Picard and weakly Picard operators technique for nonlinear differential equations in Banach spaces

J. Math. Anal. Appl. 389 (2012) 261–274

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Picard and weakly Picard operators technique for nonlinear differential equations in Banach spaces ✩ JinRong Wang a,∗ , Yong Zhou b , Milan Medved˘ c a b c

Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, PR China Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, PR China Department of Mathematical and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 19 February 2011 Available online 25 November 2011 Submitted by J.J. Nieto

In this paper we study nonlocal Cauchy problems for differential equations in Banach spaces. Using Picard and weakly Picard operators technique and suitable Bielecki norms, some existence, uniqueness and data dependence results are obtained under some mild conditions. As an application, we also discuss a class of impulsive Cauchy problems by adapting the same methods. © 2011 Elsevier Inc. All rights reserved.

Keywords: Nonlocal Cauchy problems Impulsive Picard operators Weakly Picard operators

1. Introduction Throughout this paper, ( X ,  · ) will be a Banach spaces, and J := [0, T ], T > 0. We firstly consider the following nonlocal Cauchy problem of differential equation



x˙ (t ) = f (t , x(t )),

t ∈ J,

x(0) = x0 + g (x),

x0 ∈ X ,

(1)

where f : J × X → X is continuous function and the nonlocal term g : C ( J , X ) → X is a given function satisfying some assumptions that will be specified later. It is well known that a function x ∈ C 1 ( J , X ) is a solution of the nonlocal Cauchy problem (1) if and only if x is a solution in C ( J , X ) of the integral equation

t x(t ) = x0 + g (x) +





f s, x(s) ds.

(2)

0

Differential equations arise in a wide variety of scientific and technical applications, including the modelling of problems from the natural and social sciences such as physics, biological sciences and economics. Nonlocal Cauchy problems for differential equations were initiated by Byszewski [5]. As remarked by Byszewski and Lakshmikantham [6], the nonlocal

✩ Research supported by Tianyuan Special Funds of the National Natural Science Foundation of China (11026102), Key Projects of Science and Technology Research in the Ministry of Education (211169), Natural Science Foundation of Guizhou Province (2010, No. 2142), National Natural Science Foundation of China (10971173) and the Slovak Research and Development Agency under the contract No. APVV-0134-10 and by the Slovak Grant Agency VEGA No. 1/0507/11. Corresponding author. ˘ E-mail addresses: [email protected] (J.R. Wang), [email protected] (Y. Zhou), [email protected] (M. Medved).

*

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.11.059

©

2011 Elsevier Inc. All rights reserved.

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conditions can be more useful than the standard initial condition to describe some physical phenomena. For more recent related works, one can refer to Ahmad and Nieto [1], Chen [4], Dubey [10], Vrabie [23] and Zhou and Jiao [34]. To obtain the existence results for solutions of the nonlocal Cauchy problems for some nonlinear differential equations, all kinds of fixed point theorems via Gronwall inequality [15,16,33,34] and Kuratowski measure of noncompactness [7,11], upper and lower solutions method via monotone iterative technique and topological degree theory [9,13,14,17] are widely used. Recently, Muresan ¸ [18] and Serban ¸ et al. [29] discuss some abstract integrodifferential equations by using Picard and weakly Picard operators technique due to Rus et al. [24–28] which is much different from the method used in the papers quoted above. Moreover, Brestovanská [3] proved an existence theorem for an integral equation which is a generalization of that studied by Gripenberg [12]. By the technique of integral inequalities a sufficient condition for approaching all solution of the equation to zero for t → ∞ is also proved there. The proof of the existence theorem is based on the Banach fixed point theorem in a convenient space with suitable Bielecki norm and on a gadget helping the calculation of the constant of contractivity. This equation is related to an epidemic model of the spread of an infectious disease that does not induce the permanent immunity (see [12]). Using the Brestovanská method and Picard operators technique Olaru [20,21] proved existence results for more general type of integral equations. Motivated by [3,18,20,21,29,33], we apply Picard and weakly Picard operators technique to study the nonlocal Cauchy problem (1) under some mild conditions. We consider suitable Bielecki norms in convenient space and obtain some new existence, uniqueness and data dependence results for the solutions of the nonlocal Cauchy problem (1). On the other hand, many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. These processes are subject to short term perturbations whose duration is negligible in comparison with the duration of the processes. Consequently, it is natural to assume that these perturbations act instantaneously, that is in the form of impulses. For more details on theory and applications of impulsive differential equations, see the monograph of Benchohra et al. [2] and papers of Chang and Nieto [8], Peng and Xiang [22] where numerous properties of their solutions and impulsive controls are studied and detailed bibliographies are given. Particularly, by constructing impulsive periodic evolution operators and introducing some generalized Gronwall inequalities, Wang et al. studied the impulsive periodic evolution equations and optimal controls in Banach spaces (see [30–32]). By adapting the Picard operator and weakly Picard operator techniques and ideas established, we discuss a Cauchy problem of impulsive differential equations

