Positive solutions to periodic boundary value problems involving nonlinear operators of Meir-Keeler-type

Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)

Avery, R.I., Peterson, A.C.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42(3–5), 313–322 (2001)

Bai, Z.: The upper and lower solution method for some fourth-order boundary value problems. Nonlinear Anal. 67, 1704–1709 (2007)

Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)

Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)

Chen, Y.Z.: A variant of the Meir-Keeler-Type theorem in ordered Banach spaces. J. Math. Anal. Appl. 236, 585–593 (1999)

Ćirić, Lj.: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 45, 267–273 (1974)

Samet, B.: Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 72, 4508–4517 (2010)

Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

Dugundji, J., Granas, A.: Fixed Point Theory. Springer, Berlin (2003)

Edelstein, M.: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. 12, 7–10 (1961)

Graef, J.R., Yang, B.: Existence and nonexistence of positive solutions of fourth-order nonlinear boundary value problems. Appl. Anal. 74(1–2), 201–214 (2000)

Guo, D.J., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988)

Jachymski, J.R.: Equivalent conditions and the Meir-Keeler type theorems. J. Math. Anal. Appl. 194, 293–303 (1995)

Jachymski, J.R.: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125, 2327–2335 (1997)

Lakshmikantham, V., Leela, S.: Remarks on first and second order periodic boundary value problem. Nonlinear Anal. 8, 281–287 (1984)

Leela, S.: Monotone method for second periodic boundary value problems. Nonlinear Anal. 7, 349–355 (1983)

Meir, A., Keeler, E.: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)

Nashine, H.K., Samet, B., Vetro, C.: Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces. Math. Comput. Model. 54, 712–720 (2011)

Nieto, J.J.: Nonlinear second order periodic value problems with Caratheodory functions. Appl. Anal. 34, 111–128 (1989)

Nieto, J.J.: Periodic boundary value problem for second order integro-ordinary differential equations with general kernel and Caratheodory nonlinearities. Int. J. Math. Math. Sci. 18, 757–764 (1985)

Nussbaum, R.D.: Iterated Nonlinear Maps and Hilbert’s Projective Metric. Mem. Amer. Math. Soc., vol. 75. American Mathematical Society, Providence (1988)

Ping, S.Y.: Existence and multiplicity of positive solutions for an elastic beam equation. Appl. Math. J. Chin. Univ. 26(3), 253–264 (2011)

Rakotch, E.: A note on contractive mappings. Proc. Am. Math. Soc. 13, 459–465 (1962)

Suzuki, T.: A generalized Banach contraction principle which characterizes metric completeness. Proc. Am. Math. Soc. 136, 1861–1869 (2008)

Thompson, A.C.: On certain contraction mappings in a partially ordered vector space. Proc. Am. Math. Soc. 14, 438–443 (1963)

Vetro, C.: On Branciari’s theorem for weakly compatible mappings. Appl. Math. Lett. 23(6), 700–705 (2010)

Yang, Y.R.: Triple positive solutions of a class of fourth-order two-point boundary value problems. Appl. Math. Lett. 23, 366–370 (2010)

Yao, Q.: Positive solutions of nonlinear second-order periodic boundary value problems. Appl. Math. Lett. 20, 583–590 (2007)

Zeidler, E.: Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems. Springer, Berlin (1986)

Zhai, C., Anderson, D.R.: A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations. J. Math. Anal. Appl. 375, 388–400 (2011)