SOLVABILITY FOR A CLASS OF NONLINEAR CAPUTO-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN OPERATOR IN BANACH SPACES

Choukri Derbazi

DOI Number
https://doi.org/10.22190/FUMI2003693D
First page
693
Last page
711

Abstract


This paper is devoted to the existence of solutions for certain classes of nonlinear differential equations involving the Caputo-Hadamard fractional-order with $\mathrm{p}$-Laplacian operator in Banach spaces. The arguments are based on M\"{o}nch's fixed point theorem combined with the technique of measures of noncompactness. An example is also presented to illustrate the effectiveness of the main results. 

Keywords

Banach spaces; differential equations; Caputo-Hadamard fractional-order; Laplacian operator; M\"{o}nch's fixed point theorem.

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References


bibitem{Abas}

{sc S. Abbas, M. Benchohra and J. Henderson}: {it Weak Solutions for Implicit Fractional Differential Equations of Hadamard Type}. Adv. Dyn. Syst. Appl. {bf 11} 1 (2016) 1--13.

bibitem{Hamidi}

{sc S. Abbas, M. Benchohra, N. Hamidi and J Henderson}: {it Caputo--Hadamard fractional differential equations in Banach spaces}. Fract. Calc. Appl. Anal. {bf 21} 4 (2018) 1027--1045.

bibitem{Jarad2}

{sc Y. Adjabi, F. Jarad, D. Baleanu, T. Abdeljawad}:

{it On Cauchy problems with Caputo Hadamard fractional derivatives}.

{J. Comput. Anal. Appl.} textbf{21} 4 (2016) 661--681.

bibitem{DS}

{sc R. P. Agarwal, M. Benchohra and D. Seba}: {it On the Application of Measure of noncompactness to the existence of solutions for fractional differential equations.} Results. Math. textbf{55} (2009) 221--230.

bibitem{b}

{sc A. Ahmadkhanlu}: {it Existence and uniquensess for a class of fractional differential Equations with an integral fractional boundary condition}. Filomat {bf 31} 5 (2017) 1241--1249.

bibitem{Aubin}

{sc J. P. Aubin, I. Ekeland}: {it Applied Nonlinear Analysis.} John Wiley & Sons, New York (1984).

bibitem{Ardjouni}

{sc A. Ardjouni, A. Djoudi}: {it Positive solutions for nonlinear Caputo-Hadamard fractional differential equations with integral boundary conditions.} Open J. Math. Anal. {bf 3} 1 (2019) 62--69.

bibitem{Arioua}

{sc Y. Arioua, N. Benhamidouche}: {it Boundary value problem for Caputo--Hadamard fractional differential equations.} Surv. Math. Appl. {bf 12} (2017) 103--115.

bibitem{Bai}

{sc C. Bai}: {it Existence and uniqueness of solutions for fractional boundary value problems with p-Laplacian operator.} Adv. Difference Equ. {bf 2018}, Paper No. 4, 12 pp.

bibitem{bnas}

{sc J. Bana`{s} and K. Goebel}: {it Measures of Noncompactness in Banach Spaces}. Marcel Dekker, New York, 1980.

bibitem{Banas3}

{sc J. Bana'{s}, M. Jleli, M. Mursaleen. B. Samet, C. Vetro}: {it Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness.} Springer, singpagor 2017.

bibitem{ben}

{sc M. Benchohra, J. Henderson and D. Seba}: {it Measure of noncompactness and fractional differential equations in Banach spaces}. Commun. Appl. Anal. {bf 12} 4 (2008) 419--427.

bibitem{b6}

{sc M. Benchohra, S. Hamani, S.K. Ntouyas}: {it Boundary value problems for differential equations with fractional order and nonlocal conditions}. Nonlinear Anal. {bf 71} 7-8 (2009) 2391--2396.

bibitem{ben1}

{sc M. Benchohra and Fatima-Zohra Mostefai}: {it Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces.} Opuscula Math. {bf 32} 1 (2012) 31--40.

bibitem{Bouriah}

{sc M. Benchohra, S. Bouriah, J.J. Nieto}: {it Existence of periodic solutions for nonlinear implicit Hadamard's fractional differential equations}. Rev. R. Acad. Cienc. Exactas F'{i}s. Nat. Ser. A Mat. RACSAM {bf 112} 1 (2018) 25--35.

bibitem{Benhamida}

{sc W. Benhamida, S. Hamani, J. Henderson}: {it Boundary Value Problems For Caputo-Hadamard Fractional Differential Equations.} Adv. Theory Nonlinear Anal. Appl. {bf 2} (2018) 138-145.

bibitem{Benhamida1}

{sc W. Benhamida, J. R. Graef, S. Hamania}: {it Boundary value problems for Hadamard fractional differential equations with

nonlocal multi-point boundary conditions}. Fract. Differ. Calc. {bf 8} 1 (2018) 165--176.

bibitem{Benlabbes}

{sc A. Benlabbes, M. Benbachir, M. Lakrib}: textit{Boundary value problems for nonlinear fractional differential equations}. Facta Univ. Ser. Math. Inform. {bf 30} 2 (2015) 157--168.

bibitem{Dahmani0}

{sc Z. Dahmani, M. A. Abdellaoui}: textit{New results for a weighted nonlinear system of fractional integro-differential equations}. Facta Univ. Ser. Math. Inform. {bf 29} 3 (2014) 233--242.

bibitem{Dahmani1}

{sc Z. Dahmani, A. Taeb}: textit{New existence and uniqueness results for hight dimensional fractional differential systems}. Facta Univ. Ser. Math. Inform. {bf 30} 3 (2015) 281--293.

