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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 863561, 7 pages
http://dx.doi.org/10.1155/2013/863561
Research Article

Global Existence for Functional Differential Equations with State-Dependent Delay

1Laboratory of Mathematics, University of Sidi Bel-Abbes, P.O. Box 89, 22000 Sidi Bel-Abbes, Algeria
2Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain
3Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
4Department of Mathematics, Periyar University, Salem, Tamil Nadu 636 011, India

Received 10 May 2013; Accepted 21 September 2013

Academic Editor: Jin Liang

Copyright © 2013 Mouffak Benchohra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

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 [15]. In 1806, Poisson [6] 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 [7]). 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 [1012]. 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, 1315]. 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. [16], Ait Dads and Ezzinbi [17], Anguraj et al. [18], Hartung et al. [19, 20], Hernández et al. [21], and Li et al. [22]. 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 [23], in population models [2427], in models of commodity price fluctuations [28, 29], in models of blood cell productions [3033], and in drilling [34].

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.

2. Preliminaries

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 [1] and follow the terminology used in [3]. 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 [35]). 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 [36]). 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, [3].

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 [37]), 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 [37]) 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.

Acknowledgments

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.

References

  1. J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj, vol. 21, no. 1, pp. 11–41, 1978. View at Zentralblatt MATH · View at MathSciNet
  2. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993.
  3. Y. Hino, S. Murakami, and T. Naito, Functional-Differential Equations with Infinite Delay, vol. 1473, Springer, Berlin, Germany, 1991. View at MathSciNet
  4. H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, vol. 57, Springer, New York, NY, USA, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  5. J. Wu, Theory and Applications of Partial Functional-Differential Equations, vol. 119, Springer, New York, NY, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  6. S. D. Poisson, “Sur les équations aux différences melées,” Journal de l'École polytechnique, vol. 6, pp. 126–147, 1806.
  7. H.-O. Walther, “On Poisson's state-dependent delay,” Discrete and Continuous Dynamical Systems A, vol. 33, no. 1, pp. 365–379, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, vol. 246, John Wiley & Sons, Harlow, UK, 1991. View at MathSciNet
  9. K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, NY, USA, 2000. View at MathSciNet
  10. S. Baghli and M. Benchohra, “Uniqueness results for partial functional differential equations in Fréchet spaces,” Fixed Point Theory, vol. 9, no. 2, pp. 395–406, 2008. View at Zentralblatt MATH · View at MathSciNet
  11. S. Baghli and M. Benchohra, “Existence results for semilinear neutral functional differential equations involving evolution operators in Fréchet spaces,” Georgian Mathematical Journal, vol. 17, no. 3, pp. 423–436, 2010. View at Zentralblatt MATH · View at MathSciNet
  12. S. Baghli and M. Benchohra, “Global uniqueness results for partial functional and neutral functional evolution equations with infinite delay,” Differential and Integral Equations, vol. 23, no. 1-2, pp. 31–50, 2010. View at Zentralblatt MATH · View at MathSciNet
  13. R. D. Driver, “A neutral system with state-dependent delay,” Journal of Differential Equations, vol. 54, no. 1, pp. 73–86, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  14. E. Hernandez, A. Prokopezyk, and L. Ladeira, “A note on partial functional differential equation with state-dependent delay,” Nonlinear Analysis: Real World Applications, vol. 7, pp. 510–519, 2006.
  15. D. R. Willé and C. T. H. Baker, “Stepsize control and continuity consistency for state-dependent delay-differential equations,” Journal of Computational and Applied Mathematics, vol. 53, no. 2, pp. 163–170, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. N. Abada, R. P. Agarwal, M. Benchohra, and H. Hammouche, “Existence results for nondensely defined impulsive semilinear functional differential equations with state-dependent delay,” Asian-European Journal of Mathematics, vol. 1, no. 4, pp. 449–468, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. E. H. Ait Dads and K. Ezzinbi, “Boundedness and almost periodicity for some state-dependent delay differential equations,” Electronic Journal of Differential Equations, no. 67, pp. 1–13, 2002. View at Zentralblatt MATH · View at MathSciNet
