A Survey on Semilinear Differential Equations and Inclusions Involving Riemann-Liouville Fractional Derivative
© Copyright © 2009 2009
Received: 16 July 2008
Accepted: 5 February 2009
Published: 8 March 2009
We establish sufficient conditions for the existence of mild solutions for some densely defined semilinear functional differential equations and inclusions involving the Riemann-Liouville fractional derivative. Our approach is based on the -semigroups theory combined with some suitable fixed point theorems.
Differential equations and inclusions of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed we can find numerous applications in viscoelasticity, electrochemistry, electromagnetism, and so forth. For details, including some applications and recent results, see the monographs of Kilbas et al. , Kiryakova , Miller and Ross , Podlubny  and Samko et al. , and the papers of Agarwal et al. , Diethelm et al. [7, 8], El-Sayed [9–11], Gaul et al. , Glockle and Nonnenmacher , Lakshmikantham and Devi , Mainardi , Metzler et al. , Momani et al. [17, 18], Podlubny et al. , Yu and Gao  and the references therein. Some classes of evolution equations have been considered by El-Borai [21, 22], Jaradat et al.  studied the existence and uniqueness of mild solutions for a class of initial value problem for a semilinear integrodifferential equation involving the Caputo's fractional derivative.
In this survey paper, we give existence results for various classes of initial value problems for fractional semilinear functional differential equations and inclusions, both cases of finite and infinite delay are considered. More precisely the paper is organized as follows. In the second section we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. In the third section we will be concerned with semilinear functional differential equations with finite as well infinite delay. In the forth section, we consider semilinear functional differential equation of neutral type for the both cases of finite and infinite delay. Section 5 is devoted to the study of functional differential inclusions, we examine the case when the right-hand side is convex valued as well as nonconvex valued. In Section 6, we will be concerned with perturbed functional differential equations and inclusions. In the last section, we give some existence results of extremal solutions in ordered Banach spaces.
For more details on strongly continuous operators, we refer the reader to the books of Goldstein , Fattorini , and the papers of Travis and Webb [26, 27], and for properties on semigroup theory we refer the interested reader to the books of Ahmed , Goldstein , and Pazy .
In all our paper we adopt the following definitions of fractional primitive and fractional derivative.
where Then is a metric space and is a generalized metric space (see ).
is called upper semicontinuous (u.s.c. for short) on if for each the set is nonempty, closed subset of , and for each open set of containing , there exists an open neighborhood of such that is said to be completely continuous if is relatively compact for every If the multivalued map is completely continuous with nonempty compact valued, then is u.s.c. if and only if has closed graph, that is, imply
Essential for the main results of this paper, we state a generalization of Gronwall's lemma for singular kernels [34, Lemma 7.1.1].
In the sequel, the following fixed point theorems will be used. The following fixed point theorem for contraction multivalued maps is due to Covitz and Nadler .
The nonlinear alternative of Leray-Schauder applied to completely continuous operators .
The following is the multivalued version of the previous theorem due to Martelli .
To state existence results for perturbed differential equations and inclusions we will use the following fixed point theorem of Krasnoselskii-Scheafer type of the sum of a completely continuous operator and a contraction one due to Burton and Kirk .
Recently Dhage states the multivalued version of the previous theorem.
Hereafter are some examples of phase spaces. For other details we refer, for instance, to the book by Hino et al. .
3. Semilinear Functional Differential Equations
Functional differential and partial differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years. A good guide to the literature for functional differential equations is the books by Hale  and Hale and Verduyn Lunel , Kolmanovskii and Myshkis , and Wu  and the references therein.
In a series of papers (see [47–50]), the authors considered some classes of initial value problems for functional differential equations involving the Riemann-Liouville and Caputo fractional derivatives of order In [51, 52] some classes of semilinear functional differential equations involving the Riemann-Liouville have been considered. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types see [53, 54].
The reason for studying (3.1) is that it appears in mathematical models of viscoelasticity , and in other fields of science [54, 56]. Equation (3.1) is equivalent to solve an integral equation of convolution type. It is also of interest to explore the neighborhood of the diffusion ( ). In this survey paper, we use the fractional derivative in the Riemann-Liouville sense. The problems considered in the survey are subject to zero data, which in this case, the Riemann-Liouville and Caputo fractional derivatives coincide. From a practical point of view, in some mathematical models it is more appropriate to consider traditional initial or boundary data. This is what we are considering in this survey.
