weakly picard operators method for modified

Fixed Point Theory, 15(2014), No. 1, 297-310 http://www.math.ubbcluj.ro/∼ nodeacj/sfptcj.html

WEAKLY PICARD OPERATORS METHOD FOR MODIFIED FRACTIONAL ITERATIVE FUNCTIONAL DIFFERENTIAL EQUATIONS ∗∗ AND YONG ZHOU∗∗∗ ˘ JINRONG WANG∗ , MICHAL FECKAN

∗

Department of Mathematics, Guizhou University Guiyang, Guizhou 550025, P. R. China ∗∗ Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and Informatics, Comenius University Mlynsk´ a dolina, 842 48 Bratislava, Slovakia ∗∗∗ Department of Mathematics, Xiangtan University Xiangtan, Hunan 411105, P. R. China E-mail: [email protected]; [email protected]; [email protected] Abstract. Boundary value problems for modified fractional iterative functional differential equations is offered. Using weakly Picard operators method, some new existence and uniqueness theorems and data dependence results are presented. Further, examples are given to illustrate our results. Key Words and Phrases: Weakly Picard operators, fractional iterative functional differential equations, boundary value problems. 2010 Mathematics Subject Classification: 26A33, 34B37, 47H10.

1. Introduction Picard and weakly Picard operators methods have been a powerful tool to study the nonlinear differential equations. For more details on this novel methods to discuss existence and uniqueness and the data dependence on data of the solutions for some differential equations and integral equations, one can see Rus et al. [1, 2, 3, 4, 5, 6, 7], S ¸ erban et al. [8], Muresan [9, 10] and Olaru [11]. It is remarkable that Wang et al. [12] apply this interesting methods to study nonlocal Cauchy problems and impulsive Cauchy problems for nonlinear differential equations. On the other hand, a strong motivation for studying fractional differential equations comes from the fact they have been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics and economics. It draws a great application in nonlinear oscillations of earthquakes, many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic model. For more details on basic theory of fractional differential equations, one can see the monographs of Diethelm [13], Kilbas et al. [14], Miller and Ross [15], Podlubny [16] and Tarasov [17], and the references [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. 297

˘ JINRONG WANG, MICHAL FECKAN AND YONG ZHOU

298

However, functional differential equations with fractional derivative have not been studied extensively. In particular, weakly Picard operators methods have not been used to study such problems. Motivated by [7, 12, 31, 32], we offer to study boundary value problems for the following modified fractional iterative functional differential equations  c q  Da,t x(t) = f (t, x(t), x(xv (t))), t ∈ [a, b], v ∈ R \ {0}, q ∈ (1, 2), (1.1) x(t) = ϕ(t), t ∈ [a1 , a],  x(t) = ψ(t), t ∈ [b, b1 ], q where c Da,t is the Caputo fractional derivative of order q with the lower limit a (see Definition 2.3) and (C1 ) a, b, a1 , b1 ∈ R, a1 ≤ a < b ≤ b1 , a function Υ(z) = z v satisfies Υ ∈ C([a1 , b1 ], [a1 , b1 ]); (C2 ) f ∈ C([a, b] × [a1 , b1 ]2 , R); (C3 ) ϕ ∈ C([a1 , a], [a1 , b1 ]) and ψ ∈ C([b, b1 ], [a1 , b1 ]); (C4 ) there exists Lf > 0 such that |f (t, u1 , w1 )−f (t, u2 , w2 )| ≤ Lf (|u1 −u2 |+|w1 −w2 |) for all t ∈ [a, b], ui , wi ∈ [a1 , b1 ], i = 1, 2. A function x ∈ C([a1 , b1 ], [a1 , b1 ]) is said to be a solution of the problem (1.1) if q x satisfies the equation c Da,t x(t) = f (t, x(t), x(xv (t))) on [a, b], and the conditions x(t) = ϕ(t), t ∈ [a1 , a], x(t) = ψ(t), t ∈ [b, b1 ]. It is easy to verify that the problem (1.1) is equivalent with the following fixed point equation   ϕ(t), for t ∈ [a1 , a], Rb 1 (1.2) x(t) = G(t, s)f (s, x(s), x(xv (s)))ds, for t ∈ [a, b], w(ϕ, ψ)(t) + Γ(q) a  ψ(t), for t ∈ [b, b1 ],

and x ∈ C([a1 , b1 ], [a1 , b1 ]), where w(ϕ, ψ)(t) := ϕ(a) + ψ(b)−ϕ(a) (t − a), G is the b−a Green function defined by t−a (t − s)q−1 − b−a (b − s)q−1 , for a ≤ s ≤ t ≤ b, G(t, s) := t−a q−1 − b−a (b − s) , for a ≤ t ≤ s ≤ b, and Z

