Open Access

Existence of solutions for perturbed abstract measure functional differential equations

Advances in Difference Equations20112011:67

https://doi.org/10.1186/1687-1847-2011-67

Received: 5 August 2011

Accepted: 23 December 2011

Published: 23 December 2011

Abstract

In this article, we investigate existence of solutions for perturbed abstract measure functional differential equations. Based on the Arzelà-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 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 [26], 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 [911] and references therein, and functional differential equations were also investigated widely, such as work done in [1214]. 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 Arzelà-Ascoli theorem, the existence results of our system is derived. The perturbed abstract 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 : XX is called D-Lipschitzian, if there is a continuous and nondecreasing function ϕ T : ++ such that
T x - T y ϕ T ( x - y )

for all x, y X, ϕ T (0) = 0. T is called Lipschitzian, if ϕ T (x) = ax, where a > 0 is a Lipschitz constant. Furthermore, T is called a contraction on X, if a < 1.

Let T : XX, 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 x y ¯ is defined as
x y ¯ = { z X z = x + r ( y - x ) , 0 r 1 } .
Let x 0 X be a fixed point and z X , 0 x 0 ¯ 0 z ¯ , where 0 is the zero element of X. Then for any x x 0 z ¯ , we define the sets S x and S ̄ x as
S x = { r x | - < r < 1 } , S ̄ x = { r x | - < r 1 } .
For any x 1 , x 2 x 0 z ¯ X , we denote x 1 < x 2 if S x 1 S x 2 , or equivalently x 0 x 1 ¯ x 0 x 2 ¯ . Let ω [0, h], h > 0. For any x x 0 z ¯ , x ω is defined by
x ω < x , x - x ω = 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 π i = 1 p ( E i ) , E i X ,

where π : {E i : i } 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 M 0 denote the σ-algebra on S x 0 . For x 0 < z, M z denotes the σ-algebra on S z . It is obvious that M z contains M 0 and the sets of the form S x , x x 0 z ¯ .

Given a p ca(X, M) with p μ, consider perturbed abstract measure functional differential equation:
d p d μ = f ( x , p ( S ̄ x ω ) ) + g ( x , p ( S ̄ x ) , p ( S ̄ x ω ) ) , a . e . [ μ ] o n x 0 z ¯ ,
(1)
and
p ( E ) = q ( E ) , E M 0 .
(2)

where q is a given signed measure, d p d μ is a Radon-Nikodym derivative of p with respect to μ. f : S z × , g : S z × × . f ( x , p ( S ̄ x ω ) ) and g ( x , p ( S ̄ x ) , p ( S ̄ x ω ) ) are μ-integrable for each p ca(S z , M z ).

Define
f ( x , p ( ) ) = sup ω [ 0 , h ] f ( x , p ( S ̄ x ω ) ) , g ( x , p , p ( ) ) = sup ω [ 0 , h ] g ( x , p ( S ¯ x ) , p ( S ̄ x ω ) ) .
Definition 2.2 q is a given signed measure on M 0. A signed measure p ca(S z , M z ) is called a solution of (1)-(2), if
  1. (i)

    p(E) = q(E), E M 0,

     
  2. (ii)

    p μ on x 0 z ¯ ,

     
  3. (iii)

    p satisfies (1) a.e. [μ] on x 0 z ¯ .

     
Remark 2.1 The system (1)-(2) is equivalent to the following perturbed abstract measure functional integral system:
p ( E ) = E f ( x , p ( S ̄ x ω ) ) d μ + E g ( x , p ( S ̄ x ) , p ( S ̄ x ω ) ) d μ , E M z , E x 0 z ¯ q ( E ) , E M 0 .

We denote a solution p of (1)-(2) as p ( S ̄ x 0 , q ) .

Definition 2.3 A function β : S z × × is called Carathé odory, if
  1. (i)

    xβ(x, y, z) is μ-measurable for each (y, z) × ,

     
  2. (ii)

    (y, z) → β(x, y, z) is continuous a.e. [μ] on x 0 z ¯ .

     
The function β defined as the above is called L μ 1 -Carathé odory, further if
  1. (iii)
    for each real number r > 0, there exists a function h r ( x ) L μ 1 ( S z , + ) such that
    β ( x , y , z ) h r ( x ) a . e . [ μ ] o n x 0 z ¯
     

for each y , z with |y| ≤ r, |z| ≤ r.

Lemma 2.1[15] Let B r ( 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 : XX, B : B ̄ r ( 0 ) X are two operators satisfying the following conditions:
  1. (a)

    A is a contraction, and

     
  2. (b)

    B is completely continuous.