⎧ ⎨ x˙ (t ) = f (t , x(t )), t ∈ J \ D := {t 1 , t 2 , . . . , tm }, x(0) = x0 , ⎩ x(t i ) = I i (x(t i )), i = 1, 2, . . . , m,

where f , x0 are as in the nonlocal Cauchy problem (1), I i : X → X is a nonlinear map which determine the size of the jump at t i , 0 = t 0 < t 1 < t 2 < · · · < tm < tm+1 = T , I i (x(t i )) = x(t i+ ) − x(t i− ), x(t i+ ) = limh→0+ = x(t i + h) and x(t i− ) = x(t i ) represent respectively the right and left limits of x(t ) at t = t i . We introduce suitable piecewise Bielecki norms in another convenient space and show the existence, uniqueness and data dependence results for solutions of the above problem. 2. Preliminaries Let ( X , d) be a metric space and A : X → X an operator. We shall use the following notations: P ( X ) := {Y ⊆ X | Y = ∅}; F A := {x ∈ X | A (x) = x}-the fixed point set of A; I ( A ) := {Y ∈ P ( X ) | A (Y ) ⊆ Y }; O A (x) := {x, A (x), A 2 (x), . . . , A n (x), . . .}-the A-orbit of x ∈ X ; H : P ( X ) × P ( X ) → R+ ∪ {+∞}; H d (Y , Z ) := max{supa∈Y infb∈ Z d(a, b), supb∈ Z infa∈Y d(a, b)}-the Pompeiu–Hausdorff functional on P ( X ). Let us recall the following known definitions. For more details, see Rus [25,26]. Definition 2.1. Let ( X , d) be a metric space. An operator A : X → X is a Picard operator if there exists x∗ ∈ X such that F A = {x∗ } and the sequence ( A n (x0 ))n∈N converges to x∗ for all x0 ∈ X . Definition 2.2. Let ( X , d) be a metric space. An operator A : X → X is a weakly Picard operator if the sequence ( A n (x0 ))n∈N converges for all x0 ∈ X and its limit (which may depend on x0 ) is a fixed point of A. If A is a weakly Picard operator, then we consider the operator

A∞ : X → X ,

A ∞ (x) = lim A n (x). n→∞

The following results appeared in Rus [24,26,27] are useful in this paper.

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Theorem 2.3. Let (Y , d) be a complete metric space and A , B : Y → Y two operators. We suppose the following: (i) A is a contraction with contraction constant α and F A = {x∗A }. (ii) B has fixed points and x∗B ∈ F B . (iii) There exists γ > 0 such that d( A (x), B (x))  γ , for all x ∈ Y . Then

  d x∗A , x∗B 

γ 1−α

.

Theorem 2.4. Let ( X , d) be a complete metric space and A , B : X → X two orbitally continuous operators. We suppose the following: (i) There exists α ∈ [0, 1) such that

















d A 2 (x), A (x)  αd x, A (x) , d B 2 (x), B (x)  αd x, B (x)

for all x ∈ X . (ii) There exists γ > 0 such that d( A (x), B (x))  γ for all x ∈ X . Then

γ

Hd( F A , F B ) 

1−α

where H d denotes the Pompeiu–Hausdorff functional. Theorem 2.5. Let ( X , d) be a metric space. Then A : X → X is a weakly Picard operator if and only if there exists a partition of X , X = λ∈Λ X λ , where Λ is the indices’ set of partition, such that (i) X λ ∈ I ( A ), (ii) A | X λ : X λ → X λ is a Picard operator, for all λ ∈ Λ. Let C ( J , X ) be the space of all X -valued continuous functions from J into X . Let  ·  B and  · C be the Bielecki and Chebyshev norms on C ( J , X ) defined by

   (τ > 0) and xC := sup x(t ) x B := sup sup x(t ) e −τ t t∈ J

t∈ J

t∈ J

and denote by d B and dC their corresponding metrics. We consider the set







C L ( J , X ) := x ∈ C ( J , X ): x(τ1 ) − x(τ2 )  L |τ1 − τ2 | for all τ1 , τ2 ∈ J where L > 0, and







C L ( J , B R ) := x ∈ C ( J , B R ): x(τ1 ) − x(τ2 )  L |τ1 − τ2 | for all τ1 , τ2 ∈ J

where B R := {x ∈ X: x  R } with R > 0. Set PC ( J , X ) := {x : J → X | x is continuous at t ∈ J \ D, and x is continuous from left and has right hand limits at t ∈ D }. Let  · PB and  · PC be the piecewise Bielecki and piecewise Chebyshev norms on PC ( J , X ) defined by

   (τ > 0), xPB := sup sup x(t + 0) e −τ t , sup x(t − 0) e −τ t t∈ J

t∈ J

t∈ J



   xPC := max sup x(t + 0) , sup x(t − 0) , t∈ J

t∈ J

and denote by dPB and dPC their corresponding metrics. Set







PC L ( J , X ) := x ∈ PC ( J , X ): x(τ1 ) − x(τ2 )  L |τ1 − τ2 | for all τ1 , τ2 ∈ (t i , t i +1 ]

where L > 0, i = 0, 1, 2, . . . , m. It is clear that if d ∈ {dC , d B }, then (C ( J , X ), d) and (C L ( J , X ), d) are complete metric spaces (see [19]). If d ∈ {dPC , dPB }, one can apply the above results on each interval (t i , t i +1 ] (i = 0, 1, 2, . . . , m) to obtain that (PC ( J , X ), d) and (PC L ( J , X ), d) are complete metric spaces.

