距离空间上多值算子公共不动点
|
摘要
本文中,我们首先给出了距离空间中的一些定义和理论,利用这些定义和理论,讨论了两个多值算子的公共不动点的存在性。我们的结果是对应单值算子结果的推广和改进。特别指出,在其中的某些结果中还给出了多值算子不动点的唯一性,这些结果都是全新的。针对某些结果,我们给出了相应的例子,说明所得结果的有效性和实用性。
本文的第1章是绪论,主要是简单的介绍前人的一些工作和他们的研究方法,所用的数学工具等,并指出自己在文章中将要解决的问题。
第2章主要是给出了在距离空间的一些定义和理论,其中包含锥距离空间的一些基本概念,还包括在锥距离空间中定义的积分算子。给出了多值算子的一些基础知识。例如多值算子Hausdorff距离的概念,这不仅只是比单值问题的距离复杂,而且有些困难难以克服。我们通过引入偏序,便能解决这些问题。我们列出了前人做的一些结论,他们都是关于单值算子的一些理论,并且得出了令人满意的结果。我们是在原有结论的基础上,将其条件进行修改和推广进一步得出我们想要的结果,其中有些压缩条件不是一般的压缩条件。比如利用锥可积函数构成的压缩条件,作者得出的是单值算子的公共不动点。而我们的目的是得出多值算子公共不动点的存在性和唯一性。公共不动点的唯一性是个很好的结论,而且我们在得出唯一性的同时并没有利用另外的压缩条件。
第3章我们主要在完备锥距离空间中,研究了两个广义压缩的多值算子的公共不动点。本章本质上是利用前人在单值算子中已经获得的结果,加上一些限定条件平推到多值问题中去。由于作者在引进新条件的同时,也去掉或减弱了一些已有的单值算子的相应结果时需要的条件,因此本章的结果即使对单值算子而言,也是新的。我们的结果不仅有一定的理论意义,而且有实用价值。这一点可以从最后举出的应用实例中看出。
第4章主要在完备的锥距离空间中得出利用锥可积函数构成的压缩多值算子公共不动点的存在性和唯一性。由于我们所考虑的是多值算子的公共不动点。在本章中,我们不仅定义了锥距离空间中的强极小锥,而且给出了在强极小锥中的Hausdorff距离的概念。我们还证明了:若在偏序的情况下得出了公共不动点的存在性,当偏序条件加强成全序条件时,则不仅可得到公共不动点的存在性,而且还可得到公共不动点的唯一性结果。
本文最后我们结合自己的想法对我们的研究方向作了展望,同时指出了自身理论的有待改进之处。
In this paper, we first give some definitions and theories in the metric space. According to these definitions and theories, we discuss the existence of the common fixed point about the multivalued operators. Our results are the extension and improvement of the corresponding single-valued operator results. In particular, some results in which also gives the operator more than the value of the uniqueness of fixed points, these results are new. For some results, we give a corresponding example of the effectiveness and practical of the results.
First chapter is introduction, mainly introduce some results of former researcher and their methods and mathematical tools which deal with these problems, and we enumerate the problems of which will be solved in our paper.
In the second chapter, we mainly give some definitions and theories, containing the basic concepts of the cone metric space. Furthermore, we also define the concept of the integral operator in cone metric space and give some basic knowledge of the multi-valued operators. For example, the concept of the Hausdorff metric. Not only are the problem of the multi-valued operators more difficult than the single-valued operators, but also some problems can not resolve. By introducing the notion of order, we get the conclusion of the existence of common fixed point under the assumption of the partial order. We list some conclusions that have been done by professors. They gave the theories of the single-valued operators and obtained the satisfied conclusions. On the basic of the original conclusions, we modified and promoted the conditions of the original conclusions and got the conclusion that we wanted. Some of the contractive conditions are not the general contractive conditions. For example the contractive conditions that are constituted by the cone integrable function. They got the common fixed point of the single-valued operators. It is the aim that we get the existence and uniqueness of the common fixed point about the multi-valued operators. The uniqueness of the common fixed point is a very good conclusion and we did not add the additional contractive conditions.
In the third chapter, we mainly research the common fixed point of the general contractive multi-valued operators in the complete cone metric space. Essentially, we construct the common fixed point of multi-valued operators by the previous result in the single-valued operators and add some qualifiers. So the result is new to the single-valued operators also. Our result not only have the meaning of the theory but also have the practical value. Last we cite the example .