b v

G(t, s)f (s, x(s), x(x (s)))ds a

Z t−a b = − (b − s)q−1 f (s, x(s), x(xv (s)))ds b−a a Z t + (t − s)q−1 f (s, x(s), x(xv (s)))ds. a

t−a (b − t)q−1 < 0 Remark 1.1. Note G(t, s) ≤ 0 for a ≤ t ≤ s ≤ b, while G(t, t) = − b−a ∂ q−2 q−2 and G(t, a) = (t − a) (t − a) − (b − a) > 0 for a < t < b. Next ∂s G(t, s) = q−2

q−2

(b−a) (q − 1) (b−s) (t−a)−(t−s) < 0 for a < s < t < b. So for any t ∈ (a, b) there b−a is a unique s(t) ∈ (a, t) such that G(t, s(t)) = 0, G(t, s) < 0 for s(t) < s < b and G(t, s) > 0 for s(t) > s ≥ a. Note G(t, b) = 0 and 1

s(t) = b +

(b − a) q−1 (b − t) 1

1

(t − a) q−1 − (b − a) q−1

.

FRACTIONAL ITERATIVE FUNCTIONAL DIFFERENTIAL EQUATIONS

299

Furthermore, s(t) = a for q = 2, and this is a great difference for the Green function when q ∈ (1, 2). Since we cannot expect monotonicity of the integral operator Bf defined in (3.1) below. On the other hand, we derive Z b t−a G(t, s)ds = (t − a)q−1 − (b − a)q−1 . q a Rb Rb Hence a G(t, s)ds ≤ 0 for t ∈ [a, b] and a G(t, s)ds < 0 for t ∈ (a, b], which holds also for q = 2. So G(t, ·) is nonpositive in average on [a, b]. On the other hand, the first equation of the problem (1.1) is equivalent with  x(t), for t ∈ [a1 , a],    w(x| [a1 ,a] , x|[b,b1 ] )(t) Rb (1.3) x(t) := 1 G(t, s)f (s, x(s), x(xv (s)))ds, for t ∈ [a, b], +   Γ(q) a  x(t), for t ∈ [b, b1 ], and x ∈ C([a1 , b1 ], [a1 , b1 ]). We will apply a new method to study the equations (1.2) and (1.3). More precisely, we will use the weakly Picard operator technique to obtain some new existence, uniqueness and data dependence results for the solution of the problem (1.1). 2. Notation, definitions and auxiliary facts To end this section, we recall some basic definitions of the fractional calculus theory which are used further in this paper. For more details, see Kilbas et al. [14]. Definition 2.1. The fractional order integral of the function h ∈ L1 ([a, b], R) of order q ∈ R+ is defined by Z t (t − s)q−1 q Ia,t h(t) = h(s)ds Γ(q) a where Γ is the Gamma function. Definition 2.2. For a function h given on the interval [a, b], the qth RiemannLiouville fractional order derivative of h, is defined by n Z t 1 d q L (Da,t h)(t) = (t − s)n−q−1 h(s)ds, Γ(n − q) dt a here n = [q] + 1 and [q] denotes the integer part of q. Definition 2.3. The Caputo derivative of order q for a function f : [a, b] → R can be written as n−1 X tk (k) c q L q Da,t h(t) = Da,t h(t) − h (a) , t > 0, n − 1 < q < n. k! k=0

We need some notions and results from the weakly Picard operator theory (for more details see Rus [5, 6]). Let (X, d) be a metric space and A : X → X an operator. We shall use the following notations:

300


FA = {x ∈ X | A(x) = x}−the fixed point set of A; I(A) = {Y ∈ P (X) | A(Y ) ⊆ Y, Y 6= ∅}; An+1 = An ◦ A, A1 = A, A0 = I, n ∈ N P (X) = {Y ⊆ X | Y 6= ∅}; OA (x) = {x, A(x), A2 (x), · · · , An (x), · · · }−the A−orbit of x ∈ X; H : P (X) × P (X) → R+ ∪ {+∞}; H(Y, Z) = max supy∈Y inf z∈Z d(y, z), supz∈Z inf y∈Y d(y, z) −the PompeiuHausdorff functional on P (X) × P (X). Definition 2.4. Let (X, d) be a metric space. An operator A : X → X is a Picard operator if there exists x∗ ∈ X such that FA = {x∗ } and the sequence (An (x0 ))n∈N converges to x∗ for all x0 ∈ X. Theorem 2.5. (Contraction principle) Let (X, d) be a complete metric space and A : X → X a γ−contraction. Then (i) FA = {x∗ }; (ii) (An (x0 ))n∈N converges to x∗ for all x0 ∈ X; γn d(x0 , A(x0 )), for all n ∈ N . (iii) d(x∗ , An (x0 )) ≤ 1−γ Remark 2.6. Accordingly to the Definition 2.4, the contraction principle insures that, if A : X → X is a γ−contraction on the complete metric space X, then it is a Picard operator. Theorem 2.7. Let (X, d) be a complete metric space and A, B : X → X two operators. We suppose the following: (i) A is a contraction with contraction constant γ and FA = {x∗A }. (ii) B has fixed points and x∗B ∈ FB . (iii) There exists η > 0 such that d(A(x), B(x)) ≤ η, for all x ∈ X. η Then d(x∗A , x∗B ) ≤ 1−γ . Definition 2.8. Let (X, d) be a metric space. An operator A : X → X is a weakly Picard operator if the sequence (An (x0 ))n∈N converges for all x0 ∈ X and its limit (which may depend on x0 ) is a fixed point of A. Theorem 2.9. Let (X, d) be a metric space. ThenSA : X → X is a weakly Picard operator if and only if there exists a partition X = λ∈Λ Xλ of X such that (i) Xλ ∈ I(A), for all λ ∈ Λ; (ii) A |Xλ : Xλ → Xλ is a Picard operator, for all λ ∈ Λ. Definition 2.10. If A is a weakly Picard operator, then we consider the operator A∞ defined by A∞ : X → X, A∞ (x) = limn→∞ An (x). It is clear that A∞ (X) = FA and ωA (x) = {A∞ (x)} where ωA (x) is the ω−limit point set of mapping A for point x. Definition 2.11. Let A be a weakly Picard operator and c > 0. The operator A is c−weakly Picard operator if d(x, A∞ (x)) ≤ cd(x, A(x)), ∀ x ∈ X. Remark 2.12. Let (X, d) be a complete metric space and A : X → X a continuous operator. We suppose that there exists γ ∈ [0, 1) such that d(A2 (x), A(x)) ≤ γd(x, Ax), ∀ x ∈ X.


Then A is c−weakly Picard operator with c =

301

1 1−γ .

Theorem 2.13. Let (X, d) be a complete metric space and Ai : X → X, i = 1, 2. We suppose that (i) Ai is ci −weakly Picard operator, i = 1, 2; (ii) There exists α > 0 such that d(A1 (x), A2 (x)) ≤ α, ∀ x ∈ X. Then H(FA1 , FA2 ) ≤ α max(c1 , c2 ). 3. Existence In what follows we consider the fixed point equation (1.2). Consider the operator Bf : C([a1 , b1 ], [a1 , b1 ]) → C([a1 , b1 ], [a1 , b1 ]) where   ϕ(t), for t ∈ R[a1 , a], b Bf (x)(t) := w(ϕ, ψ)(t) + a G(t, s)f (s, x(s), x(xv (s)))ds, for t ∈ [a, b],  ψ(t), for t ∈ [b, b1 ].

(3.1)

It is clear that x is a solution of the problem (1.1) if and only if x is a fixed point of the operator Bf . So, the problem is to study the fixed point equation x = Bf (x). Let L > 0 and introduce the following notation: CL ([a1 , b1 ], [a1 , b1 ]) = {x ∈ C([a1 , b1 ], [a1 , b1 ]) : |x(t1 ) − x(t2 )| ≤ L|t1 − t2 |} , for all t1 , t2 ∈ [a1 , b1 ]. Remark that CL ([a1 , b1 ], [a1 , b1 ]) ⊆ C([a1 , b1 ], R) is also a complete metric space with respect to the metric, d(x1 , x2 ) := max |x1 (t) − x2 (t)|. a1 ≤t≤b1

