Show Summary Details
More options …

# Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

IMPACT FACTOR 2018: 3.514
5-year IMPACT FACTOR: 3.524

CiteScore 2018: 3.44

SCImago Journal Rank (SJR) 2018: 1.891
Source Normalized Impact per Paper (SNIP) 2018: 1.808

Mathematical Citation Quotient (MCQ) 2018: 1.08

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 22, Issue 2

# On fractional differential inclusions with Nonlocal boundary conditions

Charles Castaing
/ C. Godet-Thobie
• Université de Bretagne Occidentale Laboratoire de Mathématiques de Bretagne Atlantique, CNRS UMR 6205, 6, avenue Victor Le Gorgeu, CS 9387, F-29238, Brest, Cedex 3, France
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Phan D. Phung
/ Le X. Truong
Published Online: 2019-05-11 | DOI: https://doi.org/10.1515/fca-2019-0027

## Abstract

The main purpose of this paper is to study a class of boundary value problem governed by a fractional differential inclusion in a separable Banach space E

$Dαu(t)+λDα−1u(t)∈F(t,u(t),Dα−1u(t)),t∈[0,1]I0+βu(t)t=0=0,u(1)=I0+γu(1)$

in both Bochner and Pettis settings, where α ∈ ]1, 2], β ∈ [0, 2 – α], λ ≥ 0, γ > 0 are given constants, Dα is the standard Riemann-Liouville fractional derivative, and F : [0, 1] × E × E → 2E is a closed valued multifunction. Topological properties of the solution set are presented. Applications to control problems and subdifferential operators are provided.

MSC 2010: 26A33; 34A60; 34B10; 34A08; 47N70

Key Words and Phrases: fractional differential inclusion; Young measures; Bolza and relaxation problem; subdifferential operators

## References

• [1]

C. Angosto and B. Cascales, Measures of weak noncompactness in Banach spaces. Topology Appl. 156 (2009), 1412–1421.

• [2]

R.P. Agarwal, S. Arshad, D. O’Regan and V. Lupulescu, Fuzzy fractional integral equations under compactness type condition. Fract. Calc. Appl. Anal. 15, No 4 (2012), 572–590: ; https://www.degruyter.com/view/j/fca.2012.15.issue-4/issue-files/fca.2012.15.issue-4.xml

• [3]

R.P. Agarwal, B. Ahmad, A. Alsaedi and N. Shahzad, Dimension ot the solution set for fractional differential inclusion. J. Nonlinear Convex Anal. 14, No 2 (2013), 314–323.Google Scholar

• [4]

B. Ahmad and J. Nieto, Riemann-Liouville fractional differential equations with fractional boundary conditions. J. Fixed Point Theory Appl. 13, No 2 (2012), 329–336.Google Scholar

• [5]

E.J. Balder, Lectures on Young Measure Theory and its Applications in Economics. Rend. Istit. Mat. Univ. Trieste, 31, Suppl.: 1–69 (2000). Workshop di Teoria della Misura et Analisi Reale Grado (1997).Google Scholar

• [6]

M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for fractional functional differential inclusions with infinite delay and applications to control theory. Fract. Calc. Appl. Anal. 11, No 1 (2008), 35–56.Google Scholar

• [7]

M. Benchohra, J. Graef and F-Z. Mostefai, Weak solutions for boundary-value problems with nonlinear fractional differential inclusions. Nonlinear Dyn. Syst. Theory 11, No 3 (2011), 227–237.Google Scholar

• [8]

F.S. De Blasi, On a property of the unit sphere in a Banach space. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 21, No 64 (1977), 3–4.Google Scholar

• [9]

A. Bressan, A. Cellina and A. Fryszkowski, A class of absolute retracts in spaces of integrable functions. Proc. A. M. S. 112 (1991), 413–418.

