Báo cáo sinh học: " Existence of solutions for perturbed abstract measure functional differential equations"

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Advances in Difference Equations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Existence of solutions for perturbed abstract measure functional differential equations Advances in Difference Equations 2011, 2011:67 doi:10.1186/1687-1847-2011-67 Xiaojun Wan (wanxiaojun508@sina.com) Jitao Sun (sunjt@sh163.net) ISSN 1687-1847 Article type Research Submission date 5 August 2011 Acceptance date 23 December 2011 Publication date 23 December 2011 Article URL http://www.advancesindifferenceequations.com/content/2011/1/67 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Wan and Sun ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Existence of solutions for perturbed abstract measure func- tional diﬀerential equations Xiaojun Wan and Jitao Sun∗ Department of Mathematics, Tongji University, Shanghai 200092, China Email: XW: wanxiaojun508@sina.com; ∗ sunjt@sh163.net; ∗ Corresponding author Abstract In this article, we investigate existence of solutions for perturbed abstract measure functional diﬀerential equations. Based on the Arzela–Ascoli theorem and the ﬁxed point theorem, we give suﬃcient conditions for existence of ` solutions for a class of perturbed abstract measure functional diﬀerential equations. Our system includes the systems studied in some previous articles as special cases and our suﬃcient conditions for existence of solutions are less conservative. An example is given to illustrate the eﬀectiveness of our existence theorem of solutions. 1 Introduction Abstract measure diﬀerential equations are more general than diﬀerence equations, diﬀerential equations, and diﬀerential equations with impulses. The study of abstract measure diﬀerential equations was initiated by Sharma [1] in 1970s. From then on, properties of abstract measure diﬀerential equations have been 1
researched by various authors. But up to now, there were only some limited results on abstract measure diﬀerential equations can be found, such as existence [2–6], uniqueness [2, 3, 5], and extremal solutions [3, 4, 6]. There were also several researches on abstract measure integro-diﬀerential equations [7, 8]. The study on abstract measure diﬀerential equations is still rare. Recently, there were a number of focuses on existence problems, for example, see [9–11] and references therein, and functional diﬀerential equations were also investigated widely, such as work done in [12–14]. However, there were only very few results on existence of solutions for abstract measure functional diﬀerential equations. There were some consideration on abstract measure delay diﬀerential equations [2] and perturbed abstract measure diﬀerential equations [4]. However, to the best of authors’ knowledge, there were not any results dealing with perturbed abstract measure functional diﬀerential equations. In this article, we investigate the existence of solutions for perturbed abstract measure functional diﬀerential equations. This is a problem of diﬃculty and challenge. Based on the Leray–Schauder alternative involving