     
Then either
  1. (i)

    the operator equation Ax + Bx = x has a solution x in B ̄ r ( 0 ) , or

     
  2. (ii)

    there exists an element u B ̄ r ( 0 ) such that λ A ( u λ ) + λ B u = u for some λ (0, 1).

     

3 Main results

We consider the following assumptions:

(A 0) For any z X satisfies x 0 < z, the σ-algebra M z is compact with respect to the topology generated by the pseudo-metric d defined by
d ( E 1 , E 2 ) = μ ( E 1 Δ E 2 ) , E 1 , E 2 M z .
  • (A 1) μ({x 0}) = 0.

  • (A 2) q is continuous on M z with respect to the pseudo-metric d defined in (A 0).

  • (A 3) There exists a μ-integrable function α : S z + such that
    f ( x , y 1 ( ) ) - f ( x , y 2 ( ) ) α ( x ) y 1 ( ) - y 2 ( ) a . e . [ μ ] on x 0 z ¯ .
  • (A 4) g(x, y, z(·)) is L μ 1 -Carathé odory.

Theorem 3.1 Suppose that the assumptions (A 0)-(A 4) hold. Further if α L μ 1 < 1 and there exists a real number r > 0 such that
r > F 0 + q | | + | | h r L μ 1 1 - | | α | | L μ 1
(3)

where F 0 = x 0 z ¯ f ( x , 0 ) d μ . Then the system (1)-(2) has a solution on x 0 z ¯ .

Proof: Consider the open ball B r ( 0 ) and the closed ball B ̄ r ( 0 ) in ca(S z , M z ), with r satisfying the inequality (3). Define two operators A : ca(S z , M z ) → ca(S z , M z ), B : B ̄ r ( 0 ) c a ( S z , M z ) as:
A p ( E ) = E f ( x , p ( S ̄ x ω ) ) d μ , E M z , E x 0 z ¯ 0 , E M 0 . B p ( E ) = E g ( x , p ( S ̄ x ) , p ( S ̄ x ω ) ) d μ , E M z , E x 0 z ¯ q ( E ) , E M 0 .

Now we prove the operators A and B satisfy conditions that are given in Lemma 2.1 on ca(S z , M z ) and B ̄ r ( 0 ) , respectively.

Step I. A is a contraction on ca(S z , M z ).

Let p 1, p 2 ca(S z , M z ). Then by assumption (A 3)
A p 1 ( E ) - A p 2 ( E ) = E f ( x , p 1 ( S ̄ x ω ) ) d μ - E f ( x , p 2 ( S ̄ x ω ) ) d μ E α ( x ) sup ω p 1 ( S ̄ x ω ) - p 2 ( S ̄ x ω ) d μ E α ( x ) p 1 - p 2 ( S ̄ x ) d μ α L μ 1 p 1 - p 2 ( E )

for all E M z .

Considering the definition of norm on ca(S z , M z ), we have
A p 1 - A p 2 α L μ 1 p 1 - p 2 ,

for all p 1, p 2 ca(S z , M z ). So A is a contraction on ca(S z , M z ).

Step II. B is continuous on B ̄ r ( 0 ) .

Let {p n } nbe a sequence of signed measures in B ̄ r ( 0 ) , and {p n } nconverges to a signed measure p. In case E M z , E x 0 z ¯ , using dominated convergence theorem
lim B p n ( E ) = lim n E g ( x , p n ( S ̄ x ) , p n ( S ̄ x ω ) ) d μ = E g ( x , p ( S ̄ x ) , p ( S ̄ x ω ) ) d μ = B p ( E ) .

In case E M 0, lim n B p n ( E ) = q ( E ) = B p ( E ) . Obviously, B is a continuous operator on B ̄ r ( 0 ) .

Step III. B is a totally bounded operator on B ̄ r ( 0 ) .

Let {p n } nbe a sequence of signed measures in B ̄ r ( 0 ) , then ||p n || ≤ r(n ). Next we show that {Bp n } nare uniformly bounded and equicontinuous.

First, {Bp n } nare uniformly bounded. Let E M z , and E = F G, where F M 0 and G M z , G x 0 z ¯ . FG = . Hence,
B p n ( E ) q ( F ) + G g ( x , p n ( S ̄ x ) , p n ( S ̄ x ω ) ) d μ q ( F ) + G h r ( x ) d μ ,
consequently,
B p n = B p n ( S z ) = sup i = 1 B p n ( E i ) q + h r L μ 1 ,

for every p n B ̄ r ( 0 ) . Then {Bp n } nare uniformly bounded.

Second, {Bp n } nis an equicontinuous sequence in ca(S z , M z ). Let E i M z , and E i = F i G i , where F i M 0 and G i M z , G i x 0 z ¯ , and F i G i = . i = 1, 2.