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3. The first results via using Picard operators technique We introduce the following hypotheses: (C1) f : J × X → X is continuous. (C2) There exists a function m(t ) ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that

f (t , x)  m(t ),

for all x ∈ X and all t ∈ J . (C3) There exists a constant L > 0 such that



L  M := sup m(t ) . t∈ J

(C4) There exists a function l1 (t ) ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that

f (t , u 1 ) − f (t , u 2 )  l1 (t )u 1 − u 2 

for all u i ∈ X (i = 1, 2) and all t ∈ J . (C5) There exists a function l2 (t ) ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that

g (u ) − g ( v )  l2 (t )u − v C .

τ > 0 such that

(C6) There exists a constant

Lg +

L f (1 − e −τ T )

τ

where



L f := max l1 (t ) t∈ J

< 1, 

L g := max l2 (t ) .

and

t∈ J

We have the following existence results. Theorem 3.1. Let hypotheses (C1)–(C6) be satisfied. Then nonlocal Cauchy problem (1) has a unique solution x∗ in C L ( J , X ), and this solution can be obtained by the successive approximation method, starting from any element of C L ( J , X ). Proof. Problem (1) is equivalent with the integral equation (2). Consider the operator







A : C L ( J , X ),  ·  B → C L ( J , X ),  ·  B



defined by

t A (x)(t ) := x0 + g (x) +





f s, x(s) ds. 0

It is easy to see the operator A is well defined due to (C1). Firstly, A (x) ∈ C ( J , X ) for every x ∈ C L ( J , X ). For any δ > 0, every x ∈ C L ( J , X ), by (C2), we obtain

A (x)(t + δ) − A (x)(t ) 

t +δ t +δ   f s, x(s) ds  m(s) ds  M δ. t

t

It is easy to see that the right-hand side of the above inequality tends to zero as δ → 0. Therefore A (x) ∈ C ( J , X ). Secondly, A (x) ∈ C L ( J , X ). Without lose of generality, for any τ1 < τ2 , τ1 , τ2 ∈ J , applying (C2), we have

A (x)(τ2 ) − A (x)(τ1 ) 

τ2 τ2   f s, x(s) ds  m(s) ds  M |τ1 − τ2 |. τ1

τ1

Similarly, for any τ1 > τ2 , τ1 , τ2 ∈ J , we also have the above inequality. This implies that A (x) is belong to C L ( J , X ) due to (C3). Thirdly, A is continuous.

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For that, let {xn } be a sequence of B R such that xn → x in B R . Then, f (s, xn (s)) → f (s, x(s)) as n → ∞ due to (C1). On the one other hand using (C2), we get for each s ∈ [0, t ],  f (s, xn (s)) − f (s, x(s))  2m(s). On the other hand, using the fact that the functions s → m(s) is integrable on [0, t ], by means of the Lebesgue Dominated Convergence Theorem yields

t

    f s, xn (s) − f s, x(s) ds → 0.

0

For all t ∈ J , we have

A (xn )(t ) − A (x)(t )  l2 (t )xn − xC +

t

    f s, xn (s) − f s, x(s) ds.

0

The above inequality yields that

A (xn ) − A (x)  L g xn − xC + C

t

    f s, xn (s) − f s, x(s) ds → 0.

0

Thus, A (xn ) → A (x) as n → ∞ which implies that A is continuous. Finally, A is a Picard operator. In fact, for all x, z ∈ C L ( J , X ), using (C4) and (C5) we have

A (x)(t ) − A ( z)(t )  g (x) − g ( z) +

t

  f (s, x(s) − f s, z(s) ds

0

t  l2 (t )x − zC +





l1 (s) x(s) − z(s) ds

0

 L g x − z C + L f

t 







sup x(s) − z(s) e −τ s e τ s ds

s∈[0,t ]

0

t e τ s ds

 L g x − z C + L f x − z  B 0

 L g x − z C + L f

eτ t − 1

τ

x − z  B .

