Exact Controllability for Hilfer Fractional Differential Inclusions Involving Nonlocal Initial Conditions

Du, Jun; Jiang, Wei; Pang, Denghao; Niazi, Azmat Ullah Khan

doi:https://doi.org/10.1155/2018/9472847

Complexity

On this page

Abstract Introduction Preliminaries Applications Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 9472847 | https://doi.org/10.1155/2018/9472847

Exact Controllability for Hilfer Fractional Differential Inclusions Involving Nonlocal Initial Conditions

Jun Du,^1,2Wei Jiang ,¹Denghao Pang,¹and Azmat Ullah Khan Niazi¹

Academic Editor: Dimitri Volchenkov

Received27 Mar 2018

Accepted11 Jul 2018

Published19 Aug 2018

Abstract

The exact controllability results for Hilfer fractional differential inclusions involving nonlocal initial conditions are presented and proved. By means of the multivalued analysis, measure of noncompactness method, fractional calculus combined with the generalized Mnch fixed point theorem, we derive some sufficient conditions to ensure the controllability for the nonlocal Hilfer fractional differential system. The results are new and generalize the existing results. Finally, we talk about an example to interpret the applications of our abstract results.

1. Introduction

Fractional calculus generalizes the standard integer calculus to arbitrary order. It provides a valuable tool for the description of memory and hereditary properties of diversified materials and processes. In the past twenty years, the subject of the fractional calculus is picking up considerable popularity and importance. We can refer to the monographs of Diethelm and Freed [1], Kilbas et al. [2], Miller and Ross [3], Podlubny [4], and Zhou [5]. Fractional differential equations and inclusions involving Caputo derivative or Riemann-Liouville derivative have obtained more and more results (see [6–15]). Recently, Hilfer [16] initiated an extended Riemann-Liouville fractional derivative, named Hilfer fractional derivative, which interpolates Caputo fractional derivative and Riemann-Liouville fractional derivative. This operator appeared in the theoretical simulation of dielectric relaxation in glass forming materials. Hilfer et al. [17] initially presented linear differential equations with the new Hilfer fractional derivative and applied operational calculus to solve such generalized fractional differential equations. Subsequently, Furati et al. [18] and Gu and Trujillo [19] generalized to consider nonlinear problems and proved the existence, nonexistence, and stability results for initial value problems of nonlinear fractional differential equations with Hilfer fractional derivative in a suitable weighted space of continuous functions.

Control theory is an interdisciplinary branch of engineering and mathematics that deals with influence behavior of dynamical systems. Controllability is one of the fundamental concepts in mathematical control theory, it means that it is possible to steer a dynamical system from an arbitrary initial state to arbitrary final state using the set of admissible controls. Recently, the controllability conditions for various linear and nonlinear integer or fractional order systems have been considered in many papers by using different methods [20–33] and the references. There have also been some results [20–24, 32, 33] about the investigations of the exact controllability of systems represented by nonlinear evolution equations in infinite dimensional space. But when the semigroup or the control action operator B is compact, then the controllability operator is also compact and the applications of exact controllability results is just restricted to the finite dimensional space [20]. Therefore, we investigate the exact controllability of the fractional evolution systems only involving noncompact semigroups.

The nonlocal initial problems have been initially proposed by Byszewski et al. [34, 35] to generalize the study of the canonical initial problem, comes from physical science. For instance, it used to determine the unknown physical parameters in some inverse heat condition problems. It has been found that the nonlocal initial condition is more exact to describe the nature phenomena than the classical initial condition, since more data is taken into account, therefore abating the negative influences induced by a possible inaccurate single estimation taken at the start time. For more discussion on this type of differential equations and inclusions, we can see papers [36–42] and references given therein.

Boucherif and Precup [36] proved the existence for mild solutions to the following nonlocal initial problem for first-order evolution equations using Schaefer fixed point theorem: where is the infinitesimal generator of a C₀-semigroup on a Banach space and is a known function.

