{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:10:23Z","timestamp":1760231423438,"version":"build-2065373602"},"reference-count":58,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,18]],"date-time":"2022-09-18T00:00:00Z","timestamp":1663459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Technology of the Republic of China","award":["MOST 111-2115-M-017-002","174024"],"award-info":[{"award-number":["MOST 111-2115-M-017-002","174024"]}]},{"name":"Ministry of Science and Technological Development, Republic of Serbia and Bilateral project between MANU and SANU","award":["MOST 111-2115-M-017-002","174024"],"award-info":[{"award-number":["MOST 111-2115-M-017-002","174024"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we introduce the notion of a (weak) proportional Caputo fractional derivative of order \u03b1\u2208(0,1) for a continuous (locally integrable) function u:[0,\u221e)\u2192E, where E is a complex Banach space. In our definition, we do not require that the function u(\u00b7) is continuously differentiable, which enables us to consider the wellposedness of the corresponding fractional relaxation problems in a much better theoretical way. More precisely, we systematically investigate several new classes of (degenerate) fractional solution operator families connected with the use of this type of fractional derivatives, obeying the multivalued linear approach to the abstract Volterra integro-differential inclusions. The quasi-periodic properties of the proportional fractional integrals as well as the existence and uniqueness of almost periodic-type solutions for various classes of proportional Caputo fractional differential inclusions in Banach spaces are also considered.<\/jats:p>","DOI":"10.3390\/sym14091941","type":"journal-article","created":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T04:28:55Z","timestamp":1663648135000},"page":"1941","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Proportional Caputo Fractional Differential Inclusions in Banach Spaces"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2316-4361","authenticated-orcid":false,"given":"Abdelkader","family":"Rahmani","sequence":"first","affiliation":[{"name":"Laboratory of Mathematics, Modeling and Applications (LaMMA), University of Adrar, National Road No. 06, Adrar 01000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8996-2270","authenticated-orcid":false,"given":"Wei-Shih","family":"Du","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3373-6578","authenticated-orcid":false,"given":"Mohammed Taha","family":"Khalladi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Sciences, University of Adrar, National Road No. 06, Adrar 01000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0392-4976","authenticated-orcid":false,"given":"Marko","family":"Kosti\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovi\u0107a 6, 21125 Novi Sad, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8811-4288","authenticated-orcid":false,"given":"Daniel","family":"Velinov","sequence":"additional","affiliation":[{"name":"Department for Mathematics and Informatics, Faculty of Civil Engineering, Ss. Cyril and Methodius University in Skopje, Partizanski Odredi 24, 1000 Skopje, North Macedonia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,18]]},"reference":[{"unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations. North\u2013Holland Mathematics Studies, Elsevier Science B. V.","key":"ref_1"},{"unstructured":"Lakshmikantham, V., Leela, S., and Devi, J.V. (2009). Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers.","key":"ref_2"},{"unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press.","key":"ref_3"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"981728","DOI":"10.1155\/2009\/981728","article-title":"A survey on semilinear differential equations and inclusions involving Riemann\u2013Liouville fractional derivative","volume":"2009","author":"Agarwal","year":"2009","journal-title":"Adv. Differ. Equ."