Theorem 3.1. We suppose that (i) the conditions (C1 )–(C4 ) are satisfied but in addition v ≥ 1; (ii) ϕ ∈ CL ([a1 , a], [a1 , b1 ]), ψ ∈ CL ([b, b1 ], [a1 , b1 ]); (iii) there are mf , Mf ∈ R such that mf ≤ f (t, u, w) ≤ Mf , ∀ t ∈ [a, b], u, w ∈ [a1 , b1 ], and moreover, mf (b − a)q Mf (b − a)q + min 0, , a1 ≤ min(ϕ(a), ψ(b)) − max 0, Γ(q + 1) Γ(q + 1) mf (b − a)q Mf (b − a)q max(ϕ(a), ψ(b)) − min 0, + max 0, ≤ b1 ; Γ(q + 1) Γ(q + 1) |ψ(b) − ϕ(a)| (b − a)q−1 (1 + q) max{|mf |, |Mf |} (iv) + < L; b−a Γ(q + 1) q v−1 2(b − a) Lf (Lv max{|a1 |, |b1 |} + 2) (v) < 1. Γ(q + 1) Then the problem (1.1) has in CL ([a1 , b1 ], [a1 , b1 ]) a unique solution. Moreover, the operator Bf , Bf : CL ([a1 , b1 ], [a1 , b1 ]) → CL ([a, b], [a1 , b1 ])


302

is a c−Picard operator with c :=

Γ(q + 1) Γ(q + 1) − 2(b −

a)q Lf (Lv max{|a1 |, |b1 |}v−1

+ 2)

.

Proof. First of all we remark that the condition (iii) and (iv) imply that CL ([a1 , b1 ], [a1 , b1 ]) is an invariant subset for Bf . Indeed, for t ∈ [a1 , a] ∪ [b, b1 ], we have Bf (x)(t) ∈ [a1 , b1 ]. Furthermore, we obtain a1 ≤ Bf (x)(t) ≤ b1 , ∀ t ∈ [a, b], if and only if a1 ≤ min Bf (x)(t) (3.2) t∈[a,b]

and max Bf (x)(t) ≤ b1

(3.3)

t∈[a,b]

hold. Since min Bf (x)(t) ≥

t∈[a,b]

min(ϕ(a), ψ(b)) mf (b − a)q Mf (b − a)q + min 0, , − max 0, Γ(q + 1) Γ(q + 1)

respectively max Bf (x)(t) ≤

t∈[a,b]

max(ϕ(a), ψ(b)) Mf (b − a)q mf (b − a)q + max 0, , − min 0, Γ(q + 1) Γ(q + 1)

the requirements (3.2) and (3.3) are equivalent with the conditions appearing in (iii). Now, consider a1 ≤ t1 < t2 ≤ a. Then, |Bf (x)(t2 ) − Bf (x)(t1 )| = |ϕ(t2 ) − ϕ(t1 )| ≤ L|t1 − t2 | as ϕ ∈ CL ([a1 , a], [a1 , b1 ]), due to (ii). Similarly, for b ≤ t1 < t2 ≤ b1 , |Bf (x)(t2 ) − Bf (x)(t1 )| = ≤

|ψ(t2 ) − ψ(t1 )| L|t1 − t2 |

that follows from (ii), too. On the other hand, for a ≤ t1 < t2 ≤ b, |Bf (x)(t2 ) − Bf (x)(t1 )| ≤ |w(ϕ, ψ)(t2 ) − w(ϕ, ψ)(t1 )| Z b 1 + |G(t2 , s) − G(t1 , s)||f (s, x(s), x(xv (s))))|ds Γ(q) a Z b |ψ(b) − ϕ(a)| |t2 − t1 | ≤ |t2 − t1 | + (b − s)q−1 |f (s, x(s), x(xv (s)))|ds b−a (b − a)Γ(q) a Z t1 1 + [(t2 − s)q−1 − (t1 − s)q−1 ]|f (s, x(s), x(xv (s)))|ds Γ(q) a


1 + Γ(q)

Z

303

t2

(t2 − s)q−1 |f (s, x(s), x(xv (s)))|ds

t1

|ψ(b) − ϕ(a)| (b − a)q−1 max{|mf |, |Mf |}|t2 − t1 | |t2 − t1 | + b−a Γ(q + 1) max{|mf |, |Mf |} max{|mf |, |Mf |} + ((t2 − a)q − (t1 − a)q − (t2 − t1 )q ) + (t2 − t1 )q Γ(q + 1) Γ(q + 1) |ψ(b) − ϕ(a)| (b − a)q−1 (1 + q) max{|mf |, |Mf |} ≤ + |t1 − t2 |. b−a Γ(q + 1) where we use the inequality ≤

rq − sq ≤ qrq−1 (r − s) for all r ≥ s ≥ 0. Therefore, due to (iv), the function Bf (x) is L–Lipschitz in t. Thus, according to the above, we have CL ([a1 , a], [a1 , b1 ]) ∈ I(Bf ). From the condition (v) it follows that Bf is an LBf –contraction with LBf :=