Before stating our main results in this section for problem (3.1) and (3.2) we give the definition of the mild solution.
Definition 3.1 (see ).
3.2. Existence Results for Finite Delay
By using the Banach's contraction principle, we get the following existence result for problem (3.1) and (3.2).
The following existence result is based upon Theorem 2.9.
As and sufficiently small, 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 . By the Arzelá-Ascoli theorem it suffices to show that maps into a precompact set in .
3.3. An Example
3.4. Existence Results for Infinite Delay
The first existence result is based on Banach's contraction principle.
which yields the contraction of for sufficiently large values of . Therefore, by the Banach's contraction principle has a unique fixed point . Then is a fixed point of the operator , which gives rise to a unique mild solution of the problem (3.34).
Next we give an existence result based upon the nonlinear alternative of Leray-Schauder type.
Then there exists a constant such that This shows that the set is bounded. As a consequence of the Leray-Schauder Theorem, we deduce that the operator has a fixed point, then has one which gives rise to a mild solution of the problem (3.34).
3.5. An Example
Then generates a semigroup (see ).
Then, problem (3.65) takes the abstract neutral evolution form (3.34).
4. Semilinear Functional Differential Equations of Neutral Type
Neutral differential equations arise in many areas of applied mathematics, an extensive theory is developed, we refer the reader to the book by Hale and Verduyn Lunel  and Kolmanovskii and Myshkis . The work for neutral functional differential equations with infinite delay was initiated by Hernández and Henríquez in [57, 58]. In the following, we will extend such results to arbitrary order functional differential equations of neutral type with finite as well as infinite delay. We based our main results upon the Banach's principle and the Leray-Schauder theorem.
4.2. Existence Results for the Finite Delay
Our first existence result is based on the Banach's contraction principle.
Next we give an existence result using the nonlinear alternative of Leray-Schauder.
is continuous and completely continuous. This can be done following the proof of Theorem 3.3.
4.3. Existence Results for the Infinite Delay
Our first existence result is based on the Banach's contraction principle.
which yields the contraction of for sufficiently large values of . Therefore, by the Banach's contraction principle has a unique fixed point . Then is a fixed point of the operator , which gives rise to a unique mild solution of the problem (4.15).
Next we give an existence result based upon the the nonlinear alternative of Leray-Schauder.
Then there exists a constant such that This shows that the set is bounded. As a consequence of the Leray-Schauder Theorem, we deduce that the operator has a fixed point which gives rise to a mild solution of the problem (4.15).
5. Semilinear Functional Differential Inclusions
Differential inclusions are generalization of differential equations, therefore all problems considered for differential equations, that is, existence of solutions, continuation of solutions, dependence on initial conditions and parameters, are present in the theory of differential inclusions. Since a differential inclusion usually has many solutions starting at a given point, new issues appear, such as investigation of topological properties of the set of solutions, and selection of solutions with given properties.
Functional differential inclusions with fractional order are first considered by El Sayed and Ibrahim . Very recently Benchohra et al. , and Ouahab  have considered some classes of ordinary functional differential inclusions with delay, and in [6, 61] Agarwal et al. considered a class of boundary value problems for differential inclusion involving Caputo fractional derivative of order . Chang and Nieto  considered a class of fractional differential inclusions of order . Here we continue this study by considering partial functional differential inclusions involving the Riemann-Liouville derivative of order . The both cases of convex valued and nonconvex valued of the right-hand side are considered, and where the delay is finite as well as infinite. Our approach is based on the -semigroups theory combined with some suitable fixed point theorems.
In the following, we give our first existence result for problem (5.1) with a convex valued right-hand side. Our approach is based upon Theorem 2.10.
Then the problem (5.1) has at least one mild solution.
Now we will be concerned with existence results for problem (5.1) with nonconvex valued right-hand side. Our approach is based on the fixed point theorem for contraction multivalued maps due to Covitz and Nadler Jr. .
Assume that (H19) holds.
In the following, we give an existence result for problem (5.46) with convex valued right-hand side. Our approach is based upon Theorem 2.10.
Then the problem (5.46) has at least one mild solution.
Now we give an existence result for problem (5.46) with nonconvex valued right-hand side by using the fixed point Theorem 2.8.
As the previous theorem and following steps of the proof of Theorem 5.3.
