Abstract
In this paper, we study a class of nonlinear operator equations x = Ax + x 0 on ordered Banach spaces, where A is a monotone generalized concave operator. Using the properties of cones and monotone iterative technique, we establish the existence and uniqueness of solutions for such equations. In particular, we do not demand the existence of upper-lower solutions and compactness and continuity conditions. As applications, we study first-order initial value problems and two-point boundary value problems with the nonlinear term is required to be monotone in its second argument. In the end, applications to nonlinear systems of equations and to nonlinear matrix equations are also considered.
Similar content being viewed by others
References
Agarwal R.P., Wong P.J.Y., Regan D.O.: Positive Solutions of Differential, Difference, Integral Equations. Kluwer Academic, Boston (1999)
Agarwal R.P., Regan D.O.: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem. Appl. Math. Comput. 161, 433–439 (2005)
Amann H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18(4), 620–709 (1976)
Avery R.I., Henderson J.: Three symmetric positive solutions for a second order boundary value problem. Appl. Math. Lett. 13(3), 1–7 (2000)
Avery R.I., Peterson A.C.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42, 313–322 (2001)
Bushell P.J.: On a class of Volterra and Fredholm nonlinear integral equations. Math. Proc. Camb. Phil. Soc. 79, 329–335 (1976)
Chen Y.Z.: Continuation method for α-sublinear mappings. Proc. Am. Math. Soc. 129, 203–210 (2001)
Deimling K.: Nonlinear Functional Analysis. Springer-Varlag, Berlin (1985)
Du S.W., Lakshmikantham V.: Monotone iterative technique for differential equations in Banach space. J. Math. Anal. Appl. 87, 454–459 (1982)
Guo D.J.: Fixed points and eigenelements of a class of concavex and convex operator (in Chinese). Chinese Sci. Bull. 15, 1132–1135 (1985)
Guo D.J.: Existence and uniqueness of positive fixed points for mixed monotone operators and applications. Appl. Anal. 46, 91–100 (1992)
Guo D.J.: Existence and nuiqueness of positive fixed points for noncompact decreasing operators. Indian J. Pure. Appl. Math. 31(5), 551–562 (2000)
Guo D.J., Lakshmikantham V.: Nonlinear Problems in Abstract Cones. Academic Press Inc (2), Boston (1988)
Guo D.J., Lakshmikantham V., Liu X.Z.: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic Publishers, Dordrecht (1996)
Krasnosel’skii M.A.: Positive Solutions of Operators Equations. Noordoff, Groningen (1964)
Ladde, G.S., Lakshmikantham, V., Vatsala, A.S.: Monotone Iterative Techniques for Differential Equations. Pitman (1985)
Lakshmikantham V., Leela S., Oguztoreli M.N.: Quasi-solutions, veter Lyapunov functions and monotone methods. IEEE. Trans. Autom. Control. 26, 1149–1153 (1981)
Li F.Y.: Existence and uniqueness of positive solutions of some nonlinear equations. Acta. Math. Appl. Sinica. 20(4), 609–615 (1997)
Li K., Liang J., Xiao T.J.: Positive fixed points for nonlinear operators. Comput. Math. Appl. 50, 1569–1578 (2005)
Li K., Liang J., Xiao T.J.: A fixed point theorem for convex and decreasing operators. Nonlinear Anal. 63, e209–e206 (2005)
Liang Z.D., Wang W.X., Li S.J.: On concave operators. Acta. Math. Sinica (Engl. Ser.) 22(2), 577–582 (2006)
Loewner C.: Uber monotone matrixfunktionen. Math. Z. 38, 177–216 (1934)
Ly I., Seck D.: Isoperimetric inequality for an interior free boundary problem with p-Laplacian operator. Electron. J. Differ. Equ. 109, 1–12 (2004)
Ma D., Du Z., Ge W.: Existence and iteration of monotone positive solutions for multipoint boundary value problem with p-Laplacian operator. Comput. Math. Appl. 50, 729–739 (2005)
Regan D.O.: Some general existence principles results for (ψ p (y′))′ = q(t) f (t, y, y′), 0 < t < 1. SIAM J. Math. Anal. 24(3), 648–668 (1993)
Potter A.J.B.: Applications of Hilbert’s projective metric to certain class of nonhomogeneous operators. Quart. J. Math. Oxford. 28(2), 93–99 (1977)
Wan W.X.: Conditions for contraction of mappings and Banach type fixed point theorem (in Chinese). Acta. Math. Sinica. 27, 35–52 (1984)
Wang W.X., Liang Z.D.: Fixed point theorems of a class of nonlinear operators and applications (in Chinese). Acta. Math. Sinica. 48(4), 789–800 (2005)
Wang Y., Ge W.: Positive solutions for multipoint boundary value problems with a one-dimensional p-Laplacian. Nonlinear Anal. 66, 1246–1256 (2007)
Yang C., Zhai C.B., Yan J.R.: Positive solutions of three-point boundary value problem for second order differential equations with an advanced argument. Nonlinear Anal. 65, 2013–2023 (2006)
Zhai C.B., Guo C.M.: On α-convex operators. J. Math. Anal. Appl. 316, 556–565 (2006)
Zhai C.B., Yang C., Guo C.M.: Positive solutions of operator equation on ordered Banach spaces and applications. Comput. Math. Appl. 56, 3150–3156 (2008)
Zhai C.B., Wang W.X., Zhang L.L.: Generalization for a class of concave and convex operators (in Chinese). Acta. Math. Sinica. 51(3), 529–540 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to professor Zhandong Liang and Jurang Yan on their 70th birthdays.
Rights and permissions
About this article
Cite this article
Zhai, CB., Yang, C. & Zhang, XQ. Positive solutions for nonlinear operator equations and several classes of applications. Math. Z. 266, 43–63 (2010). https://doi.org/10.1007/s00209-009-0553-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-009-0553-4
Keywords
- Positive solution
- Nonlinear operator equation
- Normal cone
- Initial value problem
- Boundary value problem
- Nonlinear algebra systems