Chapter 6 Nonlocal Initial and Boundary Value Problems: A Survey

Handbook of Differential Equations: Ordinary Differential Equations

Volumn 2, Issue , 2006, Pages 461-557

  Ntouyas, S K ^a  

^a UNIVERSITY OF IOANNINA   (Greece)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 66049155323     PISSN: 18745725     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5725(05)80008-2     Document Type: Article

References (95)

1. Aftabizadeh A., Gupta C., and Xu J.-M. Existence and uniqueness theorems for three-point boundary value problems. SIAM J. Math. Anal. 20 (1989) 716-726
2. Avery R. A generalization of the Legget-Willaimas fixed point theorem. Math. Res. Hot-Line 2 (1998) 9-14
3. Avery R., and Henderson J. Existence of three positive pseudo-symmetric solutions for a one-dimensional p-Laplacian. J. Math. Anal. Appl. 277 (2003) 395-404
4. Bai C., and Fang J. Existence of multiple positive solutions for nonlinear m -point boundary value problems. J. Math. Anal. Appl. 281 (2003) 76-85
5. Bai C., and Fang J. Existence of multiple positive solutions for nonlinear m -point boundary-value problems. Appl. Math. Comput. 140 (2003) 297-305
6. Bai C., and Fang J. Existence of positive solutions for three-point boundary value problems at resonance. J. Math. Anal. Appl. 291 (2004) 538-549
7. Bitsadze A. On the theory of nonlocal boundary value problems, Russian Acad. Sci. Dokl. Math. 30 (1984) 8-10
8. Bitsadze A., and Samarskii A. On some simple generalizations of linear elliptic boundary problems. Russian Acad. Sci. Dokl. Math. 10 (1969) 398-400
9. Boucherif A., and Bouguima S. Nonlinear second order ordinary differential equations with nonlocal boundary conditions. Comm. Appl. Nonlinear Anal. 5 (1998) 73-85
10. Byszewski L. Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162 (1991) 494-505
11. Byszewski L. Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem. J. Appl. Math. Stochastic Anal. 12 (1999) 91-97
12. Byszewski L., and Lakshmikantham V. Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40 (1990) 11-19
13. Calvert B., and Gupta C. Multiple solutions for a super-linear three-point boundary value problem. Nonlinear Anal. 50 (2002) 115-128
14. Dugundji J., and Granas A. Fixed Point Theory. Monographic Matematyczne (1982), PNW
15. Part 6 (Athens, 1996). Feng W., and Webb J.R.L. Solvability of three-point boundary value problems at resonance, Proceedings of the Second World Congress of Nonlinear Analysts. Nonlinear Anal. 30 (1997) 3227-3238
16. Feng W., and Webb J.R.L. Solvability of m-point boundary value problems with nonlinear growth. J. Math. Anal. Appl. 212 (1997) 467-480
17. Garcia-Huidobro M., Gupta C.P., and Manasevich R. An m-point boundary value problem of Neumann type for a p-Laplacian like operator. Nonlinear Anal. 56 (2004) 1071-1089
18. Graef J., Qian C., and Yang B. A three-point boundary value problem for nonlinear fourth order differential equations. J. Math. Anal. Appl. 287 (2003) 217-233
19. Guo Y., and Ge W. Positive solutions for three-point boundary value problems with dependence on the first order derivative. J. Math. Anal. Appl. 290 (2004) 291-301
20. Guo Y., Ge W., and Gao Y. Twin positive solutions for higher order m-point boundary value problems with sign changing nonlinearities. Appl. Math. Comput. 146 (2003) 299-311
21. Guo D., and Lakshmikantham V. Nonlinear Problems in Abstract Cones (1985), Springer-Verlag, Warsawa
22. Guo Y., Liu X., and Qiu J. Three positive solutions for higher order m-point boundary value problems. J. Math. Anal. Appl. 289 (2004) 545-553
23. Guo Y., Shan W., and Ge W. Positive solutions for second-order m-point boundary value problems. J. Corn-put. Appl. Math. 151 (2003) 415-424
24. Gupta C. On a third-order three-point boundary value problem at resonance. Differential Integral Equations 2 (1989) 1-12
25. Gupta C. Solvability of a three-point nonlinear boundary value problem for second order ordinary differential equations. J. Math. Anal. Appl. 168 (1992) 540-551
26. Gupta C. A note on a second order three-point boundary value problem. J. Math. Anal. Appl. 186 (1994) 277-281
27. Gupta C. A second order m-point boundary value problem at resonance. Nonlinear Anal. 24 (1995) 1483-1489
28. Atlanta, GA, 1995. Gupta C. Solvability of a generalized multi-point boundary value problem of mixed type for second order ordinary differential equations. Proceedings of Dynamic Systems and Applications (1996), Dynamic Publishers, New York 215-222
29. Gupta C. A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. Math. Comput. 89 (1998) 133-146
30. Edmond, OK, 2001. Gupta C. A new a priori estimate for multi-point boundary-value problems, Proceedings of the 16th Conference on Applied Mathematics. electronic. Electron. J. Differ. Equ. Conf. 7 (2001) 47-59