In the Forth chapter, we got the common fixed point of the multi-valued operators in the complete cone metric space. The contractive condition is constructed by the cone integaral function. Because we consider the common fixed point of multi-valued operators. In this chapter, we not only defined the strong small cone in the cone metric space, but also gave the concept of the Hausdorff metric about the strong small cone. By introducing the notion of order, we get the conclusion of the existence of common fixed point under the assumption of the partial order, also, the uniqueness of common fixed point under the assumption of total order.
In the end of the paper we combine with our thoughts to get the expectation of our research realm, while point out the amelioration of our theories.
本文的第1章是绪论,主要是简单的介绍前人的一些工作和他们的研究方法,所用的数学工具等,并指出自己在文章中将要解决的问题。
第2章主要是给出了在距离空间的一些定义和理论,其中包含锥距离空间的一些基本概念,还包括在锥距离空间中定义的积分算子。给出了多值算子的一些基础知识。例如多值算子Hausdorff距离的概念,这不仅只是比单值问题的距离复杂,而且有些困难难以克服。我们通过引入偏序,便能解决这些问题。我们列出了前人做的一些结论,他们都是关于单值算子的一些理论,并且得出了令人满意的结果。我们是在原有结论的基础上,将其条件进行修改和推广进一步得出我们想要的结果,其中有些压缩条件不是一般的压缩条件。比如利用锥可积函数构成的压缩条件,作者得出的是单值算子的公共不动点。而我们的目的是得出多值算子公共不动点的存在性和唯一性。公共不动点的唯一性是个很好的结论,而且我们在得出唯一性的同时并没有利用另外的压缩条件。
第3章我们主要在完备锥距离空间中,研究了两个广义压缩的多值算子的公共不动点。本章本质上是利用前人在单值算子中已经获得的结果,加上一些限定条件平推到多值问题中去。由于作者在引进新条件的同时,也去掉或减弱了一些已有的单值算子的相应结果时需要的条件,因此本章的结果即使对单值算子而言,也是新的。我们的结果不仅有一定的理论意义,而且有实用价值。这一点可以从最后举出的应用实例中看出。
第4章主要在完备的锥距离空间中得出利用锥可积函数构成的压缩多值算子公共不动点的存在性和唯一性。由于我们所考虑的是多值算子的公共不动点。在本章中,我们不仅定义了锥距离空间中的强极小锥,而且给出了在强极小锥中的Hausdorff距离的概念。我们还证明了:若在偏序的情况下得出了公共不动点的存在性,当偏序条件加强成全序条件时,则不仅可得到公共不动点的存在性,而且还可得到公共不动点的唯一性结果。
本文最后我们结合自己的想法对我们的研究方向作了展望,同时指出了自身理论的有待改进之处。
In this paper, we first give some definitions and theories in the metric space. According to these definitions and theories, we discuss the existence of the common fixed point about the multivalued operators. Our results are the extension and improvement of the corresponding single-valued operator results. In particular, some results in which also gives the operator more than the value of the uniqueness of fixed points, these results are new. For some results, we give a corresponding example of the effectiveness and practical of the results.
First chapter is introduction, mainly introduce some results of former researcher and their methods and mathematical tools which deal with these problems, and we enumerate the problems of which will be solved in our paper.
In the second chapter, we mainly give some definitions and theories, containing the basic concepts of the cone metric space. Furthermore, we also define the concept of the integral operator in cone metric space and give some basic knowledge of the multi-valued operators. For example, the concept of the Hausdorff metric. Not only are the problem of the multi-valued operators more difficult than the single-valued operators, but also some problems can not resolve. By introducing the notion of order, we get the conclusion of the existence of common fixed point under the assumption of the partial order. We list some conclusions that have been done by professors. They gave the theories of the single-valued operators and obtained the satisfied conclusions. On the basic of the original conclusions, we modified and promoted the conditions of the original conclusions and got the conclusion that we wanted. Some of the contractive conditions are not the general contractive conditions. For example the contractive conditions that are constituted by the cone integrable function. They got the common fixed point of the single-valued operators. It is the aim that we get the existence and uniqueness of the common fixed point about the multi-valued operators. The uniqueness of the common fixed point is a very good conclusion and we did not add the additional contractive conditions.
In the third chapter, we mainly research the common fixed point of the general contractive multi-valued operators in the complete cone metric space. Essentially, we construct the common fixed point of multi-valued operators by the previous result in the single-valued operators and add some qualifiers. So the result is new to the single-valued operators also. Our result not only have the meaning of the theory but also have the practical value. Last we cite the example .
In the Forth chapter, we got the common fixed point of the multi-valued operators in the complete cone metric space. The contractive condition is constructed by the cone integaral function. Because we consider the common fixed point of multi-valued operators. In this chapter, we not only defined the strong small cone in the cone metric space, but also gave the concept of the Hausdorff metric about the strong small cone. By introducing the notion of order, we get the conclusion of the existence of common fixed point under the assumption of the partial order, also, the uniqueness of common fixed point under the assumption of total order.
