On the attractivity of solutions for a class of multi-term fractional functional differential equations

Accepted Manuscript On the attractivity of solutions for a class of multi-term fractional functional differential equations J. Losada, J.J. Nieto, E. Pourhadi PII: DOI: Reference:

S0377-0427(15)00376-3 http://dx.doi.org/10.1016/j.cam.2015.07.014 CAM 10232

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Received date: 20 May 2015 Please cite this article as: J. Losada, J.J. Nieto, E. Pourhadi, On the attractivity of solutions for a class of multi-term fractional functional differential equations, Journal of Computational and Applied Mathematics (2015), http://dx.doi.org/10.1016/j.cam.2015.07.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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On the attractivity of solutions for a class of multi-term fractional functional differential equations J. Losadaa , J. J. Nietoa,b , E. Pourhadic a Departamento

de An´ alise Matem´ atica, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain b Faculty of Science, King Abdulaziz University, P.O. Box 80203, 21589, Jeddah, Saudi Arabia c School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran

Abstract In this paper, we present some alternative results concerning with the existence and attractivity dependence of solutions for a class of nonlinear fractional functional differential equations. In our consideration, we apply the well-known Schauder fixed point theorem in conjunction with the technique of measure of noncompactness. Moreover, we provide some examples to illustrate the effectiveness of the obtained results. Mathematics Subject Classification (2010): 47H10; 34A08. Keywords: Schauder fixed point theorem; attractive solutions; fractional functional differential equations; measure of noncompactness.

1. Introduction During the last two decades, a great interest has been devoted to the study of fractional differential equations and such class pulled the interest of so many authors towards itself, motivated by their extensive use in mathematical modelling. Fractional calculus, which is as old as classical calculus, has found important applications in the study of problems in acoustics, thermal systems, rheology or modelling of materials and mechanical systems. Moreover, in some areas of science like signal processing, identification systems, control theory or robotics, fractional differential operators seem more suitable to model than the classical integer order operators. Due to this, fractional differential equations have also been used in models about biochemistry (modelling of polymers and proteins), electrical engineering (transmission of ultrasound waves), medicine (modelling of human tissue under mechanical loads), etc. Thus, differential and integral equations of fractional order play nowdays a very important role in describing some real world phenomena. In the recent years, the theory of fractional differential equations has been analytically investigated by a big number of very interesting and novel papers (see [4, 5, 6, 16, 19, 20, 23, 24, 25]). Moreover, fractional order differential equations, as functional differential equations [15], are related with causality principles and recently attractivity of solutions of fractional differential equations and fractional functional equations has been intensively investigated (see [8, 9, 10, 11, 12, 13, 17] for more details). The aim of this paper is to study the existence of solutions of a class of multi-term fractional functional equations in the space of bounded and continuous functions on an unbounded interval. Moreover, we investigate some important properties of the solutions related with the concept of attractivity of solutions. Email addresses: [email protected] (J. Losada), [email protected] (J. J. Nieto), [email protected] (E. Pourhadi)

May 20, 2015

Consider the initial value problem (IVP for short) of the following fractional functional differential equation:    c Dα u (t) = Pm c Dαi fi (t, ut ) + f0 (t, ut ), t > t0 , i=1 (1.1)   u(t) = ϕ(t), t0 − σ ≤ t ≤ t0 ,

where c Dα denotes Caputo’s fractional derivative of order α > 0, σ is a positive constant, ϕ ∈ C([t0 − σ, t0 ], R) and for each i = 1, 2, . . . , m, c Dαi is the Caputo fractional derivative of order 0 < αi < α and fi : I × C ([−σ, 0], R) −→ R, such that I = [t0 , ∞), is a given function. We also consider for any t ∈ I the function ut : [−σ, 0] −→ R given by ut (s) = u(t + s) for each s ∈ [−σ, 0]. By using the classical Schauder fixed point principle and the concept of measure of noncompactness, we show that Eq. (1.1) has attractive solutions under rather general and convenient assumptions. We note that employing classical Schauder fixed point principle, we give an alternative result using a control function and new imposed conditions without applying the concept of measure of noncompactness.

