Existence of three solutions to a non-homogeneous multi-point BVP of second order diﬀerential equations

This paper is concerned with a non-homogeneous multi-point boundary value problem of second order diﬀerential equation with one-dimensional p-Laplacian. Using multiple ﬁxed point theorems, suﬃcient conditions to guarantee the existence of at least three solutions of this kind of BVP are established. Two examples are presented to illustrate the main results.

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[1] Avery R.: A generalization of the Leggett-Williams ﬁxed point theorem. MSR Hot-Line. 3, 9-14 (1999).
[2] Anderson D., Avery R., Henderson J.: Corollary to the ﬁve functionals ﬁxed point theorem. Journal of Nonlinear Studies. 8, 451-464 (2001).
[3] Avery R., Peterson A.: Three Positive Fixed Points of Nonlinear Operators on Ordered Banach Spaces. Comput. Math. Appl. 42, 313-322 (2001).
[4] Feng M., Ge W.: Existence of three positive solutions for m-point boundary-value problems with one-dimensional p-Laplacian. Nonl. Anal. 68, 2017-2026 (2008).
[5] Feng M., Ge W., Jiang M.: Multiple positive solutions for m-point boundary-value problems with a one-dimensional p-Laplacian. Nonl. Anal. 68, 2269-2279 (2008).
[6] Ge W.: Boundary Value Problems for Ordinary Diﬀerential Equations. Beijing. Science Press 2007.
[7] He X., Ge W., Feng M.: Multiple positive solutions for one-dimensional p-Laplacian boundary value problems. Appl. Math. Letters. 15, 937-943 (2002).
[8] He X., Ge, W., Peng H.: Multiple positive solutions for one-dimensional p-Laplacian boundary value problems. Appl. Math. Letters. 15, 937-943 (2002).
[9] Ji D., Feng M., Ge W.: Multiple positive solutions for multi-point boundary value problems with sign changing nonlinearity. Appl. Math. Comput. 196, 511-520 (2008).
[10] Kong L., Kong Q.: Second-order boundary value problems with nonhomogeneous boundary conditions(II). J. Math. Anal. Appl. 330, 1393-1411 (2007).
[11] Kong L., Kong Q.: Multi-point boundary value problems of second order diﬀerential equations(I). Nonl. Anal. 58, 909-931 (2004).
[12] Kong L., Kong Q.: Second-order boundary value problems with nonhomogeneous boundary conditions. Mathema- tische Nachrichten. 278, 173-193 (2005).
[13] Liu Y.: The existence of multiple positive solutions of p-Laplacian boundary value problems. Mathematica Slovaca. 57, 225-242 (2007).
[14] Liu Y.: Solutions of Sturm-Liouville type multi-point boundary value problems for higher-order diﬀerential equa- tions. J. Appl. Math. Computing. 23, 167-182 (2007).
[15] Liu Y.: Solutions of Sturm-Liouville boundary value problems for higher-order diﬀerential equations. J. Appl. Math. Computing. 24, 231-243 (2007).
[16] Liu Y.: Solutions to second order non-homogeneous multi-point BVPs using ﬁve-functionals ﬁxed-point theorem. Electronic Journal of Diﬀ. Eqns. 96, 1-52 (2008).
[17] Liu Y.: Positive solutions of mixed type multi-point non-homogeneous BVPs for p-Laplacian equations, Appl. Math. Comput., 206, 796-805 (2008).
[18] Lian H., Ge W.: Positive solutions for a four-point boundary value problem with the p-Laplacian. Nonl. Anal. 68, 3493-3503 (2008).
[19] Liu Y., Ge W.: Multiple positive solutions to a three-point boundary value problem with p-Laplacian. J. Math. Anal. Appl. 277, 293-302 (2003).
[20] Liu B., Zhao Z.: A note on multi-point boundary value problems. Nonl. Anal. 67, 2680-2689 (2007).
[21] Ma R: Multiple positive solutions for nonlinear m-point boundary value problems. Appl. Math. Comput. 148, 249-262 (2004).
[22] Ma R., Thompson B: Positive solutions for nonlinear m-point eigenvalue problems. J. Math. Anal. Appl. 297, 24-37 (2004).
[23] Pang H., Ge W.: Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line. J. Math. Anal. Appl. 321,781-792 (2006).
[24] Pang H., Lian H., Ge W.: Multiple positive solutions for second-order four-point boundary value problem. Comput. Math. Appl. 54, 1267-1275 (2007).
[25] Sun B., Ge W.: Existence and iteration of positive solutions for some p-Laplacian boundary value problems. Nonl. Anal. 67, 1820-1830 (2007).
[26] Sun B., Qu Y., Ge W.: Existence and iteration of positive solutions for a multipoint one-dimensional p-Laplacian boundary value problem. Appl. Math. Comput. 197, 389-398 (2008).
[27] Sun Y., Zhang X.: Existence of symmetric positive solutions for an m-point boundary value problem. Boundary Value Problems. doi:10.1155/2007/79090 (2007).
[28] Wang Y., Ge W.: Existence of triple positive solutions for multi-point boundary value problems with a one dimensional p-Laplacian. Comput. Math. Appl. 54, 793-807 (2007).
[29] Wang Y., Ge W.: Existence of multiple positive solutions for multi-point boundary value problems with a one- dimensional p-Laplacian. Nonl. Anal. 67, 476-485 (2007).