Darboux problem for fractional-order discontinuous hyperbolic partial differential equations in Banach algebras

References

• Kilbas , AA and Marzan , SA . 2005 . Nonlinear differential. equations with the Caputo fractional derivative in the space of continuously differentiable functions . Differ. Eqns. , 41 : 84 – 89 .
   Web of Science ®Google Scholar
• Vityuk , AN . 1997 . Existence of solutions of partial differential inclusions of fractional order . Izv. Vyssh. Uchebn., Ser. Mat. , 8 : 13 – 19 .
   Google Scholar
• Kilbas , AA , Bonilla , B and Trujillo , J . 2000 . Nonlinear differential equations of fractional order in a space of integrable functions . Dokl. Ross. Akad. Nauk , 374 ( 4 ) : 445 – 449 .
   Google Scholar
• Semenchuk , NP . 1982 . On one class of differential equations of noninteger order . Differents. Uravn. , 10 : 1831 – 1833 .
   Google Scholar
• De Blasi , F and Myjak , J . 1986 . On the structure of the set of solutions of the Darboux problem for hyperbolic equations . Proc. Edinb. Math. Soc. , 29 : 7 – 14 .
   Web of Science ®Google Scholar
• Lakshmikantham , V and Pandit , SG . 1985 . The Method of upper, lower solutions and hyperbolic partial differential equations . J. Math. Anal. Appl. , 105 : 466 – 477 .
   Web of Science ®Google Scholar
• Pandit , SG . 1997 . Monotone methods for systems of nonlinear hyperbolic problems in two independent variables . Nonlinear Anal. , 30 : 2735 – 2742 .
   Web of Science ®Google Scholar
• Diethelm , K and Freed , AD . 1999 . “ 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 , Edited by: Keil , F , Mackens , W , Voss , H and Werther , J . 217 – 224 . Heidelberg : Springer–Verlag .
   Google Scholar
• Gaul , L , Klein , P and Kempfle , S . 1991 . Damping description involving fractional operators . Mech. Syst. Signal Process. , 5 : 81 – 88 .
   Web of Science ®Google Scholar
• Glockle , WG and Nonnenmacher , TF . 1995 . A fractional calculus approach of selfsimilar protein dynamics . Biophys. J. , 68 : 46 – 53 .
   PubMed Web of Science ®Google Scholar
• Hilfer , R . 2000 . Applications of Fractional Calculus in Physics , Singapore : World Scientific .
   Google Scholar
• Mainardi , F . 1997 . “ Fractional calculus: Some basic problems in continuum and statistical mechanics ” . In Fractals and Fractional Calculus in Continuum Mechanics , Edited by: Carpinteri , A and Mainardi , F . 291 – 348 . Wien : Springer-Verlag .
   Google Scholar
• Metzler , F , Schick , W , Kilian , HG and Nonnenmacher , TF . 1995 . Relaxation in filled polymers: A fractional calculus approach . J. Chem. Phys. , 103 : 7180 – 7186 .
   Web of Science ®Google Scholar
• Kilbas , AA , Srivastava , HM and Trujillo , JJ . 2006 . Theory and Applications of Fractional Differential Equations , Amsterdam : Elsevier Science B.V. .
   Google Scholar
• Lakshmikantham , V , Leela , S and Vasundhara , J . 2009 . Theory of Fractional Dynamic Systems , Cambridge : Cambridge Academic Publishers .
   Google Scholar
• Miller , KS and Ross , B . 1993 . An Introduction to the Fractional Calculus and Differential Equations , New York : John Wiley .
   Google Scholar
• Podlubny , I . 1999 . Fractional Differential Equation , San Diego : Academic Press .
   Google Scholar
• Samko , SG , Kilbas , AA and Marichev , OI . 1993 . Fractional Integrals and Derivatives. Theory and Applications , Yverdon : Gordon and Breach .
   Google Scholar
• Abbas , S and Benchohra , M . 2009 . Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative . Commun. Math. Anal. , 7 : 62 – 72 .