the sum of two operators [15] and the Arzel`–Ascoli theorem, the existence results of our system is derived. The perturbed abstract a measure functional diﬀerential system researched in this paper includes the systems studied in [2, 4] as special cases. Additionally, considering appropriate degeneration, our suﬃcient conditions for existence of solutions are also less conservative than those in [2, 4], respectively. The study in the previous articles are improved. The content of this article is organized as follows: In Section 2, some preliminary fact is recalled; the perturbed abstract measure functional diﬀerential equation is proposed, as well as some relative notations. In Section 3, the existence theorem is given and strict proof is shown; two remarks are given to analyze that our existence results are less conservative. In Section 4, an example is used to illustrate the eﬀectiveness of our results for existence of solutions. 2 Preliminary Deﬁnition 2.1 Let X be a Banach space, a mapping T : X → X is called D -Lipschitzian, if there is a continuous and nondecreasing function φT : R+ → R+ such that T x − T y ≤ φT ( x − y ) for all x, y ∈ X , φT (0) = 0. T is called Lipschitzian, if φT (x) = ax, where a > 0 is a Lipschitz constant. Furthermore, T is called a contraction on X , if a < 1. 2
Let T : X → X , where X is a Banach space. T is called totally bounded, if T (M ) is totally bounded for any bounded subset M of X . T is called completely continuous, if T is continuous and totally bounded on X . T is called compact, if T (X ) is a compact subset of X . Every compact operator is a totally bounded operator. Deﬁne any convenient norm · on X. Let x, y be two arbitrary points in X , then segment xy is deﬁned as xy = {z ∈ X |z = x + r(y − x), 0 ≤ r ≤ 1}. Let x0 ∈ X be a ﬁxed point and z ∈ X , 0x0 ⊂ 0z , where 0 is the zero element of X. Then for any x ∈ x0 z , we deﬁne the sets Sx and S x as Sx = {rx| − ∞ < r < 1}, S x = {rx| − ∞ < r ≤ 1}. For any x1 , x2 ∈ x0 z ⊂ X , we denote x1 < x2 if Sx1 ⊂ Sx2 , or equivalently x0 x1 ⊂ x0 x2 . Let ω ∈ [0, h], h > 0. For any x ∈ x0 z , xω is deﬁned by xω < x, x − xω = ω. Let M denote the σ -algebra which generated by all subsets of X , so that (X, M ) is a measurable space. Let ca(X, M ) be the space consisting of all signed measures on M . The norm · on ca(X, M ) is deﬁned as: p = |p|(X ), where |p| is a total variation measure of p, ∞ |p|(X ) = sup |p(Ei )|, Ei ⊂ X , π i=1 where π : {Ei : i ∈ N} is any partition of X . Then ca(X, M ) is a Banach space with the norm deﬁned above. Let µ be a σ -ﬁnite positive measure on X. p ∈ ca(X, M ) is called absolutely continuous with respect to the measure µ, if µ(E ) = 0 implies p(E ) = 0 for some E ∈ M . And we denote p µ. Let M0 denote the σ -algebra on Sx0 . For x0 < z , Mz denotes the σ -algebra on Sz . It is obvious that Mz contains M0 and the sets of the form Sx , x ∈ x0 z . Given a p ∈ ca(X, M ) with p µ, consider perturbed abstract measure functional diﬀerential equation: (1) dp = f (x, p(S xω )) + g (x, p(S x ), p(S xω )), a.e. [µ] on x0 z, dµ 3