2(b − a)q Lf (Lv max{|a1 |, |b1 |}v−1 + 2) Γ(q + 1)

Indeed, for all t ∈ [a1 , a] ∪ [b, b1 ], we have |Bf (x1 )(t) − Bf (x2 )(t)| = 0. Moreover, for t ∈ [a, b] we get |Bf (x1 )(t) − Bf (x2 )(t)| Z

≤ ≤

b

1 |G(t, s)f (s, x1 (s)) − G(t, s)f (s, x2 (s))|ds Γ(q) a Z b (b − s)q−1 |f (s, x1 (s), x1 (xv1 (s))) − f (s, x2 (s), x2 (xv2 (s)))| ds

t−a (b − a)Γ(q) a Z t 1 + (t − s)q−1 |f (s, x1 (s), x1 (xv1 (s))) − f (s, x2 (s), x2 (xv2 (s)))| ds Γ(q) a Z b Lf ≤ (b − s)q−1 [|x1 (s) − x2 (s)| + |x1 (xv1 (s)) − x2 (xv2 (s))|] ds Γ(q) a Z t Lf + (t − s)q−1 [|x1 (s) − x2 (s)| + |x1 (xv1 (s)) − x2 (xv2 (s))|] ds Γ(q) a Z b Lf q−1 (b − s) |x1 (s) − x2 (s)| + |x1 (xv1 (s)) − x1 (xv2 (s))| ≤ Γ(q) a v v + |x1 (x2 (s)) − x2 (x2 (s))| ds +

Lf Γ(q)

Z a

t

(t − s)q−1 |x1 (s) − x2 (s)| + |x1 (xv1 (s)) − x1 (xv2 (s))| + |x1 (xv2 (s)) − x2 (xv2 (s))| ds


304

Lf ≤ Γ(q) + ≤

Lf Γ(q)

+

Lf Γ(q)

q−1

a

b

(b − s)q−1 2kx1 − x2 kC + Lv max{|a1 |, |b1 |}v−1 |x1 (s) − x2 (s)| ds a Z t (t − s)q−1 2kx1 − x2 kC + Lv max{|a1 |, |b1 |}v−1 |x1 (s) − x2 (s)| ds Z

a

Lf ≤ Γ(q) +

Lf Γ(q)

b

v v (b − s) 2kx1 − x2 kC + L |x1 (s) − x2 (s)| ds a Z t (t − s)q−1 2kx1 − x2 kC + L |xv1 (s) − xv2 (s)| ds Z

Lf Γ(q)

b

v−1 (b − s) 2kx1 − x2 kC + Lv max{|a1 |, |b1 |} kx1 − x2 kC ds a Z t q−1 v−1 (t − s) 2kx1 − x2 kC + Lv max{|a1 |, |b1 |} kx1 − x2 kC ds Z

q−1

a

2(b − a)q Lf (Lv max{|a1 |, |b1 |}v−1 + 2) kx1 − x2 kC . Γ(q + 1) So, Bf is a c−Picard operator, with 1 c= . 2(b−a)q Lf (Lv max{|a1 |,|b1 |}v−1 +2) 1− Γ(q+1) ≤

This completes the proof. In what follows, consider the following operator

Ef : CL ([a1 , b1 ], [a1 , b1 ]) → CL ([a1 , b1 ], [a1 , b1 ]) where  x(t), for t ∈ [a1 , a],    w(x| [a1 ,a] , x|[b,b1 ] )(t) Rb (Ef x)(t) := 1 + G(t, s)f (s, x(s), x(xv (s))))ds, for t ∈ [a, b],   Γ(q) a  x(t), for t ∈ [b, b1 ]. Theorem 3.2. Under the conditions of Theorem 3.1, the operator Ef CL ([a1 , b1 ], [a1 , b1 ]) → CL ([a1 , b1 ], [a1 , b1 ]) is a weakly Picard operator.

:

Proof. The operator Ef is a continuous operator but it is not a contraction operator. Let us take the following notation: Xϕ,ψ := {x ∈ CL ([a1 , b1 ], [a1 , b1 ]) : x|[a1 ,a] = ϕ, x|[b,b1 ] = ψ}. Then we can write CL ([a1 , b1 ], [a1 , b1 ]) =

[

Xϕ,ψ .

ϕ∈CL ([a1 ,a],[a1 ,b1 ]);ψ∈CL ([a1 ,a],[a1 ,b1 ])

We have that Xϕ,ψ ∈ I(Ef ) and Ef |Xϕ,ψ is a Picard operator, because it is the operator which appears in the proof of Theorem 3.1. By applying Theorem 2.9, we obtain that Ef is weakly Picard operator. This completes the proof. Finally, in general, we have immediately from the proof of Theorem 3.1 and Schauder fixed point theorem the following existence result.