The organization of this paper is as follows. In Section 2, we recall some useful preliminaries. In Section 3, we provide some assumptions and a lemma to present the main result of such section for Eq. (1.1) using the Schauder fixed point principle. In Section 4, we first recall some auxilary facts about the concept of measure of noncompactness and related notations, then we study the existence of solution for Eq. (1.1) applying a generalized version of the well-known Darbo type fixed point theorem together with the technique of measure of noncompactness. Finally, in Section 5 some illustrative examples are given to show the practibility of the obtained results. 2. Preliminaries This section is devoted to recall some essential definitions and auxiliary facts in fractional calculus. We also define some concepts related to attractivity of Eq. (1.1) together with the Schauder fixed point theorem, which will be needed further on (c.f. [20]). Definition 2.1. [18, 21] The Riemann-Liouville fractional integral of order γ > 0 with the lower limit t0 ∈ R for a function f is defined as: Z t f (s) 1 rl γ ds, t > t0 , I f (t) = Γ(γ) t0 (t − s)1−γ provided that the right-hand side is point-wise defined on [t0 , ∞), where Γ(·) is the gamma function. Definition 2.2. [18, 21] The Riemann-Liouville fractional derivative of order n < γ < n + 1 with the lower limit t0 ∈ R for a function f ∈ C n ([t0 , ∞), R) can be written as: rl

dn 1 D f (t) = Γ(n − γ) dtn γ

Z

t

t0

f (s) ds, (t − s)γ+1−n

t > t0 ,

n ∈ N ∪ {0}.

Definition 2.3. [18] Caputo’s fractional derivative of order n < γ < n + 1 for a function f ∈ C n ([t0 , ∞), R) can be written as ! n (k) X (t ) f 0 c γ D f (t) = rl Dγ f (s) − (s − t0 )k−γ (t), t > t0 , n ∈ N ∪ {0}. Γ(k − γ + 1) k=0

Definition 2.4. [13, Definition 2.5] The solution u(t) of IVP (1.1) is attractive if there exist a constant b0 (t0 ) such that |ϕ(s)| ≤ b0 (for all s ∈ [t0 − σ, t0 ]) implies that u(t) → 0 as t → ∞. 2

Definition 2.5. [17, Definition 1] The solution u(t) of IVP (1.1) is said to be globally attractive, if there are lim (u(t) − v(t)) = 0, t→∞

for any solution v = v(t) of IVP (1.1). Remark 2.6. In Section 4 we consider another definition related to concept of attractivity of solutions (see Definition 4.5). Theorem 2.7 (Schauder Fixed Point Theorem). [22, Theorem 4.1.1] Let U be a nonempty and convex subset of a normed space B. Let T be a continuous mapping of U into a compact set K ⊂ U . Then T has a fixed point. 3. Attractivity of Solutions with Schauder fixed point principle Throughout this section we investigate Eq. (1.1) using the Schauder fixed point theorem under the following assumptions: (H0 ) For each i = 0, 1, . . . , m, the function fi (t, ut ) is Lebesgue measurable with respect to t on [t0 , ∞) and fi (t, ψ) is continuous with respect to ψ on C([−σ, 0], R). (H1 ) There exists an strictly decreasing function H : R+ −→ R+ which vanishes at infinity such that,  Z t m  X 1 α−αi −1 ϕ(t0 ) + (t − s) fi (s, us )ds ≤ H(t − t0 ) Γ(α − αi ) i=0

t0

for all

t ∈ I = [t0 , ∞).


(H2 ) There is a constant β such that for each i = 1, 2, . . . , m, we have that fi ∈ L1/β I, C([−σ, 0], R) with   β ∈ 0, min α − αi . 0≤i≤m

Under the condition (H0 ), the equivalent representation for Eq. (1.1) is  Rt P 1   ϕ(t0 ) + m (t − s)α−αi −1 fi (s, us ) ds, t > t0 , i=0 Γ(α − αi ) t0 u(t) =   ϕ(t), t 0 − σ ≤ t ≤ t0 ,

(3.1)

where α0 = 0 and 0 < αi < α for i = 1, 2, . . . , m. Now we define the operator F as  Rt P 1   ϕ(t0 ) + m (t − s)α−αi −1 fi (s, us ) ds, t > t0 , i=0 t0 Γ(α − α ) i [Fu] (t) =   ϕ(t), t0 − σ ≤ t ≤ t0 ,

for every u ∈ C([t0 − σ, ∞), R). It is clear that u(t) is a solution of Eq. (1.1) if it is a fixed point of the operator F. Now applying the operator F and the imposed conditions as above we have the following lemma. Lemma 3.1. Suppose that fi (t, ut ) satisfies conditions (H0 )–(H2 ). Then Eq. (1.1) has at least one solution in C([t0 − σ, ∞), R). 3

Proof. Let us define the set S ⊂ C([t0 − σ, ∞), R) by S = {u : u ∈ C([t0 − σ, ∞), R)

and |u(t)| ≤ H(t − t0 )

for all

t ≥ t0 }.