and (2) p(E ) = q (E ), E ∈ M0 . dp where q is a given signed measure, is a Radon-Nikodym derivative of p with respect to µ. f : Sz × R → R, dµ g : Sz × R × R → R. f (x, p(S xω )) and g (x, p(S x ), p(S xω )) are µ-integrable for each p ∈ ca(Sz , Mz ). Deﬁne |f (x, p(·))| = sup |f (x, p(S xω ))|, ω ∈[0,h] |g (x, p, p(·))| = sup |g (x, p(S x ), p(S xω ))|. ω ∈[0,h] Deﬁnition 2.2 q is a given signed measure on M0 . A signed measure p ∈ ca(Sz , Mz ) is called a solution of (1)–(2), if (i) p(E ) = q (E ), E ∈ M0 , (ii) p µ on x0 z , (iii) p satisﬁes (1) a.e. [µ] on x0 z . Remark 2.1 The system (1)–(2) is equivalent to the following perturbed abstract measure functional integral system:     f (x, p(S xω ))dµ + g (x, p(S x ), p(S xω ))dµ, E ∈ Mz , E ⊂ x0 z E E p(E ) =   q (E ),  E ∈ M0 . We denote a solution p of (1)–(2) as p(S x0 , q ). Deﬁnition 2.3 A function β : Sz × R × R → R is called Carath´odory, if e (i) x → β (x, y, z ) is µ-measurable for each (y, z ) ∈ R × R, (ii) (y, z ) → β (x, y, z ) is continuous a.e. [µ] on x0 z . The function β deﬁned as the above is called L1 -Carath´odory, further if e µ (iii) for each real number r > 0, there exists a function hr (x) ∈ L1 (Sz , R+ ) such that µ |β (x, y, z )| ≤ hr (x) a.e. [µ] on x0 z for each y ∈ R, z ∈ R with |y | ≤ r, |z | ≤ r. 4
Lemma 2.1 [15] Let Br (0) and B r (0) denote, respectively, the open and closed balls in a Banach algebra X with center 0 and radius r for some real number r > 0. Suppose A : X → X , B : B r (0) → X are two operators satisfying the following conditions: (a) A is a contraction, and (b) B is completely continuous. Then either (i) the operator equation Ax + Bx = x has a solution x in B r (0), or u (ii) there exists an element u ∈ ∂ B r (0) such that λA( λ ) + λBu = u for some λ ∈ (0, 1). 3 Main results We consider the following assumptions: (A0 ) For any z ∈ X satisﬁes x0 < z , the σ -algebra Mz is compact with respect to the topology generated by the pseudo-metric d deﬁned by d(E1 , E2 ) = |µ|(E1 ∆E2 ), E1 , E2 ∈ Mz . (A1 ) µ({x0 }) = 0. (A2 ) q is continuous on Mz with respect to the pseudo-metric d deﬁned in (A0 ). (A3 ) There exists a µ-integrable function α : Sz → R+ such that |f (x, y1 (·)) − f (x, y2 (·))| ≤ α(x)|y1 (·) − y2 (·)| a.e.[µ] on x0 z. (A4 ) g (x, y, z (·)) is L1 -Carath´odory. e µ Theorem 3.1 Suppose that the assumptions (A0 )–(A4 ) hold. Further if α < 1 and there exists a real L1 µ number r > 0 such that F0 + q + hr L1 (3) r> µ 1− α L1 µ where F0 = |f (x, 0)|dµ. Then the system (1)–(2) has a solution on x0 z . x0 z 5
Proof: Consider the open ball Br (0) and the closed ball B r (0) in ca(Sz , Mz ), with r satisfying the inequality (3). Deﬁne two operators A : ca(Sz , Mz ) → ca(Sz , Mz ), B : B r (0) → ca(Sz , Mz ) as:     f (x, p(S xω ))dµ, E ∈ Mz , E ⊂ x0 z E Ap(E ) =   0,  E ∈ M0 .     