305

Theorem 3.3. Suppose that the conditions (C1 )–(C4 ) are satisfied together with assumptions (ii)–(iv) of Theorem 3.1. Then the problem (1.1) has a solution in CL ([a1 , b1 ], [a1 , b1 ]). We do not know about uniqueness. But this is not so surprising, since Bf is not Lipschitzian in general. So we cannot apply metric fixed point theorems, only topological one. This can be simply illustrated on the problems x0 (t) = Ax(x2 (t)),

x(0) = 0,

(3.4)

and p x0 (t) = Ax( 4 x(t)),

x(0) = 0, (3.5) Rt for A > 0. Rewriting (3.4) as x(t) = B1 (x)(t) = A 0 x(x2 (s))ds, and applying the above procedure to B1 , it follows that B1 : CAb1 ([0, b1 ], [0, b1 ]) → CAb1 ([0, b1 ], [0, b1 ]), Ab1 ≤ 1 0 < b1 ≤ 1 is Ab1 (1 + 2Ab21 )–Lipschitzian, so its only solution is x(t) = 0 in that space when Ab1 (1 +p2Ab21 ) < 1. On the other hand, rewriting (3.5) as Rt x(t) = B2 (x)(t) = A 0 x( 4 x(s))ds, it follows that B2 : CAb1 ([0, b1 ], [0, b1 ]) → CAb1 ([0, b1 ], [0, b1 ]), b1 ≥ 1, Ab1 ≤ 1 satisfies p kB2 (x1 ) − B2 (x2 )kC ≤ Ab1 kx1 − x2 kC + Ab1 4 kx1 − x2 kC , so it is not Lipschitzian. Hence (3.5) should have a nonzero solution, and it does have x(t) = A4 t2 . 4. Data dependence In this section, we consider the problem (1.1) and suppose the conditions of Theorem 3.1 are satisfied. Denote by x(·; ϕ, ψ, f ) the solution of this problem. Theorem 4.1. Let ϕi , ψi , fi , i = 1, 2, be as in Theorem 3.1. Furthermore, we suppose that (i) there exists η1 > 0, such that |ϕ1 (t) − ϕ2 (t)| ≤ η1 , t ∈ [a1 , a], and |ψ1 (t) − ψ2 (t)| ≤ η1 , t ∈ [b, b1 ]; (ii) there exists η2 > 0 such that |f1 (t, u, w) − f2 (t, u, w)| ≤ η2 , ∀ t ∈ [a, b], u, w ∈ [a1 , b1 ]. Then |x(t; ϕ1 , ψ1 , f1 ) − x(t; ϕ2 , ψ2 , f2 )| where Lf = min{Lf1 , Lf2 }.

≤

2(b−a)q Γ(q+1) η2 2(b−a)q Lf (Lv max{|a1 |,|b1 |}v−1 +2) Γ(q+1)

3η1 +

1−

,


306

Proof. Consider the operators Bϕi ,ψi ,fi , i = 1, 2. From Theorem 3.1 these operators are contractions. Additionally, kBϕ1 ,ψ1 ,f1 (x) − Bϕ2 ,ψ2 ,f2 (x)kC ≤

|w(ϕ1 , ψ1 )(t) − w(ϕ2 , ψ2 )(t)| Z b 1 G(t, s)|f1 (s, x(s), x(xv (s)))) − f2 (s, x(s), x(xv (s))))|ds + Γ(q) a 2(b − a)q ≤ 3η1 + η2 . Γ(q + 1)

Now, the proof follows from Theorem 2.7, with A := Bϕ1 ,ψ1 ,f1 , B := Bϕ2 ,ψ2 ,f2 , η := 3η1 +

2(b − a)q η2 , Γ(q + 1)

and

2(b − a)q Lf (Lv max{|a1 |, |b1 |}v−1 + 2) , Γ(q + 1) where we can suppose that Lf = Lf1 = min{Lf1 , Lf2 }. γ := LA =

Remark 4.2. Let ϕi , ψi , fi , i ∈ N , and ϕ, ψ, f be as in Theorem 3.1. We suppose that unif.

unif.

unif.