},{"doi-asserted-by":"crossref","unstructured":"Diethelm, K. (2010). The Analysis of Fractional Differential Equations, Springer. Lecture Notes in Mathematics.","key":"ref_5","DOI":"10.1007\/978-3-642-14574-2"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3457","DOI":"10.1140\/epjst\/e2018-00021-7","article-title":"Fractional derivatives generated by a class of local proportional derivatives","volume":"226","author":"Jarad","year":"2017","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_7","first-page":"709","article-title":"Fractional derivatives and Laplace transform","volume":"13","author":"Jarad","year":"2020","journal-title":"Discret. Contin. Dyn. Syst. Ser. S"},{"doi-asserted-by":"crossref","unstructured":"Abbas, M.I., and Ragusa, M.A. (2021). On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry, 13.","key":"ref_8","DOI":"10.3390\/sym13020264"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18514\/MMN.2021.3470","article-title":"Controllability and Hyers-Ulam stability results of initial value problems for fractional differential equations via proportional-Caputo fractional derivative","volume":"22","author":"Abbas","year":"2021","journal-title":"Miskolc Math. Notes"},{"key":"ref_10","first-page":"48","article-title":"Existence results and the Ulam stability for fractional differential equations with hybrid proportional-Caputo derivatives","volume":"2020","author":"Abbas","year":"2020","journal-title":"J. Nonlinear Funct. Anal."},{"doi-asserted-by":"crossref","unstructured":"Hristova, S., and Abbas, M.I. (2021). Explicit solutions of initial value problems for fractional proportional differential equations with and without impulses. Symmetry, 13.","key":"ref_11","DOI":"10.3390\/sym13060996"},{"doi-asserted-by":"crossref","unstructured":"Agarwal, R., Hristova, S., and O\u2019Regan, D. (2022). Proportional Caputo fractional differential equations with noninstantaneous impulses: Concepts, integral representations, and Ulam-type stability. Mathematics, 10.","key":"ref_12","DOI":"10.3390\/math10132315"},{"key":"ref_13","first-page":"197","article-title":"Qualitative analysis of a proportional Caputo fractional pantograph differential equation with mixed nonlocal conditions","volume":"26","author":"Khaminsou","year":"2021","journal-title":"Nonlinear Funct. Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1186\/s13661-020-01473-7","article-title":"Nonlocal boundary value problems for integro-differential Langevin equation via the Caputo proportional fractional derivative","volume":"2020","author":"Khaminsou","year":"2020","journal-title":"Bound. Value Probl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1186\/s13662-019-2038-z","article-title":"Existence results for nonlinear fractional boundary value problem involving proportional derivative","volume":"2019","author":"Shammakh","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"415","DOI":"10.3934\/eect.2021006","article-title":"(\u03c9,T)-Periodic solutions of impulsive evolution equations","volume":"11","author":"Liu","year":"2022","journal-title":"Evol. Equ. Control Theory"},{"doi-asserted-by":"crossref","unstructured":"Almeida, R., Agarwal, R.P., Hristova, S., and O\u2019Regan, D. (2022). Stability of gene regulatory networks modeled by generalized proportional Caputo fractional differential equations. Entropy, 24.","key":"ref_17","DOI":"10.3390\/e24030372"},{"doi-asserted-by":"crossref","unstructured":"Agarwal, R., Hristova, S., and O\u2019Regan, D. (2022). Stability of generalized proportional Caputo fractional differential equations by Lyapunov functions. Fractal Fract., 6.","key":"ref_18","DOI":"10.3390\/fractalfract6010034"},{"key":"ref_19","first-page":"345","article-title":"On the new concept of solutions and existence results for impulsive fractional evolution equations","volume":"8","author":"Wang","year":"2011","journal-title":"Dyn. PDE"},{"doi-asserted-by":"crossref","unstructured":"Kosti\u0107, M. (2019). Almost Periodic and Almost Automorphic Type Solutions of Abstract Volterra Integro-Differential Equations, W. de Gruyter.","key":"ref_20","DOI":"10.1515\/9783110641851"},{"doi-asserted-by":"crossref","unstructured":"Diagana, T. (2013). Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer.","key":"ref_21","DOI":"10.1007\/978-3-319-00849-3"},{"doi-asserted-by":"crossref","unstructured":"N\u2019Gu\u00e9r\u00e9kata, G.M. (2001). Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic Publishers.","key":"ref_22","DOI":"10.1007\/978-1-4757-4482-8"},{"unstructured":"Levitan, M. (1959). Almost Periodic Functions, G.I.T.T.L.. (In Russian).","key":"ref_23"},{"unstructured":"Zaidman, S. (1985). Almost-Periodic Functions in Abstract Spaces, Pitman. Pitman Research Notes in Mathematics.","key":"ref_24"},{"doi-asserted-by":"crossref","unstructured":"Kosti\u0107, M. (2022). Selected Topics in Almost Periodicity, W. de Gruyter.","key":"ref_25","DOI":"10.1515\/9783110763522"},{"unstructured":"Shcherbakov, B.A. (1972). Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Stiinta. (In Russian).","key":"ref_26"},{"doi-asserted-by":"crossref","unstructured":"Akhmet, M., Tleubergenova, M., and Zhamanshin, A. (2021). Modulo periodic Poisson stable solutions of quasilinear differential equations. Entropy, 23.","key":"ref_27","DOI":"10.3390\/e23111535"},{"key":"ref_28","first-page":"898","article-title":"Poisson asymptotic stability of motions of dynamical systems and their comparability with regard to the recurrence property in the limit","volume":"13","author":"Cheban","year":"1977","journal-title":"Diff. Equ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1007\/s11425-018-9407-8","article-title":"Poisson stable motions of monotone nonautonomous dynamical systems","volume":"62","author":"Cheban","year":"2019","journal-title":"Sci. China Math."},{"unstructured":"Chaouchi, B., Kosti\u0107, M., and Velinov, D. (2021, November 25). Metrical Almost Periodicity: Levitan and Bebutov Concepts. Preprint. Available online: https:\/\/arxiv.org\/abs\/2111.14614.","key":"ref_30"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2018.1.16","article-title":"(\u03c9,c)-Periodic functions and mild solution to abstract fractional integro-differential equations","volume":"16","author":"Alvarez","year":"2018","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_32","first-page":"1","article-title":"(\u03c9,c)-Pseudo periodic functions, first order Cauchy problem and Lasota-Wazewska model with ergodic and unbounded oscillating production of red cells","volume":"106","author":"Alvarez","year":"2019","journal-title":"Bound. Value Probl."},{"key":"ref_33","first-page":"149","article-title":"Existence and uniqueness of (\u03c9,c)-periodic solutions of semilinear evolution equations","volume":"10","author":"Agaoglou","year":"2020","journal-title":"Int. J. Dyn. Syst. Differ. Equ."},{"doi-asserted-by":"crossref","unstructured":"Ren, L., and Wang, J.R. (2022). (\u03c9,c)-Periodic solutions to fractional differential equations with impulses. Axioms, 11.","key":"ref_34","DOI":"10.3390\/axioms11030083"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/10652469.2015.1087400","article-title":"On quasi-periodicity properties of fractional integrals and fractional derivatives of periodic functions","volume":"27","author":"Area","year":"2016","journal-title":"Integral Transforms Spec. Funct."},{"unstructured":"Kosti\u0107, M. (2020). Abstract Degenerate Volterra Integro-Differential Equations, Mathematical Institue of SANU.","key":"ref_36"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"392598","DOI":"10.1155\/2014\/392598","article-title":"On fractional derivatives and primitives of periodic functions","volume":"2014","author":"Area","year":"2014","journal-title":"Abstr. Appl. Anal."},{"unstructured":"Bazhlekova, E. (2001). Fractional Evolution Equations in Banach Spaces. [Ph.D. Thesis, Eindhoven University of Technology].","key":"ref_38"},{"unstructured":"Cross, R. (1998). Multivalued Linear Operators, Marcel Dekker Inc.","key":"ref_39"},{"doi-asserted-by":"crossref","unstructured":"Favini, A., and Yagi, A. (1998). Degenerate Differential Equations in Banach Spaces, Chapman and Hall\/CRC Pure and Applied Mathematics.","key":"ref_40","DOI":"10.1201\/9781482276022"},{"doi-asserted-by":"crossref","unstructured":"Kosti\u0107, M. (2015). Abstract Volterra Integro-Differential Equations, Taylor and Francis Group\/CRC Press\/Science Publishers.","key":"ref_41","DOI":"10.1201\/b18463"},{"doi-asserted-by":"crossref","unstructured":"Xue, D. (2017). Appendix A. Inverse Laplace transforms involving fractional and irrational operations. Fractional-Order