分数微分方程的若干问题
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摘要
近年来,对于分数积分方程和发展方程的研究获得了许多新的结果.但是,相对于整数阶微分方程而言,分数阶方程在理论研究方面还很不完善,有许多领域尚未涉及,需要我们进一步研究.
本文讨论了一类积分方程的解的存在性、极值解的存在性问题和一类分数发展方程的非局部Cauchy问题.在第2章,我们首先利用Krasnoselskii不动点理论研究了一类特定的Volterra积分方程解的存在性问题;然后,我们利用混合不动点理论研究了这类特定积分方程极值解的存在性问题;最后,我们将结果应用到分数微分方程中并得到相应的结论.在第3章,我们采用引入概率密度函数和算子半群给出的分数发展方程适度解的定义,通过运用算子半群的不动点理论和泛函分析方法给出了适度解的存在性法则.
In recent years, many new papers appeared about fractional integral equa-tions and fractional evolution equations, but, compared with the di?erential equa-tions of integer order, fractional order is far from perfect in the theory, many areasare not involved, we need to do further study.
In this paper, we discuss the existence of solutions and the extremal solutionsfor integral equations and a class of fractional evolution equations with nonlocalCauchy conditions. In Chapter 2, firstly, by the use of Krasnoselskii’s fixed pointtheorem we prove the existence of positive solutions for certain Volterra integralequations; then by the use of the hybrid fixed point theorem we prove the ex-istence of extremal solutions for certain Volterra integral equations, and finally,the results are applied to a variety of fractional di?erential equations. In Chapter3, by considering probability density and semigroup, we give definitions of mildsolutions for fractional evolution equations with nonlocal conditions; by using thefunctional analysis concerning to the semigroup of operators and some fixed pointtheorems e?ectively, we give the criteria on existence of mild solutions.
本文讨论了一类积分方程的解的存在性、极值解的存在性问题和一类分数发展方程的非局部Cauchy问题.在第2章,我们首先利用Krasnoselskii不动点理论研究了一类特定的Volterra积分方程解的存在性问题;然后,我们利用混合不动点理论研究了这类特定积分方程极值解的存在性问题;最后,我们将结果应用到分数微分方程中并得到相应的结论.在第3章,我们采用引入概率密度函数和算子半群给出的分数发展方程适度解的定义,通过运用算子半群的不动点理论和泛函分析方法给出了适度解的存在性法则.
In recent years, many new papers appeared about fractional integral equa-tions and fractional evolution equations, but, compared with the di?erential equa-tions of integer order, fractional order is far from perfect in the theory, many areasare not involved, we need to do further study.
In this paper, we discuss the existence of solutions and the extremal solutionsfor integral equations and a class of fractional evolution equations with nonlocalCauchy conditions. In Chapter 2, firstly, by the use of Krasnoselskii’s fixed pointtheorem we prove the existence of positive solutions for certain Volterra integralequations; then by the use of the hybrid fixed point theorem we prove the ex-istence of extremal solutions for certain Volterra integral equations, and finally,the results are applied to a variety of fractional di?erential equations. In Chapter3, by considering probability density and semigroup, we give definitions of mildsolutions for fractional evolution equations with nonlocal conditions; by using thefunctional analysis concerning to the semigroup of operators and some fixed pointtheorems e?ectively, we give the criteria on existence of mild solutions.
引文
[1] A. A. Kilbas, Hari M. Srivastava and J. Juan Trujillo, Theory and Applicationsof Fractional Di?erential Equations, in: North-Holland Mathematics Studies, vol.204, Elsevier Science B.V., Amsterdam, 2006.
[2] I. Podlubny, Fractional Di?erential Equations, Academic Press, San Diego, 1999.
[3] Wei Lin, Global existence theory and chaos control of fractional di?erential equa-tions, J. Math. Anal. Appl. 332(2007) 709-726.
[4] B. C. Dhage, Existence of extremal solutions for discontinuous functional integralequations, Applied Math. Lett. 19(2006) 881-886.
