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Existence of solutions for perturbed abstract measure functional differential equations
Advances in Difference Equations volume 2011, Article number: 67 (2011)
Abstract
In this article, we investigate existence of solutions for perturbed abstract measure functional differential equations. Based on the ArzelàAscoli 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 [2–6], uniqueness [2, 3, 5], and extremal solutions [3, 4, 6]. There were also several researches on abstract measure integrodifferential 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 LeraySchauder alternative involving the sum of two operators [15] and the ArzelàAscoli 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 : X → X is called DLipschitzian, if there is a continuous and nondecreasing function ϕ _{ T } : ℝ^{+} → ℝ^{+} such that
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 : 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 $\overline{T(X)}$ 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 $\overline{xy}$ is defined as
Let x _{0} ∈ X be a fixed point and $z\in X,\phantom{\rule{2.77695pt}{0ex}}\overline{0{x}_{0}}\subset \overline{0z}$, where 0 is the zero element of X. Then for any $x\in \overline{{x}_{0}z}$, we define the sets S _{ x } and ${\stackrel{\u0304}{S}}_{x}$ as
For any ${x}_{1},\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\in \overline{{x}_{0}z}\subset X$, we denote x _{1} < x _{2} if ${S}_{{x}_{1}}\subset {S}_{{x}_{2}}$, or equivalently $\overline{{x}_{0}{x}_{1}}\subset \overline{{x}_{0}{x}_{2}}$. Let ω ∈ [0, h], h > 0. For any $x\in \overline{{x}_{0}z},\phantom{\rule{2.77695pt}{0ex}}{x}_{\omega}$ is defined by
Let M denote the σalgebra 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:
where p is a total variation measure of p,
where π : {E _{ i } : i ∈ ℕ} is any partition of X. Then ca(X, M) is a Banach space with the norm defined above.
Let μ be a σfinite 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 σalgebra on ${S}_{{x}_{0}}$. For x _{0} < z, M _{ z } denotes the σalgebra on S _{ z } . It is obvious that M _{ z } contains M _{0} and the sets of the form ${S}_{x},\phantom{\rule{2.77695pt}{0ex}}x\in \overline{{x}_{0}z}$.
Given a p ∈ ca(X, M) with p ≪ μ, consider perturbed abstract measure functional differential equation:
and
where q is a given signed measure, $\frac{dp}{d\mu}$ is a RadonNikodym derivative of p with respect to μ. f : S _{ z } × ℝ → ℝ, g : S _{ z } × ℝ × ℝ → ℝ. $f\left(x,p\left({\stackrel{\u0304}{S}}_{{x}_{\omega}}\right)\right)$ and $g\left(x,p\left({\stackrel{\u0304}{S}}_{x}\right),p\left({\stackrel{\u0304}{S}}_{{x}_{\omega}}\right)\right)$ are μintegrable for each p ∈ ca(S _{ z } , M _{ z } ).
Define
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

(i)
p(E) = q(E), E ∈ M _{0},

(ii)
p ≪ μ on $\overline{{x}_{0}z}$,

(iii)
p satisfies (1) a.e. [μ] on $\overline{{x}_{0}z}$.
Remark 2.1 The system (1)(2) is equivalent to the following perturbed abstract measure functional integral system:
We denote a solution p of (1)(2) as $p\left({\stackrel{\u0304}{S}}_{{x}_{0}},\phantom{\rule{2.77695pt}{0ex}}q\right)$.
Definition 2.3 A function β : S _{ z } × ℝ × ℝ → ℝ is called Carathé odory, if

(i)
x → β(x, y, z) is μmeasurable for each (y, z) ∈ ℝ × ℝ,

(ii)
(y, z) → β(x, y, z) is continuous a.e. [μ] on $\overline{{x}_{0}z}$.
The function β defined as the above is called ${L}_{\mu}^{1}$Carathé odory, further if