Liang and Yang [33] concerned the controllability for the following fractional integrodifferential evolution equations involving nonlocal conditions using the Mnch fixed point theorem: where is the Caputo derivative of order , is the infinitesimal generator of -semigroup of uniformly bounded linear operator, the control function is known in ; is a Banach space, is a linear bounded operator from to ; is a known function and is a Volterra integral operator.

Du et al. [43] generalized the results of [33] and gave the controllability for a new class of fractional neutral integrodifferential evolution equations with infinite delay and nonlocal conditions using Mönch fixed point theorem. However, it should be emphasized that to the best of our knowledge, the exact controllability of Hilfer fractional differential system has not been investigated yet. Motivated by [19, 30, 33, 36, 43], in this paper, we concern the controllability of the following fractional differential inclusions involving a more general fractional derivative with nonlocal initial conditions: where is the Hilfer fractional derivative of order ( obeys ) and type ( obeys ) which will be given in Section 2; is separable and is bounded, so is a uniformly continuous semigroup and . The nonlinear term is multivalued function. Let are two finite intervals of ; The control function takes values in , with as a Banach space; is a linear bounded operator from to .

In this paper, by means of a concrete nonlocal function, we do not have to suppose the compactness and Lipschitz conditions on the nonlocal function but only assume that satisfy the hypothesis (H0) (see Section 3). Furthermore, the proofs of our main results are based on fractional calculus theory, the multivalued analysis, measure of noncompactness method, in addition to the O’Regan-Precup fixed point theorem, which is an extension of the Mönch fixed point theorem.

2. Preliminaries and Notations

Let and denote the space of -valued continuous functions from to and from to , respectively; Let .

Define exists and is finite}, involving the norm defined by . Then, is a Banach space. We also note that (1)When , then and ;(2)Let for , if and only if and .

For , define Thus is a bounded closed and convex subset of .

Let , Then is a closed ball of the space with the radius and center at 0. And is also a bounded closed and convex subset of .

Next, we list some definitions and properties in fractional calculus, multivalued analysis, semigroup theory, and measure of noncompactness.

The following definitions concerning fractional calculus can be found in the books [2–4, 16].

Definition 1. The fractional integral for function from lower limit and order can be expressed by where is the gamma function, and right side of upper equality is point-wise defined on .

Definition 2. The Riemann-Liouville derivative of order with the lower limit 0 for function can be expressed by

Definition 3. The Caputo derivative of order for function can be denoted by

Definition 4. The left Hilfer derivative of order and type of function is defined by where

Remark 1. (i)The operator can be written as (ii)When and , the Hilfer fractional derivative coincides with the Riemann-Liouville derivative: (iii)When and , the Hilfer fractional derivative coincides with the Caputo derivative: Let be the set of all nonempty subsets of . We will use the following notations:

Remark 2. (i)A measurable function is Bochner integrable if and only if is Lebesgue integrable.(ii)A multivalued map is said to be convex valued (closed valued) if is convex (closed) for all is said to be bounded on bounded sets if is bounded in for all (iii)A multivalued map is said to be upper semicontinuous (u.s.c.) on if for each , the set is a nonempty closed subset of , and if for each open subset of containing there exists an open neighborhood of such that (iv)A multivalued map is said to be 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, that is, imply We say that has a fixed point if there is such that (v)A multivalued map is said to be measurable if for each , the function defined by is measurable.

Lemma 1 (see [44]). Let be a Banach space. The multivalued map satisfies the following: for each is u.s.c.; for each the function is strongly measurable and the set is nonempty. Let be a linear continuous mapping from to then the operator is a closed graph operator in
Consider where and and is a probability density function defined on , that is

Remark 3 (see [2]). As we all know from [2] that can be denoted by the Mittag-Leffler functions: The following essential propositions can be found in the papers [19, 38].

Lemma 2. If , then for each (i)(ii) and .