Obviously, the set S is a nomempty, closed, bounded and convex subset of C([t0 − σ, ∞), R). To prove that Eq. (1.1) has a solution it only needs to show that the operator F has a fixed point in S. First, we show that S is F-invariant. This is easily obtained by condition (H1 ). Now we should prove that F is continuous. To do this, let be (un )n∈N a sequence of functions such that un ∈ S for all n ∈ N and un → u as n → ∞. Obviously, by the continuity of fi (t, ut ) we have lim fi (t, un t ) = fi (t, ut ),

for all

n→∞

t > t0

and i = 0, 1, . . . , m.

Suppose that ε > 0 is given. Since H is strictly decreasing, then for some T > t0 we get H(t − t0 ) < For t0 < t ≤ T , we obtain |[Fu ] (t) − [Fu] (t)| ≤ n

≤ ≤ ≤

m X

i=0 m X i=0

m X i=0 m X i=0

1 Γ(α − αi ) 1 Γ(α − αi )

Z

t

t0

ε , 2

for all

(3.2)

t > T.

(t − s)α−αi −1 |fi (s, un s ) − fi (s, us )| ds

Z

t

t0

(t − s)

α−αi −1 1−β

1−β Z t β 1 ds |fi (s, un s ) − fi (s, us )| β ds t0

(1 − β)1−β (T − t0 )α−αi −β Γ(α − αi )(α − αi − β)1−β (1 − β)1−β (T − t0 )α−αi Γ(α − αi )(α − αi − β)1−β

Z

T

t0

1 β

!β

|fi (s, u s ) − fi (s, us )| ds n

sup |fi (s, un s ) − fi (s, us )|,

t0
which vanishes as n → ∞. On the other hand, since S is F-invariant, then (3.2) yields |[Fun ] (t) − [Fu] (t)| ≤ 2 H(t − t0 ) < ε,

for all

t > T.

Consequently, for t > t0 we infer that |[Fun ] (t) − [Fu] (t)| → 0

as

n → ∞.

If t ∈ [t0 − σ, t0 ], we obviously have that |[Fun ] (t) − [Fu] (t)| = 0. Hence, the continuity of F has been proved. Next, we show that F(S) is equicontinuous. Suppose that ε > 0 is given, t1 , t2 > t0 and t1 < t2 . Let t1 , t2 ∈ (t0 , T ] where T > t0 is chosen such that (3.2) holds. Applying condition (H2 ) we have [Fu] (t2 ) − [Fu] (t1 ) Z t2 Z t1 m X 1 α−αi −1 α−αi −1 (t2 − s) fi (s, us ) ds − (t1 − s) fi (s, us ) ds ≤ Γ(α − αi ) ≤

i=0 m X i=0

t0

1 Γ(α − αi )

Z

t0

+

t0

t1

Z

|(t2 − s)α−αi −1 − (t1 − s)α−αi −1 | |fi (s, us )| ds

t2

t1

(t2 − s)

α−αi −1

 |fi (s, us )| ds 4

≤

m X i=0

1 Γ(α − αi )

 Z

≤

i=0

1 Γ(α − αi )

Z

 Z

t2

t1

|(t1 − s)

Z

t2

t1

≤

m X i=0

(t2 − s)

t1

t0

+

α−αi −1

|(t1 − s)

t0

+ m X

t1

α−αi −1 1−β

1−β  Z ds

t2

t1

α−αi −1 1−β

(t2 − s)

− (t2 − s)

α−αi −1

− (t2 − s)

α−αi −1 1−β

|

1 1−β

1−β  Z ds

t0

β  1 β |fi (s, us )| ds

α−αi −1 1−β

1−β  Z | ds

t1

t0

1−β  Z ds

t2

t1

t1

β |fi (s, us )| ds 1 β

β 1 |fi (s, us )| β ds

β  1 |fi (s, us )| β ds

 1−β α−αi −β α−αi −β α−αi −β (1 − β)1−β 1−β 1−β 1−β |t − t | + |(t − t ) − (t − t ) | 2 1 1 0 2 0 Γ(α − αi )(α − αi − β)1−β Z T β Z T β  1 1 |fi (s, us )| β ds + (t2 − t1 )α−αi −β × |fi (s, us )| β ds , t0

t0

which tends to zero when t1 → t2 . Now, consider t1 , t2 > T . Then, taking into account the fact that S is F-invariant and (3.2), we obtain that |[Fu] (t2 ) − [Fu] (t1 )| ≤

m X i=0

Z t2 Z t1 1 α−αi −1 α−αi −1 (t − s) f (s, u ) ds − (t − s) f (s, u ) ds 2 i s 1 i s Γ(α − αi ) t0 t0

≤ H(t1 − t0 ) + H(t2 − t0 ) < ε.

