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Wen-Xue Zhou, Hai-Zhong Liu, "Existence of Weak Solutions for Nonlinear Fractional Differential Inclusion with Nonseparated Boundary Conditions", Journal of Applied Mathematics, vol. 2012, Article ID 530624, 13 pages, 2012. https://doi.org/10.1155/2012/530624
Existence of Weak Solutions for Nonlinear Fractional Differential Inclusion with Nonseparated Boundary Conditions
We discuss the existence of solutions, under the Pettis integrability assumption, for a class of boundary value problems for fractional differential inclusions involving nonlinear nonseparated boundary conditions. Our analysis relies on the Mönch fixed point theorem combined with the technique of measures of weak noncompactness.
This paper is mainly concerned with the existence results for the following fractional differential inclusion with non-separated boundary conditions: where is a real number, is the Caputo fractional derivative. is a multivalued map, is a Banach space with the norm , and is the family of all nonempty subsets of .
Recently, fractional differential equations have found numerous applications in various fields of physics and engineering [1, 2]. It should be noted that most of the books and papers on fractional calculus are devoted to the solvability of initial value problems for differential equations of fractional order. In contrast, the theory of boundary value problems for nonlinear fractional differential equations has received attention quite recently and many aspects of this theory need to be explored. For more details and examples, see [3–18] and the references therein.
To investigate the existence of solutions of the problem above, we use Mönch’s fixed point theorem combined with the technique of measures of weak noncompactness, which is an important method for seeking solutions of differential equations. This technique was mainly initiated in the monograph of Banaś and Goebel  and subsequently developed and used in many papers; see, for example, Banaś and Sadarangani , Guo et al. , Krzyśka and Kubiaczyk , Lakshmikantham and Leela , Mönch’s , O’Regan [25, 26], Szufla [27, 28], and the references therein.
In 2007, Ouahab  investigated the existence of solutions for -fractional differential inclusions by means of selection theorem together with a fixed point theorem. Very recently, Chang and Nieto  established some new existence results for fractional differential inclusions due to fixed point theorem of multivalued maps. Problem (1.1) was discussed for single valued case in the paper ; some existence results for single- and multivalued cases for an extension of (1.1) to non-separated integral boundary conditions were obtained in the article  and . About other results on fractional differential inclusions, we refer the reader to . As far as we know, there are very few results devoted to weak solutions of nonlinear fractional differential inclusions. Motivated by the above mentioned papers, the purpose of this paper is to establish the existence results for the boundary value problem (1.1) by virtue of the Mönch fixed point theorem combined with the technique of measures of weak noncompactness.
The remainder of this paper is organized as follows. In Section 2, we present some basic definitions and notations about fractional calculus and multivalued maps. In Section 3, we give main results for fractional differential inclusions. In the last section, an example is given to illustrate our main result.
2. Preliminaries and Lemmas
In this section, we introduce notation, definitions, and preliminary facts that will be used in the remainder of this paper. Let be a real Banach space with norm and dual space , and let denote the space with its weak topology. Here, let be the Banach space of all continuous functions from to with the norm and let denote the Banach space of functions that are the Lebesgue integrable with norm We let to be the Banach space of bounded measurable functions equipped with the norm Also, will denote the space of functions that are absolutely continuous and whose first derivative, , is absolutely continuous.
Let be a Banach space, and let , , , and . A multivalued map is convex (closed) valued if is convex (closed) for all . We say that is bounded on bounded sets if is bounded in for all (i.e., . The mapping is called upper semicontinuous (u.s.c.) on if for each , the set is a nonempty closed subset of and if for each open set of containing , there exists an open neighborhood of such that . We say that is completely continuous if is relatively compact for every . If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph (i.e., imply ). The mapping has a fixed point if there is such that . The set of fixed points of the multivalued operator will be denoted by . A multivalued map is said to be measurable if for every , the function is measurable. For more details on multivalued maps, see the books of Aubin and Cellina , Aubin and Frankowska , Deimling , Hu and Papageorgiou , Kisielewicz , and Covitz and Nadler .
Moreover, for a given set of functions , let us denote by , , and .
For any , let be the set of selections of defined by
Definition 2.1. A function is said to be weakly sequentially continuous if takes each weakly convergent sequence in to a weakly convergent sequence in (i.e., for any in with in then in for each ).
Definition 2.2. A function has a weakly sequentially closed graph if for any sequence , for with in for each and in for each , then .
Definition 2.3 (see ). The function is said to be the Pettis integrable on if and only if there is an element corresponding to each such that for all , where the integral on the right is supposed to exist in the sense of Lebesgue. By definition, .
Let be the space of all -valued Pettis integrable functions in the interval .
Lemma 2.4 (see ). If is Pettis’ integrable and is a measurable and essentially bounded real-valued function, then is Pettis’ integrable.