6. Perturbed Semilinear Differential Equations and Inclusions
In this section, we will be concerned with semilinear functional differential equations and inclusion of fractional order and where a perturbed term is considered. Our approach is based upon Burton-Kirk fixed point theorem (Theorem 2.11).
Our first main result in this section reads as follows.
where and . We will show that the operator is closed, convex, and bounded valued and it is a contraction. Let such that in . Using (H31), we can show that the values of Niemysky operator are closed in , and hence is closed for each
This shows that the set is bounded. As a result, the conclusion (ii) of Theorem 2.12 does not hold. Hence, the conclusion (i) holds and consequently has a fixed point which is a mild solution of problem (6.18).
7. Some Existence Results in Ordered Banach Spaces
In this section, we present some existence results in ordered Banach spaces using the method of upper and lower mild solutions. Before stating our main results let us introduce some preliminaries.
The following fixed point theorem is crucial for our existence result.
Theorem 7.5 (see ).
Let be a normal cone in a partially ordered Banach space . Let be increasing on the interval and transform into itself, that is, and . Assume further that is continuous and completely continuous. Then has at least one fixed point .
Our first main result reads as follows.
Now reconsider the perturbed initial value problem (6.1). To state our second main result in this section we use the following fixed point theorem due to Dhage and Henderson.
Theorem 7.7 (see ).
We need the following definitions in the sequel.
The authors thank the referees for their comments and remarks.
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science, Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
- Kiryakova V: Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series. Volume 301. Longman Scientific & Technical, Harlow, UK; John Wiley & Sons, New York, NY, USA; 1994:x+388.Google Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.Google Scholar
- Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.Google Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
- Agarwal RP, Benchohra M, Hamani S: Boundary value problems for fractional differential equations. to appear in Georgian Mathematical JournalGoogle Scholar
- Diethelm K, Freed AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. Edited by: Keil F, Mackens W, Voß H, Werther J. Springer, Heidelberg, Germany; 1999:217-224.View ArticleGoogle Scholar
- Diethelm K, Ford NJ: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 2002,265(2):229-248. 10.1006/jmaa.2000.7194MATHMathSciNetView ArticleGoogle Scholar
- El-Sayed AMA: Fractional order evolution equations. Journal of Fractional Calculus 1995, 7: 89-100.MATHMathSciNetGoogle Scholar
- El-Sayed AMA: Fractional-order diffusion-wave equation. International Journal of Theoretical Physics 1996,35(2):311-322. 10.1007/BF02083817MATHMathSciNetView ArticleGoogle Scholar
- El-Sayed AMA: Nonlinear functional-differential equations of arbitrary orders. Nonlinear Analysis: Theory, Methods & Applications 1998,33(2):181-186. 10.1016/S0362-546X(97)00525-7MATHMathSciNetView ArticleGoogle Scholar
- Gaul L, Klein P, Kempfle S: Damping description involving fractional operators. Mechanical Systems and Signal Processing 1991,5(2):81-88. 10.1016/0888-3270(91)90016-XView ArticleGoogle Scholar
- Glockle WG, Nonnenmacher TF: A fractional calculus approach to self-similar protein dynamics. Biophysical Journal 1995,68(1):46-53. 10.1016/S0006-3495(95)80157-8View ArticleGoogle Scholar
- Lakshmikantham V, Devi JV: Theory of fractional differential equations in a Banach space. European Journal of Pure and Applied Mathematics 2008,1(1):38-45.MATHMathSciNetGoogle Scholar
- Mainardi F: Fractional calculus: some basic problems in continuum and statistical mechanis. In Fractals and Fractional Calculus in Continuum Mechanics. Edited by: Carpinteri A, Mainard F. Springer, Vienna, Austria; 1997:291-348.View ArticleGoogle Scholar
- Metzler F, Schick W, Kilian HG, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. Journal of Chemical Physics 1995,103(16):7180-7186. 10.1063/1.470346View ArticleGoogle Scholar
- Momani SM, Hadid SB: Some comparison results for integro-fractional differential inequalities. Journal of Fractional Calculus 2003, 24: 37-44.MATHMathSciNetGoogle Scholar