Lemma 3 (Example 2.1.3 [45]).
For each and define If is equicontinuous and bounded, then is continuous on and Here, is the Hausdorff noncompact measure on defined on every bounded subset of Banach space by

Lemma 4 (see Lemma 5 [46]). Let be a countable subset with for almost everywhere and any , where . ThenTo end this section, we reintroduce the O’Regan-Precup fixed point theorem.

Lemma 5 (see Theorem 3.2 [47]). Let be a subset of Banach space which is closed and convex. is a relatively open subset of , and Suppose graph () is closed, maps compact sets into relatively compact sets, and that for some , the following two conditions are satisfied: Then has a fixed point.

3. Controllability Results

We first consider linear Hilfer fractional differential equations of the form where .

Assume that there exists the bounded operator given by where .

By means of [48], we can present the sufficient conditions for the existence and boundedness of the operator .

Lemma 6. If the hypothesis (H0) holds, the operator defined in (21) exists and is bounded.

Proof 1. From the hypothesis , we have By operator spectrum theorem, the operator exists and is bounded. Furthermore, by Neumann expression, we get Using Lemma 6 and [19], we give the following definition of mild solution for the Hilfer fractional system (20) involving nonlocal initial conditions.

Definition 5. A function is called a mild solution for the Hilfer fractional system (20) if it satisfies the following equation: where .

Remark 4. By virtue of [19], a mild solution for Hilfer fractional evolution (20) with the initial condition is Specially, Using (20) and (26), we get Since exists a bounded inverse operator which is denoted by so And hence, it is indeed (24).

Using Definition 5, we give a new definition of the mild solution for the Hilfer fractional nonlocal differential inclusions (3) as follows:

Definition 6. A function is called a mild solution of the Hilfer fractional nonlocal differential inclusions (3) if for any , the following integral equation is satisfied: where , and .

To present and prove the main results of this paper, we enumerate the following hypotheses: (H1) is a separable Banach space, is bounded, hence is a uniformly continuous semigroup.(H2)The multivalued map satisfies the following: (2a)For every , is u.s.c., for each , the function is strongly measurable. The set is nonempty;(2b)There exists a function , and a continuous nondecreasing function such that for any we have ;(2c)There exists a constant and a function s.t. for any countable subset .(H3)(3a) Linear operator defined by is reversible, the inverse operator denoted by and takes values in , and there exist two constants such that (3b) There exists a constant and such that for any countable subset (H4) where , and , , ,

For any define an operator as follows: where .

It is evident to see that .

For any , let for , then . Define as follows

Clearly, is a mild solution of (3) in if and only if has a solution

For brevity, let us take the following notations

In view of Lemma 2, we obtain the following lemma that will be useful in the proof of the main results.

Lemma 7. Under the hypothesis (H2) (2b), (H3) (3a), for each set , we have

Proof 2. By Lemma 2, for each set , it is easy to get Hence, we easily see that This completes the proof.
Next, we derive the controllability results for the Hilfer fractional nonlocal differential inclusions (3).

Theorem 1. Assume the hypotheses (H0)–(H4) hold, then the Hilfer fractional nonlocal differential inclusions (3) are exact controllable on provided that

Proof 3. According to (H2) (2a) and [22], for each , the multivalued function has a measurable selection, and in view of (H2) (2b), this selection belongs to . Thus, we can define a multivalued function as follows. For every , let and a function if and only if where and .
Note that according to (H4), there is such that for all , Denote and , where and We just prove that the multivalued operator meets the conditions of Lemma 5. Obviously, since the values of are convex, the values of are also convex.

Claim 1. Each solution to the inclusion satisfies .
Let be a solution of (44). Then, by reminding the definition of , the hypothesis (H2) (2b) and recalling also Lemma 2, for each , we derive This inequality with (43) deduces So .

Claim 2. Every function in is equicontinuous.
For each , set such that By (42), there is for , we get where From (H1), one can deduce that as From Lemma 7, we have which implies that as Noting that and exists, then by Lebesgue’s dominated convergence theorem, we obtain as .
For , we get The assumption (H1) guarantees that as and We prove that is uniformly continuous on Consequently, is equicontinuous on .