It follows that





sup A (x)(t ) − A ( z)(t ) e −τ t  L g x − zC e −τ t + L f

eτ t − 1

τ

t∈ J

 L g x − z  B + L f

1 − e −τ T

τ

 x − z  B e −τ t

x − z  B

for all t ∈ J . So we have

 −τ T )  A (x) − A ( z)  L g + L f (1 − e x − z  B B

τ

for all x, z ∈ C L ( J , X ). The operator A is of Lipschitz type with constant

L A := L g +

L f (1 − e −τ T )

τ

(3)

and 0 < L A < 1 due to (C6). By applying the Contraction Principle to this operator we obtain that A is contraction, so A is a Picard operator. This completes the proof. 2

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Example 3.2. Consider the nonlocal Cauchy problem

⎧ x˙ (t ) = sin x(t ), ⎪ ⎪ ⎨ 3  x ( 0 ) = c j x(t j ), ⎪ ⎪ ⎩ j =1

on [0, 1], where c j > 0, j = 1, 2, 3. Set T = 1, M = 1, L = 2, L f = 1, L g =

⇐⇒

the condition (C6)

1 − e −τ

τ

3

j =1 c j

= 12 . In fact,

1

< . 2

If we choose τ∗ = 2 then all the assumptions in Theorem 3.1 are satisfied. By Theorem 3.1, we have following two results. Corollary 3.3. Suppose the following conditions hold: (C1 ) (C2 ) (C3 ) (C4 ) (C5 )

f ∈ C ( J × X , X ). There exists a constant There exists a constant There exists a constant There exists a constant

M > 0 such that  f (t , x)  M for all x ∈ X and all t ∈ J . L > 0 such that L  M. L f > 0 such that  f (t , u 1 ) − f (t , u 2 )  L f u 1 − u 2  for all u i ∈ X (i = 1, 2) and all t ∈ J . L g > 0 such that  g (u ) − g ( v )  L g u − v C for all u , v ∈ C ( J , X ). L

(C6 ) There exists a constant τ > 0 such that L A = L g + τf < 1. Then the nonlocal Cauchy problem (1) has a unique solution x∗ in C L ( J , X ), and this solution can be obtained by the successive approximation method, starting from any element of C L ( J , X ). Corollary 3.4. Suppose the following conditions hold: (C1 ) f ∈ C ( J × B R , X ). (C2 ) There exists a constant M ( R ) > 0 such that  f (t , x)  M ( R ) for all x ∈ B R and all t ∈ J with R  x0  + M g + M ( R ) T where M g   g (x) for all x ∈ C ( J , B R ). (C3 ) There exists a constant L > 0 such that L  M ( R ). (C4 ) There exists a constant L f > 0 such that  f (t , u 1 ) − f (t , u 2 )  L f u 1 − u 2  for all u i ∈ B R (i = 1, 2) and all t ∈ J . (C5 ) There exists a constant L g > 0 such that  g (u ) − g ( v )  L g u − v C for all u , v ∈ C ( J × B R ). L

(C6 ) There exists a constant τ > 0 such that L A = L g + τf < 1. Then the nonlocal Cauchy problem (1) has a unique solution x∗ in C L ( J , B R ), and this solution can be obtained by the successive approximation method, starting from any element of C L ( J , B R ). Now we turn to consider another nonlocal Cauchy problem



x˙ (t ) = h(t , x(t )),

t ∈ J, ¯ x(0) = y 0 + g (x), y 0 ∈ X ,

(4)

where h : J × X → X is continuous, g¯ : C ( J , X ) → X is another nonlocal term. A function x ∈ C 1 ( J , X ) is a solution of nonlocal Cauchy problem (4) if and only if x ∈ C ( J , X ) is a solution of the integral equation

t x(t ) = y 0 + g¯ (x) +





h s, x(s) ds.

(5)

0

Now, we consider both integral equations (2) and (5). We have Theorem 3.5. Suppose the following: (D1) (D2) (D3) (D4)

All conditions in Theorem 3.1 are satisfied and x∗ ∈ C L ( J , X ) is the unique solution of the integral equation (2). With the same function m(t ) as in Theorem 3.1, h(t , x)  m(t ) for all x ∈ X and all t ∈ J . With the same function l1 (t ) as in Theorem 3.1, h(t , u 1 ) − h(t , u 2 )  l1 (t )u 1 − u 2  for all u i ∈ X (i = 1, 2) and all t ∈ J . With the same function l2 (t ) as in Theorem 3.1,  g¯ (u 1 ) − g¯ (u 2 )  l2 (t )u 1 − u 2  for all u i ∈ X (i = 1, 2) and all t ∈ J .

J.R. Wang et al. / J. Math. Anal. Appl. 389 (2012) 261–274

267

(D5) L  M. (D6) There exists a function η(t ) ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that  f (t , u ) − h(t , u )  η(t ) for all u ∈ X and all t ∈ J . (D7) There exists a function μ(t ) ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that  g (u ) − g¯ (u )  μ(t ) for all u ∈ C ( J , X ). Then, if y ∗ is the solution of the integral equation (5),

x∗ − y ∗  B  where

L η := max t∈ J



x0 − y 0  + L μ + L η T 1 − LA 

L μ := max

η(t ) ,

t∈ J

(6)

μ(t ) ,

and L A is given by (3) with a τ = τ∗ > 0 such that 0 < L A < 1. Proof. Consider the following two continuous operators







A , B : C L ( J , X ),  ·  B → C L ( J , X ),  ·  B



defined by

t A (x)(t ) := x0 + g (x) +









f s, x(s) ds, 0

t B (x)(t ) := y 0 + g¯ (x) +

h s, x(s) ds, 0

on J . We have

A (x)(t ) − B (x)(t )  x0 − y 0  + g (x) − g¯ (x) +

t

    f s, x(s) − h s, x(s) ds

0

t  x0 − y 0  + μ(t ) +

η(s) ds 0

 x0 − y 0  + L μ + L η T , for t ∈ J . It follows that

A (x) − B (x)  x0 − y 0  + L μ + L η T . B

So we can apply Theorem 2.3 to obtain the inequality (6) which completes the proof.