g (x, p(S x ), p(S xω ))dµ, E ∈ Mz , E ⊂ x0 z E Bp(E ) =   q (E ),  E ∈ M0 . Now we prove the operators A and B satisfy conditions that are given in Lemma 2.1 on ca(Sz , Mz ) and B r (0), respectively. Step I. A is a contraction on ca(Sz , Mz ). Let p1 , p2 ∈ ca(Sz , Mz ). Then by assumption (A3 ) |Ap1 (E ) − Ap2 (E )| =| f (x, p1 (S xω ))dµ − f (x, p2 (S xω ))dµ| E E ≤ α(x) sup |p1 (S xω ) − p2 (S xω )|dµ E ω ≤ α(x)|p1 − p2 |(S x )dµ E ≤α L1 |p1 − p2 |(E ) µ for all E ∈ Mz . Considering the deﬁnition of norm on ca(Sz , Mz ), we have Ap1 − Ap2 ≤ α p1 − p2 , L1 µ for all p1 , p2 ∈ ca(Sz , Mz ). So A is a contraction on ca(Sz , Mz ). Step II. B is continuous on B r (0). Let {pn }n∈N be a sequence of signed measures in B r (0), and {pn }n∈N converges to a signed measure p. In case E ∈ Mz , E ⊂ x0 z , using dominated convergence theorem lim Bpn (E ) = lim g (x, pn (S x ), pn (S xω ))dµ n→∞ E = g (x, p(S x ), p(S xω ))dµ E = Bp(E ). In case E ∈ M0 , lim Bpn (E ) = q (E ) = Bp(E ). Obviously, B is a continuous operator on B r (0). n→∞ 6
Step III. B is a totally bounded operator on B r (0). Let {pn }n∈N be a sequence of signed measures in B r (0), then pn ≤ r(n ∈ N). Next we show that {Bpn }n∈N are uniformly bounded and equicontinuous. First, {Bpn }n∈N are uniformly bounded. Let E ∈ Mz , and E = F G, where F ∈ M0 and G ∈ Mz , G ⊂ x0 z . F G = ∅. Hence, |Bpn (E )| ≤ |q (F )| + |g (x, pn (S x ), pn (S xω ))|dµ G ≤ |q (F )| + hr (x)dµ, G consequently, ∞ B pn = |Bpn |(Sz ) = sup |Bpn (Ei )| ≤ q + hr L1 , µ i=1 for every pn ∈ B r (0). Then {Bpn }n∈N are uniformly bounded. Second, {Bpn }n∈N is an equicontinuous sequence in ca(Sz , Mz ). Let Ei ∈ Mz , and Ei = Fi Gi , where Fi ∈ M0 and Gi ∈ Mz , Gi ⊂ x0 z , and Fi Gi = ∅. i = 1, 2. Considering assumption (A4 ), then |Bpn (E1 ) − Bpn (E2 )| ≤ |q (F1 ) − q (F2 )| + | g (x, pn (S x ), pn (S xω ))dµ G1 − g (x, pn (S x ), pn (S xω ))dµ| G2 ≤ |q (F1 ) − q (F2 )| + |g (x, pn (S x ), pn (S xω ))|dµ G1 ∆G2 ≤ |q (F1 ) − q (F2 )| + hr (x)dµ. G1 ∆G2 when d(E1 , E2 ) → 0, E1 → E2 . Then, F1 → F2 , and |µ|(G1 ∆G2 ) = d(G1 ∆G2 ) → 0. Considering assumption (A2 ), q is continuous on compact Mz implies it is uniformly continuous on Mz . so |Bpn (E1 ) − Bpn (E2 )| → 0, as d(E1 , E2 ) → 0 for every pn ∈ B r (0). {Bpn }n∈N is an equicontinuous sequence in ca(Sz , Mz ). According to the Arzel`–Ascoli theorem, there is a subset {Bpnk }n,k∈N of {Bpn }n∈N that converges a uniformly. Thus, operator B is compact on B r (0). Then, B is a totally bounded operator on B r (0). From steps II and III, the operator B is completely continuous on B r (0). Step IV. (1)–(2) has a solution on x0 z . 7
Now, by applying Lemma 2.1, we show that (i) holds. Otherwise, there exists an element u ∈ ca(Sz , Mz ) u with u = r such that λA( λ ) + λBu = u for some λ ∈ (0, 1). If it is true, we have    λ f (x, u(S xω ) )dµ + λ  g (x, u(S x ), u(S xω ))dµ, E ∈ Mz , E ⊂ x0 z λ E E u(E ) =   λq (E ),  E ∈ M0 . for some λ ∈ (0, 1). Then |u(E )| ≤ |λA( u(λ ) )| + |λB (u(E ))| E u(S xω ) ≤ λ|q (F )| + λ [|f (x, ) − f (x, 0)| + |f (x, 0)|]dµ λ G +λ |g (x, u(S x ), u(S xω ))|dµ G ≤ |q (F )| + α(x)|u(S xω )|dµ + |f (x, 0)|dµ + hr (x)dµ G G G ≤ |q (F )| + α L1 |u(E )| + |f (x, 0)|dµ + hr (x)dµ. G G µ so we get |q (F )| + |f (x, 0)|dµ + hr (x)dµ G G |u(E )| ≤ , 1 − α L1 µ for all E ∈ Mz . By the deﬁnition of the norm on ca(Sz , Mz ), q + F0 + hr L1 µ u≤ . 1− α L1 µ As u = r, we have q + F0 + hr L1 µ r≤ . 1− α L1 µ This is a contradiction. Consequently, the equation p(E ) = Ap(E ) + Bp(E ) has a solution p(S x0 , q ) ∈ B r (0) ⊂ ca(Sz , Mz ). It is said that (1)–(2) has a solution on x0 z . The proof of Theorem 3.1 is completed. Remark 3.1 If f (x, y ) = 0 and ω is a given constant, then system (1)–(2) degenerates into (4) dp = g (x, p(S x ), p(S xω )), a.e. [µ] on x0 z, dµ and (5) p(E ) = q (E ), E ∈ M0 , 8