Claim 3. The inference (13) holds with .
Let with countable. We assert that is relatively compact. In fact, since is countable and , we can chase down a countable set with . Then, there exists with . This means that there is such that for where From , for , . By utilizing Lemmas 2, 4, (H2) (2c), (H3) (3b), and the properties of the noncompact measure, we can derive and Reminding , by Claim 2, is equicontinuous. So we find from Lemma 3 that Since we obtain That is, is compact.

Claim 4. maps compact sets into relatively compact sets.
Let be a compact subset of . From Claim 2, is equicontinuous. Let by the definition of , for any and , there is such that where
Therefore Then, Lemma 3 indicates that is, the set is relatively compact.

Claim 5. The graph is closed.
For any , let and , . Let , We will show that . Since there is such that for any , So we just need to demonstrate the existence of such that for any

Take into account the linear continuous operator where

Clearly, we can get from Lemma 1 that the operator is a closed graph.

Since , we can obtain that as

In addition, we get

Since , we can obtain from Lemma 1 that

For some , this infers that Hence, has a closed graph. Thus, Claims 1–5 are completed. By use of Lemma 5, we know operator has a fixed point in . Let , then is a mild solution of (3) and it satisfies . Therefore, the Hilfer fractional nonlocal differential inclusions (3) are exact controllable on

4. Applications

Consider the following partial differential system where is constants, is continuous.

Let and is defined by

As we all know that generates an equicontinuous semigroup in and it satisfies for Thus, is not compact in and Take

Then, for any where let then and we obtain

Thus, the hypothesis holds for and for all By we verify that the hypothesis (H0) holds.

For the operator is defined as where and satisfy

If satisfies the hypothesis (H3), from Theorem 1, we get that the Hilfer fractional differential inclusion (66) involving nonlocal initial conditions is exact controllable on provided that (H4) and (41) are satisfied.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 1371027, 11471015, and 1601003), Natural Science Foundation of Anhui Province (no. 1708085MA15), and Natural Science Fund of Colleges and Universities in Anhui Province (no. KJ2018A0470).