Note that for the case t0 < t1 < T < t2 we have the following implication (t1 → t2 ) =⇒ (t1 → T ) ∧ (t2 → T ). This, together with the above discussion implies |[Fu] (t2 ) − [Fu] (t1 )| ≤ |[Fu] (t2 ) − [Fu] (T )| + |[Fu] (T ) − [Fu] (t1 )| → 0

as

t1 → t2 .

Consequently, we conclude that F(S) is equicontinuous on each compact interval [t0 , T ] for all T > 0. Moreover, since F(S) ⊂ S and from the definition of S it is clear that lim

sup ( sup {|u(t)| : t > T }) = 0.

T →∞ u∈F (S)

Therefore, F(S) is a relatively compact set in C([t0 − σ, ∞), R) and all the conditions of Schauder fixed point theorem are fulfilled. Hence the operator F, as a self map on S, has a fixed point in this set. This fact shows that Eq. (1.1) has at least one solution in S. Now we are prepared to formulate our main existence result. Theorem 3.2. Suppose that conditions (H0 )–(H1 ) are satisfied, then IVP (1.1) admits at least one attractive solution in the sense of Definition 2.4. Proof. According to the previous lemma, there is at least ine solution of Eq. (1.1) belonging to S. On the other hand, in order to prove the attractivity, using the property of function H, we infer that all functions in S vanish at infinity and hence the solution of Eq. (1.1) tends to zero as t → ∞. This makes the proof completed. Remark 3.3. Note that conclusion of Theorem 3.2 does not imply globally attractivity of solutions in the sense of Definition 2.5. 5

4. Uniform local attractivity of solutions with measure of noncompactness This section is devoted to the study of solutions of Eq. (1.1) in Banach space BC(Rt0 −σ ) consisting of all real functions defined, continuous and bounded on the interval Rt0 −σ = [t0 − σ, ∞), via the technique of measure of noncompactness. It is focused on an alternative way to construct some sufficient conditions (quite distinct from the ones in previous section) for solvability of Eq. (1.1). More precisely, we look for assumptions concerning the functions involved in Eq. (1.1) which guarantee that this equation has solutions belonging to BC(Rt0 −σ ) and also being locally attractive on Rt0 −σ . In the following we gather some definitions and auxiliary facts which will be needed further on. Let E be a Banach space, X and Conv X stand for the closure and the convex closure of X as a subset of E, respectively. Further, denote by ME the family of all nonempty bounded subsets of E and by NE its subfamily consisting of all relatively compact sets. Also suppose that B(x, r) is the closed ball centered at x with radius r and the symbol Br stands for the ball B(θ, r) such that θ is the zero element of the Banach space E. In the following definition we recall the notion of measure of noncompactness which has been initially introduced by Bana´s and Goebel [8]. Definition 4.1. [8, Definition 3.1.3] A mapping µ : ME −→ R+ is said to be a measure of noncompactness in E if it satisfies the following conditions: (i) The family ker µ = {X ∈ ME : µ(X) = 0} is nonempty and ker µ ⊂ NE . (ii) X ⊂ Y ⇒ µ(X) ≤ µ(Y ). (iii) µ(X) = µ(X). (iv) µ(Conv X) = µ(X). (v) For all λ ∈ [0, 1],

µ(λ X + (1 − λ) Y ) ≤ λ µ(X) + (1 − λ) µ(Y ).

(vi) If (Xn )n∈N is a sequence of closed sets from ME such that Xn+1 ⊂ Xn

for all

then the intersection set X∞ =

n = 1, 2, . . . ∞ \

Xn

and

lim µ(Xn ) = 0,

n→∞

is nonempty.

n=1

The family ker µ described in (i) is said to be the kernel of the measure of noncompactness µ. Definition 4.2. Let µ be a measure of noncompactness in E. The mapping T : C ⊆ E −→ E is said to be a µE -contraction if there exists a constant 0 < k < 1 such that µ(T (W )) ≤ k µ(W ),