2

4. The second results via using weakly Picard operators technique Now, we consider another integral equation

t x(t ) = x(0) + g (x) +





f s, x(s) ds

(7)

0

on J , where f , g are as in the nonlocal Cauchy problem (1). Theorem 4.1. Suppose that for the integral equation (7) the same conditions as in Theorem 3.1 are satisfied. Then this equation has solutions in C L ( J , X ). If S ⊂ C L ( J , X ) is its solutions set, then card S = card X . Proof. Consider the operator







A ∗ : C L ( J , X ),  ·  B → C L ( J , X ),  ·  B



defined by

t A ∗ (x)(t ) := x(0) + g (x) +





f s, x(s) ds. 0

(8)

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This is a continuous operator, but not a Lipschitz one. We can write



CL( J , X) =



X α = x ∈ C L ( J , X ): x(0) = α .

Xα ,

α∈ X

We have that X α is an invariant set of A ∗ and we apply Theorem 3.1 to A ∗ | X α . By using Theorem 2.5 we obtain that A ∗ is a weakly Picard operator. Consider the operator

A∞ ∗ : C L ( J , X ) → C L ( J , X ),

A∞ A n (x). ∗ (x) = nlim →∞ ∗

From A n∗+1 (x) = A ∗ ( A n∗ (x)) and the continuity of A ∗ , A ∞ ∗ (x) ∈ F A ∗ . Then





A∞ ∗ C L ( J , X ) = F A∗ = S So, card S = card X .

S = ∅.

and

2

In order to study data dependence for the solutions set of the integral equation (7) we consider both (7) and the following integral equation

t



x(t ) = x(0) + g¯ (x) +



h s, x(s) ds 0

on J where g¯ , h are as in the nonlocal Cauchy problem (4). Let S1 be the solutions set of this equation. We make the following assumptions. (E1) There exists a function l∗ (t ) ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that

f (t , u 1 ) − f (t , u 2 )  l∗ (t )u 1 − u 2  and h(t , u 1 ) − h(t , u 2 )  l∗ (t )u 1 − u 2 

for all u i ∈ X (i = 1, 2) and all t ∈ J . (E2) There exists a function ¯l∗ (t ) ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that

g (u ) − g ( v )  ¯l∗ (t )u − v C

and

g¯ (u ) − g¯ ( v )  ¯l∗ (t )u − v C

for all u , v ∈ C ( J , X ). (E3) There exists a function m∗ (t ) ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that

f (t , x)  m∗ (t ) and h(t , x)  m∗ (t )

for all x ∈ X and all t ∈ J . (E4) There exists a constant L > 0 such that L  M ∗ := supt ∈ J {m∗ (t )}.

η∗ ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that f (t , u ) − h(t , u )  η∗ (t )

(E5) There exists a function

for all u ∈ X and all t ∈ J . (E6) There exists a function

g (u ) − g¯ (u )  μ∗ (t )

for all u ∈ C ( J , X ). (E7) L ∗ + L ∗ T < 1 where



L ∗ := max l∗ (t ) , t∈ J

μ∗ (t ) ∈ L 1 ( J , R+ ) ∩ C ( J , R+ ) such that



L ∗ := max ¯l∗ (t ) . t∈ J

Theorem 4.2. Suppose the assumptions (E1)–(E7) are satisfied. Then

H ·C (S , S1 ) 

L ∗μ + L ∗η T 1 − L∗ − L∗ T

where by H ·C we denote the Pompeiu–Hausdorff functional with respect to  · C on C L ( J , X ) and

L ∗η = max t∈ J



η∗ (t ) ,

L ∗μ = max t∈ J



μ∗ (t ) .

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269

Proof. Consider the operator A ∗ defined by (8) in Theorem 4.1 and another operator







B ∗ : C L ( J , X ),  ·  B → C L ( J , X ),  ·  B



defined by

t B ∗ (x)(t ) := x(0) + g¯ (x) +





for t ∈ J .

h s, x(s) ds, 0

It is easy to see A ∗ , B ∗ : (C L ( J , X ),  ·  B ) → (C L ( J , X ),  ·  B ) are two orbitally continuous operators. Moreover, we have

2 A (x)(t ) − A ∗ (x)(t )  L ∗ A ∗ (x) − x + L ∗

t

∗

  A ∗ (x)(s) − x(s) ds  L ∗ + L ∗ T A ∗ (x) − x

C

0

for all x ∈ C L ( J , X ). Similarly,

2   B (x)(t ) − B ∗ (x)(t )  L ∗ + L ∗ T B ∗ (x) − x ∗

C

for all x ∈ C L ( J , X ). It follows that

2   A (x) − A ∗ (x)  L ∗ + L ∗ T A ∗ (x) − x , ∗ C C 2   B (x) − B ∗ (x)  L ∗ + L ∗ T B ∗ (x) − x . ∗ C C

Because of (E5)–(E6),

A ∗ (x) − B ∗ (x)  μ∗ (t ) + C

t

η∗ (s) ds  L ∗μ + L ∗η T

0

for all x ∈ C L ( J , X ). By (E7) and applying Theorem 2.4 we obtain

H ·C ( F A ∗ , F B ∗ )  This completes the proof.