[5] B. C. Dhage, Hybrid fixed point theory and existence of extremal solutionsfor perturbed neutral functional di?erential equations, Bull. Korean Math. Soc.44(2)(2007) 315-330.
[6] B. C. Dhage, Hybrid fixed point theory for strictly monotone increasing multi-valued mapping with applications, Comput. Math. Appl. 53(2007) 803-824.
[7] R. P. Agarwal, M. Belmekki and M. Benchohra, Existence results for semilin-ear functional di?erential inclusions involving Riemann-Liouville derivative, Dyn.Continuous Discrete Impuls. Syst. 17(2010) 347-361.
[8] R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results forboundary value problems of nonlinear fractional di?erential equations, Acta Appl.Math. 109(2010) 973-1033.
[9] Domenico Delbosco and Luigi Rodino, Existence and Uniqueness for a NonlinearFractional Di?erential Equation, J. Math. Anal. Appl. 204(1996) 609-625.
[10] M. M. El-Borai, Semigroups and some nonlinear fractional di?erential equations,Applied Math. Comput. 149(2004) 823-831.
[11] A. M. A. El-Sayed, Nonlinear functional di?erential equations of arbitrary orders,Nonlinear Anal. 33(1998) 181-186.
[12] R. W. Ibrahim, S. Momani, On the existence and uniqueness of solutions of a classof fractional di?erential equations, J. Math. Anal. Appl. 334(1)(2007) 1-10.
[13] H. Jafari, S. Seifi, Solving a system of nonlinear fractional partial di?erentialequations using homotopy analysis method, Communications in Nonlinear Scienceand Numerical Simulation, 14(2009) 1962-1969.
[14] O. K. Jaradat, A. Al-Omari, S. Momani, Existence of the mild solution for frac-tional semilinear initial value problems, Nonlinear Anal. 69(9)(2008) 3153-3159.
[15] V. Lakshmikantham, Theory of fractional functional di?erential equations, Non-linear Anal. 69(2008) 3337-3343.
[16] V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterativetechnique for fractional di?erential equations, Appl. Math. Lett. 21(2008) 828-834.
[17] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional di?erential equa-tions, Nonlinear Anal. 69(2008) 2677-2682.
[18] Shuqin Zhang, Existence of positive solutions for some class of nonlinear fractionaldi?erential equations, J. Math. Anal. Appl. 278(2003) 136-148.
[19] Shuqin Zhang, The Existence of a Positive Solution for a Nonlinear FractionalDi?erential Equation, J. Math. Anal. Appl. 252(2000) 804-812.
[20] Zhanbin Bai, Haishen Lv, Positive solutions for boundary value problem of non-linear fractional di?erential equation, J. Math. Anal. Appl. 311(2005) 495-505.
[21] Chuanzhi Bai, Positive solutions for nonlinear fractional di?erential equations withcoe?cient that changes sign, Nonlinear Anal. 64(2006) 677-685.
[22] K.S. Miller and B.Ross, An Introduction to the Fractional Calculus and FractionalDi?erential Equations, Wiley, New York, 1993.
[23] I. Podlubny, Fractional Di?erential Equations, Academic Press, New York, 1993.
[24] M. Rivero, L. R Germ′a, J. J. Trujillo, Linear fractional di?erential equations withvariable coe?cients, Appl. Math. Lett. 21(2008) 892-897.
[25] H. A. H. Salem, On the existence of continuous solutions for a singular system ofnon-linear fractional di?erential equations, Appl. Math. Comput. 198(2008) 445-452.
[26] Yong Zhou, Feng Jiao and Jing Li, Existence and uniqueness for fractional neutraldi?erential equations with infinite delay, Nonlinear Anal. 71(7-8)(2009) 3249-3256.
[27] Yong Zhou, Feng Jiao and Jing Li, Existence and uniqueness for p?type fractionalneutral di?erential equations, Nonlinear Anal. 71(7-8)(2009) 2724-2733.
[28] Yong Zhou and Feng Jiao, Existence of mild solutions for fractional neutral evo-lution equations, Comput. Math. Apple. 59(2010) 1063-1077.