- Momani SM, Hadid SB, Alawenh ZM: Some analytical properties of solutions of differential equations of noninteger order. International Journal of Mathematics and Mathematical Sciences 2004,2004(13–16):697-701.MATHMathSciNetView ArticleGoogle Scholar
- Podlubny I, Petráš I, Vinagre BM, O'Leary P, Dorčák L': Analogue realizations of fractional-order controllers. Fractional order calculus and its applications. Nonlinear Dynamics 2002,29(1–4):281-296.MATHMathSciNetView ArticleGoogle Scholar
- Yu C, Gao G: Existence of fractional differential equations. Journal of Mathematical Analysis and Applications 2005,310(1):26-29. 10.1016/j.jmaa.2004.12.015MATHMathSciNetView ArticleGoogle Scholar
- El-Borai MM: On some fractional evolution equations with nonlocal conditions. International Journal of Pure and Applied Mathematics 2005,24(3):405-413.MATHMathSciNetGoogle Scholar
- El-Borai MM: The fundamental solutions for fractional evolution equations of parabolic type. Journal of Applied Mathematics and Stochastic Analysis 2004,2004(3):197-211. 10.1155/S1048953304311020MATHMathSciNetView ArticleGoogle Scholar
- Jaradat OK, Al-Omari A, Momani S: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):3153-3159. 10.1016/j.na.2007.09.008MATHMathSciNetView ArticleGoogle Scholar
- Goldstein JA: Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. Clarendon Press/Oxford University Press, New York, NY, USA; 1985:x+245.Google Scholar
- Fattorini HO: Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies. Volume 108. North-Holland, Amsterdam, The Netherlands; 1985:xiii+314.Google Scholar
- Travis CC, Webb GF: Second order differential equations in Banach spaces. In Nonlinear Equations in Abstract Spaces (Proc. Internat. Sympos., Univ. Texas, Arlington, Tex., 1977). Academic Press, New York, NY, USA; 1978:331-361.Google Scholar
- Travis CC, Webb GF: Cosine families and abstract nonlinear second order differential equations. Acta Mathematica Academiae Scientiarum Hungaricae 1978,32(1-2):75-96. 10.1007/BF01902205MATHMathSciNetView ArticleGoogle Scholar
- Ahmed NU: Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series. Volume 246. Longman Scientific & Technical, Harlow, UK; John Wiley & Sons, New York, NY, USA; 1991:x+282.Google Scholar
- Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.View ArticleGoogle Scholar
- Kisielewicz M: Differential Inclusions and Optimal Control, Mathematics and Its Applications. Volume 44. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xx+240.Google Scholar
- Deimling K: Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications. Volume 1. Walter de Gruyter, Berlin, Germany; 1992:xii+260.Google Scholar
- Górniewicz L: Topological Fixed Point Theory of Multivalued Mappings, Mathematics and Its Applications. Volume 495. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:x+399.View ArticleGoogle Scholar
- Hu S, Papageorgiou NS: Handbook of Multivalued Analysis. Volume I: Theory, Mathematics and Its Applications. Volume 419. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xvi+964.View ArticleGoogle Scholar
- Henry D: Geometric Theory of Semilinear Parabolic Partial Differential Equations. Springer, Berlin, Germany; 1989.Google Scholar
- Covitz H, Nadler SB Jr.: Multi-valued contraction mappings in generalized metric spaces. Israel Journal of Mathematics 1970,8(1):5-11. 10.1007/BF02771543MATHMathSciNetView ArticleGoogle Scholar
- Granas A, Dugundji J: Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2003:xvi+690.View ArticleGoogle Scholar
- Martelli M: A Rothe's type theorem for non-compact acyclic-valued maps. Bollettino della Unione Matematica Italiana. Serie 4 1975,11(3, supplement):70-76.MATHMathSciNetGoogle Scholar
- Burton TA, Kirk C: A fixed point theorem of Krasnoselskii-Schaefer type. Mathematische Nachrichten 1998, 189: 23-31. 10.1002/mana.19981890103MATHMathSciNetView ArticleGoogle Scholar
- Dhage BC: Multi-valued mappings and fixed points. I. Nonlinear Functional Analysis and Applications 2005,10(3):359-378.MATHMathSciNetGoogle Scholar
- Dhage BC: Multi-valued mappings and fixed points. II. Tamkang Journal of Mathematics 2006,37(1):27-46.MATHMathSciNetGoogle Scholar
- Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 1978,21(1):11-41.MATHMathSciNetGoogle Scholar
- Hino Y, Murakami S, Naito T: Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics. Volume 1473. Springer, Berlin, Germany; 1991:x+317.Google Scholar