References

K. Diethelm and A. D. Freed, “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, F. Keil, W. Mackens, H. Voss, and J. Werther, Eds., pp. 217–224, Springer-Verlag, Heidelberg, 1999.
View at: Publisher Site | Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland and Mathematics Studies, Elsevier Science BV, Netherlands Amsterdam, 2006.
View at: Publisher Site
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, USA, 1999.
Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.
View at: Publisher Site
M. M. el-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos Solitons Fractals, vol. 14, no. 3, pp. 433–440, 2002.
View at: Publisher Site | Google Scholar
Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1063–1077, 2010.
View at: Publisher Site | Google Scholar
S. Liu, X. Wu, X. F. Zhou, and W. Jiang, “Asymptotical stability of Riemann-Liouville fractional nonlinear systems,” Nonlinear Dynamics, vol. 86, no. 1, pp. 65–71, 2016.
View at: Publisher Site | Google Scholar
S. Liu, X. F. Zhou, X. Y. Li, and W. Jiang, “Asymptotical stability of Riemann-Liouville fractional singular systems with multiple time-varying delays,” Applied Mathematics Letters, vol. 65, pp. 32–39, 2017.
View at: Publisher Site | Google Scholar
K. X. Li, J. G. Peng, and J. X. Jia, “Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives,” Journal of Functional Analysis, vol. 263, no. 2, pp. 476–510, 2012.
View at: Publisher Site | Google Scholar
R. P. Agarwal and A. Özbekler, “Lyapunov type inequalities for mixed nonlinear Riemann-Liouville fractional differential equations with a forcing term,” Journal of Computational and Applied Mathematics, vol. 314, pp. 69–78, 2017.
View at: Publisher Site | Google Scholar
R. P. Agarwal, V. Lupulescu, D. O’Regan, and G. ur Rahman, “Fractional calculus and fractional differential equations in nonreflexive Banach spaces,” Communications in Nonlinear Science Numerical Simulation, vol. 20, no. 1, pp. 59–73, 2015.
View at: Publisher Site | Google Scholar
H. Liu, Y. Pan, S. Li, and Y. Chen, “Adaptive fuzzy backstepping control of fractional-order nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 8, pp. 2209–2217, 2017.
View at: Publisher Site | Google Scholar
H. Liu, S. Li, J. Cao, G. Li, A. Alsaedi, and F. E. Alsaadi, “Adaptive fuzzy prescribed performance controller design for a class of uncertain fractional-order nonlinear systems with external disturbances,” Neurocomputing, vol. 219, pp. 422–430, 2017.
View at: Publisher Site | Google Scholar
H. Liu, S. Li, G. Li, and H. Wang, “Adaptive controller design for a class of uncertain fractional-order nonlinear systems: an adaptive fuzzy approach,” International Journal of Fuzzy Systems, vol. 20, no. 2, pp. 366–379, 2018.
View at: Publisher Site | Google Scholar
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
View at: Publisher Site
R. Hilfer, Y. Luchko, and Ž. Tomovski, “Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives,” Fractional Calculus and Applied Analysis, vol. 12, pp. 289–318, 2009.
View at: Google Scholar
K. M. Furati, M. D. Kassim, and N. E. Tatar, “Existence and uniqueness for a problem involving Hilfer fractional derivative,” Computers and Mathematics with Applications, vol. 64, no. 6, pp. 1616–1626, 2012.
View at: Publisher Site | Google Scholar
H. B. Gu and J. J. Trujillo, “Existence of mild solution for evolution equation with Hilfer fractional derivative,” Applied Mathematics and Computation, vol. 257, pp. 344–354, 2015.
View at: Publisher Site | Google Scholar
M. Eduardo Hernández and D. O’Regan, “Controllability of Volterra–Fredholm type systems in Banach spaces,” Journal of the Franklin Institute, vol. 346, no. 2, pp. 95–101, 2009.
View at: Publisher Site | Google Scholar
S. C. Ji, G. Li, and M. Wang, “Controllability of impulsive differential systems with nonlocal conditions,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6981–6989, 2011.
View at: Publisher Site | Google Scholar
Y.-K. Chang, A. Anguraj, and M. Mallika Arjunan, “Controllability of impulsive neutral functional differential inclusions with infinite delay in Banach spaces,” Chaos Solitons Fractals, vol. 39, no. 4, pp. 1864–1876, 2009.
View at: Publisher Site | Google Scholar
V. Obukhovski and P. Zecca, “Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup,” Nonlinear Analysis, vol. 70, no. 9, pp. 3424–3436, 2009.
View at: Publisher Site | Google Scholar
N. Abada, M. Benchohra, and H. Hammouche, “Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions,” Journal of Differential Equations, vol. 246, no. 10, pp. 3834–3863, 2009.
View at: Publisher Site | Google Scholar
X.-F. Zhou, J. Wei, and L.-G. Hu, “Controllability of a fractional linear time-invariant neutral dynamical system,” Applied Mathematics Letters, vol. 26, no. 4, pp. 418–424, 2013.
View at: Publisher Site | Google Scholar