(iii)
for each real number r > 0, there exists a function ${h}_{r}\left(x\right)\in {L}_{\mu}^{1}\left({S}_{z},\phantom{\rule{2.77695pt}{0ex}}{\mathbb{R}}^{+}\right)$ such that
$$\mid \beta \left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}z\right)\mid \le {h}_{r}\left(x\right)\phantom{\rule{2.77695pt}{0ex}}a.e.\phantom{\rule{2.77695pt}{0ex}}\left[\mu \right]\phantom{\rule{2.77695pt}{0ex}}on\phantom{\rule{2.77695pt}{0ex}}\overline{{x}_{0}z}$$
for each y ∈ ℝ, z ∈ ℝ with y ≤ r, z ≤ r.
Lemma 2.1[15] Let ${\mathcal{B}}_{r}\left(0\right)$ and ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$ 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, $B:\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)\to X$ 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 ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$, or

(ii)
there exists an element $u\in \partial {\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$ such that $\lambda A\left(\frac{u}{\lambda}\right)+\lambda Bu=u$ for some λ ∈ (0, 1).
3 Main results
We consider the following assumptions:
(A _{0}) For any z ∈ X satisfies x _{0} < z, the σalgebra M _{ z } is compact with respect to the topology generated by the pseudometric d defined by

(A _{1}) μ({x _{0}}) = 0.

(A _{2}) q is continuous on M _{ z } with respect to the pseudometric d defined in (A _{0}).