[29] Yong Zhou and Feng Jiao, Nonlocal Cauchy problem for fractional evolution equa-tions. (in press).
[30] K. Maleknejad, K. Nouri and R. Mollapourasl, Existence of solutions for somenonlinear integral equations, Commun Nonlinear Sci Numer Simulat, 14(2009)2559–2564.
[31] Jozef Banas and Donal ORegan, On existence and local attractivity of solutionsof a quadratic Volterra integral equation of fractional order, J. Math. Anal. Appl.345(2008) 573–582.
[32] Abderrazek Karoui, On the existence of continuous solutions of nonlinear integralequations, Appl. Math. Lett. 18(2005) 299–305.
[33] B. C. Dhage, A general multi-valued hybrid fixed point theorem and perturbeddi?erential inclusions, Nonlinear Anal. 64(2006) 2747-2772.
[34] S. D. Eidelman and A.N. Kochubei, Cauchy problem for fractional di?usion equa-tions, J. Di?. Equ. 199(2004) 211-255.
[35] A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional di?erential equations,Appl. Math. Comput. 68(1995) 15-25.
[36] A. N. Kochubei, A Cauchy problem for evolution equations of fractional order,Di?erential Equations, 25(1989) 967-974.
[37] Qing Liu and Rong Yuan, Existence of mild solutions for semilinear evolutionequations with non-local initial conditions, Nonliear Analysis, 71(2009) 4177-4184.
[38] E. G. Bajlekova, Fractional Evolution Equations in Banach Space, University PressFacilities, Eindhoven University of Technology, 2001.
[39] L. Byszewski, Theorems about existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162(1991) 494-505.
[40] X. Fu and K. Ezzinbi, Existence of solutions for neutral di?erential evolutionequations with nonlocal conditions, Nonlinear Anal. 54(2003) 215-227.
[2] I. Podlubny, Fractional Di?erential Equations, Academic Press, San Diego, 1999.
[3] Wei Lin, Global existence theory and chaos control of fractional di?erential equa-tions, J. Math. Anal. Appl. 332(2007) 709-726.
[4] B. C. Dhage, Existence of extremal solutions for discontinuous functional integralequations, Applied Math. Lett. 19(2006) 881-886.
[5] B. C. Dhage, Hybrid fixed point theory and existence of extremal solutionsfor perturbed neutral functional di?erential equations, Bull. Korean Math. Soc.44(2)(2007) 315-330.
[6] B. C. Dhage, Hybrid fixed point theory for strictly monotone increasing multi-valued mapping with applications, Comput. Math. Appl. 53(2007) 803-824.
[7] R. P. Agarwal, M. Belmekki and M. Benchohra, Existence results for semilin-ear functional di?erential inclusions involving Riemann-Liouville derivative, Dyn.Continuous Discrete Impuls. Syst. 17(2010) 347-361.
[8] R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results forboundary value problems of nonlinear fractional di?erential equations, Acta Appl.Math. 109(2010) 973-1033.
[9] Domenico Delbosco and Luigi Rodino, Existence and Uniqueness for a NonlinearFractional Di?erential Equation, J. Math. Anal. Appl. 204(1996) 609-625.
[10] M. M. El-Borai, Semigroups and some nonlinear fractional di?erential equations,Applied Math. Comput. 149(2004) 823-831.
[11] A. M. A. El-Sayed, Nonlinear functional di?erential equations of arbitrary orders,Nonlinear Anal. 33(1998) 181-186.
[12] R. W. Ibrahim, S. Momani, On the existence and uniqueness of solutions of a classof fractional di?erential equations, J. Math. Anal. Appl. 334(1)(2007) 1-10.
[13] H. Jafari, S. Seifi, Solving a system of nonlinear fractional partial di?erentialequations using homotopy analysis method, Communications in Nonlinear Scienceand Numerical Simulation, 14(2009) 1962-1969.
[14] O. K. Jaradat, A. Al-Omari, S. Momani, Existence of the mild solution for frac-tional semilinear initial value problems, Nonlinear Anal. 69(9)(2008) 3153-3159.