ϕi −→ ϕ, ψi −→ ψ, fi −→ f. Then unif.

x(·; ϕi , ψi , fi ) −→ x(·; ϕ, ψ, f ), as i → ∞. Theorem 4.3. Let f1 and f2 be as in Theorem 3.1. Let FEfi be the solution set of the first equation of the problem (1.1) corresponding to fi , i = 1, 2. Suppose that there exists η > 0 such that |f1 (t, u1 , w1 ) − f2 (t, u2 , w2 )| ≤ η, ∀ t ∈ [a, b], ui , wi ∈ [a1 , b1 ], i = 1, 2. Then Hk·kC (FEf1 , FEf2 ) ≤

Γ(q + 1) − 2(b −

η2(b − a)q , v−1 + 2) f (Lv max{|a1 |, |b1 |}

a)q L

where Lf = max{Lf1 , Lf2 } and Hk·kC denotes the Pompeiu-Hausdorff functional with respect to k · kC on CL ([a1 , b1 ], [a1 , b1 ]). Proof. We will look for those ci , for which in condition of Theorem 3.1 the operators Efi , i = 1, 2, are ci −weakly Picard operators. Set Xϕ,ψ := {x ∈ CL ([a1 , b1 ], [a1 , b1 ]) : x|[a1 ,a] = ϕ, x|[b,b1 ] = ψ}. It is clear that Efi |Xϕ,ψ = Bfi . So, from Theorem 2.9 and Theorem 3.1 we have kEf2i (x) − Efi (x)kC ≤

2(b − a)q Lfi (Lv max{|a1 |, |b1 |}v−1 + 2) kEfi (x) − xkC Γ(q + 1)

for all x ∈ CL ([a1 , b1 ], [a1 , b1 ]) and i = 1, 2.


307

Now, choosing λi =

2(b − a)q Lfi (Lv max{|a1 |, |b1 |}v−1 + 2) , Γ(q + 1)

1 , i = 1, 2. we get that Efi are ci −weakly Picard operators, with ci = 1−λ i Next, we obtain 2(b − a)q kEf1 (x) − Ef2 (x)kC ≤ η , Γ(q + 1) for all x ∈ CL ([a1 , b1 ], [a1 , b1 ]). Applying Theorem 2.13 we have that

Hk·kC (FEf1 , FEf2 ) ≤

Γ(q + 1) − 2(b −

η2(b − a)q . v−1 + 2) f (Lv max{|a1 |, |b1 |}

a)q L

The proof is completed.

5. Examples

Consider the following problem:  3   c D 22 ,t x(t) = µx(x(t)), t ∈ [ 52 , 35 ], µ > 0, 5 x(t) = 21 , t ∈ [ 15 , 25 ],   x(t) = 12 , t ∈ [ 35 , 45 ],

(5.1)

where x ∈ CL ([ 15 , 54 ], [ 15 , 45 ]). Proposition 5.1. Consider the problem (5.1). We suppose that √ √ 5π µ < 3L√ for 0 < L ≤ 6 − 1, 8 √ 15 5π 6 − 1 ≥ L. µ < 8(L+2) for

(5.2)

Then the problem (5.1) has in CL ([ 51 , 45 ], [ 15 , 45 ]) a unique solution. Proof. First of all notice that accordingly to Theorem 3.1 we have v = 1, q = 32 , a = 25 , b = 35 , ψ( 35 ) = 12 , ϕ( 25 ) = 21 , a1 = 15 , b1 = 45 . Observe that the Lipschitz constant for the function f (t, u1 , u2 ) = µu2 is Lf = µ and |f (t, u1 , u2 )−f (t, w1 , w2 )| ≤ µ|u2 − w2 |, ui , wi ∈ [ 15 , 54 ]. So we choose mf = µ5 and Mf = 4µ 5 . By a common check in the conditions of Theorem 3.1 we can make sure that √ 45 5π µ≤ 32  q q  a1 ≤ min(ϕ(a), ψ(b)) − max 0, Mf (b−a) + min 0, mf (b−a) , Γ(q+1) Γ(q+1) ⇐⇒ q q  max(ϕ(a), ψ(b)) − min 0, mf (b−a) + max 0, Mf (b−a) ≤ b1 ; Γ(q+1) Γ(q+1) 8 |ψ(b) − ϕ(a)| (b − a)q−1 (1 + q) max{|mf |, |Mf |} √ µ < L ⇐⇒ + < L, b Γ(q + 1) 3 5π and 8 √

15 5π

µ
0, 5 x(t) = ϕ1 , [ 51 , 52 ],   x(t) = ψ1 , [ 35 , 45 ],

(5.3)

and  3   c D 22 ,t x(t) = µ2 x(x(t)), t ∈ [ 52 , 35 ], µ2 > 0, 5 x(t) = ϕ2 , [ 51 , 52 ],   x(t) = ψ2 , [ 35 , 45 ].