Considering assumption (A 4), then
| B p n ( E 1 ) - B p n ( E 2 ) | | q ( F 1 ) - q ( F 2 ) | + | G 1 g ( x , p n ( S ̄ x ) , p n ( S ̄ x ω ) ) d μ - G 2 g ( x , p n ( S ̄ x ) , p n ( S ̄ x ω ) ) d μ | | q ( F 1 ) - q ( F 2 ) | + G 1 Δ G 2 | g ( x , p n ( S ̄ x ) , p n ( S ̄ x ω ) ) | d μ | q ( F 1 ) - q ( F 2 ) | + G 1 Δ G 2 h r ( x ) d μ .

when d(E 1, E 2) → 0, E 1E 2. Then, F 1F 2, and |μ|(G 1ΔG 2) = d(G 1ΔG 2) → 0.

Considering assumption (A 2), q is continuous on compact M z implies it is uniformly continuous on M z . so
B p n ( E 1 ) - B p n ( E 2 ) 0 , a s d ( E 1 , E 2 ) 0

for every p n B ̄ r ( 0 ) .

{Bp n } nis an equicontinuous sequence in ca(S z , M z ).

According to the Arzelà-Ascoli theorem, there is a subset { B p n k } n , k of {Bp n } nthat converges 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 x 0 z ¯ .

Now, by applying Lemma 2.1, we show that (i) holds. Otherwise, there exists an element u ca(S z , M z ) with ||u|| = r such that λ A ( u λ ) + λ B u = u for some λ (0, 1).

If it is true, we have
u ( E ) = λ E f ( x , u ( S ̄ x ω ) λ ) d μ + λ E g ( x , u ( S ̄ x ) , u ( S ̄ x ω ) ) d μ , E M z , E x 0 z ¯ λ q ( E ) , E M 0 .
for some λ (0, 1). Then
u ( E ) λ A ( u ( E ) λ ) + λ B ( u ( E ) ) λ q ( F ) + λ G [ | f ( x , u ( S ̄ x ω ) λ ) - f ( x , 0 ) + f ( x , 0 ) ] d μ + λ G g ( x , u ( S ̄ x ) , u ( S ̄ x ω ) ) d μ q ( F ) + G α ( x ) u ( S ̄ x ω ) d μ + G f ( x , 0 ) d μ + G h r ( x ) d μ q ( F ) | + α L μ 1 u ( E ) + G f ( x , 0 ) d μ + G h r ( x ) d μ .
so we get
u ( E ) q ( F ) + G f ( x , 0 ) d μ + G h r ( x ) d μ 1 - α L μ 1 ,

for all E M z .

By the definition of the norm on ca(S z , M z ),
u q + F 0 + h r L μ 1 1 - α L μ 1 .
As ||u|| = r, we have
r q + F 0 + h r L μ 1 1 - α L μ 1 .

This is a contradiction. Consequently, the equation p(E) = Ap(E) + Bp(E) has a solution p ( S ̄ x 0 , q ) B ̄ r ( 0 ) c a ( S z , M z ) . It is said that (1)-(2) has a solution on x 0 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
d p d μ = g ( x , p ( S ̄ x ) , p ( S ̄ x ω ) ) , a . e . [ μ ] o n x 0 z ¯ ,
(4)
and
p ( E ) = q ( E ) , E M 0 ,
(5)

obviously, (4)-(5) is the system (4) considered in [2]. Additionally, our degenerated assumptions for the existence theorem equal to (A 1)-(A 4) in [2], the more complex assumption (A 5) [2] is not necessary. So our results are less conservative.

Remark 3.2 If ω = 0, then system (1)-(2) degenerates into
d p d μ = f ( x , p ( S ̄ x ) ) + g ( x , p ( S ̄ x ) ) , a . e . [ μ ] o n x 0 z ¯ ,
(6)
and
p ( E ) = q ( E ) , E M 0 .
(7)

obviously, (6)-(7) is the system (3.6)-(3.7) studied in [4]. Additionally, our degenerated assumptions for the existence theorem equal to (A 0)-(A 2) and (B 0)-(B 1) in [4], the more complex assumption (B 2) [4] is not necessary. So, our results are less conservative.