(4.1)

for any bounded closed subset W ⊆ C. Theorem 4.3 (Darbo-Sadovskii). [8] Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and let the continuous mapping T : C −→ C be a µE -contraction. Then T has at least one fixed point in C. From the point of view of historical remarks, we note that G. Darbo [14] initially introduced condition (4.1) for any arbitrary measure of noncompactness µ and he presented a similar result if the continuous mapping T is being a µ-contraction. Very recently Aghajani, Bana´s and Pourhadi (see [1, 2]) have extended 6

the Darbo’s fixed point theorem using control functions and presented new results which one of them is applied in this section. Measure of noncompactness has been applied in some class of fractional differential equations in several papers. For instance, Aghajani, Trujillo and one of the authors have used this tool for Cauchy problem as a classic fractional differential equations in Banach spaces in [3], Bana´s and O´Regan in [9] have studied existence of a solutions for a nonlinear quadratic Volterra equation of fractional order and Balachandran, Park and Diana Julie [7] have done something similar for a nonlinear fractional functional integral equation of fractional order with deviating arguments. We will use the following fixed point theorem. Theorem 4.4. [1, Theorem 2.2] Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and let T : C −→ C be a continuous function satisfying µ(T (W )) ≤ φ(µ(W ))

(4.2)

for each W ⊆ C, where µ is an arbitrary measure of noncompactness and φ : R+ −→ R+ is a monotone increasing (not necessarily continuous) function with limn→∞ φn (t) = 0 for all t ∈ R+ . Then T has at least one fixed point in C. In what follows, we will work in the Banach space BC(Rt0 −σ ), where t0 and σ are given in (1.1). Such functional space is furnished with the standard norm kuk = sup {|u(t)| : t ≥ t0 − σ}. For further purposes, we introduce a measure of noncompactness in the space BC(Rt0 −σ ), which is constructed similar to the one in the space BC(R+ ) (for more information see [8, Chapter 9] and references therein). To do this, let B be a bounded subset of BC(Rt0 −σ ) and T > t0 − σ given. For u ∈ B and ε > 0 we denote by ωtT0 −σ (u, ε) the modulus of continuity of the function u on the interval [t0 − σ, T ], i.e. ωtT0 −σ (u, ε) = sup {|u(t) − u(s)| : t, s ∈ [t0 − σ, T ], |t − s| ≤ ε}. Now, let us take ωtT0 −σ (B, ε) = sup {ωtT0 −σ (u, ε) : u ∈ B},

ωtT0 −σ (B) = lim ωtT0 −σ (B, ε), ε→0

ωt0 −σ (B) = lim ωtT0 −σ (B). T →∞

If t ≥ t0 − σ is a fixed number, let us denote B(t) = {u(t) : u ∈ B} and

diam B(t) = sup {|u(t) − v(t)| : u, v ∈ B}.

Finally, consider the mapping µ defined on the family MBC(Rt0 −σ ) by the formula µ(B) = ωt0 −σ (B) + lim sup diam B(t).

(4.3)

t→∞

Similarly to the measure of noncompactness constructed for BC(R+ ), one can show that the mapping µ is a measure of noncompactness in the space BC(Rt0 −σ ) (see also [8]). Let us point out that, as we will show, information about ker µ is very useful. In this case, the kernel ker µ consists of non empty and bounded sets X of functions such that functions belonging to X are locally equicontinuous on R+ and the thickness of the boundle formed by functions from X tends to 0 at infinity. Now, let us assume that Ω is a nonempty subset of the space BC(Rt0 −σ ) and Q is an operator defined on Ω with values in BC(R+ ). 7

Consider the following operator equation: x(t) = [Qx] (t),

for all

t ∈ Rt0 −σ .

(4.4)

Definition 4.5. [9, Definition 2] We say that solutions of (4.4) are locally attractive if there exists a closed ball B(u0 , r) in the space BC(Rt0 −σ ) such that for arbitrary solutions u = u(t) and v = v(t) of (4.4) belonging to B(u0 , r) ∩ Ω we have that lim (u(t) − v(t)) = 0.

(4.5)

t→∞

In the case when limit (4.5) is uniform respect to set B[u0 , r] ∩ Ω, i.e., when for each ε > 0 there exists T > 0 such that |u(t) − v(t)| ≤ ε for all u, v ∈ B[u0 , r] ∩ Ω and t ≥ T, we will say that solutions of IVP (1.1) are uniformly locally attractive.

Remark 4.6. As has been remarked in [9], observe that global attractivity of solutions implies local attractivity, but the comverse implication is not true. Remark 4.7. Let us note that the concept of uniform local attractivity of solutions is equivalent to the concept of asymptotic stability introduced in [10, 11] limited to the space BC(Rt0 −σ ). So we can use these concepts interchangeably. Our considerations are based on the following hypothesis: (H3 ) For each i = 0, 1, . . . , m, function fi : Rt0 −σ × C([t0 − σ, t0 ], R) −→ R is continuous and there exists a continuous function hi : Rt0 −σ −→ R+ such that |fi (t, u) − fi (t, v)| ≤ hi (t) H(ku − vk),

(4.6)

where H : R+ −→ R+ is a super-additive function, i.e., H(a) + H(b) ≤ H(a + b) for all a, b ≥ 0.

