Weak Solutions for a Coupled System of Partial Pettis Hadamard Fractional Integral Equations
Year 2017,
Volume: 1 Issue: 2, 136 - 146, 20.12.2017
Said Abbas
Mouffak Benchohra
,
Johnny Henderson
Jamal E. Lazreg
Abstract
In this paper we investigate the existence of weak solutions under the Pettis integrability assumption for a coupled system of partial integral equations via Hadamard’s fractional integral, by applying the technique of measure of weak noncompactness and Mönch’s fixed point theorem.
References
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- S. Abbas, M. Benchohra and G.M. N’Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science
Publishers, New York, 2015.
- S. Abbas, M. Benchohra and A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential
equations, Frac. Calc. Appl. Anal. 15 (2012), 168–182.
- R.R. Akhmerov, M.I. Kamenskii, A.S. Patapov, A.E. Rodkina and B.N. Sadovskii, Measures of Noncompactness and
Condensing Operators. Birkhauser Verlag, Basel, 1992.
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Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid 79 (1985), 53–66.
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- M. Benchohra, J.R. Graef and F-Z. Mostefai, Weak solutions for nonlinear fractional differential equations on reflexive
Banach spaces, Electron. J. Qual. Theory Differ. Equ. 54 (2010), 1–10.
- M. Benchohra, J. Graef and F-Z. Mostefai,Weak solutions for boundary-value problems with nonlinear fractional differential
inclusions, Nonlinear Dyn. Syst. Theory 11 (3) (2011), 227–237.
- M. Benchohra, J. Henderson and F-Z. Mostefai, Weak solutions for hyperbolic partial fractional differential inclusions in
Banach spaces, Comput. Math. Appl. 64 (2012), 3101–3107.
- M. Benchohra, J. Henderson and D. Seba, Measure of noncompactness and fractional differential equations in Banach
spaces, Commun. Appl. Anal. 12 (4) (2008), 419–428.
- M. Benchohra and F-Z. Mostefai, Weak solutions for nonlinear differential equations with integral boundary conditions in
Banach spaces, Opuscula Math. 32 (1) (2012), 31–40.
- M. Benchohra, J.J. Nieto and D. Seba, Measure of noncompactness and hyperbolic partial fractional differential equations
in Banach spaces, PanAmer. Math. J. 20 (3) (2010), 27–37.
- D. Bugajewski and S. Szufla, Kneser’s theorem for weak solutions of the Darboux problem in a Banach space, Nonlinear
Anal. 20 (2) (1993), 169–173.
- P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals.
J. Math. Anal. Appl. 269 (2002), 1–27.
- P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional
integrals. J. Math. Anal. Appl. 270 (2002), 1–15.
- M.A. Darwish, On integral equations of UrysohnâASVolterra type, Appl. Math. Comput. 136 (1) (2003), 93–98.
- M.A. Darwish, J. Henderson and D. O’Regan, Existence and asymptotic stability of solutions of a perturbed fractional
functional-integral equation with linear modification of the argument, Bull. Korean Math. Soc. 48 (3) (2011), 539–553.
- M.A. Darwish and S.K. Ntouyas, On a quadratic fractional HammersteinâASVolterra integral equation with linear modification
of the argument, Nonlinear Anal. 74 (11) (2011), 3510–3517.
- M.A. Darwish, On a perturbed functional integral equation of Urysohn type, Appl. Math. Comput. 218 (2012), 8800-
8805.
- M.A. Darwish and J. Henderson, Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type, Rocky
Mountain J. Math. 42 (2) (2012), 545–566.
- M.A. Darwish and J. Banas, Existence and characterization of solutions of nonlinear Volterra-Stieltjes integral equations
in two vriables, Abstr. Appl. Anal. 2014, Art. ID 618434, 11 pp.
- F.S. De Blasi, On the property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977),
259–262.
- D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers,
Dordrecht, 1996.
- J. Hadamard, Essai sur l’étude des fonctions données par leur développment de Taylor, J. Pure Appl. Math. 4 (8) (1892),
101–186.
- R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- A.A. Kilbas, Hari M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier
Science B.V., Amsterdam, 2006.