Definition 2.5 (see ). Let be a Banach space, the set of all bounded subsets of , and the unit ball in . The De Blasi measure of weak noncompactness is the map defined by
Lemma 2.6 (see ). The De Blasi measure of noncompactness satisfies the following properties:(a);(b) is relatively weakly compact;(c);(d), where denotes the weak closure of ;(e);(f);(g);(h).
The following result follows directly from the Hahn-Banach theorem.
Lemma 2.7. Let be a normed space with . Then there exists with and .
For completeness, we recall the definitions of the Pettis-integral and the Caputo derivative of fractional order.
Definition 2.8 (see ). Let be a function. The fractional Pettis integral of the function of order is defined by where the sign “” denotes the Pettis integral and is the gamma function.
Definition 2.9 (see ). For a function , the Caputo fractional-order derivative of is defined by where and denotes the integer part of .
Lemma 2.10 (see ). Let be a Banach space with a nonempty, bounded, closed, convex, equicontinuous subset of . Suppose has a weakly sequentially closed graph. If the implication holds for every subset of , then the operator inclusion has a solution in .
3. Main Results
Let us start by defining what we mean by a solution of problem (1.1).
Definition 3.1. A function is said to be a solution of (1.1), if there exists a function with for a.e. , such that
and satisfies conditions .
To prove the main results, we need the following assumptions:(H1) has weakly sequentially closed graph;(H2)for each continuous , there exists a scalarly measurable function with a.e. on and is Pettis integrable on ;(H3)there exist and a continuous nondecreasing function such that (H4)for each bounded set , and each , the following inequality holds: (H5)there exists a constant such that where and are defined by (3.9).
Theorem 3.2. Let be a Banach space. Assume that hypotheses (H1)–(H5) are satisfied. If then the problem (1.1) has at least one solution on .
Proof. Let be a given function; it is obvious that the boundary value problem 
has a unique solution
where is defined by the formula
From the expression of and , it is obvious that is continuous on and is continuous on . Denote by
We transform the problem (1.1) into fixed point problem by considering the multivalued operator defined by and refer to  for defining the operator . Clearly, the fixed points of are solutions of Problem (1.1). We first show that (3.10) makes sense. To see this, let ; by (H2) there exists a Pettis’ integrable function such that for a.e. . Since , then is Pettis integrable and thus is well defined.
Let , and consider the set clearly, the subset is a closed, convex, bounded, and equicontinuous subset of . We shall show that satisfies the assumptions of Lemma 2.10. The proof will be given in four steps.
Step 1. We will show that the operator is convex for each .
Indeed, if and belong to , then there exists Pettis’ integrable functions , such that, for all , we have Let . Then, for each , we have Since has convex values, and we have .
Step 2. We will show that the operator maps into .
To see this, take . Then there exists with and there exists a Pettis integrable function with for a.e. . Without loss of generality, we assume for all . Then, there exists with and . Hence, for each fixed , we have
Therefore, by (H5), we have
Next suppose and , with so that . Then, there exists such that . Hence, this means that .
Step 3. We will show that the operator has a weakly sequentially closed graph.
Let be a sequence in with in for each , in for each , and for . We will show that . By the relation , we mean that there exists such that
We must show that there exists such that, for each ,
Since has compact values, there exists a subsequence such that Since has a weakly sequentially closed graph, . The Lebesgue dominated convergence theorem for the Pettis integral then implies that for each , that is, in . Repeating this for each shows .
Step 4. The implication (2.9) holds. Now let be a subset of such that . Clearly, for all . Hence, , is bounded in .
Since function is continuous on , the set is compact, so . By assumption (H4) and the properties of the measure , we have for each which gives
This means that By (3.5) it follows that ; that is, for each , and then is relatively weakly compact in . In view of Lemma 2.10, we deduce that has a fixed point which is obviously a solution of Problem (1.1). This completes the proof.
In the sequel we present an example which illustrates Theorem 3.2.
4. An Example
Example 4.1. We consider the following partial hyperbolic fractional differential inclusion of the form
Set , , , then . So .
Let with the norm Set For each and , we have
Hence conditions , , and hold with , and . For any bounded set , we have Hence (H4) is satisfied. From (3.8), we have So, we get A simple computation gives We shall check that condition (3.5) is satisfied. Indeed which is satisfied for some , and (H5) is satisfied for . Then by Theorem 3.2, the problem (4.1) has at least one solution on for values of satisfying (4.10).
The first author’s work was supported by NNSF of China (11161027), NNSF of China (10901075), and the Key Project of Chinese Ministry of Education (210226). The authors are grateful to the referees for their comments according to which the paper has been revised.
- R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advances in Fractional Calculus, Springer, 2007.