31. Gupta C., and Lakshmikantham V.T. Existence and uniqueness theorems for a third-order three-point boundary value problem. Nonlinear Anal. 16 (1991) 949-957
32. Gupta C., Ntouyas S.K., and Tsamatos P.Ch. Existence results for m-point boundary value problems. Differential Equations Dynam. Systems 2 (1994) 289-298
33. Gupta C., Ntouyas S.K., and Tsamatos P.Ch. On an m-point boundary-value problem for second-order ordinary differential equations. Nonlinear Anal. 23 (1994) 1427-1436
34. Gupta C., Ntouyas S.K., and Tsamatos P.Ch. Solvability of an m-point boundary value problem for second order ordinary differential equations. J. Math. Anal. Appl. 189 (1995) 575-584
35. Gupta C., Ntouyas S.K., and Tsamatos P.Ch. On the solvability of some multi-point boundary value problems. Appl. Math. 41 (1996) 1-17
36. Gupta C., Ntouyas S.K., and Tsamatos P.Ch. Existence results for multi-point boundary value problems for second order ordinary differential equations. Bull. Greek Math. Soc. 43 (2000) 105-123
37. Gupta C., and Trofimchuk S. Solvability of a multi-point boundary value problem and related a priori estimates, Geoffrey J. Butler Memorial Conference in Differential Equations and Mathe. Edmonton, AB, 1996. Canad. Appl. Math. Quart. 6 (1998) 45-60
38. Gupta C., and Trofimchuk S. Existence of a solution of a three-point boundary value problem and the spectral radius of a related linear operator. Nonlinear Anal. 34 (1998) 489-507
39. Gupta C., and Trofimchuk S. A Wirtinger type inequality and a three-point boundary value problem. Dynam. Systems Appl. 8 (1999) 127-132
40. Gupta C., and Trofimchuk S. An interesting example for a three-point boundary value problem. Bull. Belg. Math. Soc. Simon Stevin 7 (2000) 291-302
41. He X., and Ge W. A remark on some three-point boundary value problems for the one-dimensional p-Laplacian. Z. Angew. Math. Mech. 82 (2002) 728-731
42. He X., and Ge W. Triple solutions for second-order three-point boundary value problems. J. Math. Anal. Appl. 268 (2002) 256-265
43. Il'in V., and Moiseev E. Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations 23 (1987) 803-810
44. Il'in V., and Moiseev E. Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differential Equations 23 (1987) 979-987
45. Infante G. Eigenvalues of some nonlocal boundary-value problems. Proc. Edinburgh Math. Soc. 2 (2003) 75-86 46
46. Infante G., and Webb J.R.L. Nonzero solutions of Hammerstein integral equations with discontinuous kernels. J. Math. Anal. Appl. 272 (2002) 30-42
47. Infante G., and Webb J.R.L. Positive solutions of some nonlocal boundary value problems. Abstr. Appl. Anal. 18 (2003) 1047-1060
48. 8 pp. electronic. Karakostas G.L., and Tsamatos P.Ch. Positive solutions for a nonlocal boundary-value problem with increasing response. Electron. J. Differential Equations 73 (2000)
49. 9 pp. electronic. Karakostas G.L., and Tsamatos P.Ch. Positive solutions of a boundary-value problem for second order ordinary differential equations. Electron. J. Differential Equations 49 (2000)
50. Karakostas G.L., and Tsamatos P.Ch. On a nonlocal boundary value problem at resonance. J. Math. Anal. Appl. 259 (2001) 209-218
51. 10 pp. electronic. Karakostas G.L., and Tsamatos P.Ch. Multiple positive solutions for a nonlocal boundary-value problem with response function quiet at zero. Electron. J. Differential Equations 13 (2001)
52. Karakostas G.L. RCh. Tsamatos, Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem. Appl. Math. Lett. 15 (2002) 401-407
53. Karakostas G.L. RCh. Tsamatos, Nonlocal boundary vector value problems for ordinary differential systems of higher order. Nonlinear Anal. 51 (2002) 1421-1427
54. Krasnosel'skii M. Positive Solutions of Operator Equations (1964), Noordhoff, Atlanta, GA
55. Legget R., and Williams L. Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28 (1979) 673-688
56. Liu B. Positive solutions of a nonlinear three-point boundary value problem. Appl. Math. Comput. 132 (2002) 11-28
57. Liu B. Positive solutions of a nonlinear three-point boundary value problem. Comput. Math. Appl. 44 (2002) 201-211
58. Liu B. Positive solutions of a nonlinear three-point boundary value problem. Appl. Math. Comput. 132 (2002) 11-28
59. Liu B. Solvability of multi-point boundary value problem at resonance IV. Appl. Math. Comput. 143 (2003) 275-299
60. Liu B. Solvability of multi-point boundary value problem at resonance II. Appl. Math. Comput. 136 (2003) 353-377
61. Liu B., and Yu J. Solvability of multi-point boundary value problems at resonance I. Indian J. Pure Appl. Math. 33 (2002) 475-494
