Fractional sewage treatment models with impulses at variable times

References

• Lakshmikantham , V , Bainov , DD and Simeonov , PS . 1989 . Theory of Impulsive Differential Equations , Singapore, London : World Scientific .
   Google Scholar
• Yang , T . 2001 . Impulsive Control Theory , Berlin, Heidelberg : Springer-Verlin .
   Google Scholar
• Benchohra , M , Henderson , J and Ntouyas , S . 2006 . Impulsive Differential Equations and Inclusions , New York : Hindawi Publishing Corporation .
   Google Scholar
• Xiang , X and Wei , W . 2006 . Mild solution for a class of nonlinear impulsive evolution inclusion on Banach space . Southeast Asian Bull. Math. , 30 : 367 – 376 .
   Google Scholar
• Wei , W , Xiang , X and Peng , Y . 2006 . Nonlinear impulsive integro-differential equation of mixed type and optimal controls . Optimization , 55 : 141 – 156 .
   Web of Science ®Google Scholar
• Wang , J , Xiang , X and Peng , Y . 2009 . Periodic solutions of semilinear impulsive periodic system on Banach space . Nonlinear Anal.: TMA , 71 : e1344 – e1353 .
   Web of Science ®Google Scholar
• Wang , J and Wei , W . 2010 . A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces . Results. Math. , 58 : 379 – 397 .
   Web of Science ®Google Scholar
• Wang , J , Xiang , X and Wei , W . 2011 . Stability analysis and periodic optimal control problems for impulsive periodic system . Math. Commun. , 16 : 235 – 250 .
   Web of Science ®Google Scholar
• K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010
   Google Scholar
• Kilbas , AA , Srivastava , HM and Trujillo , JJ . 2006 . “ Theory and Applications of Fractional Differential Equations ” . In North-Holland Mathematics Studies , Vol. 204 , Amsterdam : Elsevier Science B.V. .
   Google Scholar
• Lakshmikantham , V , Leela , S and Vasundhara Devi , J . 2009 . Theory of Fractional Dynamic Systems , Cambridge : Cambridge Scientific Publishers .
   Google Scholar
• Miller , KS and Ross , B . 1993 . An Introduction to the Fractional Calculus and Differential Equations , New York : Wiley .
   Google Scholar
• Podlubny , I . 1999 . Fractional Differential Equations , San Diego : Academic Press .
   Google Scholar
• Tarasov , VE . 2010 . Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media , New York : Springer, HEP .
   Google Scholar
• Agarwal , RP , Benchohra , M and Hamani , S . 2010 . A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions . Acta. Appl. Math. , 109 : 973 – 1033 .
   Web of Science ®Google Scholar
• Ahmad , B and Nieto , JJ . 2010 . Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory . Topol. Methods Nonlinear Anal. , 35 : 295 – 304 .
   Web of Science ®Google Scholar
• Bai , ZB . 2010 . On positive solutions of a nonlocal fractional boundary value problem . Nonlinear Anal.: TMA , 72 : 916 – 924 .
   Web of Science ®Google Scholar
• Benchohra , M , Henderson , J , Ntouyas , SK and Ouahab , A . 2008 . Existence results for fractional order functional differential equations with infinite delay . J. Math. Anal. Appl. , 338 : 1340 – 1350 .
   Web of Science ®Google Scholar
• Mophou , GM and N'Guérékata , GM . 2010 . Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay . Appl. Math. Comput. , 216 : 61 – 69 .
   Web of Science ®Google Scholar
• Wang , J and Zhou , Y . 2011 . A class of fractional evolution equations and optimal controls . Nonlinear Anal.: RWA , 12 : 262 – 272 .
   Web of Science ®Google Scholar
• Wang , J , Zhou , Y and Wei , W . 2011 . A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces . Commun. Nonlinear Sci. Numer. Simul. , 16 : 4049 – 4059 .