(H4 ) Suppose that for each i = 0, 1, . . . , m, the following constants exist: Ai := sup t∈I

Z

and, in addition,

t

t0

(t − s)α−αi −1 hi (s) ds < ∞,

Bi := sup t∈I

lim λA n Hn (t) = 0

n→∞

where λA =

m X i=0

Z

t

t0

(t − s)α−αi −1 |fi (s, 0)| ds < ∞;

for all t > 0,

Ai < 1. Γ(α − αi )

(H5 ) There exists a positive solution r0 of the inequality sup t∈[t0 −σ,t0 ]

|ϕ(t)| + λA H(r) + λB ≤ r,

where λB =

m X i=0

(4.7)

Bi . Γ(α − αi )

Now we are prepared to formulate our main result as follows. Theorem 4.8. Under the assumptions (H3 )–(H5 ) Eq. (1.1) has at least one solution in BC(Rt0 −σ ). Moreover, solutions of (1.1) are uniformly locally attractive. 8

Proof. First of all, we consider the operator F as defined in the former section by the formula  Rt P 1   ϕ(t0 ) + m (t − s)α−αi −1 fi (s, us ) ds, t > t0 , i=0 t0 Γ(α − α ) i [Fu] (t) =   ϕ(t), t 0 − σ ≤ t ≤ t0 ,

for all u ∈ BC(Rt0 −σ ). Observe that in view of conditions (H3 )–(H5 ) the function Fu is continuous on Rt0 −σ . Also, we see that BC(Rt0 −σ ) is F-invariant. Indeed, for any u ∈ BC(Rt0 −σ ) and t > t0 we get |[Fu] (t)| ≤ |ϕ(t0 )| + ≤ |ϕ(t0 )| + ≤ |ϕ(t0 )| + +

m X i=0

m X i=0

m X

i=0 m X i=0

1 Γ(α − αi ) 1 Γ(α − αi ) 1 Γ(α − αi ) 1 Γ(α − αi )

Z

t

t0

Z

t

t0

Z

t

t0

Z

t

t0

(t − s)α−αi −1 |fi (s, us )| ds
(t − s)α−αi −1 |fi (s, us ) − fi (s, 0)| + |fi (s, 0)| ds (t − s)α−αi −1 hi (s) H(kus k) ds (t − s)α−αi −1 |fi (s, 0)| ds

≤ |ϕ(t0 )| + λA H(kuk) + λB , which shows that Fu is bounded on [t0 , ∞) and linking with the fact that ϕ ∈ C([t0 − σ, t0 ], R) we infer that Fu ∈ BC(Rt0 −σ ) and so F transforms BC(Rt0 −σ ) into itself. On the other hand, using condition (H5 ) there exists a number r0 > 0 which enjoys in (4.7). For such number, the operator F transforms the ball Br0 of BC(Rt0 −σ ) into itself. Let us now take a nonempty subset X of the ball Br0 and fix x, y ∈ X quite arbitrarily. Then, for fixed t > t0 we obtain |[Fu] (t) − [Fv] (t)| ≤ ≤

m X i=0

m X i=0

1 Γ(α − αi ) 1 Γ(α − αi )

Z

t

t0

Z

t

t0

(t − s)α−αi −1 |fi (s, us ) − fi (s, vs )| ds (t − s)α−αi −1 hi (s) H(kus − vs k) ds

≤ λA H(ku − vk), which also implies that

lim sup diam (F(X))(t) ≤ λA H(lim sup diam X(t)). t→∞

(4.8)

t→∞

Further, let us take T > t0 as fixed and ε > 0. Also, suppose that u ∈ X is chosen and let t1 , t2 ∈ (t0 , T ] such that |t1 − t2 | ≤ ε. Without loss of generality we may assume that t1 < t2 . Then, considering our hypothesis, we get Z t2 Z t1 m X 1 α−αi −1 α−αi −1 |[Fu] (t2 ) − [Fu] (t1 )| ≤ (t − s) f (s, u ) ds − (t − s) f (s, u ) ds 2 i s 1 i s Γ(α − α ) i t t 0 0 i=0  Z t1 m X 1 ≤ |(t1 − s)α−αi −1 − (t2 − s)α−αi −1 | |fi (s, us )| ds Γ(α − α ) i t 0 i=0  Z t2 + (t2 − s)α−αi −1 |fi (s, us )| ds t1