In the end of the paper we combine with our thoughts to get the expectation of our research realm, while point out the amelioration of our theories.
引文
[1] Georgi V. Smirnov. Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics [J]. American Mathematical Society. 2002.
[2] K. Deimiling. Multivalued Differential Equations [M]. Berlin, Walter & Gruyter, 1992.
[3] A. Tolstonogov. Differential Inclusions in Banach Spaces [J]. Kluwer Academic Publishers, 2000.
[4] Aubin J. P. and Cellina A. Differential Inclusions [M]. Berlin, Springer-Verlag, 1984.
[5] Ravi P. Agarwal and D. O’Regan. A note on the existence of multiple fixed points for multivalued maps with applications [J]. J. Differential Equations, 2000, 160: 389-403.
[6] D. O’Regan. Comtinuation methods based on essential and 0-epi maps [J]. Acta Applicandae Mathematicae. 1998,54:319-330.
[7] B. C. Dhage. Multivalued mappings and fixed points II [J]. Nonlinear Funct. Aal. Appl. 2005, 10:359-378.
[8] L.Gorniewicz. Topological fixed point theory of multivalued mappings [J]. Math. Appl. 1999.
[9] Sh. Hu, N. Papageorgiou. Handbook of multivalued analysis[J]. Theory, Kluwer Academic-Dordrecht 1997.
[10] Chang, K. Infinite Dimensional Morse Theory and Multiple Solution Problems[J]. Birkhauser,1993.
[11] Chang, K. The Obstacle Problem and Partial Differential Equations with Discontinuous Nonlinear Term, Comm [J]. Pure & Appl. Math. 1980,3:117-146.
[12] Chang, K. Variational Methods for Non-differentiable Functionals [J]. J. Math. Anal. Appl. 1981, 80: 102– 128.
[13] X. Xue, J. Yu. Periodic solutions for semi-linear evolution inclusions [J] .J. Math. Anal. Appl. 2007,331: 1246-1262.
[14]王志华.非线性微分包含的极点解[J].数学年刊.1998,19A:417-422.
[15] S.S.Chang,Y.H.Ma. Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of function equations arising in dynamic programming [J]. J.Math.Anal.Appl. 1991, 160:468-479.
[16] S. H. Hong. Fixed points for mixed monotone multivalued operators in Banach spaces with applications [J]. Journal of Mathematical Analysis and Applications. 2008, 337 : 333–342.
[17] S. H. Hong. D. Guan, W. Li. Hybrid fixed points of multivalued operators in metric spaces withapplications [J].Nonlinear Analysis.2008.
[18] S. H. Hong. Multiple positive solutions for a class of integral inclusions [J].Journaal of Computational and Applied Mathematics.2008, 214: 19-29.
[19] S. H. Hong, The method of upper and lower solutions for nth order nonlinear impulsive differential inclusions, Dynamics of Continuous, Discrete and Impulsive Systems [J]. Series A: Math. Anal. 2007, 14: 739-753.
[20]洪世煌,章春国,谢素英,何泽荣.序Banach空间中一类算子方程的可解性[J].数学学报.2007, 50(4): 823-830.
[21]洪世煌,微分包含的非线性边值问题[J].数学物理学报.2007, 27(4), 3: 711-719.
[22] S. H. Hong. Existence of solutions for integral inclusions [J]. Journal of Mathematical Analysis and Applications. 2006, 317: 429–441.
[23] S. H. Hong. Existence Results for Functional Differential Inclusions with Infinite Delay [J]. Acta Mathematica Sinica, English Series. 2005 , 22 : 773–780.
[24] S. H. Hong. Existence of solutions to initial value problems for the second order mixed monotone type of impulsive differential inclusions [J].J. Math. Kyoto Univ. 2005, 45 ,2 :329-341.
[25] S. H. Hong. Solvability of nonlinear impulsive Volterra integral inclusions and functional differential inclusions1 [J],.Journal of Mathematical Analysis and Applications. 2004 , 295: 331–340.
[26]洪世煌.多值微分系统及动态规划中的泛函方程的可解性[J].数学学报. 2004, 47, 3 :479-486.
[27] S. H. Hong. Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications [J]. J.Math.Anal.Appl. 2003, 282: 151-162.
[28] S. H. Hong. Boundary-value problems for first and second order functional differential inclusions [J]. Electronic Journal of Differential Equations. 2003, 1-10.