L ∗μ + L ∗η T 1 − L∗ − L∗ T

.

2

5. Application to impulsive Cauchy problems The results obtained in Section 3 and Section 4 can be applied to the following Cauchy problem of impulsive differential equation

⎧ ⎨ x˙ (t ) = f (t , x(t )), t ∈ J \ D := {t 1 , t 2 , . . . , tm }, x(0) = x0 , ⎩ x(t i ) = I i (x(t i )), i = 1, 2, . . . , m,

(9)

where f , x0 are as in nonlocal Cauchy problem (1), I i : X → X is a nonlinear map which determine the size of the jump at t i , 0 = t 0 < t 1 < t 2 < · · · < tm < tm+1 = T , I i (x(t i )) = x(t i+ ) − x(t i− ), x(t i+ ) = limh→0+ = x(t i + h) and x(t i− ) = x(t i ) represent respectively the right and left limits of x(t ) at t = t i . It is well known that a function x ∈ P C 1 ( J , X ) is a solution of the impulsive Cauchy problem (1) if and only if x ∈ PC ( J , X ) is a solution of the integral equation

t x(t ) = x0 +





f s, x(s) ds + 0







I i x(t i ) .

(10)

0
We introduce the following hypotheses: (IC1) f : J × X → X is continuous. (IC2) There exists a constant M > 0 such that  f (t , x)  M, for all x ∈ X and all t ∈ J . (IC3) There exists a constant L > 0 such that L  M . (IC4) There exists a constant L f > 0 such that  f (t , u 1 ) − f (t , u 2 )  L f u 1 − u 2  for all u i ∈ X (i = 1, 2) and all t ∈ J . (IC5) I i : X → X and there exists a constant L I > 0 such that  I i (u ) − I i ( v )  L I u − v , for all u , v ∈ X and i = 1, 2, . . . , m. (IC6) There exists a constant τ > 0 such that We give the first main results in this section.

L f (1−e −τ T )

τ

+ mL I < 1.

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Theorem 5.1. Let hypotheses (IC1)–(IC6) be satisfied. Then impulsive Cauchy problem (9) has a unique solution x∗ in PC L ( J , X ). Proof. Consider the operator A 1 : (PC L ( J , X ),  · PB ) → (PC L ( J , X ),  · PB ) defined by

t A 1 (x)(t ) := x0 +







f s, x(s) ds +





I i x(t i ) .

0
0

Step 1. A 1 (x) ∈ PC ( J , X ) for every x ∈ PC L ( J , X ). In fact, for 0  t  t + δ  t 1 , any δ > 0, by (IC2), we obtain

A 1 (x)(t + δ) − A 1 (x)(t ) 

t +δ   f s, x(s) ds  M δ. t

It is easy to see that the right-hand side of the above inequality tends to zero as δ → 0. Therefore A 1 (x) ∈ C ( J , X ). Thus, we can deduce that A 1 (x) ∈ C ([0, t 1 ], X ). Similarly we can also obtain that A 1 (x) ∈ C ((t 1 , t 2 ], X ), A 1 (x) ∈ C ((t 2 , t 3 ], X ), . . . , A 1 (x) ∈ C ((tm , T ], X ). That is, A 1 (x) ∈ PC ( J , X ). Step 2. A 1 x ∈ PC L ( J , X ). Without lose of generality, for any τ1 < τ2 , τ1 , τ2 ∈ (t i , t i +1 ], i = 0, 1, 2, . . . , m, applying (IC2), we have

A 1 (x)(τ2 ) − A 1 (x)(τ1 ) 

τ2   f s, x(s) ds  M (τ2 − τ1 ) = M |τ1 − τ2 |. τ1

Similarly, for any τ1 > τ2 , τ1 , τ2 ∈ (t i , t i +1 ], i = 0, 1, 2, . . . , m, we also have the above inequality. This implies that A 1 x is belong to PC L ( J , X ) due to (IC3). Step 3. A 1 is continuous. For that, let {xn } be a sequence of B R such that xn → x in B R where B R is a bounded set of PC L ( J , X ). For all t ∈ J , we have