J. Wei, “The controllability of fractional control systems with control delay,” Computers and Mathematics with Applications, vol. 64, no. 10, pp. 3153–3159, 2012.
View at: Publisher Site | Google Scholar
K. Balachandran, S. Divya, L. Rodríguez-Germá, and J. J. Trujillo, “Relative controllability of nonlinear neutral fractional integro-differential systems with distributed delays in control,” Mathematical Methods in the Applied Sciences, vol. 39, no. 2, pp. 214–224, 2016.
View at: Publisher Site | Google Scholar
N. I. Mahmudov and S. Zorlu, “On the approximate controllability of fractional evolution equations with compact analytic semigroup,” Journal of Computational and Applied Mathematics, vol. 259, pp. 194–204, 2014.
View at: Publisher Site | Google Scholar
Z. Liu and X. Li, “Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives,” SIAM Journal on Control and Optimization, vol. 53, no. 4, pp. 1920–1933, 2015.
View at: Publisher Site | Google Scholar
M. Yang and Q. R. Wang, “Approximate controllability of Riemann-Liouville fractional differential inclusions,” Applied Mathematics and Computation, vol. 274, pp. 267–281, 2016.
View at: Publisher Site | Google Scholar
M. Yang and Q. R. Wang, “Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions,” Mathematical Methods in the Applied Sciences, vol. 40, no. 4, pp. 1126–1138, 2017.
View at: Publisher Site | Google Scholar
J. R. Wang and Y. Zhou, “Complete controllability of fractional evolution systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4346–4355, 2012.
View at: Publisher Site | Google Scholar
J. Liang and H. Yang, “Controllability of fractional integro-differential evolution equations with nonlocal conditions,” Applied Mathematics and Computation, vol. 254, pp. 20–29, 2015.
View at: Publisher Site | Google Scholar
L. Byszewski and V. Lakshmikantham, “Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space,” Applicable Analysis, vol. 40, no. 1, pp. 11–19, 1991.
View at: Publisher Site | Google Scholar
L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 162, no. 2, pp. 494–505, 1991.
View at: Publisher Site | Google Scholar
A. Boucherif and R. Precup, “Semilinear evolution equations with nonlocal initial conditions,” Dynamic Systems and Applications, vol. 16, no. 3, pp. 507–516, 2007.
View at: Google Scholar
Q. Liu, “Existence results for a class of semilinear nonlocal evolution equations,” Mathematical Methods in the Applied Sciences, vol. 35, no. 11, pp. 1356–1364, 2012.
View at: Publisher Site | Google Scholar
Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4465–4475, 2010.
View at: Publisher Site | Google Scholar
F. Li, J. Liang, and H.-K. Xu, “Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 391, no. 2, pp. 510–525, 2012.
View at: Publisher Site | Google Scholar
A. Chauhan and J. Dabas, “Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 4, pp. 821–829, 2014.
View at: Publisher Site | Google Scholar
J. R. Wang, A. G. Ibrahim, and M. Fečkan, “Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces,” Applied Mathematics and Computation, vol. 257, pp. 103–118, 2015.
View at: Publisher Site | Google Scholar
J. R. Wang, A. G. Ibrahim, and M. Fečkan, “Nonlocal Cauchy problems for semilinear differential inclusions with fractional order in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 27, no. 1–3, pp. 281–293, 2015.
View at: Publisher Site | Google Scholar
J. Du, W. Jiang, D. Pang, and A. U. K. Niazi, “Controllability for a new class of fractional neutral integro-differential evolution equations with infinite delay and nonlocal conditions,” Advances in Difference Equations, vol. 2017, no. 1, 2017.
View at: Publisher Site | Google Scholar
A. Lasota and Z. Opial, “An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations,” Bulletin of Academy of Polish Sciences, Serial Sciences of Mathematics Astronomy Physics, vol. 13, pp. 781–786, 1965.
View at: Google Scholar
M. Kamenskii, V. V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin, NY, USA, 2001.
View at: Publisher Site
J. Banaś, “On measures of noncompactness in Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 21, no. 1, pp. 131–143, 1980.
View at: Google Scholar
D. O’Regan and R. Precup, “Fixed point theorems for set-valued maps and existence principles for integral inclusions,” Journal of Mathematical Analysis and Applications, vol. 245, no. 2, pp. 594–612, 2000.
View at: Publisher Site | Google Scholar
J. R. Wang, Y. Zhou, and M. Fečkan, “Alternative results and robustness for fractional evolution equations with periodic boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, no. 97, pp. 1–15, 2011.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2018 Jun Du 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.

PDF Download Citation

Download other formats

Order printed copies

Views

1373

Downloads

1365

Citations