  18. A. Anguraj, M. M. Arjunan, and E. Hernández M., “Existence results for an impulsive neutral functional differential equation with state-dependent delay,” Applicable Analysis, vol. 86, no. 7, pp. 861–872, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  19. F. Hartung, “Linearized stability in periodic functional differential equations with state-dependent delays,” Journal of Computational and Applied Mathematics, vol. 174, no. 2, pp. 201–211, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. F. Hartung, T. Krisztin, H. O. Walther, and J. Wu, “Functional differential equations with state-dependent delays: theory and applications,” in Handbook of Differential Equations: Ordinary Diffrential Equations, A. Canada, P. Drabek, and A. Fonda, Eds., vol. 3, Elsevier, 2006.
  21. E. Hernández, R. Sakthivel, and S. T. Aki, “Existence results for impulsive evolution differential equations with state-dependent delay,” Electronic Journal of Differential Equations, vol. 2008, no. 28, pp. 1–11, 2008. View at Scopus
  22. W.-S. Li, Y.-K. Chang, and J. J. Nieto, “Solvability of impulsive neutral evolution differential inclusions with state-dependent delay,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1920–1927, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. R. D. Driver and M. J. Norris, “Note on uniqueness for a one-dimensional two-body problem of classical electrodynamics,” Annals of Physics, vol. 42, pp. 347–351, 1967. View at Publisher · View at Google Scholar · View at MathSciNet
  24. W. G. Aiello, H. I. Freedman, and J. Wu, “Analysis of a model representing stage-structured population growth with state-dependent time delay,” SIAM Journal on Applied Mathematics, vol. 52, no. 3, pp. 855–869, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. J. Bélair, “Population models with state-dependent delays,” in Mathematical Population Dynamics, vol. 131 of Lecture Notes in Pure and Applied Mathematics, pp. 165–176, Dekker, New York, NY, USA, 1991. View at Zentralblatt MATH · View at MathSciNet
  26. Y. Cao, J. Fan, and T. C. Gard, “The effects of state-dependent time delay on a stage-structured population growth model,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 19, no. 2, pp. 95–105, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. F. Chen, D. Sun, and J. Shi, “Periodicity in a food-limited population model with toxicants and state dependent delays,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 136–146, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. J. Bélair and M. C. Mackey, “Consumer memory and price fluctuations in commodity markets: an integrodifferential model,” Journal of Dynamics and Differential Equations, vol. 1, no. 3, pp. 299–325, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  29. M. C. Mackey, “Commodity price fluctuations: price dependent delays and nonlinearities as explanatory factors,” Journal of Economic Theory, vol. 48, no. 2, pp. 497–509, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J. Bélair, “Age-structured and two-delay models for erythropoiesis,” Mathematical Biosciences, vol. 128, no. 1-2, pp. 317–346, 1995. View at Publisher · View at Google Scholar · View at Scopus
  31. C. Colijn and M. C. Mackey, “Bifurcation and bistability in a model of hematopoietic regulation,” SIAM Journal on Applied Dynamical Systems, vol. 6, no. 2, pp. 378–394, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. F. Crauste, “Delay model of hematopoietic stem cell dynamics: asymptotic stability and stability switch,” Mathematical Modelling of Natural Phenomena, vol. 4, no. 2, pp. 28–47, 2009. View at Zentralblatt MATH · View at MathSciNet
  33. M. C. Mackey and J. Milton, “Feedback delays and the origin of blood cell dynamics,” Comments on Theoretical Biology, vol. 1, pp. 299–372, 1990.
  34. K. Nandakumar and M. Wiercigroch, “Galerkin projections for state-dependent delay differential equations with applications to drilling,” Applied Mathematical Modelling, vol. 37, no. 4, pp. 1705–1722, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  35. A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2003. View at MathSciNet
  36. C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, NY, USA, 1973. View at MathSciNet
  37. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. View at Publisher · View at Google Scholar · View at MathSciNet