- Hale JK: Theory of Functional Differential Equations, Applied Mathematical Sciences. Volume 3. 2nd edition. Springer, New York, NY, USA; 1977:x+365.View ArticleGoogle Scholar
- Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.View ArticleGoogle Scholar
- Kolmanovskii V, Myshkis A: Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and Its Applications. Volume 463. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:xvi+648.View ArticleGoogle Scholar
- Wu J: Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences. Volume 119. Springer, New York, NY, USA; 1996.View ArticleGoogle Scholar
- Belarbi A, Benchohra M, Hamani S, Ntouyas SK: Perturbed functional differential equations with fractional order. Communications in Applied Analysis 2007,11(3-4):429-440.MATHMathSciNetGoogle Scholar
- Belarbi A, Benchohra M, Ouahab A: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Applicable Analysis 2006,85(12):1459-1470. 10.1080/00036810601066350MATHMathSciNetView ArticleGoogle Scholar
- Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional functional differential inclusions with infinite delay and applications to control theory. Fractional Calculus & Applied Analysis 2008,11(1):35-56.MATHMathSciNetGoogle Scholar
- Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2008,338(2):1340-1350. 10.1016/j.jmaa.2007.06.021MATHMathSciNetView ArticleGoogle Scholar
- Belmekki M, Benchohra M: Existence results for fractional order semilinear functional differential equations. Proceedings of A. Razmadze Mathematical Institute 2008, 146: 9-20.MATHMathSciNetGoogle Scholar
- Belmekki M, Benchohra M, Górniewicz L: Functional differential equations with fractional order and infinite delay. Fixed Point Theory 2008,9(2):423-439.MATHMathSciNetGoogle Scholar
- Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta 2006,45(5):765-772. 10.1007/s00397-005-0043-5View ArticleGoogle Scholar
- Podlubny I: Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus & Applied Analysis 2002,5(4):367-386.MATHMathSciNetGoogle Scholar
- Prüss J: Evolutionary Integral Equations and Applications, Monographs in Mathematics. Volume 87. Birkhäuser, Basel, Switzerland; 1993:xxvi+366.View ArticleGoogle Scholar
- Hilfe R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463.Google Scholar
- Hernández E, Henríquez HR: Existence results for partial neutral functional differential equations with unbounded delay. Journal of Mathematical Analysis and Applications 1998,221(2):452-475. 10.1006/jmaa.1997.5875MATHMathSciNetView ArticleGoogle Scholar
- Hernández E, Henríquez HR: Existence of periodic solutions of partial neutral functional differential equations with unbounded delay. Journal of Mathematical Analysis and Applications 1998,221(2):499-522. 10.1006/jmaa.1997.5899MATHMathSciNetView ArticleGoogle Scholar
- El-Sayed AMA, Ibrahim A-G: Multivalued fractional differential equations. Applied Mathematics and Computation 1995,68(1):15-25. 10.1016/0096-3003(94)00080-NMATHMathSciNetView ArticleGoogle Scholar
- Ouahab A: Some results for fractional boundary value problem of differential inclusions. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):3877-3896. 10.1016/j.na.2007.10.021MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Benchohra M, Hamani S: Boundary value problems for differential inclusions with fractional order. Advanced Studies in Contemporary Mathematics 2008,16(2):181-196.MATHMathSciNetGoogle Scholar
- Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009,49(3-4):605-609. 10.1016/j.mcm.2008.03.014MATHMathSciNetView ArticleGoogle Scholar
- Yosida K: Functional Analysis, Grundlehren der Mathematischen Wissenschaften. Volume 123. 6th edition. Springer, Berlin, Germany; 1980:xii+501.Google Scholar
- Castaing C, Valadier M: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics. Volume 580. Springer, Berlin, Germany; 1977:vii+278.View ArticleGoogle Scholar
- Heikkilä S, Lakshmikantham V: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 181. Marcel Dekker, New York, NY, USA; 1994:xii+514.Google Scholar
- Joshi MC, Bose RK: Some Topics in Nonlinear Functional Analysis, A Halsted Press Book. John Wiley & Sons, New York, NY, USA; 1985:viii+311.Google Scholar
- Dhage BC, Henderson J: Existence theory for nonlinear functional boundary value problems. Electronic Journal of Qualitative Theory of Differential Equations 2004,2004(1):1-15.MathSciNetGoogle Scholar
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