- K. Latrach and M.A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L1 spaces. Nonlinear
Anal. 66 (2007), 2325–2333.
- K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York,
1993.
- A. R. Mitchell and Ch. Smith, Nonlinear Equations in Abstract Spaces. In: Lakshmikantham, V. (ed.) An existence
theorem for weak solutions of differential equations in Banach spaces, pp. 387âAS403. Academic Press, New York (1978)
- H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4 (1980), 985–999.
- H. Mönch and G.F. Von Harten, On the Cauchy problem for ordinary differential equations in Banach spaces, Archiv.
Math. Basel 39 (1982), 153–160.
- D. O’Regan, Fixed point theory for weakly sequentially continuous mapping, Math. Comput. Model. 27 (5) (1998), 1–14.
- D. O’Regan, Weak solutions of ordinary differential equations in Banach spaces, Appl. Math. Lett. 12 (1999), 101–105.
- A. Petrusel, G. Petrusel, A study of a general system of operator equations in b-metric spaces via the vector approach in
fixed point theory. J. Fixed Point Theory Appl. 19 (2017), 1793-1814.
- B.J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277–304.
- S. Pooseh, R. Almeida, and D. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional
integral and derivative. Numer. Funct. Anal. Optim. 33 (3) (2012), 301–319.
- S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and
Breach, Yverdon, 1993.
- S. Szufla, On the application of measure of noncompactness to existence theorems, Rend. Sem. Mat. Univ. Padova 75
(1986), 1–14.
- M.A. Taoudi, Integrable solutions of a nonlinear functional integral equation on an unbounded interval, Nonlinear Anal.
71 (2009), 4131–4136.
- V.E. Tarasov, Fractional dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media,
Springer, Heidelberg; Higher Education Press, Beijing, 2010.
- A.N. Vityuk, On solutions of hyperbolic differential inclusions with a nonconvex right-hand side. (Russian) Ukran. Mat.
Zh. 47 (4) (1995), 531–534; translation in Ukrainian Math. J. 47 (4) (1996), 617–621.
- A.N. Vityuk and A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order,
Nonlinear Oscil. 7 (2004), 318–325.
Year 2017,
Volume: 1 Issue: 2, 136 - 146, 20.12.2017
Said Abbas
Mouffak Benchohra
,
Johnny Henderson
Jamal E. Lazreg
References
- S. Abbas, M. Benchohra and G.M. N’Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012.
- S. Abbas, M. Benchohra and G.M. N’Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science
Publishers, New York, 2015.
- S. Abbas, M. Benchohra and A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential
equations, Frac. Calc. Appl. Anal. 15 (2012), 168–182.
- R.R. Akhmerov, M.I. Kamenskii, A.S. Patapov, A.E. Rodkina and B.N. Sadovskii, Measures of Noncompactness and
Condensing Operators. Birkhauser Verlag, Basel, 1992.
- J.C. Alvàrez, Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces,
Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid 79 (1985), 53–66.
- J. Bana`s and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.
- M. Benchohra, J.R. Graef and F-Z. Mostefai, Weak solutions for nonlinear fractional differential equations on reflexive
Banach spaces, Electron. J. Qual. Theory Differ. Equ. 54 (2010), 1–10.
- M. Benchohra, J. Graef and F-Z. Mostefai,Weak solutions for boundary-value problems with nonlinear fractional differential
inclusions, Nonlinear Dyn. Syst. Theory 11 (3) (2011), 227–237.
- M. Benchohra, J. Henderson and F-Z. Mostefai, Weak solutions for hyperbolic partial fractional differential inclusions in
Banach spaces, Comput. Math. Appl. 64 (2012), 3101–3107.
- M. Benchohra, J. Henderson and D. Seba, Measure of noncompactness and fractional differential equations in Banach
spaces, Commun. Appl. Anal. 12 (4) (2008), 419–428.
- M. Benchohra and F-Z. Mostefai, Weak solutions for nonlinear differential equations with integral boundary conditions in
Banach spaces, Opuscula Math. 32 (1) (2012), 31–40.