• [10]

C. Castaing, Quelques résultats de compacité liés a ľintégration. C. R. Acad. Sc. Paris 270 (1970), 1732–1735; and Bull. Soc. Math. France 31 (1972), 73–81.Google Scholar

• [11]

C. Castaing, Topologie de la convergence uniforme sur les parties uniformément intégrables de $\begin{array}{}{L}_{E}^{1}\end{array}$ et théorèmes de compacité faible dans certains espaces du type Köthe-Orlicz. Sém. Anal. Convexe 10 (1980), 5.1–5.27.Google Scholar

• [12]

C. Castaing, Weak compactness and convergences in Bochner and Pettis integration. Vietnam J. Math. 24, No 3 (1996), 241–286.Google Scholar

• [13]

C. Castaing, P. Raynaud de Fitte and M. Valadier, Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory. Kluwer Academic Publishers, Dordrecht (2004).Google Scholar

• [14]

C. Castaing, L X. Truong and P.D. Phung, On a fractional differential inclusion with integral boundary condition in Banach spaces. J. Nonlinear Convex Anal. 17, No 3 (2016), 441–471.Google Scholar

• [15]

C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-Heidelberg-New York (1977).Google Scholar

• [16]

A. Cernea, On a fractional differential inclusion with boundary condition. Stud. Univ. Babes-Bolyai Math. LV (2010), 105–113.Google Scholar

• [17]

J.P.R. Christensen, Topology and Borel Structure. Math. Studies 10, Notas de Mathematica (1974).Google Scholar

• [18]

F. Clarke, Generalized gradients and applications. Trans. Amer. Math. Soc. 205 (1975), 247–262.

• [19]

H. Covitz, S.B. Nadler, Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8 (1970), 5–11.

• [20]

J. Diestel and J.J. Uhl, Vector Measures. Math. Surveys 15, A.M.S. Providence, Rhode Island (1977).Google Scholar

• [21]

A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33 (1998), 181–186.

• [22]

A.M.A. El-Sayed, Sh.A. Abd El-Salam, Nonlocal boundary value problem of a fractional-order functional differential equation. Int. J. Nonlinear Sci. 7 (2009), 436–442.Google Scholar

• [23]

A.M.A. El-Sayed, A.G. Ibrahim, Set-valued integral equations of arbitrary (fractional) order. J. Appl. Math. Comput. 118 (2001), 113–121.

• [24]

L.C. Florescu and C. Godet-Thobie, Young Measures and Compactness in Measure Spaces. De Gruyter, Berlin (2012).Google Scholar

• [25]

A. Grothendieck, Espaces Vectoriels Topologiques. Publicacão da Sociedade de Matemática de Sao Paulo (1964).Google Scholar

• [26]

A.G. Ibrahim and A.M.A. El-Sayed, Definite integral of fractional order for set-valued function. J. Frac. Calculus 11 (1997), 81–87.Google Scholar

• [27]

A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Math. Stud. 204, North Holland (2006).Google Scholar

• [28]

K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993).Google Scholar

• [29]

A. Ouahab, Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 69 (2008), 3877–3896.

• [30]

I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).Google Scholar

• [31]

P.D. Phung and L.X. Truong, On a fractional differential inclusion with integral boundary conditions in Banach space. Fract. Calc. Appl. Anal. 16, No 3 (2013), 538–558; ; https://www.degruyter.com/view/j/fca.2013.16.issue-3/issue-files/fca.2013.16.issue-3.xml.

• [32]

D. O’Regan, Fixed point theorem for weakly sequentially closed maps. Arch. Math. (Brno) 36 (2000), 61–70.Google Scholar

• [33]

H.A.H. Salem, A.M.A. El-Sayed, O.L. Moustafa, A note on the fractional calculus in Banach spaces. Studia Sci. Math. Hungar. 42 (2005), 115–130.Google Scholar

• [34]

S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993).Google Scholar

• [35]

L. Thibault, Propriétés des sous-différentiels de fonctions localement Lipschitziennes définies sur un espace de Banach séparable. Applications. Thèse, Université Montpellier (1976).Google Scholar

## About the article

Accepted: 2019-02-04

Published Online: 2019-05-11

Published in Print: 2019-04-24

Citation Information: Fractional Calculus and Applied Analysis, Volume 22, Issue 2, Pages 444–478, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454,

Export Citation

© 2019 Diogenes Co., Sofia.