Control Systems: Fundamentals and Numerical Implementations, W. de Gruyter.","key":"ref_42","DOI":"10.1515\/9783110497977"},{"doi-asserted-by":"crossref","unstructured":"Arendt, W., Batty, C.J.K., Hieber, M., and Neubrander, F. (2001). Vector-valued Laplace Transforms and Cauchy Problems, Birkh\u00e4user\/Springer Basel AG. Monographs in Mathematics.","key":"ref_43","DOI":"10.1007\/978-3-0348-5075-9"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1797","DOI":"10.1007\/s11425-012-4477-9","article-title":"Abstract Volterra equations in locally convex spaces","volume":"55","year":"2012","journal-title":"Sci. China Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"179750","DOI":"10.1186\/1687-1847-2010-179750","article-title":"On type of periodicity and ergodicity to a class of fractional order differential equations","volume":"2010","author":"Agarwal","year":"2010","journal-title":"Adv. Difference Equ."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1016\/j.jmaa.2011.04.078","article-title":"Existence of asymptotically almost periodic solutions for damped wave equations","volume":"382","author":"Lizama","year":"2011","journal-title":"J. Math. Anal. Appl."},{"doi-asserted-by":"crossref","unstructured":"Zhou, Y. (2017). Basic Theory of Fractional Differential Equations, World Scientific.","key":"ref_47","DOI":"10.1142\/10238"},{"unstructured":"Larrouy, J., and N\u2019Gu\u00e9r\u00e9kata, G.M. (2021, September 02). (\u03c9,c)-Periodic and asymptotically (\u03c9,c)-periodic mild solutions to fractional Cauchy problems. Appl. Anal. 2021. Available online: https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/00036811.2021.1967332.","key":"ref_48"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.amc.2015.09.082","article-title":"On quasi-periodic properties of fractional sums and fractional differences of periodic functions","volume":"273","author":"Area","year":"2016","journal-title":"Appl. Math. Comp."},{"key":"ref_50","first-page":"339","article-title":"Quasi-periodic solutions of fractional nabla difference systems","volume":"7","author":"Jonnalagadda","year":"2017","journal-title":"Fract. Diff. Calc."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1016\/j.jmaa.2008.02.023","article-title":"On S-asymptotically \u03c9-periodic functions on Banach spaces and applications","volume":"343","author":"Pierri","year":"2008","journal-title":"J. Math. Appl. Anal."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1186\/s13662-020-02767-x","article-title":"More properties of the proportional fractional integrals and derivatives of a function with respect to another function","volume":"2020","author":"Jarad","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.3934\/math.2022075","article-title":"On a Langevin equation involving Caputo fractional proportional derivatives with respect to another function","volume":"7","author":"Laadjal","year":"2022","journal-title":"AIMS Math."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1515\/math-2020-0014","article-title":"On more general forms of proportional fractional operators","volume":"18","author":"Jarad","year":"2020","journal-title":"Open Math."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1186\/s13662-020-02792-w","article-title":"On Hilfer generalized proportional fractional derivative","volume":"2020","author":"Ahmed","year":"2020","journal-title":"Adv. Diff. Equ."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"2973","DOI":"10.2298\/FIL2109973N","article-title":"Minkowski-Type inequalities using generalized proportional Hadamard fractional integral operators","volume":"35","author":"Nale","year":"2021","journal-title":"Filomat"},{"doi-asserted-by":"crossref","unstructured":"Rahman, G., Nisar, K.S., and Abdejawad, T. (2020). Certain Hadamard proportional fractional integral inequalities. Mathematics, 8.","key":"ref_57","DOI":"10.3390\/math8040504"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1186\/s13662-019-2381-0","article-title":"Certain inequalities via generalized proportional Hadamard fractional integral operators","volume":"2019","author":"Rahman","year":"2019","journal-title":"Adv. Diff. 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