   Web of Science ®Google Scholar
• Wang , J and Zhou , Y . 2011 . Existence of mild solutions for fractional delay evolution systems . Appl. Math. Comput. , 218 : 357 – 367 .
   Web of Science ®Google Scholar
• Dong , X , Wang , J and Zhou , Y . 2011 . On nonlocal problems for fractional differential equations in Banach spaces . Opuscula Math. , 31 : 341 – 357 .
   Google Scholar
• Zhang , S . 2003 . Existence of positive solution for some class of nonlinear fractional differential equations . J. Math. Anal. Appl. , 278 : 136 – 148 .
   Web of Science ®Google Scholar
• Zhou , Y and Jiao , F . 2010 . Nonlocal Cauchy problem for fractional evolution equations . Nonlinear Anal.: RWA , 11 : 4465 – 4475 .
   Web of Science ®Google Scholar
• Zhou , Y , Jiao , F and Li , J . 2009 . Existence and uniqueness for p-type fractional neutral differential equations . Nonlinear Anal.: TMA , 71 : 2724 – 2733 .
   Web of Science ®Google Scholar
• Zhou , Y , Jiao , F and Li , J . 2009 . Existence and uniqueness for fractional neutral differential equations with infinite delay . Nonlinear Anal.: TMA , 71 : 3249 – 3256 .
   Web of Science ®Google Scholar
• Agarwal , RP , Benchohra , M and Hamani , S . 2010 . A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions . Acta. Appl. Math. , 109 : 973 – 1033 .
   Web of Science ®Google Scholar
• Ahmad , B and Sivasundaram , S . 2009 . Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations . Nonlinear Anal.: HS , 3 : 251 – 258 .
   Web of Science ®Google Scholar
• Mophou , GM . 2010 . Existence and uniqueness of mild solution to impulsive fractional differential equations . Nonlinear Anal.: TMA , 72 : 1604 – 1615 .
   Web of Science ®Google Scholar
• Shu , X , Lai , Y and Chen , Y . 2011 . The existence of mild solutions for impulsive fractional partial differential equations . Nonlinear Anal.: TMA , 74 : 2003 – 2011 .
   Web of Science ®Google Scholar
• Benchohra , M , Henderson , J and Ntouyas , S . 2010 . Impulsive fractional differential equations with variable times . Comput. Math. Appl. , 59 : 1245 – 1252 .
   Web of Science ®Google Scholar
• Peng , Y , Xiang , X and Wang , C . 2009 . A class of differential equations with impulses at variable times on Banach spaces . Nonlinear Anal.: TMA , 71 : e1312 – e1319 .
   Web of Science ®Google Scholar
• Wang , J , Zhou , Y and Wei , W . 2011 . Impulsive problems for fractional evolution equations and optimal controls in infinite dimensional spaces . Topol. Methods Nonlinear Anal. , 38 : 17 – 43 .
   Web of Science ®Google Scholar
• Zhou , Y and Jiao , F . 2010 . Existence of mild solutions for fractional neutral evolution equations . Comput. Math. Appl. , 59 : 1063 – 1077 .
   Web of Science ®Google Scholar
• Michalski , MW . 1993 . “ Derivatives of noninteger order and their applications, Dissertationes Mathematicae, Polska Akademia Nauk ” . In Instytut Matematyczny Warszawa
   Google Scholar
• Henry , D . 1981 . Geometric Theory of Semilinear Parabolic Equations , Berlin, Heidelberg , New York : LNM 840, Springer-Verlag .
   Google Scholar
• Ye , H , Gao , J and Ding , Y . 2007 . A generalized Gronwall inequality and its application to a fractional differential equation . J. Math. Anal. Appl. , 328 : 1075 – 1081 .
   Web of Science ®Google Scholar
• Ahmed , NU . 1991 . Semigroup Theory with Application to Systems and Control , New York : Longman Science and Technical .
   Google Scholar