A 1 (xn ) − A 1 (x)

t  PC

         f s, xn (s) − f s, x(s) ds + I i xn (t i ) − I i x(t i ) 0
0

t 

     f s, xn (s) − f s, x(s) ds + L I xn − xPC . 0
0

By Lebesgue Dominated Convergence Theorem,  A 1 (xn ) − A 1 (x)PC → 0. Thus, A 1 (xn ) → A 1 (x) as n → ∞ which implies that A 1 is continuous. Finally, A 1 is a Picard operator. In fact, for all x, z ∈ PC L ( J , X ), using (IC4) and (IC5) we have

A 1 (x)(t ) − A 1 ( z)(t ) 

t

       f (s, x(s) − f s, z(s) ds + I i x(t i ) − I i z(t i ) 0
0

t  Lf

 x(s) − z(s) ds + L I x(t i ) − z(t i ) 0
0

t  Lf

     x(s) − z(s) e −τ s e τ s ds + L I x(t i ) − z(t i ) e −τ t i e τ t i 0
0

t e τ s dsx − zPB + L I

 Lf

It follows that

e τ t i x − zPB

0
0

 Lf



eτ t − 1

τ

x − zPB + L I

 0
e τ t i x − zPB .

J.R. Wang et al. / J. Math. Anal. Appl. 389 (2012) 261–274

271

τt  A 1 (x)(t ) − A 1 ( z)(t ) e −τ t  L f e −τ t e − 1 x − zPB + L I e −τ t e τ t i x − zPB

τ

 Lf

1 − e −τ T

τ

0
x − zPB + L I



e τ t i x − zPB

0
for all t ∈ J . So we have

A 1 (x) − A 1 ( z)

   1 − e −τ T τ t i x − z   L + L e I PB f PB

τ

0
for all x, z ∈ PC L ( J , X ). The operator A 1 is of Lipschitz type with constant  L f (1 − e −τ T ) L f (1 − e −τ T ) L = +L eτ ti  + mL A1

I

τ

I

τ

0
(11)

and 0 < L A 1 < 1 due to (IC6). By applying the Contraction Principle to this operator we obtain that A 1 is a Picard operator. This completes the proof. 2 Now we turn to consider another impulsive Cauchy problem

⎧ t ∈ J \ D := {t 1 , t 2 , . . . , tm }, ⎨ x˙ (t ) = h(t , x(t )), x(0) = y 0 , ⎩ x(t i ) = I i (x(t i )), i = 1, 2, . . . , m,

(12)

where h, y 0 are as in nonlocal Cauchy problem (4), I : X → X is another impulsive term. A function x ∈ P C 1 ( J , X ) is a solution of nonlocal Cauchy problem (12) if and only if x ∈ PC ( J , X ) is a solution of the integral equation

t x(t ) = y 0 +





h s, x(s) ds +







I i x(t i ) ,

(13)

0
0

where P C 1 ( J , X ) ≡ {x ∈ PC ( J , X ): x˙ ∈ PC ( J , X )}, endowed with the norm x P C 1 = xPC + ˙xPC , ( P C 1 ( J , X ),  ·  P C 1 ) is a Banach space. Now, we consider both integral equations (10) and (13). We have Theorem 5.2. Suppose the following: (ID1) (ID2) (ID3) (ID4) (ID5) (ID6) (ID7)

All conditions in Theorem 5.1 are satisfied and x∗ ∈ PC L ( J , X ) is the unique solution of the integral equation (10). With the same constant M as in Theorem 5.1, h(t , x)  M for all x ∈ X and all t ∈ J . With the same constant L f as in Theorem 5.1, h(t , u 1 ) − h(t , u 2 )  L f u 1 − u 2  for all u i ∈ X (i = 1, 2) and all t ∈ J . With the same constant L I as in Theorem 5.1,  I i (u ) − I i ( v )  L I u − v  for all u , v ∈ X . L  M. There exists a constant L η > 0 such that  f (t , u ) − h(t , u )  L η for all u ∈ X and t ∈ J . There exists a constant L μ > 0 such that  I (u ) − I (u )  L μ for all u ∈ X .

Then, if y ∗ is the solution of the integral equation (5),

∗ x − y ∗

PB



x0 − y 0  + L η T + mL μ 1 − L A1

(14)

and L A 1 is given by (11) with a τ = τ∗ > 0 such that 0 < L A 1 < 1. Proof. Consider the following two continuous operators







A 1 , B 1 : PC L ( J , X ),  · PB → PC L ( J , X ),  · PB



defined by

t A 1 (x)(t ) := x0 +





f s, x(s) ds +

B 1 (x)(t ) := y 0 +





h s, x(s) ds + 0









I i x(t i ) ,

0
0

t



 0
I i x(t i ) ,

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J.R. Wang et al. / J. Math. Anal. Appl. 389 (2012) 261–274

on J . We have

A 1 (x)(t ) − B 1 (x)(t )  x0 − y 0  +

t

         f s, x(s) − h s, x(s) ds + I i x(t i ) − I i x(t i )

0



 x0 − y 0  + L η T +

0
Lμ,

0
for t ∈ J . It follows that

A 1 (x) − B 1 (x)

PB

 x0 − y 0  + L η T + mL μ .