[29] D.O’Regan. Topological transversality: Application to third-order boundary value problem [J]. SIAM J. Math. Anal. 1987, 19: 630-541.
[30] D.O’Regan. Fixed points for set valued mappings in locally convex linear topological spaces [J]. Math. Comput. Modelling . 1998, 28, 1:45-55.
[31] R.P.Agarwal, D.O’Regan. Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease [J]. Nonlinear Anal. 2000,40 :21-35.
[32] R. P. Agarwal, Nan-Ding Huang, Man-Yi Tan. Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions [J].Appl. Math. Lett. 2004,17: 345-352.
[33] R. P. Agarwal . A selection of oscillation criteria for second-order differential inclusions [J]. Applied Mathematics Letters.2008.
[34] R. P. Agarwal. Leray–Schauder and Krasnoselskii results for multivalued maps defined on pseudo-open subsets of a Fréchet space. Applied Mathematics Letters. 2006, 19: 1327–1334.
[35] R. P. Agarwal. Fixed points of cone compression and expansion multimaps defined on Fréchet spaces: The projective limit approach [J]. Journal of Applied Mathematics and Stochastic Analysis, 2006.
[36] Ravi P. Agarwal and D. O’Regan. A note on the existence of multiple fixed points for multivalued maps with applications [J]. J. Differential Equations, 2000, 160:389-403.
[37] Nguyen Bich Huy, Nguyen Huu Khanh. Note of fixed point for multivalued increasing operators [J]. J.Math.Anal.Appl. 2000,250, 368-371.3929-3942.
[38] S.Banach.Sueles operations dans les ensembles abstraits et ieue application aux equations integrals[J].Fund.Math.1922,3:133-181.
[39] R.Caccioppoli.Un teorema generale sull’esistenza uniti in una trasformazionale funzionale[J].Rend.Accad.dei Lincei.1930,11:794-799.
[40] Xian Zhang.Common fixed point theorems for some new generalized contractive type mappings[J].J.Math.Anal.Appl.2007,333:780-786.
[41] G.Jungck,P.P.Murthy,Y.J.Cho.Compatible mappings of type (A) and common fixed points[J].Math.Japon.1993,38(2):381-390.
[42] S.Sessa.On a weak commutativity condition of mappings in fixed point considerations[J].Pupl.Inst.Math.Beograd.1982,32:149-153.
[43] A.Djoudi,L.Nisse.Gregus type fixed points for weakly compatible mappings[J]. Bull.Bleg.Math.Soc.2003,10:369-378.
[44] V.Berinde.A common fixed point theorem for compatible quasi contractive self mappings in metric spaces[J].Appl.Math.Comp.2009,213:348-354.
[45] A.Djoudi,F.Merghadi.Common fixed point theorems for maps under a contractive condition of integral type[J]. J.Math.Anal.Appl.2008,341:953-960.
[46] S.H.Hong.Fixed points for mixed monotone multivalued operators in Banach spaces with applications[J]. J.Math.Anal.Appl.2008,337:333-342.
[47] M.Abbas,B.E.Rhoades.Fixed point theorems for two new classes of multivalued mappings[J]. Applied Mathematics Letters.2009,22:1364-1368.
[48] T.Husain,A.Latif.Fixed points of multivalued nonexpansive maps[J].Math.Japonica.1988,33: 385-391.
[49] T.Husain,E.Tarafdar.Fixed point theorems for multivalued mappings of nonexpansive type[J].Yokohama.Math.J.1980,28:1-6.
[50]A.Latuf,I.Beg.Geometric fixed poonts for single valued and multivalued mappings[J].Demonstratio Math.1997,30(4):791-800.
[51] S.B.Nadler.Multivalued contraction mappings[J].Pacific J.Math.1969.30:475-488.
[52] H.L.Guang,X.Zhang.Cone metric spaces and fixed point theorems of contractive mappings[J]. J.Math.Anal.Appl.2007,332:1468-1476.
[53] S.Radenovic,B.E.Rhoades.Fixed point theorem for two non-self mappings in cone metric spaces[J].Comp.Math.Appl.2009,57:1701-1707.
[54] D.Ilic,V.Rakocevic.Common fixed points for maps on cone metric space[J]. J.Math.Anal.Appl.2008,341:876-882.
[55] S.Radenovic.Common fixed points under contractive conditions in cone metric space[J]. Comp.Math.Appl.2009,58:1273-1278.
[56] B.S.Choudhury,N.Metiya.Fixed points of weak contractions in cone metric spaces[J].Nonlinear Analysis.2010,72:1589-1593.