4 Example

Let p ca(S z , M z ) with p μ. Consider the equation as follows:
d p d μ = α ( x ) p ( S ¯ x ω ) + h r ( x ) p ( S ̄ x ) + p ( S ̄ x ω ) 1 + p ( S ̄ x ) + p ( S ̄ x ω ) , a . e . [ μ ] on x 0 z ¯ ,
(8)
and
p ( E ) = q ( E ) , E M 0 .
(9)
where h r ( x ) L μ 1 ( S z , + ) , α L μ 1 < 1 and 0 ≤ ωh(h > 0). f : S z × and g : S z × × are defined as
f ( x , y ( ) ) = α ( x ) p ( S ̄ x ω ) , g ( x , y , z ( ) ) = h r ( x ) p ( S ̄ x ) + p ( S ̄ x ω ) 1 + p ( S ̄ x ) + p ( S ̄ x ω ) .

It is obvious that the assumptions (A 0) - (A 2) hold. Then, we show that f and g satisfy the assumptions (A 3) and (A 4), respectively.

First, f is continuous on ca(S z , M z ).
f ( x , y 1 ( ) ) - f ( x , y 2 ( ) ) α ( x ) sup ω p 1 ( S ̄ x ω ) - p 2 ( S ̄ x ω ) α ( x ) sup ω p 1 ( S ̄ x ω ) - p 2 ( S ̄ x ω ) = α ( x ) y 1 ( ) - y 2 ( ) ,

f(x, y(·)) satisfies (A 3).

Second, |g(x, y, z(·))| ≤ h r (x). g(x, y, z(·)) satisfies the assumption (A 4).

Thus, if there exists r satisfies r > F 0 + q + h r L μ 1 1 - α L μ 1 with F 0 = x 0 z ¯ f ( x , 0 ) d μ , all conditions in

Theorem 3.1 are satisfied. So, (8)-(9) has a solution p ( S ̄ x 0 , q ) on x 0 z ¯ .

Declarations

Acknowledgements

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.

Authors’ Affiliations

(1)
Department of Mathematics, Tongji University

References

  1. Sharma RR: An abstract measure differential equation. Proc Am Math Soc 1972, 32: 503-510. 10.1090/S0002-9939-1972-0291600-3View ArticleMathSciNetMATHGoogle Scholar
  2. Dhage BC, Chate DN, Ntouyas SK: A system of abstract measure delay differential equations. Electron J Qual Theory Diff Equ 2003, 8: 1-14.MathSciNetMATHGoogle Scholar
  3. Dhage BC, Chate DN, Ntouyas SK: Abstract measure differential equations. Dyn Syst Appl 2004, 13: 105-117.MathSciNetMATHGoogle Scholar
  4. Dhage BC, Bellale SS: Existence theorems for perturbed abstract measure differential equations. Nonlinear Anal 2009, 71: e319-e328. 10.1016/j.na.2008.11.057MathSciNetView ArticleMATHGoogle Scholar
  5. Joshi SR, Kasaralikar SN: Differential inequalities for a system of abstract measure delay differential equations. J Math Phys Sci 1982, 16: 515-523.MathSciNetMATHGoogle Scholar
  6. Shendge GR, Joshi SR: Abstract measure differential inequalities and applications. Acta Mathematica Hungarica 1983, 41: 53-59. 10.1007/BF01994061MathSciNetView ArticleMATHGoogle Scholar
  7. Dhage BC: On abstract measure integro-differential equations. J Math Phys Sci 1986, 20: 367-380.MathSciNetMATHGoogle Scholar
  8. Dhage BC: Existence theory for quadratic perturbations of abstract measure integro-differential equations. Diff Equ Appl 2009, 1: 307-323.MathSciNetMATHGoogle Scholar
  9. Chen FL, Luo XN, Zhou Y: Existence results for nonlinear fractional difference equation. Adv Diff Equ 2011, 713201.Google Scholar
  10. Han XL, Gao HL, Xu J: Existence of positive solutions for nonlocal fourth-order boundary value problem with variable parameter. Fixed Point Theory Appl 2011, 604046.Google Scholar
  11. Plubtieng S, Sitthithakerngkiet K: Existence result of generalized vector quasiequilibrium problems in locally G-convex spaces. Fixed Point Theory Appl 2011, 967515.Google Scholar
  12. Wang LG: Intuitionistic fuzzy stability of a quadratic functional equation. Fixed Point Theory Appl 2010, 107182.Google Scholar
  13. Park C: Fixed points, inner product spaces, and functional equations. Fixed Point Theory Appl 2010, 713675.Google Scholar
  14. Zhang JJ, Liao LW: Further extending results of some classes of complex difference and functional equations. Adv Diff Equ 2010, 404582.Google Scholar
  15. Dhage BC, O'Regan D: A fixed point theorem in Banach algebras with applications to functional integral equations. Funct Diff Equ 2000, 7: 259-267.MathSciNetMATHGoogle Scholar

Copyright

© Wan and Sun; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. 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.