- M. Benchohra, J.J. Nieto and D. Seba, Measure of noncompactness and hyperbolic partial fractional differential equations
in Banach spaces, PanAmer. Math. J. 20 (3) (2010), 27–37.
- D. Bugajewski and S. Szufla, Kneser’s theorem for weak solutions of the Darboux problem in a Banach space, Nonlinear
Anal. 20 (2) (1993), 169–173.
- P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals.
J. Math. Anal. Appl. 269 (2002), 1–27.
- P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional
integrals. J. Math. Anal. Appl. 270 (2002), 1–15.
- M.A. Darwish, On integral equations of UrysohnâASVolterra type, Appl. Math. Comput. 136 (1) (2003), 93–98.
- M.A. Darwish, J. Henderson and D. O’Regan, Existence and asymptotic stability of solutions of a perturbed fractional
functional-integral equation with linear modification of the argument, Bull. Korean Math. Soc. 48 (3) (2011), 539–553.
- M.A. Darwish and S.K. Ntouyas, On a quadratic fractional HammersteinâASVolterra integral equation with linear modification
of the argument, Nonlinear Anal. 74 (11) (2011), 3510–3517.
- M.A. Darwish, On a perturbed functional integral equation of Urysohn type, Appl. Math. Comput. 218 (2012), 8800-
8805.
- M.A. Darwish and J. Henderson, Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type, Rocky
Mountain J. Math. 42 (2) (2012), 545–566.
- M.A. Darwish and J. Banas, Existence and characterization of solutions of nonlinear Volterra-Stieltjes integral equations
in two vriables, Abstr. Appl. Anal. 2014, Art. ID 618434, 11 pp.
- F.S. De Blasi, On the property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977),
259–262.
- D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers,
Dordrecht, 1996.
- J. Hadamard, Essai sur l’étude des fonctions données par leur développment de Taylor, J. Pure Appl. Math. 4 (8) (1892),
101–186.
- R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- A.A. Kilbas, Hari M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier
Science B.V., Amsterdam, 2006.
- K. Latrach and M.A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L1 spaces. Nonlinear
Anal. 66 (2007), 2325–2333.
- K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York,
1993.
- A. R. Mitchell and Ch. Smith, Nonlinear Equations in Abstract Spaces. In: Lakshmikantham, V. (ed.) An existence
theorem for weak solutions of differential equations in Banach spaces, pp. 387âAS403. Academic Press, New York (1978)
- H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4 (1980), 985–999.
- H. Mönch and G.F. Von Harten, On the Cauchy problem for ordinary differential equations in Banach spaces, Archiv.
Math. Basel 39 (1982), 153–160.
- D. O’Regan, Fixed point theory for weakly sequentially continuous mapping, Math. Comput. Model. 27 (5) (1998), 1–14.
- D. O’Regan, Weak solutions of ordinary differential equations in Banach spaces, Appl. Math. Lett. 12 (1999), 101–105.
- A. Petrusel, G. Petrusel, A study of a general system of operator equations in b-metric spaces via the vector approach in
fixed point theory. J. Fixed Point Theory Appl. 19 (2017), 1793-1814.
- B.J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277–304.
- S. Pooseh, R. Almeida, and D. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional
integral and derivative. Numer. Funct. Anal. Optim. 33 (3) (2012), 301–319.
- S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and
Breach, Yverdon, 1993.
- S. Szufla, On the application of measure of noncompactness to existence theorems, Rend. Sem. Mat. Univ. Padova 75
(1986), 1–14.
- M.A. Taoudi, Integrable solutions of a nonlinear functional integral equation on an unbounded interval, Nonlinear Anal.
71 (2009), 4131–4136.
- V.E. Tarasov, Fractional dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media,
Springer, Heidelberg; Higher Education Press, Beijing, 2010.
- A.N. Vityuk, On solutions of hyperbolic differential inclusions with a nonconvex right-hand side. (Russian) Ukran. Mat.
Zh. 47 (4) (1995), 531–534; translation in Ukrainian Math. J. 47 (4) (1996), 617–621.
- A.N. Vityuk and A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order,
Nonlinear Oscil. 7 (2004), 318–325.