So we can apply Theorem 2.3 to obtain the inequality (14) which completes the proof.

2

Next, we consider another integral equation

t x(t ) = x(0) +







f s, x(s) ds +





I i x(t i )

(15)

0
0

on J , where f , I i are as in the impulsive Cauchy problem (9). Theorem 5.3. Suppose that for the integral equation (15) the same conditions as in Theorem 5.1 are satisfied. Then this equation has  ⊂ PC L ( J , X ) is its solutions set, then card S = card X . solutions in PC L ( J , X ). If S Proof. Consider the operator A 1∗ : (PC L ( J , X ),  · PB ) → (PC L ( J , X ),  · PB ) defined by

t A 1∗ (x)(t ) := x(0) +





f s, x(s) ds +







I i x(t i ) .

(16)

0
0



This is a continuous operator, but not a Lipschitz one. We can write PC L ( J , X ) = α ∈ X X α , X α = {x ∈ PC L ( J , X ): x(0) = α }. We have that X α is an invariant set of A 1∗ and we apply Theorem 3.1 to A 1∗ | X α . By using Theorem 2.5 we obtain that A 1∗ n+1 n ∞ is a weakly Picard operator. Consider the operator A ∞ 1∗ : PC L ( J , X ) → PC L ( J , X ), A 1∗ (x) = limn→∞ A 1∗ (x). From A 1∗ (x) = n ∞ ∞    A 1∗ ( A 1∗ (x)) and the continuity of A 1∗ , A 1∗ (x) ∈ F A 1∗ . Then A 1∗ (PC L ( J , X )) = F A 1∗ = S and S = ∅. So, card S = card X . 2 In order to study data dependence for the solutions set of the integral equation (15) we consider both (15) and the following integral equation

t x(t ) = x(0) +





h s, x(s) ds +







I i x(t i )

0
0

1 be the solutions set of this equation. on J where h, I are as in the impulsive Cauchy problem (12). Let S We make the following assumptions. (IE1) There exists a constant L ∗ > 0 such that  f (t , u 1 ) − f (t , u 2 )  L ∗ u 1 − u 2  and h(t , u 1 ) − h(t , u 2 )  L ∗ u 1 − u 2  for all u i ∈ X (i = 1, 2) and all t ∈ J . (IE2) There exists a constant L ∗ > 0 such that  I i (u ) − I i ( v )  L ∗ u − v  and  I i (u ) − I i ( v )  L ∗ u − v . for all u , v ∈ X . (IE3) There exists a constant M ∗ > 0 such that  f (t , x)  M ∗ and h(t , x)  M ∗ for all x ∈ X and all t ∈ J . (IE4) There exists a constant L > 0 such that L  M ∗ . (IE5) There exists a constant L ∗η > 0 such that  f (t , u ) − h(t , u )  L ∗η for all u ∈ X and all t ∈ J . (IE6) There exists a constant L ∗μ > 0 such that  I (u ) − I (u )  L ∗μ for all u ∈ X . (IE7) L ∗ T + mL ∗μ < 1.

Theorem 5.4. Suppose the assumptions (IE1)–(IE7) are satisfied. Then

, S1 )  H ·PC (S

L ∗μ + L ∗η T 1 − L∗ − L∗ T

where by H ·PC we denote the Pompeiu–Hausdorff functional with respect to  · PC on PC L ( J , X ).

J.R. Wang et al. / J. Math. Anal. Appl. 389 (2012) 261–274

273

Proof. Consider the operator B 1∗ : (PC L ( J , X ),  · PB ) → (PC L ( J , X ),  · PB ) defined by

t B 1∗ (x)(t ) := x(0) +





h s, x(s) ds +







I i x(t i ) ,

for t ∈ J .

0
0

Moreover, we have

2 A (x)(t ) − A 1∗ (x)(t )  L ∗ 1∗ 

t

   A 1∗ (x)(s) − x(s) ds + I i A 1∗ (x)(t i ) − I i (x(t i )

0



 L∗ T +

L ∗μ



0
A 1∗ (x) − x

PC

0
for all x ∈ PC L ( J , X ). Similarly,

   2 ∗ B (x)(t ) − B 1∗ (x)(t )  L ∗ T + L B 1∗ (x) − x PC μ 1∗ 0
for all x ∈ PC L ( J , X ). It follows that

2   A (x) − A 1∗ (x)  L ∗ T + mL ∗ A 1∗ (x) − x , μ 1∗ PC PC 2   B (x) − B 1∗ (x)  L ∗ T + mL ∗ B 1∗ (x) − x . 1∗

μ

PC

PC

Because of (IE5)–(IE6),

A 1∗ (x) − B 1∗ (x)

PC

 L ∗η T + mL ∗μ

for all x ∈ PC L ( J , X ). By (IE7) and applying Theorem 2.4 we obtain

H ·PC ( F A ∗ , F B ∗ )  This completes the proof.

L ∗η T + mL ∗μ 1 − L ∗ T − mL ∗μ

.

2

Acknowledgments The authors thank the referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improve the presentation of the paper.

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