(A _{3}) There exists a μintegrable function α : S _{ z } → ℝ^{+} such that
$$\mid \phantom{\rule{0.3em}{0ex}}f\left(x,\phantom{\rule{2.77695pt}{0ex}}{y}_{1}\left(\cdot \right)\right)f\left(x,\phantom{\rule{2.77695pt}{0ex}}{y}_{2}\left(\cdot \right)\right)\mid \le \alpha \left(x\right)\mid \phantom{\rule{0.3em}{0ex}}{y}_{1}\left(\cdot \right){y}_{2}\left(\cdot \right)\mid \phantom{\rule{2.77695pt}{0ex}}a.e.\left[\mu \right]\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\overline{{x}_{0}z}.$$ 
(A _{4}) g(x, y, z(·)) is ${L}_{\mu}^{1}$Carathé odory.
Theorem 3.1 Suppose that the assumptions (A _{0})(A _{4}) hold. Further if $\parallel \alpha {\parallel}_{{L}_{\mu}^{1}}<1$ and there exists a real number r > 0 such that
where ${F}_{0}={\int}_{\overline{{x}_{0}z}}\mid \phantom{\rule{0.3em}{0ex}}f\left(x,\phantom{\rule{2.77695pt}{0ex}}0\right)\mid d\mu $. Then the system (1)(2) has a solution on $\overline{{x}_{0}z}$.
Proof: Consider the open ball ${\mathcal{B}}_{r}\left(0\right)$ and the closed ball ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$ 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 } ), $B:\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)\to ca\left({S}_{z},\phantom{\rule{2.77695pt}{0ex}}{M}_{z}\right)$ as:
Now we prove the operators A and B satisfy conditions that are given in Lemma 2.1 on ca(S _{ z } , M _{ z } ) and ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$, 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})
for all E ∈ M _{ z } .
Considering the definition of norm on ca(S _{ z } , M _{ z } ), we have
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 ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$.
Let {p _{ n } }_{ n∈ℕ}be a sequence of signed measures in ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$, and {p _{ n } }_{ n∈ℕ}converges to a signed measure p. In case E ∈ M _{ z } , $E\subset \overline{{x}_{0}z}$, using dominated convergence theorem
In case E ∈ M _{0}, $\underset{n\to \infty}{lim}B{p}_{n}\left(E\right)=q\left(E\right)=Bp\left(E\right)$. Obviously, B is a continuous operator on ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$.
Step III. B is a totally bounded operator on ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$.
Let {p _{ n } }_{ n∈ℕ}be a sequence of signed measures in ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$, then p _{ n }  ≤ r(n ∈ ℕ). Next we show that {Bp _{ n } }_{ n∈ℕ}are uniformly bounded and equicontinuous.
First, {Bp _{ n } }_{ n∈ℕ}are uniformly bounded. Let E ∈ M _{ z } , and E = F ∪ G, where F ∈ M _{0} and G ∈ M _{ z } , $G\subset \overline{{x}_{0}z}$. F ∩ G = ∅. Hence,
consequently,
for every ${p}_{n}\in \phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$. Then {Bp _{ n } }_{ n∈ℕ}are uniformly bounded.
Second, {Bp _{ n } }_{ n∈ℕ}is 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 ${G}_{i}\in {M}_{z},\phantom{\rule{2.77695pt}{0ex}}{G}_{i}\subset \overline{{x}_{0}z}$, and F _{ i } ∩ G _{ i } = ∅. i = 1, 2.
Considering assumption (A _{4}), then
when d(E _{1}, E _{2}) → 0, E _{1} → E _{2}. Then, F _{1} → F _{2}, and μ(G _{1}ΔG _{2}) = d(G _{1}ΔG _{2}) → 0.
Considering assumption (A _{2}), q is continuous on compact M _{ z } implies it is uniformly continuous on M _{ z } . so
for every ${p}_{n}\in \phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$.
{Bp _{ n } }_{ n∈ℕ}is an equicontinuous sequence in ca(S _{ z } , M _{ z } ).
According to the ArzelàAscoli theorem, there is a subset ${\left\{B{p}_{{n}_{k}}\right\}}_{n,k\in \mathbb{N}}$ of {Bp _{ n } }_{ n∈ℕ}that converges uniformly. Thus, operator B is compact on ${\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)$. Then, B is a totally bounded operator on ${\mathcal{B}}_{r}\left(0\right)$.
From steps II and III, the operator B is completely continuous on ${\mathcal{B}}_{r}\left(0\right)$.
Step IV. (1)(2) has a solution on $\overline{{x}_{0}z}$.
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 $\lambda A\left(\frac{u}{\lambda}\right)+\lambda Bu=u$ for some λ ∈ (0, 1).
If it is true, we have
for some λ ∈ (0, 1). Then
so we get
for all E ∈ M _{ z } .
By the definition of the norm on ca(S _{ z } , M _{ z } ),
As u = r, we have
This is a contradiction. Consequently, the equation p(E) = Ap(E) + Bp(E) has a solution $p\left({\stackrel{\u0304}{S}}_{{x}_{0}},q\right)\in \phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0304}{\mathcal{B}}}_{r}\left(0\right)\subset ca\left({S}_{z},{M}_{z}\right)$. It is said that (1)(2) has a solution on $\overline{{x}_{0}z}$. 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
and
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
and
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:
and
where ${h}_{r}\left(x\right)\in {L}_{\mu}^{1}\left({S}_{z},\phantom{\rule{2.77695pt}{0ex}}{\mathbb{R}}^{+}\right),\phantom{\rule{2.77695pt}{0ex}}\parallel \alpha {\parallel}_{{L}_{\mu}^{1}}<1$ and 0 ≤ ω ≤ h(h > 0). f : S _{ z } × ℝ → ℝ and g : S _{ z } × ℝ × ℝ → ℝ are defined as
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(·)) 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 ∈ ℝ satisfies $r>\frac{{F}_{0}\phantom{\rule{2.77695pt}{0ex}}+\phantom{\rule{2.77695pt}{0ex}}\parallel q\parallel \phantom{\rule{2.77695pt}{0ex}}+\phantom{\rule{2.77695pt}{0ex}}\parallel {h}_{r}\mid {\mid}_{{L}_{\mu}^{1}}}{1\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\parallel \alpha \mid {\mid}_{{L}_{\mu}^{1}}}$ with ${F}_{0}={\int}_{\overline{{x}_{0}z}}\mid \phantom{\rule{0.3em}{0ex}}f\left(x,\phantom{\rule{2.77695pt}{0ex}}0\right)\mid d\mu $, all conditions in
Theorem 3.1 are satisfied. So, (8)(9) has a solution $p\left({\stackrel{\u0304}{S}}_{{x}_{0}},\phantom{\rule{2.77695pt}{0ex}}q\right)$ on $\overline{{x}_{0}z}$.
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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.
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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.
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Wan, X., Sun, J. Existence of solutions for perturbed abstract measure functional differential equations. Adv Differ Equ 2011, 67 (2011). https://doi.org/10.1186/16871847201167
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Keywords
 Banach Space
 Signed Measure
 Abstract Measure
 Fixed Point Theorem
 Existence Theorem