obviously, (4)–(5) is the system (4) considered in [2]. Additionally, our degenerated assumptions for the existence theorem equal to (A1 )–(A4 ) in [2], the more complex assumption (A5 ) [2] is not necessary. So our results are less conservative. Remark 3.2 If ω = 0, then system (1)–(2) degenerates into (6) dp = f (x, p(S x )) + g (x, p(S x )), a.e. [µ] on x0 z, dµ and (7) p(E ) = q (E ), E ∈ M0 . obviously, (6)–(7) is the system (3.6)–(3.7) studied in [4]. Additionally, our degenerated assumptions for the existence theorem equal to (A0 )–(A2 ) and (B0 )–(B1 ) in [4], the more complex assumption (B2 ) [4] is not necessary. So, our results are less conservative. 4 Example Let p ∈ ca(Sz , Mz ) with p µ. Consider the equation as follows: hr (x)|p(S x )+p(S xω )| (8) dp = α(x)|p(S xω )| + , a.e. [µ] on x0 z, dµ 1+|p(S x )+p(S xω )| and (9) p(E ) = q (E ), E ∈ M0 . where hr (x) ∈ L1 (Sz , R+ ), α < 1 and 0 ≤ ω ≤ h(h > 0). f : Sz × R → R and g : Sz × R × R → R are L1 µ µ deﬁned as f (x, y (·)) = α(x)|p(S xω )|, hr (x)|p(S x ) + p(S xω )| g (x, y, z (·)) = . 1 + |p(S x ) + p(S xω )| It is obvious that the assumptions (A0 ) − (A2 ) hold. Then, we show that f and g satisfy the assumptions (A3 ) and (A4 ), respectively. First, f is continuous on ca(Sz , Mz ). |f (x, y1 (·)) − f (x, y2 (·))| ≤ |α(x)| sup ||p1 (S xω )| − |p2 (S xω )|| ω ≤ |α(x)| sup |p1 (S xω ) − p2 (S xω )| ω = |α(x)||y1 (·) − y2 (·)|, 9
f (x, y (·)) satisﬁes (A3 ). Second, |g (x, y, z (·))| ≤ hr (x). g (x, y, z (·)) satisﬁes the assumption (A4 ). F0 + q + hr L1 Thus, if there exists r ∈ R satisﬁes r > with F0 = |f (x, 0)|dµ, all conditions in µ 1− α x0 z L1 µ Theorem 3.1 are satisﬁed. So, (8)–(9) has a solution p(S x0 , q ) on x0 z . Competing interests The authors declare that they have no competing interests. Authors’ contributions JS directed the study and helped inspection. XW carried out the main results of this paper, including the existence theorem and the example. All the authors read and approved the ﬁnal manuscript. Acknowledgments This study was supported by the National Science Foundation of China under grant 61174039, and by the Fundamental Research Funds for the Central Universities of China. The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the article. References 1. Sharma, RR: An abstract measure diﬀerential equation. Proc. Am. Math. Soc. 32, 503–510 (1972) 2. Dhage BC, Chate DN, Ntouyas SK: A system of abstract measure delay diﬀerential equations. Electron. J. Qual. Theory Diﬀ. Equ. 8, 1–14 (2003) 3. Dhage BC, Chate DN, Ntouyas SK: Abstract measure diﬀerential equations. Dyn. Syst. Appl. 13, 105–117 (2004) 4. Dhage BC, Bellale SS: Existence theorems for perturbed abstract measure diﬀerential equations. Nonlin- ear Anal. 71, e319–e328 (2009) 5. Joshi SR, Kasaralikar SN: Diﬀerential inequalities for a system of abstract measure delay diﬀerential equations. J. Math. Phys. Sci. 16, 515–523 (1982) 10
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