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Global Existence for Functional Differential Equations with State-Dependent Delay
Our aim in this work is to study the existence of solutions of a functional differential equation with state-dependent delay. We use Schauder's fixed point theorem to show the existence of solutions.
The theory of functional differential equations has emerged as an important branch of nonlinear analysis. Differential delay equations, and functional differential equations, have been used in modeling scientific phenomena for many years. Often, it has been assumed that the delay is either a fixed constant or is given as an integral in which case it is called a distributed delay [1–5]. In 1806, Poisson  published one of the first papers on functional differential equations and studied a geometric problem leading to an example with a state-dependent delay (see also ). An extensive theory is developed for evolution equations [8, 9]. Uniqueness and existence results have been established recently for different evolution problems in the papers by Baghli and Benchohra for finite and infinite delay in [10–12]. However, complicated situations in which the delay depends on the unknown functions have been considered in recent years. These equations are frequently called equations with state-dependent delay: see, for instance, [3, 13–15]. Existence results were derived recently for functional differential equations when the solution is depending on the delay for impulsive problems. We refer the reader to the papers by Abada et al. , Ait Dads and Ezzinbi , Anguraj et al. , Hartung et al. [19, 20], Hernández et al. , and Li et al. . Over the past several years it has become apparent that equations with state-dependent delay arise also in several areas such as in classical electrodynamics , in population models [24–27], in models of commodity price fluctuations [28, 29], in models of blood cell productions [30–33], and in drilling .
In this work, we prove the existence of solutions of a class of functional differential equations. Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the real axis . We will use Schauder's fixed point theorem combined with the semigroup theory to have the existence of solutions of the following functional differential equation with state-dependent delay: where is a given function, is the infinitesimal generator of a strongly continuous semigroup is the phase space to be specified later, , , and is a real Banach space. For any function defined on and any we denote by the element of defined by . Here represents the history of the state from time up to the present time . We assume that the histories belong to some abstract phase space , to be specified later. To our knowledge, the literature on the global existence of evolution equations with delay is very limited, so the present paper can be considered as a contribution to this question.
In this section, we present briefly some notations, a definition and a theorem which are used throughout this work.
In this paper, we will employ an axiomatic definition of the phase space introduced by Hale and Kato in  and follow the terminology used in . Thus, will be a seminormed linear space of functions mapping into and satisfying the following axioms. If , is continuous on and , then for every the following conditions hold:(i);(ii)there exists a positive constant such that ;(iii)there exist two functions independent of with continuous and bounded and locally bounded such that For the function in , is a -valued continuous function on . The space is complete. Denote
Remark 1. (ii) is equivalent to for every .
Since is a seminorm, two elements can verify without necessarily for all .
From the equivalence in the first remark, we can see that, for all such that . We necessarily have that .
By BUC we denote the space of bounded uniformly continuous functions defined from to .
By we denote the Banach space of all bounded and continuous functions from into equipped with the standard norm Finally, by we denote the Banach space of all bounded and continuous functions from into equipped with the standard norm
Definition 2. A map is said to be Carathéodory if(i) is measurable for all ;(ii) is continuous for almost each .
Theorem 3 (see Schauder fixed point ). Let be a closed, convex, and nonempty subset of a Banach space . Let be a continuous mapping such that is a relatively compact subset of . Then has at least one fixed point in . That is, there exists an such that .
Lemma 4 (see Corduneanu ). Let . Then is relatively compact if the following conditions hold.(a) is bounded in BC. (b)The function belonging to is almost equicontinuous on , that is, equicontinuous on every compact of .(c)The set is relatively compact on every compact of .(d)The function from is equiconvergent that is, given corresponds a such that , for any and .
3. Existence of Mild Solutions
Now we give our main existence result for problem (1). Before starting and proving this result, we give the definition of the mild solution.
Definition 5. We say that a continuous function is a mild solution of problem (1) if , and the restriction of to the interval is continuous and satisfies the following integral equation: Set
We always assume that is continuous. Additionally, we introduce the following hypothesis: the function is continuous from into , and there exists a continuous and bounded function such that
Remark 6. The condition is frequently verified by functions continuous and bounded. For more details, see, for instance, .
Lemma 7 ([21, Lemma 2.4]). If is a function such that , then where .
Let us introduce the following hypotheses. is the infinitesimal generator of a strongly continuous semigroup , which is compact for in the Banach space . Let . The function is Carathéodory. There exists a continuous function such that The function with .
Theorem 8. Assume that − hold. If , then the problem (1) has at least one mild solution on .
Proof. Transform problem (1) into a fixed point problem. Consider the operator defined by
Let be the function defined by Then . For each with , we denote by the function If satisfies , we can decompose it as , which implies for every , and the function satisfies Set and let is a Banach space with the norm . We define the operator by We will show that the operator satisfies all conditions of Schauder's fixed point theorem. The operator maps into ; indeed, the map is continuous on for any , and for each we have Set Then, we have Hence, .
Moreover, let be such that and let be the closed ball in centered at the origin and of radius . Let , and let . Then, Thus, which means that the operator transforms the ball into itself.
Now we prove that satisfies the assumptions of Schauder's fixed theorem. The proof will be given in several steps.
Step 1. is continuous in . Let be a sequence such that in . At first, we study the convergence of the sequences .
If is such that , then we have which proves that in as for every such that . Similarly, if , we get which also shows that in as for every such that . Combining the pervious arguments, we can prove that for every such that . Finally, Then by , we have and by the Lebesgue dominated convergence theorem we get Thus, is continuous.
Step 2. which is clear.
Step 3. is equicontinuous on every compact interval of for . Let with ; we have When , the right-hand side of the above inequality tends to zero; since is a strongly continuous operator, and the compactness of for implies the continuity in the uniform operator topology (see ), this proves the equicontinuity.
Step 4. is relatively compact on every compact interval of . Let for , and let be a real number satisfying . For , we define Note that the set is bounded. Since is a compact operator for , the set is precompact in for every , . Moreover, for every we have
Therefore, the set is precompact, that is, relatively compact.
Step 5 ( is equiconvergent). Let and ; we have Then by (37), we have Hence,
As a consequence of Steps 1–4, with Lemma 4, we can conclude that is continuous and compact. From Schauder's theorem, we deduce that has a fixed point . Then is a fixed point of the operators , which is a mild solution of problem (1).
4. An Example
Consider the following functional partial differential equation: where . Set where , and are continuous functions.
Take and define by with domain
Then where , is the orthogonal set of eigenvectors in . It is well known (see ) that is the infinitesimal generator of an analytic semigroup , in and is given by Since the analytic semigroup is compact, there exists a positive constant such that Let , and let then .
The function is Carathéodory, and Thus, ; moreover, we have Then problem (1) in an abstract formulation of the problem (37) and conditions are satisfied. Theorem 8 implies that the problem (37) has at least one mild solution on BC.
This work has been completed during the visits of Mouffak Benchohra and P. Prakash to the USC and has been partially supported by Ministerio de Economia y Competitividad (Spain), Project MTM2010-15314, and cofinanced by the European Community fund FEDER. The authors are grateful to the referee for the helpful remarks.
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