bibitem{Dahmani2}

{sc Z. Dahmani, A. Taieb, N. Bedjaoui}: textit{Solvability and Stability for nonlinear fractional integro-differential systems of hight fractional orders}. Facta Univ. Ser. Math. Inform. {bf 31} 3 (2016) 629--644.

bibitem{Dong}

{sc X. Dong, Z. Bai and S. Zhang}: {it Positive solutions to boundary value problems of $mathrm{p}$--Laplacian with fractional derivative}. Bound. Value Probl. {bf 2017}, Paper No. 5, 15 pp.

bibitem{222}

{sc F. T. Fen, I. Y. Karacac, O. B. Ozenc}: { it Positive Solutions of Boundary Value Problems for p-Laplacian Fractional Differential Equations.} Filomat {bf 31} 5 (2017) 1265--1277

bibitem{Gambo}

{sc Y. Y. Gambo, F. Jarad, D. Baleanu, T. Abdeljawad}:

{it On Caputo modification of the Hadamard fractional derivatives}. Adv. Difference Equ. {bf 2014}, 2014:10, 12 pp.

bibitem{Graef}

{sc J. R. Graef, N. Guerraiche and S. Hamani}, {it Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces}. Stud. Univ. Babec{s}-Bolyai Math. {bf 62} 4 (2017) 427--438.

bibitem{b10}

{sc A. Guezane-Lakoud, R. Khaldi}: {it Solvability of a fractional boundary value problem with integral condition}. Nonlinear Anal. {bf 75} 4 (2012) 2692--2700.

bibitem{Hadamard}

{sc J. Hadamard}: {it Essai sur l'etude des fonctions donnees par leur developpment de Taylor}. J. Mat. Pure Appl. Ser. {bf 8} (1892) 101--186.

bibitem{Hilfer}

{sc R. Hilfer}: {it Application of fractional calculus in physics}. New Jersey: World Scientific, (2001).

bibitem{Houas}

{sc M. Houas and M. Bezziou}: {it Existence and stability resuts for fractional differential equations with two Caputo

fractional derivatives}. Facta Univ. Ser. Math. Inform. {bf 34} 2 (2019) 341--357.

bibitem{Jarad}

{sc F. Jarad, D. Baleanu and A. Abdeljawad}: {it Caputo-type modification of the Hadamard fractional derivatives}. Adv. Difference Equ. {bf 2012}, 2012:142, 8 pp.

bibitem{Leibenson}

{sc LS Leibenson}: {it General problem of the movement of a compressible fluid in a porous medium}. Izv. Akad. Nauk Kirg.SSR, Ser. Biol. Nauk. {bf 9} (1983) 7--10.

bibitem{Khan}

{sc N. A. Khan,A. Ara , A. Mahmood}: {it Approximate solution of time-fractional chemical engineering equations: a comparative study.} Int. J. Chem. Reactor Eng. {bf 8} (2010) Article A19.

bibitem{b0}

{sc A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo}: {it Theory and Applications of Fractional Differential Equations}, vol. 204 of North-Holland Mathematics Sudies Elsevier Science B.V. Amsterdam the Netherlands, 2006.

bibitem{X.Liu}

{sc X. Liu, M. Jia, X. Xiang}: {it On the solvability of a fractional differential equation model involving the p-Laplacian operator}. Comput. Math. Appl. {bf 64} 10 (2012) 3267--3275.

bibitem{Lu}

{sc H. Lu, Z. Han, S. Sun} {it Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with $mathrm{p}$--Laplacian.} Bound. Value Probl. 2014, 2014:26, 17 pp.

bibitem{DO}

{sc H. M"onch}: {it Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces.} Nonlinear Anal. {bf 4} 5 (1980) 985--999.

bibitem{Oldham}

{sc K. B. Oldham}: {it Fractional differential equations in electrochemistry}. Adv. Eng. Softw. {bf 41} 1 (2010) 9--12.

bibitem{b2}

{sc I. Podlubny}: {it Fractional Differential Equations}, Academic Press, San Diego (1993).

bibitem{hamza}

{sc H. Rebai, D. Seba}: {it Weak Solutions for Nonlinear Fractional Differential Equation with Fractional Separated Boundary Conditions in Banach Spaces}. Filomat {bf 32} 3 (2018) 1117--1125.

bibitem{Schwabik}

{sc S. Schwabik Y. Guoju}: emph{Topics in Banach Spaces Integration}, Series in Real Analysis 10, World Scientific, Singapore, 2005.

bibitem{Tan}

{sc J. Tan and M. Li}: {it Solutions of fractional differential equations with $mathrm{p}$--Laplacian operator in Banach spaces}. Bound. Value Probl. {bf 2018}, Paper No. 15, 13 pp.

bibitem{Tarasov}

V. E. Tarasov, {it Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media}. Springer, Heidelberg &

Higher Education Press, Beijing, 2010.

bibitem{Tariboon}

{sc J. Tariboon, A. Cuntavepanit, S.K. Ntouyas, W. Nithiarayaphaks}: {it Separated boundary value problems of sequential

Caputo and Hadamard fractional differential equations}. J. Funct. Spaces {bf 2018}, Art. ID 6974046, 8 pp.

bibitem{Sabatier}

{sc J. Sabatier, O.P. Agrawal, J.A.T. Machado}: {it Advances in Fractional Calculus-Theoretical Developments and Applications

in Physics and Engineering.} Dordrecht: Springer, 2007.

bibitem{Zeidler}

{sc E. Zeidler}: emph{Nonlinear functional analysis and its applications. II/B}, translated from the German by the author and Leo F. Boron, Springer-Verlag, New York, 1990.




DOI: https://doi.org/10.22190/FUMI2003693D

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