[57] R.Sumitra,V.R.Uthariaraj.Common fixed point theorem for non-self mappings satisfying gemeralized Ciric type contraction condition in cone metric space[J].Mathematics Subject Classification.2000.
[58] B.S.Choudhury,N.Metiya.Fixed points of weak contractions in cone metric spaces[J]. .Nonlinear Analysis.2010,72,1689-1593.
[59] F.Khojasteh,Z.Goodarzi.Some fixed point theorem of integral type contraction in cone metric spaces[J].Fixed Point Theorem Applications.2010.
[60] M.Abbas, G.Jungck.Common fixed point results for noncommuting mappings without continuity in cone metric spaces[J].J. Math. Anal. Appl. 2008,341:416-420.
[61] Sh.Rezapour, R.Hamlbarani. Some notes on the paper”Cone metric spaces and fixed point theorems of contractive mappings”[J].J.Math.Anal.Appl.2008,345:719-724.
[62] Lj.B.′ciri′c, J.S. Umeb,Multi-valued non-self-mappings on convex metric spaces[J]. Nonlinear Anal. 2005,60:1053-1063.
[63] D. Klim and D. Wardowski, Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces[J]. Nonlinear Anal, doi:10.1016/j.na.2009.04.001.
[64] S. H. Hong, D. Guan, L. Wang,.Hybrid fixed points of multivalued operators in metric spaces with applications[J]. Nonlinear Anal. 2009,70:4106-4117.
[65] M.J.Shen, S.H.Hong.Common fixed points for generalized contractive multivalued operators in complete metric spaces[J]. Appl.Math.Lett. 2009, 22:1864-1869.
[66] S. Banach. Surles ope′rations dans les ensembles abstraits et leur application aux e′quations integrables [J].Fund.Math. 1922,3: 131-181(in French).
[67] R.Caccioppol. Un teorema generale sull’esistenza di elementi uniti in una trasformazione funzionale [J]. Rend.Accad.dei Lincei . 1930, 11: 794-799(in Italian).
[68] R.Kannan. Some results on fixed points [J]. Bull.Calcutta Math.Soc. 1968,60:71-76.
[69] A.Branciari. A fixed point theorem for mappings satisfying a general contractive condition of integral type [J].Int.J.Math.Sci. 2002, 29 (9) :531-536.
[70] P.Vijayaraju, B.E.Rhoades, R.Mohanraj. A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type [J]. Int.J.Math.Math.Sci.2005, (15) : 2359-2364.
[71] B.E.Rhoades. A comparison of various definitions of contractive mappings [J]. Trans.Amer.Math.Soc. 1977, 266:257-290.
[72] B.E.Rhoades. Two fixed point theorems for mappings satisfying a general contractive condition of integral type [J]. Int.J.Math.Math.Sei . 2003, No.63: 4007-4013.
[1] Georgi V. Smirnov. Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics [J]. American Mathematical Society. 2002.
[2] K. Deimiling. Multivalued Differential Equations [M]. Berlin, Walter & Gruyter, 1992.
[3] A. Tolstonogov. Differential Inclusions in Banach Spaces [J]. Kluwer Academic Publishers, 2000.
[4] Aubin J. P. and Cellina A. Differential Inclusions [M]. Berlin, Springer-Verlag, 1984.
[5] Ravi P. Agarwal and D. O’Regan. A note on the existence of multiple fixed points for multivalued maps with applications [J]. J. Differential Equations, 2000, 160: 389-403.
[6] D. O’Regan. Comtinuation methods based on essential and 0-epi maps [J]. Acta Applicandae Mathematicae. 1998,54:319-330.
[7] B. C. Dhage. Multivalued mappings and fixed points II [J]. Nonlinear Funct. Aal. Appl. 2005, 10:359-378.
[8] L.Gorniewicz. Topological fixed point theory of multivalued mappings [J]. Math. Appl. 1999.
[9] Sh. Hu, N. Papageorgiou. Handbook of multivalued analysis[J]. Theory, Kluwer Academic-Dordrecht 1997.
[10] Chang, K. Infinite Dimensional Morse Theory and Multiple Solution Problems[J]. Birkhauser,1993.
[11] Chang, K. The Obstacle Problem and Partial Differential Equations with Discontinuous Nonlinear Term, Comm [J]. Pure & Appl. Math. 1980,3:117-146.
[12] Chang, K. Variational Methods for Non-differentiable Functionals [J]. J. Math. Anal. Appl. 1981, 80: 102– 128.
[13] X. Xue, J. Yu. Periodic solutions for semi-linear evolution inclusions [J] .J. Math. Anal. Appl. 2007,331: 1246-1262.