(5.4)

Suppose that we have satisfied the following assumptions (H1 ) ϕi ∈ CL ([ 15 , 25 ], [ 15 , 45 ]), ψi ∈ CL ([ 35 , 45 ], [ 15 , 45 ]) such that ϕi ( 25 ) = 12 , ψi ( 35 ) = 12 , i = 1, 2; (H2 ) we are in the conditions (5.2) of Proposition 5.1 for both of the problems (5.3) and (5.4). Let x∗1 , be the unique solution of the problem (5.3) and x∗2 be the unique solution of the problem (5.4). We are looking for an estimation for kx∗1 − x∗2 kC . Then, build upon Theorem 4.1 and Theorem 4.3, by a common substitution one can make sure that we have Proposition 5.2. Consider the problems (5.3), (5.4) and suppose the requirements (H1 )–(H2 ) hold. Additionally, (i) there exists η1 > 0 such that 1 2 |ϕ1 (t) − ϕ2 (t)| ≤ η1 , ∀ t ∈ [ , ], 5 5 and 3 4 |ψ1 (t) − ψ2 (t)| ≤ η1 , ∀ t ∈ [ , ]. 5 5 (ii) there exists η2 > 0 such that |µ1 − µ2 | ≤ Then |x∗1 (·; ψ1 , ψ1 , f1 )

−

x∗2 (·; ψ2 , ψ2 , f2 )|

≤

5 η2 . 4 √ 45 5πη1 + 8η2 √ . 15 5π − 8(L + 2) min{µ1 , µ2 }


309

Further, let FEf1 be the solution set of the first equation of the problem (5.3) and FEf2 be the solution set of the first equation of the problem (5.4). Then, √ 15 5πη2 , Hk·kC (FEf1 , FEf2 ) ≤ √ 15 5π − 8(L + 2) max{µ1 , µ2 } where Hk·kC denotes CL ([ 15 , 45 ], [ 15 , 45 ]).

the

Pompeiu-Hausdorff

functional

with

respect

to

Acknowledgement. The first author’s work was supported by Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169) and Tianyuan Special Funds of the National Natural Science Foundation of China (11026102); The second author’s work was partially supported by Grants VEGA-MS 1/0507/11 and APVV-0414-07; the third author’s work was supported by National Natural Science Foundation of China (10971173). References [1] I.A. Rus, Metrical fixed point theorems, Univ. of Cluj-Napoca, Romania, 1979. [2] I.A. Rus, Picard mappings: results and problems, Seminar on Fixed Point Theory, Preprint no. 3(1987), 55-64. [3] I.A. Rus, Weakly Picard mappings, Comment. Math. Univ. Carolinae, 34(1993), 769-773. [4] I.A. Rus, S. Mure¸san, Data dependence of the fixed points set of some weakly Picard operators, In: Proc. Itinerant Seminar (Elena Popoviciu-Ed.), Srima Publishing House, Cluj-Napoca, 2000, 201-207. [5] I.A. Rus, Functional-differential equations of mixed type, via weakly Picard operators, Seminar on Fixed Point Theory, Cluj-Napoca, 2002, 335-345. [6] I.A. Rus, Picard operators and applications, Sci. Math. Jpn., 58(2003), 191-219. [7] I.A. Rus, E. Egri, Boundary value problems for iterative functional-differential equations, Studia Univ. Babe¸s-Bolyai, Mathematica, 51(2006), 109-126. [8] M.A. S ¸ erban, I.A. Rus, A. Petru¸sel, A class of abstract Volterra equations, via weakly Picard operators technique, Math. Ineq. Appl., 13(2010), 255-269. [9] V. Mure¸san, Existence, uniqueness and data dependence for the solutions of some integrodifferential equations of mixed type in Banach space, J. Anal. Appl., 23(2004), 205-216. [10] V. Mure¸san, Volterra integral equations with iterations of linear modification of the argument, Novi Sad J. Math., 33(2003), 1-10. [11] I.M. Olaru, An integral equation via weakly Picard operators, Fixed Point Theory, 11(2010), 97-106. ˇ Picard and weakly Picard operators technique for nonlinear [12] J. Wang, Y. Zhou, M. Medved, differential equations in Banach spaces, J. Math. Anal. Appl., 389(2012), 261-274. [13] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, 2010. [14] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. [15] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [16] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [17] V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, HEP, 2010. [18] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 109(2010), 973-1033.

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Received: March 29, 2012; Accepted: June 11, 2012.