62. Liu B., and Yu J. Solvability of multi-point boundary value problem at resonance III. Appl. Math. Comput. 129 (2002) 119-143
63. Liu X. Nontrivial solutions of singular m-point boundary value problems. J. Math. Anal. Appl. 284 (2003) 576-590
64. Liu X., Qiu J., and Guo Y. Three positive solutions for second-order m-point boundary value problems. Appl. Math. Comput. 156 (2004) 733-742
65. Liu Y., and Ge W. Positive solutions for (n- 1, 1) three-point boundary value problems with coefficient that changes sign. J. Math. Anal. Appl. 282 (2003) 816-825
66. Liu Y., and Ge W. Multiple positive solutions to a three-point boundary value problem with p-Laplacian. J. Math. Anal. Appl. 277 (2003) 293-302
67. Ma R. Existence theorems for a second order m-point boundary value problem. J. Math. Anal. Appl. 211 (1997) 545-555
68. Ma R. Existence theorems for a second order three-point boundary value problem. J. Math. Anal. Appl. 212 (1997) 142-430
69. 8 pp.electronic. Ma R. Positive solutions of a nonlinear three-point boundary value problem. Electron. J. Differential Equations 34 (1999)
70. Ma R. Multiplicity of positive solutions for second-order three-point boundary value problems. Comput. Math. Appl. 40 (2000) 193-204
71. Ma R. Existence of solutions of a nonlinear m-point boundary value problems. J. Math. Anal. Appl. 256 (2001) 556-567
72. Ma R. Positive solutions of a nonlinear m-point boundary value problem. Comput. Math. Appl. 42 (2001) 755-765
73. Ma R. Positive solutions for second order functional differential equations. Dynam. Systems Appl. 10 (2001) 215-223
74. Ma R. Existence and uniqueness of solutions to first-order three-point boundary value problems. Appl. Math. Lett. 15 (2002) 211-216
75. Ma R. Existence of positive solutions for superlinear semipositone m -point boundary value problems. Proc. Edinburgh Math. Soc. 46 (2003) 279-292
76. Ma R. Multiplicity results for a three-point boundary value problem at resonance. Nonlinear Anal. 53 (2003) 777-789
77. Ma R. Multiple positive solutions for nonlinear m-point boundary value problems. Appl. Math. Comput. 148 (2004) 249-262
78. Ma R., and Castaneda N. Positive solutions for second-order three-point boundary value problems. Appl. Math. Lett. 14 (2001) 1-5
79. Ma R., and Wang H. Positive solutions of nonlinear three-point boundary-value problems. J. Math. Anal. Appl. 279 (2003) 216-227
80. Marano S. A remark on a second order three-point boundary value problem. J. Math. Anal. Appl. 183 (1994) 518-522
81. Marano S. A remark on a third-order three-point boundary value problem. Bull. Austral. Math. Soc. 49 (1994) 1-5
82. Mawhin J. Topological Degree Methods in Nonlinear Boundary Value Problems. NSF-CBMS Reg. Conf. Ser. Math. Vol. 40 (1979), Amer. Math. Soc., Groningen
83. Moshinsky M. Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas. Bol. Soc. Mat. Mexicana 7 (1950) 1-25
84. Murty K., and Sivasundaram S. Existence and uniqueness of solutions to three-point boundary value problems associated with nonlinear first order systems of differential equations. J. Math. Anal. Appl. 173 (1993) 158-164
85. 7 pp. electronic. Ni X., and Ge W. Multi-point boundary value problems for the p-Laplacian at resonance. Electron. J. Differential Equations 112 (2003)
86. Ntouyas S.K., and Tsamatos P.Ch. Global existence for semilinear evolution equations with nonlocal conditions. J. Math. Anal. Appl. 210 (1997) 679-687
87. 1-16 pp. electronic. Palamides P.K. Positive and monotone solutions of an m-point boundary value problem. Electron. J. Differential Equations 18 (2002)
88. Przeradski B., and Stanczy R. Solvability of a multi-point boundary value problem at resonance. J. Math. Anal. Appl. 264 (2001) 253-261
89. 11 pp. electronic. Raffoul Y. Positive solutions of three-point nonlinear second order boundary value problem. Electron. J. Qual. Theory Differ. Equ. 15 (2002)
90. Timoshenko S. Theory of Elastic Stability (1961), McGraw-Hill, Providence, RI
91. Wang J., and Zheng D. On the existence of positive solutions to a three-point boundary value problem for the one-dimensional p-Laplacian. Z. Angew. Math. Mech. 77 (1997) 471-479
92. Webb J.R. Positive solutions of some three-point boundary value problems via fixed point index theory. Nonlinear Anal. 47 (2001) 4319-4322
93. Zhang M., and Han Y. On the applications of Leray-Schauder continuation theorem to boundary value problems of semilinear differential equations. Ann. Differential Equations 13 (1997) 189-207
94. Zhang Z., and Wang J. The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems. J. Comput. Appl. Math. 147 (2002) 41-52
95. Zhang Z., and Wang J. Positive solutions to a second order three-point boundary value problem. J. Math. Anal. Appl. 285 (2003) 237-249