   Google Scholar
• Abbas , S and Benchohra , M . 2009 . Darboux problem for perturbed partial differential equations of fractional order with finite delay . Nonlinear Anal. Hybrid Syst. , 3 : 597 – 604 .
   Web of Science ®Google Scholar
• Agarwal , RP , Benchohra , M and Hamani , S . 2010 . A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions . Acta Appl. Math. , 109 ( 3 ) : 973 – 1033 .
   Web of Science ®Google Scholar
• Belarbi , A , Benchohra , M and Ouahab , A . 2006 . Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces . Appl. Anal. , 85 : 1459 – 1470 .
   Google Scholar
• Benchohra , M , Graef , JR and Hamani , S . 2008 . Existence results for boundary value problems with non-linear fractional differential equations . Appl. Anal. , 87 ( 7 ) : 851 – 863 .
   Web of Science ®Google Scholar
• Benchohra , M , Hamani , S and Ntouyas , SK . 2008 . Boundary value problems for differential equations with fractional order . Surv. Math. Appl. , 3 : 1 – 12 .
   Google Scholar
• Benchohra , M , Henderson , J , Ntouyas , SK and Ouahab , A . 2008 . Existence results for functional differential equations of fractional order . J. Math. Anal. Appl. , 338 : 1340 – 1350 .
   Web of Science ®Google Scholar
• Diethelm , K and Ford , NJ . 2002 . Analysis of fractional differential equations . J. Math. Anal. Appl. , 265 : 229 – 248 .
   Web of Science ®Google Scholar
• Vityuk , AN and Golushkov , AV . 2004 . Existence of solutions of systems of partial differential equations of fractional order . Nonlinear Oscil. , 7 ( 3 ) : 318 – 325 .
   Google Scholar
• Zhang , S . 2006 . Positive solutions for boundary-value problems of nonlinear fractional diffrential equations . Electron. J. Differ. Eqns. , 36 12pp
   Google Scholar
• Carl , S and Heikkila , S . 2000 . Nonlinear Differential Equations in Ordered Spaces , Boca Raton , FL : Chapman & Hall .
   Google Scholar
• Heikkila , S and Lakshmikantham , V . 1994 . Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations , New York : Pure and Applied Mathematics, Marcel Dekker .
   Google Scholar
• Dawidowski , M and Kubiaczyk , I . 1990 . An existence theorem for the generalized hyperbolic equation in Banach space . Ann. Soc. Math. Pol. Ser. I, Comment. Math. , 30 ( 1 ) : 41 – 49 .
   Google Scholar
• Kamont , Z . 1999 . Hyperbolic Functional Differential Inequalities and Applications , Dordrecht : Kluwer Academic Publishers .
   Google Scholar
• Kamont , Z and Kropielnicka , K . 2005 . Differential difference inequalities related to hyperbolic functional differential systems and applications . Math. Inequal. Appl. , 8 ( 4 ) : 655 – 674 .
   Web of Science ®Google Scholar
• Arara , A , Karande , BD , Benchohra , M and Dhage , BC . 2006 . Existence theorems for a class of hyperbolic differential equations in Banach algebras . J. Nat. Phys. Sci. , 20 ( 1–2 ) : 51 – 63 .
   Google Scholar
• Dhage , BC and Ntouyas , SK . 2006 . Existence of solutions for discontinuous hyperbolic partial differential equations in Banach algebras . Electron. J. Differ. Eqns. , 28 10pp
   Google Scholar
• Dhage , BC . 2006 . Some algebraic fixed point theorems for multi-valued mappings with applications . Diss. Math. Differ. Incl. Control Optim. , 26 : 5 – 55 .
   Google Scholar
• Dhage , BC . 2004 . A nonlinear alternative in Banach algebras with applications to functional differential equations . Nonlinear Funct. Anal. Appl. , 8 : 563 – 575 .
   Google Scholar