |x(t)−v(t)| aα (cid:0) (cid:1) <−M(α)(t) e =0. The last equality follows from Definition 15. If we set r(t) := |x(t)−v(t)|−M(t), then r(α) < 0 on A := {t ∈ [a,b] : r(t) > 0}. Moreover, since (v,M) is a tube solution to problem (1) and x satisfies|x −v(a)|≤M(a),weknowthatr(a)≤0andLemma11implies thatA=∅. Therefore, 0 x∈T(v,M) and the theorem is proved. (cid:3) 4. An Example Consider the conformable noninteger order system x(12) =a√tx3(t)+bx(t)ecx(t), t∈[1,2], 1+t (9) (x(1)=0, where a,b∈ (∞,0] and c is a real constant. According to Definition 15, (v,M) ≡(0,1) is a tube solution. It follows from our Theorem 19 that problem (9) has a solution x such that |x(t)| ≤ 1 for every t∈[1,2]. Acknowledgments This research is part of first author’s Ph.D. project, which is carried out at Sidi Bel Abbes University, Algeria. It was carriedout while Bayourwas visiting the Department of Mathematics ofUniversityofAveiro,Portugal,FebruarytoAprilof2015. Thehospitalityofthehostinstitution andthe financialsupportofUniversityofChlef,Algeria,areheregratefullyacknowledged. Torres was supported by Portuguese funds through the Center for Research and Development in Math- ematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (FCT),within projectUID/MAT/04106/2013. The authorsareverygratefulto three anonymous Referees for their valuable comments, helpful questions and suggestions. References [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015), 57–66. arXiv:1402.6892 [2] T. Abdeljawad, M. Al Horani and R. Khalil, Conformable fractional semigroups of operators, Journal of SemigroupTheoryandApplications2015(2015), Art.ID7,9pp.arXiv:1502.06014 [3] D. R. Anderson and R. I. Avery, Fractional-order boundary value problem with Sturm-Liouville boundary conditions,Electron.J.DifferentialEquations 2015(2015), no.29,10pp.arXiv:1411.5622 [4] D.R.AndersonandD.J.Ulness,PropertiesoftheKatugampolafractionalderivativewithpotentialapplication inquantum mechanics, J.Math.Phys.56(2015), no.6,063502, 18pp. [5] H.Batarfi,J.Losada,J.J.NietoandW.Shammakh,Three-PointBoundaryValueProblemsforConformable FractionalDifferentialEquations,J.Funct. Spaces 2015(2015), Art.ID706383, 6pp. [6] B.Bayour,A.HammoudiandD.F.M.Torres,Existenceofsolutiontoanonlinearfirst-orderdynamicequation ontimescales,J.Math.Anal.7(2016), no.1,31–38.arXiv:1512.00909 [7] N.Benkhettou, S.HassaniandD.F.M.Torres,Aconformablefractionalcalculusonarbitrarytimescales,J. KingSaudUniv.Sci.28(2016), no.1,93–98. arXiv:1505.03134 [8] M.BohnerandC.C.Tisdell,Secondorderdynamicinclusions,J.NonlinearMath.Phys.12(2005), suppl.2, 36–45. [9] P. L. Butzer and U. Westphal, An access to fractional differentiation via fractional difference quotients, in Fractionalcalculusanditsapplications(Proc.Internat.Conf.,Univ.NewHaven,WestHaven,Conn.,1974), 116–145. LectureNotesinMath.,457,Springer,Berlin,1975. [10] A.Cabada,M.R.GrossinhoandF.Minhós,Extremalsolutionsforthird-ordernonlinearproblemswithupper andlowersolutionsinreversedorder,NonlinearAnal.62(2005), no.6,1109–1121. 8 B.BAYOURANDD.F.M.TORRES [11] W. S. Chung, Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math. 290(2015), 150–158. [12] M.Frigon,Boundaryandperiodicvalueproblemsforsystemsofnonlinearsecondorderdifferentialequations, Topol.MethodsNonlinearAnal.1(1993), no.2,259–274. [13] M.FrigonandH.Gilbert,Existencetheoremsforsystemsofthirdorderdifferentialequations,Dynam.Systems Appl.19(2010), no.1,1–23. [14] M. Frigon and H. Gilbert, Boundary value problems for systems of second-order dynamic equations on time scaleswith∆-Carathéodoryfunctions,Abstr.Appl.Anal.2010(2010), Art.ID234015, 26pp. [15] M.FrigonandH.Gilbert,Systemsoffirstorderinclusionsontimescales,Topol.MethodsNonlinearAnal.37 (2011), no.1,147–163. [16] H. Gilbert, Existence theorems for first-order equations on time scales with∆-Carathéodory functions, Adv. DifferenceEqu.2010(2010), Art.ID650827, 20pp. [17] J. R.Graef, L.Kong, F.M.Minhós andJ. Fialho,Onthe lowerand upper solutionmethod forhigher order functionalboundaryvalueproblems,Appl.Anal.DiscreteMath.5(2011), no.1,133–146. [18] A. Granas and J. Dugundji, Fixed point theory, Springer Monographs in Mathematics, Springer, New York, 2003. [19] U.N. Katugampola, Anew approach togeneralized fractional derivatives, Bull.Math. Anal. Appl.6(2014), no.4,1–15.arXiv:1106.0965 [20] U.N.Katugampola, Anewfractionalderivativewithclassicalproperties,preprint,2014.arXiv:1410.6535 [21] R.Khalil,M.AlHorani,A.Yousef,M.Sababheh,Anewdefinitionoffractionalderivative,J.Comput.Appl. Math.264(2014), 65–70. [22] R.L.Magin,Fractional calculus in Bioengineering,CRinBiomedicalEngineering32(2004), no.1,1–104. [23] M.D.OrtigueiraandJ.A.TenreiroMachado,Whatisafractionalderivative?,J.Comput.Phys.293(2015), 4–13. [24] D.Prodanov,FractionalvariationofHölderianfunctions,Fract.Calc.Appl.Anal.18(2015), no.3,580–602. [25] M. Ruyun, Z. Jihui and F. Shengmao, The method of lower and upper solutions for fourth-order two-point boundaryvalueproblems,J.Math.Anal.Appl.215(1997), no.2,415–422. [26] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach, Yverdon,1993. [27] X.J.Yang,Advanced local fractional calculus and itsapplications, WorldSciencePublisher,NewYork,2012. [28] X.-J.Yang,D.BaleanuandJ.A.T.Machado,ApplicationofthelocalfractionalFourierseriestofractalsignals, in Discontinuity and complexity in nonlinear physical systems, 63–89, Nonlinear Syst. Complex, Springer, Cham,2014. BenaoumeurBayour University of Chlef,B.P. 151,HayEs-salem, Chlef,Algeria E-mail address: b.benaoumeur@gmail.com DelfimF. M.Torres Center forResearch andDevelopmentin MathematicsandApplications (CIDMA), Departmentof Mathematics,University of Aveiro,3810-193Aveiro,Portugal E-mail address: delfim@ua.pt

Existence of Solution to a Local Fractional Nonlinear Differential Equation

Similar Existence of Solution to a Local Fractional Nonlinear Differential Equation