[14]王志华.非线性微分包含的极点解[J].数学年刊.1998,19A:417-422.
[15] S.S.Chang,Y.H.Ma. Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of function equations arising in dynamic programming [J]. J.Math.Anal.Appl. 1991, 160:468-479.
[2] K. Deimiling. Multivalued Differential Equations [M]. Berlin, Walter & Gruyter, 1992.
[3] A. Tolstonogov. Differential Inclusions in Banach Spaces [J]. Kluwer Academic Publishers, 2000.
[4] Aubin J. P. and Cellina A. Differential Inclusions [M]. Berlin, Springer-Verlag, 1984.
[5] Ravi P. Agarwal and D. O’Regan. A note on the existence of multiple fixed points for multivalued maps with applications [J]. J. Differential Equations, 2000, 160: 389-403.
[6] D. O’Regan. Comtinuation methods based on essential and 0-epi maps [J]. Acta Applicandae Mathematicae. 1998,54:319-330.
[7] B. C. Dhage. Multivalued mappings and fixed points II [J]. Nonlinear Funct. Aal. Appl. 2005, 10:359-378.
[8] L.Gorniewicz. Topological fixed point theory of multivalued mappings [J]. Math. Appl. 1999.
[9] Sh. Hu, N. Papageorgiou. Handbook of multivalued analysis[J]. Theory, Kluwer Academic-Dordrecht 1997.
[10] Chang, K. Infinite Dimensional Morse Theory and Multiple Solution Problems[J]. Birkhauser,1993.
[11] Chang, K. The Obstacle Problem and Partial Differential Equations with Discontinuous Nonlinear Term, Comm [J]. Pure & Appl. Math. 1980,3:117-146.
[12] Chang, K. Variational Methods for Non-differentiable Functionals [J]. J. Math. Anal. Appl. 1981, 80: 102– 128.
[13] X. Xue, J. Yu. Periodic solutions for semi-linear evolution inclusions [J] .J. Math. Anal. Appl. 2007,331: 1246-1262.
[14]王志华.非线性微分包含的极点解[J].数学年刊.1998,19A:417-422.
[15] S.S.Chang,Y.H.Ma. Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of function equations arising in dynamic programming [J]. J.Math.Anal.Appl. 1991, 160:468-479.
[16] S. H. Hong. Fixed points for mixed monotone multivalued operators in Banach spaces with applications [J]. Journal of Mathematical Analysis and Applications. 2008, 337 : 333–342.
[17] S. H. Hong. D. Guan, W. Li. Hybrid fixed points of multivalued operators in metric spaces withapplications [J].Nonlinear Analysis.2008.
[18] S. H. Hong. Multiple positive solutions for a class of integral inclusions [J].Journaal of Computational and Applied Mathematics.2008, 214: 19-29.
[19] S. H. Hong, The method of upper and lower solutions for nth order nonlinear impulsive differential inclusions, Dynamics of Continuous, Discrete and Impulsive Systems [J]. Series A: Math. Anal. 2007, 14: 739-753.
[20]洪世煌,章春国,谢素英,何泽荣.序Banach空间中一类算子方程的可解性[J].数学学报.2007, 50(4): 823-830.
[21]洪世煌,微分包含的非线性边值问题[J].数学物理学报.2007, 27(4), 3: 711-719.
[22] S. H. Hong. Existence of solutions for integral inclusions [J]. Journal of Mathematical Analysis and Applications. 2006, 317: 429–441.
[23] S. H. Hong. Existence Results for Functional Differential Inclusions with Infinite Delay [J]. Acta Mathematica Sinica, English Series. 2005 , 22 : 773–780.
[24] S. H. Hong. Existence of solutions to initial value problems for the second order mixed monotone type of impulsive differential inclusions [J].J. Math. Kyoto Univ. 2005, 45 ,2 :329-341.
[25] S. H. Hong. Solvability of nonlinear impulsive Volterra integral inclusions and functional differential inclusions1 [J],.Journal of Mathematical Analysis and Applications. 2004 , 295: 331–340.
[26]洪世煌.多值微分系统及动态规划中的泛函方程的可解性[J].数学学报. 2004, 47, 3 :479-486.
[27] S. H. Hong. Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications [J]. J.Math.Anal.Appl. 2003, 282: 151-162.
[28] S. H. Hong. Boundary-value problems for first and second order functional differential inclusions [J]. Electronic Journal of Differential Equations. 2003, 1-10.
[29] D.O’Regan. Topological transversality: Application to third-order boundary value problem [J]. SIAM J. Math. Anal. 1987, 19: 630-541.
[30] D.O’Regan. Fixed points for set valued mappings in locally convex linear topological spaces [J]. Math. Comput. Modelling . 1998, 28, 1:45-55.