9

≤

m X i=0

1 Γ(α − αi )

Z

t1

t0

+

(t1 − s)α−αi −1 − (t2 − s)α−αi −1 |fi (s, us )| ds

Z

t2

t1

≤

   (t2 − s)α−αi −1 hi (s) H(kus k) + |fi (s, 0)| ds

m X ω T (fi , αi , ε) + ω T (fi , αi , ε) 1

i=0

2

Γ(α − αi )

+ λA H(ωtT0 −σ (u, ε)),

for u ∈ X ⊆ Br0 , where the notations in the last term as above are given by ω1T (fi , αi , ε) Z = sup

t1

t0

 (t1 − s)α−αi −1 − (t2 − s)α−αi −1 |fi (s, us )| ds : t1 , t2 ∈ [t0 − σ, T ], |t1 − t2 | ≤ ε, kuk ≤ r0 ,

ω2T (fi , αi , ε) = sup

Z

t2

t1

   (t2 − s)α−αi −1 hi (s) H(r0 ) + |fi (s, 0)| ds : t1 , t2 ∈ [t0 − σ, T ], |t1 − t2 | ≤ ε .

Now, taking into account that the function fi (s, us ) is uniformly continuous on the set [t0 − σ, T ] × Br0 for all i = 0, 1, . . . , m, we easily get the following inequality ωtT0 −σ (F(X)) ≤ λA H(ωtT0 −σ (X)), which together with (4.8) and super-additivity of H implies that µ (F(X)) = ωt0 −σ (F(X)) + lim sup diam (F(X)) (t) t→∞

≤ λA H(ωt0 (X)) + λA H(lim sup diam X(t)) t→∞

≤ λA H(µ(X)). Now, since µ as given by (4.3) defines a measure of noncompactness on BC(Rt0 −σ ) then, the recent inequality together with Theorem 4.4 shows that Eq. (1.1) has a solution in Banach space BC(Rt0 −σ ). To prove that all solutions of Eq. (1.1) are uniformly locally attractive in the sense of Definition 4.5 let us put Br10 = Conv F(Br0 ), Br20 = Conv F(Br10 ) and so on, where Br0 is the ball with radius r0 and center ⊂ Brn0 for n = 1, 2, . . . and zero in the space BC(Rt0 −σ ). We simply observe that Br10 ⊂ Br0 and Brn+1 0 also the sets of this sequence are closed, convex and nonempty. Besides, in view of the recent inequality we obtain that µ(Brn0 ) ≤ λA n Hn (µ(Br0 )) for any n = 1, 2, . . . Combining the fact that µ(Br0 ) ≥ 0 and condition (H4 ) with the above inequality we get lim µ(Brn0 ) = 0.

n→∞

T∞ Therefore, using the definition of measure of noncompactness we infer that the set B = n=1 Brn0 is nonempty, bounded, closed and convex. The set B is F-invariant and the operator F is continuous on such set. Moreover, keeping in mind the fact that B ∈ ker µ and the characterization of sets belonging to ker µ we conclude that all solutions of Eq. (1.1) are uniformly locally attractive in the sense of Definition 4.5. This completes the proof. Remark 4.9. It is clear that this also proves that Eq. (1.1) has at least one attractive solution in the sense of Definition 2.4. 10

5. Concrete examples In this section, we present some examples to illustrate our main results obtained in the former sections. Example 5.1. Consider the fractional functional differential equation 
  c D 12 u (t) = c D 13 (s + 1)− 12 e− sin u(s−1) (t) + sin u(t − 1) (|u(t − 1)| + t + 1)− 45 , 3   u(t) = t e−t ,

t > 0, t ∈ [−1, 0].

(5.1)

Obviously, one can show that condition (H0 ) holds. To prove that condition (H1 ) is satisfied, since u(0) = 0 we have the following relation for all t > 0, Z t Z t 1 u(s − 1) 1 − 12 − sin u(s−1) − 45 − 21 − 56 sin ds (|u(s − 1)| + s + 1) ds + 1 (t − s) (t − s) (s + 1) e Γ( 1 ) 3 Γ( 6 ) 0 0 2 Z t Z t 5 1 1 1 − 21 − 45 ≤ (t − s) (t − s)− 6 s− 2 ds s ds + (5.2) Γ( 12 ) 0 Γ( 16 ) 0 3 Z 1 Z 1 1 1 5 t− 10 1 − 4 t− 3 5 (1 − s)− 2 ds + s s− 2 (1 − s)− 6 ds. = 1 1 Γ( 2 ) 0 Γ( 6 ) 0