[15] V. Lakshmikantham, Theory of fractional functional di?erential equations, Non-linear Anal. 69(2008) 3337-3343.
[16] V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterativetechnique for fractional di?erential equations, Appl. Math. Lett. 21(2008) 828-834.
[17] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional di?erential equa-tions, Nonlinear Anal. 69(2008) 2677-2682.
[18] Shuqin Zhang, Existence of positive solutions for some class of nonlinear fractionaldi?erential equations, J. Math. Anal. Appl. 278(2003) 136-148.
[19] Shuqin Zhang, The Existence of a Positive Solution for a Nonlinear FractionalDi?erential Equation, J. Math. Anal. Appl. 252(2000) 804-812.
[20] Zhanbin Bai, Haishen Lv, Positive solutions for boundary value problem of non-linear fractional di?erential equation, J. Math. Anal. Appl. 311(2005) 495-505.
[21] Chuanzhi Bai, Positive solutions for nonlinear fractional di?erential equations withcoe?cient that changes sign, Nonlinear Anal. 64(2006) 677-685.
[22] K.S. Miller and B.Ross, An Introduction to the Fractional Calculus and FractionalDi?erential Equations, Wiley, New York, 1993.
[23] I. Podlubny, Fractional Di?erential Equations, Academic Press, New York, 1993.
[24] M. Rivero, L. R Germ′a, J. J. Trujillo, Linear fractional di?erential equations withvariable coe?cients, Appl. Math. Lett. 21(2008) 892-897.
[25] H. A. H. Salem, On the existence of continuous solutions for a singular system ofnon-linear fractional di?erential equations, Appl. Math. Comput. 198(2008) 445-452.
[26] Yong Zhou, Feng Jiao and Jing Li, Existence and uniqueness for fractional neutraldi?erential equations with infinite delay, Nonlinear Anal. 71(7-8)(2009) 3249-3256.
[27] Yong Zhou, Feng Jiao and Jing Li, Existence and uniqueness for p?type fractionalneutral di?erential equations, Nonlinear Anal. 71(7-8)(2009) 2724-2733.
[28] Yong Zhou and Feng Jiao, Existence of mild solutions for fractional neutral evo-lution equations, Comput. Math. Apple. 59(2010) 1063-1077.
[29] Yong Zhou and Feng Jiao, Nonlocal Cauchy problem for fractional evolution equa-tions. (in press).
[30] K. Maleknejad, K. Nouri and R. Mollapourasl, Existence of solutions for somenonlinear integral equations, Commun Nonlinear Sci Numer Simulat, 14(2009)2559–2564.
[31] Jozef Banas and Donal ORegan, On existence and local attractivity of solutionsof a quadratic Volterra integral equation of fractional order, J. Math. Anal. Appl.345(2008) 573–582.
[32] Abderrazek Karoui, On the existence of continuous solutions of nonlinear integralequations, Appl. Math. Lett. 18(2005) 299–305.
[33] B. C. Dhage, A general multi-valued hybrid fixed point theorem and perturbeddi?erential inclusions, Nonlinear Anal. 64(2006) 2747-2772.
[34] S. D. Eidelman and A.N. Kochubei, Cauchy problem for fractional di?usion equa-tions, J. Di?. Equ. 199(2004) 211-255.
[35] A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional di?erential equations,Appl. Math. Comput. 68(1995) 15-25.
[36] A. N. Kochubei, A Cauchy problem for evolution equations of fractional order,Di?erential Equations, 25(1989) 967-974.
[37] Qing Liu and Rong Yuan, Existence of mild solutions for semilinear evolutionequations with non-local initial conditions, Nonliear Analysis, 71(2009) 4177-4184.
[38] E. G. Bajlekova, Fractional Evolution Equations in Banach Space, University PressFacilities, Eindhoven University of Technology, 2001.
[39] L. Byszewski, Theorems about existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162(1991) 494-505.
[40] X. Fu and K. Ezzinbi, Existence of solutions for neutral di?erential evolutionequations with nonlocal conditions, Nonlinear Anal. 54(2003) 215-227.