[31] R.P.Agarwal, D.O’Regan. Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease [J]. Nonlinear Anal. 2000,40 :21-35.
[32] R. P. Agarwal, Nan-Ding Huang, Man-Yi Tan. Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions [J].Appl. Math. Lett. 2004,17: 345-352.
[33] R. P. Agarwal . A selection of oscillation criteria for second-order differential inclusions [J]. Applied Mathematics Letters.2008.
[34] R. P. Agarwal. Leray–Schauder and Krasnoselskii results for multivalued maps defined on pseudo-open subsets of a Fréchet space. Applied Mathematics Letters. 2006, 19: 1327–1334.
[35] R. P. Agarwal. Fixed points of cone compression and expansion multimaps defined on Fréchet spaces: The projective limit approach [J]. Journal of Applied Mathematics and Stochastic Analysis, 2006.
[36] Ravi P. Agarwal and D. O’Regan. A note on the existence of multiple fixed points for multivalued maps with applications [J]. J. Differential Equations, 2000, 160:389-403.
[37] Nguyen Bich Huy, Nguyen Huu Khanh. Note of fixed point for multivalued increasing operators [J]. J.Math.Anal.Appl. 2000,250, 368-371.3929-3942.
[38] S.Banach.Sueles operations dans les ensembles abstraits et ieue application aux equations integrals[J].Fund.Math.1922,3:133-181.
[39] R.Caccioppoli.Un teorema generale sull’esistenza uniti in una trasformazionale funzionale[J].Rend.Accad.dei Lincei.1930,11:794-799.
[40] Xian Zhang.Common fixed point theorems for some new generalized contractive type mappings[J].J.Math.Anal.Appl.2007,333:780-786.
[41] G.Jungck,P.P.Murthy,Y.J.Cho.Compatible mappings of type (A) and common fixed points[J].Math.Japon.1993,38(2):381-390.
[42] S.Sessa.On a weak commutativity condition of mappings in fixed point considerations[J].Pupl.Inst.Math.Beograd.1982,32:149-153.
[43] A.Djoudi,L.Nisse.Gregus type fixed points for weakly compatible mappings[J]. Bull.Bleg.Math.Soc.2003,10:369-378.
[44] V.Berinde.A common fixed point theorem for compatible quasi contractive self mappings in metric spaces[J].Appl.Math.Comp.2009,213:348-354.
[45] A.Djoudi,F.Merghadi.Common fixed point theorems for maps under a contractive condition of integral type[J]. J.Math.Anal.Appl.2008,341:953-960.
[46] S.H.Hong.Fixed points for mixed monotone multivalued operators in Banach spaces with applications[J]. J.Math.Anal.Appl.2008,337:333-342.
[47] M.Abbas,B.E.Rhoades.Fixed point theorems for two new classes of multivalued mappings[J]. Applied Mathematics Letters.2009,22:1364-1368.
[48] T.Husain,A.Latif.Fixed points of multivalued nonexpansive maps[J].Math.Japonica.1988,33: 385-391.
[49] T.Husain,E.Tarafdar.Fixed point theorems for multivalued mappings of nonexpansive type[J].Yokohama.Math.J.1980,28:1-6.
[50]A.Latuf,I.Beg.Geometric fixed poonts for single valued and multivalued mappings[J].Demonstratio Math.1997,30(4):791-800.
[51] S.B.Nadler.Multivalued contraction mappings[J].Pacific J.Math.1969.30:475-488.
[52] H.L.Guang,X.Zhang.Cone metric spaces and fixed point theorems of contractive mappings[J]. J.Math.Anal.Appl.2007,332:1468-1476.
[53] S.Radenovic,B.E.Rhoades.Fixed point theorem for two non-self mappings in cone metric spaces[J].Comp.Math.Appl.2009,57:1701-1707.
[54] D.Ilic,V.Rakocevic.Common fixed points for maps on cone metric space[J]. J.Math.Anal.Appl.2008,341:876-882.
[55] S.Radenovic.Common fixed points under contractive conditions in cone metric space[J]. Comp.Math.Appl.2009,58:1273-1278.
[56] B.S.Choudhury,N.Metiya.Fixed points of weak contractions in cone metric spaces[J].Nonlinear Analysis.2010,72:1589-1593.
[57] R.Sumitra,V.R.Uthariaraj.Common fixed point theorem for non-self mappings satisfying gemeralized Ciric type contraction condition in cone metric space[J].Mathematics Subject Classification.2000.