- A. A. Kilbas, H. H. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
- V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, UK, 2009.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
- I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
- R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for fractional differential equations,” Advanced Studies in Contemporary Mathematics, vol. 16, pp. 181–196, 2008.
- B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2009, Article ID 494720, 9 pages, 2009.
- Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
- M. El-Shahed and J. J. Nieto, “Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3438–3443, 2010.
- C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1363–1375, 2010.
- F. Jiao and Y. Zhou, “Existence of solutions for a class of fractional boundary value problems via critical point theory,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1181–1199, 2011.
- J. Wang and Y. Zhou, “Analysis of nonlinear fractional control systems in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 74, no. 17, pp. 5929–5942, 2011.
- G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 74, no. 3, pp. 792–804, 2011.
- W.-X. Zhou and Y.-D. Chu, “Existence of solutions for fractional differential equations with multi-point boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1142–1148, 2012.
- W. Zhou, Y. Chang, and H. Liu, “Weak solutions for nonlinear fractional dif-ferential equations in Banach spaces,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 527969, 13 pages, 2012.
- J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60, Marcel Dekker, New York, NY, USA, 1980.
- J. Banaś and K. Sadarangani, “On some measures of noncompactness in the space of continuous functions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 68, no. 2, pp. 377–383, 2008.
- D. Guo, V. Lakshmikantham, and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373 of Mathematics and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996.
- S. Krzyśka and I. Kubiaczyk, “On bounded pseudo and weak solutions of a nonlinear differential equation in Banach spaces,” Demonstratio Mathematica, vol. 32, no. 2, pp. 323–330, 1999.
- V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, vol. 2, Pergamon Press, New York, NY, USA, 1981.
- H. Mönch, “Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces,” Nonlinear Analysis, vol. 4, no. 5, pp. 985–999, 1980.
- D. O'Regan, “Fixed-point theory for weakly sequentially continuous mappings,” Mathematical and Computer Modelling, vol. 27, no. 5, pp. 1–14, 1998.
- D. O'Regan, “Weak solutions of ordinary differential equations in Banach spaces,” Applied Mathematics Letters, vol. 12, no. 1, pp. 101–105, 1999.
- S. Szufla, “On the application of measure of noncompactness to existence theorems,” Rendiconti del Seminario Matematico della Università di Padova, vol. 75, pp. 1–14, 1986.
- S. Szufla and A. Szukała, “Existence theorems for weak solutions of th order differential equations in Banach spaces,” Functiones et Approximatio Commentarii Mathematici, vol. 26, pp. 313–319, 1998, Dedicated to Julian Musielak.
- A. Ouahab, “Some results for fractional boundary value problem of differential inclusions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 69, no. 11, pp. 3877–3896, 2008.
- Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605–609, 2009.
- B. Ahmad, “New results for boundary value problems of nonlinear fractional differential equations with non-separated boundary conditions,” Acta Mathematica Vietnamica, vol. 36, pp. 659–668, 2011.
- B. Ahmad, J. J. Nieto, and A. Alsaedi, “Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions,” Acta Mathematica Scientia B, vol. 31, no. 6, pp. 2122–2130, 2011.
- B. Ahmad and S. K. Ntouyas, “Nonlinear fractional differential inclusions with anti-periodic typeintegral boundary conditions,” Discussiones Mathematicae Differential Inclusions, Control and Optimization. In press.
- J. Henderson and A. Ouahab, “Impulsive differential inclusions with fractional order,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1191–1226, 2010.
- J.-P. Aubin and A. Cellina, Differential Inclusions, vol. 264, Springer, New York, NY, USA, 1984.
- J.-P. Aubin and H. Frankowska, Set-Valued Analysis, vol. 2, Birkhäuser, Boston, Mass, USA, 1990.
- K. Deimling, Multivalued Differential Equations, vol. 1, Walter De Gruyter, New York, NY, USA, 1992.
- S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Theory, vol. 1, Kluwer Academic, Dordrecht, The Netherlands, 1997.
- M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic, Dordrecht, The Netherlands, 1991.
- H. Covitz and S. B. Nadler Jr., “Multi-valued contraction mappings in generalized metric spaces,” Israel Journal of Mathematics, vol. 8, pp. 5–11, 1970.
- B. J. Pettis, “On integration in vector spaces,” Transactions of the American Mathematical Society, vol. 44, no. 2, pp. 277–304, 1938.
- F. S. De Blasi, “On a property of the unit sphere in a Banach space,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 21, pp. 259–262, 1977.
- H. A. H. Salem, A. M. A. El-Sayed, and O. L. Moustafa, “A note on the fractional calculus in Banach spaces,” Studia Scientiarum Mathematicarum Hungarica, vol. 42, no. 2, pp. 115–130, 2005.
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