Fixed point results in set $P_{h,e}$ with applications to fractional differential equations

Lingling Zhang, Hui Wang, Xiaoqiang Wang

DOI: http://dx.doi.org/10.12775/TMNA.2019.052

Abstract


In this paper, without assuming operators to be continuous or compact, by employing monotone iterative technique on ordered Banach space, we at first establish new fixed point theorems for some kinds of nonlinear mixed monotone operators in set $P_{h,e}$. Then, we study a new form of fractional two point boundary value problem depending on a certain constant and give the existence and uniqueness of solutions. We also show that the unique solution exists in set $P_{h,e}$ or $P_{h}$ and can be uniformly approximated by constructing two iterative sequences for any initial values. At the end, a concrete example is given to illustrate our abstract results. The conclusions of this article enrich the fixed point theorems and provide new methods to deal with nonlinear differential equations.

Keywords


Fixed point theorem; mixed monotone operator; set $P_{h;e}$; existence and uniqueness; fractional differential equations

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References


I. Azman, M. Jlelia, B. López, K. Sadarangani and B. Samet, Positive solutions for a class of fractional boundary value problems with fractional boundary conditions, J. Nonlinear Sci. Appl. 11 (2018), 237–251.

I.J. Cabrera, B. López and K. Sadarangani, Existence of positive solutions for the nonlinear elastic beam equation via a mixed monotone operator, J. Comput. Appl. Math. 327 (2018), 306–313.

H.X. Feng and C.B. Zhai, Existence and uniqueness of positive solutions for a class of fractional differential equation with integral boundary conditions, Nonlinear Anal. Model. Control. 22 (2017), 160–172.

C.S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010), 1050–1055.

D.J. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11 (1987), 623–632.

D.J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, York, 1988.

M. Jleli and B. Samet, Existence of positive solutions to an arbitrary order fractional differential equation via a mixed monotone operator method, Nonlinear Anal. Model. Control. 20 (2015), 367–376.

V. Lakshmikantham and A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. 69 (2008), 2677–2682.

Z.D. Liang, L.L. Zhang and S.J. Li, Fixed point theorems for a class of mixed monotone operators, Z. Anal. Anwend. 22 (2003), 529–542.

X.N. Lin, X.Y. Li and D.Q. Jiang, Existence and uniqueness of solutions for singular fourth-order boundary value problems, J. Comput. Appl. Math. 196 (2006), 155–161.

L.S. Liu, X.Q. Zhang, J. Jiang and Y.H. Wu, The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems, J. Nonlinear Sci. Appl. 9 (2016), 2943–2958.

D.D. Min, L.S. Liu and Y.H. Wu, Uniqueness of positive solutions for the singular fractional differential equations involving integral boundary value conditions, Bound. Value Probl. 2018 (2018), 1–23.

H. Wang and L.L. Zhang, The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems, Bound. Value Probl. 2015 (2015), 1–16.

D. Wardowski, Mixed monotone operators and their application to integral equations, J. Fixed Point Theory Appl. 19 (2017), 1103–1117.

Y.X. Wu, New fixed point theorems and applications of mixed monotone operator, J. Math. Anal. Appl. 341 (2008), 883–893.

Y.X. Wu and Z.D. Liang, Existence and uniqueness of fixed points for mixed monotone operators with applications, Nonlinear Anal. 65 (2006), 1913–1924.

H.Y. Xu, New fixed point theorems of mixed monotone operators and applications to singular boundary value problems on time scales, Bound. Value Probl. 2011 (2011), 1–14.

C. Yang, M.R. Hao and C.B. Zhai, Uniqueness of positive solutions for several classes of sum operator equations and applications, J. Inequal. Appl. 58 (2014), 1–28.

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

C.B. Zhai and M.R. Hao, Fixed point theorems for mixed monotone operator with perturbation and application to fractional differential equation boundary value problems, Nonlinear Anal. 75 (2012), 2542–2551.

C.B. Zhai and M.R. Hao, Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann–Liouville fractional differential equation boundary value problems, Bound. Value Probl. 2013 (2013), 1–13.

C.B. Zhai and R.T. Jiang, Unique solutions for a new coupled system of fractional differential equations, Adv. Difference Equ. 2018 (2018), 1–12.

C.B. Zhai and J. Ren, Some properties of sets, fixed point theorems in ordered product spaces and applications to a nonlinear system of fractional differential equations, Topol. Methods Nonlinear Anal. 49 (2017), no. 2, 625–645.

C.B. Zhai and J. Ren, The unique solution for a fractional q-difference equation with three-point boundary conditions, Indag. Math. 29 (2018), 948–961.

C.B. Zhai and L. Wang, ϕ-(h, e)-concave operators and applications, J. Math. Anal. Appl. 454 (2017), 571–584.

C.B. Zhai, C. Yang and X.Q. Zhang, Positive solutions for nonlinear operator equations and several classes of applications, Math. Z. 266 (2010), 43–63.

C.B. Zhai and L.L. Zhang, New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems, J. Math. Anal. Appl. 382 (2011), 594–614.

L.L. Zhang, N. Yamazaki and C.B. Zhai, Optimal control problem of positive solutions to second order impulsive differential equations, Z. Anal. Anwend. 31 (2012), 237–250.

X.Q. Zhang, L.S. Liu and Y.H. Wu, Fixed point theorems for the sum of three classes of mixed monotone operators and applications, Fixed Point Theory Appl. 49 (2016), 1–22.

Z.Q. Zhao, Existence and uniqueness of fixed points for some mixed monotone operators, Nonlinear Anal. 73 (2010), 1481–1490.


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