[58] B.S.Choudhury,N.Metiya.Fixed points of weak contractions in cone metric spaces[J]. .Nonlinear Analysis.2010,72,1689-1593.
[59] F.Khojasteh,Z.Goodarzi.Some fixed point theorem of integral type contraction in cone metric spaces[J].Fixed Point Theorem Applications.2010.
[60] M.Abbas, G.Jungck.Common fixed point results for noncommuting mappings without continuity in cone metric spaces[J].J. Math. Anal. Appl. 2008,341:416-420.
[61] Sh.Rezapour, R.Hamlbarani. Some notes on the paper”Cone metric spaces and fixed point theorems of contractive mappings”[J].J.Math.Anal.Appl.2008,345:719-724.
[62] Lj.B.′ciri′c, J.S. Umeb,Multi-valued non-self-mappings on convex metric spaces[J]. Nonlinear Anal. 2005,60:1053-1063.
[63] D. Klim and D. Wardowski, Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces[J]. Nonlinear Anal, doi:10.1016/j.na.2009.04.001.
[64] S. H. Hong, D. Guan, L. Wang,.Hybrid fixed points of multivalued operators in metric spaces with applications[J]. Nonlinear Anal. 2009,70:4106-4117.
[65] M.J.Shen, S.H.Hong.Common fixed points for generalized contractive multivalued operators in complete metric spaces[J]. Appl.Math.Lett. 2009, 22:1864-1869.
[66] S. Banach. Surles ope′rations dans les ensembles abstraits et leur application aux e′quations integrables [J].Fund.Math. 1922,3: 131-181(in French).
[67] R.Caccioppol. Un teorema generale sull’esistenza di elementi uniti in una trasformazione funzionale [J]. Rend.Accad.dei Lincei . 1930, 11: 794-799(in Italian).
[68] R.Kannan. Some results on fixed points [J]. Bull.Calcutta Math.Soc. 1968,60:71-76.
[69] A.Branciari. A fixed point theorem for mappings satisfying a general contractive condition of integral type [J].Int.J.Math.Sci. 2002, 29 (9) :531-536.
[70] P.Vijayaraju, B.E.Rhoades, R.Mohanraj. A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type [J]. Int.J.Math.Math.Sci.2005, (15) : 2359-2364.
[71] B.E.Rhoades. A comparison of various definitions of contractive mappings [J]. Trans.Amer.Math.Soc. 1977, 266:257-290.
[72] B.E.Rhoades. Two fixed point theorems for mappings satisfying a general contractive condition of integral type [J]. Int.J.Math.Math.Sei . 2003, No.63: 4007-4013.
[1] Georgi V. Smirnov. Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics [J]. American Mathematical Society. 2002.
[2] K. Deimiling. Multivalued Differential Equations [M]. Berlin, Walter & Gruyter, 1992.
[3] A. Tolstonogov. Differential Inclusions in Banach Spaces [J]. Kluwer Academic Publishers, 2000.
[4] Aubin J. P. and Cellina A. Differential Inclusions [M]. Berlin, Springer-Verlag, 1984.
[5] Ravi P. Agarwal and D. O’Regan. A note on the existence of multiple fixed points for multivalued maps with applications [J]. J. Differential Equations, 2000, 160: 389-403.
[6] D. O’Regan. Comtinuation methods based on essential and 0-epi maps [J]. Acta Applicandae Mathematicae. 1998,54:319-330.
[7] B. C. Dhage. Multivalued mappings and fixed points II [J]. Nonlinear Funct. Aal. Appl. 2005, 10:359-378.
[8] L.Gorniewicz. Topological fixed point theory of multivalued mappings [J]. Math. Appl. 1999.
[9] Sh. Hu, N. Papageorgiou. Handbook of multivalued analysis[J]. Theory, Kluwer Academic-Dordrecht 1997.
[10] Chang, K. Infinite Dimensional Morse Theory and Multiple Solution Problems[J]. Birkhauser,1993.
[11] Chang, K. The Obstacle Problem and Partial Differential Equations with Discontinuous Nonlinear Term, Comm [J]. Pure & Appl. Math. 1980,3:117-146.
[12] Chang, K. Variational Methods for Non-differentiable Functionals [J]. J. Math. Anal. Appl. 1981, 80: 102– 128.
[13] X. Xue, J. Yu. Periodic solutions for semi-linear evolution inclusions [J] .J. Math. Anal. Appl. 2007,331: 1246-1262.
[14]王志华.非线性微分包含的极点解[J].数学年刊.1998,19A:417-422.
[15] S.S.Chang,Y.H.Ma. Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of function equations arising in dynamic programming [J]. J.Math.Anal.Appl. 1991, 160:468-479.