Bring to mind that for any α, β ∈ R+ we have the following identity Z 1 Γ(α)Γ(β) , sα−1 (1 − s)β−1 ds = Γ(α + β) 0 together with (5.2) implies that 1 Γ( 12 )

Z

0

t

(t − s)

− 12

4 u(s − 1) 1 (|u(s − 1)| + s + 1)− 5 ds + 1 sin 3 Γ( 6 )

where

Z

0

t

5

1

(t − s)− 6 (s + 1)− 2 e− sin u(s−1) ds ≤ H(t),

√ Γ( 51 ) − 3 1 π 10 H(t) := + 2 t− 3 , 7 t Γ( 10 ) Γ( 3 )

t > 0,

which is clearly an strictly decreasing function on R+ and shows that condition (H1 ) holds. Finally, it 1 remains to prove that condition (H2 ) holds too. To do this, let β = 10 ∈ (0, min{ 12 , 16 }), then we obtain Z

0

∞



sin

41 u(s − 1) (|u(s − 1)| + s + 1)− 5 β ds ≤ 3

Z

0

∞

(s + 1)−8 ds =

1 , 7

1 u(s − 1) 4 (|u(s − 1)| + s + 1)− 5 ∈ L β (I, C([−1, 0], R)). Similarly, for 3 e− sin u(s−1) we infer that Z ∞ Z ∞ 1  1 1 (s + 1)−5 ds = . (s + 1)− 2 e− sin u(s−1) β ds ≤ 4 0 0

which shows that f0 (s, us ) = sin 1

f1 (s, us ) = (s + 1)− 2

1

This prove that f1 (s, us ) ∈ L β (I, C([−1, 0], R)). Therefore, all conditions of Theorem 3.2 are satisfied and so the solution of Eq. (5.1) is existent and also attractive. In the following example, we present another class of fractional differential equations which shows the practicability of Theorem 4.8. 11

Example 5.2. Consider the fractional functional differential equation of the form      sin u(s − π2 ) cos u(s − π2 )  c 12 c c 13  (t) + D (t), t > 0,  Du (t) = D 2 1 2π(s + 3) 3 2π(s + 2) 2 t    u(t) = , t ∈ [− π2 , 0]. (t + 3)2

(5.3)

It is clear that condition (H3 ) holds. It only needs to take the following replacement 1 −2 t 3, 2π

h1 (t) =

h2 (t) =

1 −1 t 2, 2π

H(t) = t.

To justify condition (H4 ) we have A1 = sup t∈I

Z

1 2π

0

A2 = sup t∈I

Hence, we have that

t

1

2

(t − s)− 3 s− 3 ds =

1 2π

Z

0

t

1

1

(t − s)− 2 s− 2

    1 1 2 1 2π √ = √ < ∞, Γ = 3 3 2π 3 3     1 1 1 1 ds = Γ Γ = < ∞. 2π 2 2 2

1 Γ 2π

lim λA n Hn (t) = lim λA n t = 0

n→∞

for all

n→∞

where λA =

√

t > 0,

1 1  + √ ∼ = 0.7084 < 1. 2 2 π 3Γ 3

Finally, to show the existence of solution for (4.7) in condition (H5 ) since B1 = 0,

B2 := sup t∈I

Z

0

t

1

(t − s)− 2

1 2π(s + 2)

1 2

ds ≤

1 sup 2π t∈I

Z

0

t

1

1

(t − s)− 2 s− 2 ds =

1 1 =⇒ λB ≤ , 2 2

we obtain a positive solution r0 for t + λA r + λB ≤ 0.25 + 0.8r + 0.5 ≤ r sup 2 t∈[−1,0] (t + 3)

(5.4)

by considering r0 ≥ 3.75. Now the solution of Eq. (5.3) is existent and uniformly locally attractive since all conditions of Theorem 4.8 are satisfied. Remark 5.3. We note that verification of the existence of solution for Eq. (5.3) using Theorem 3.2 seems very unlikely since finding a vanishing and strictly decreasing function H in (H1 ) is difficult for such equation.

Acknowledgements This work was supported by the Ministerio de Econom´ıa y Competitividad of Spain under grants MTM2010–15314 and MTM2013–43014–P, Xunta de Galicia under grant R2014/002, and co-financed by the European Community fund FEDER. J. Losada also acknowledge financial support by the Xunta de Galicia under Grant Plan I2C ED481A-2015/272. Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this paper. 12

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