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Unique Solution for Multi-point Fractional Integro-Differential Equations

  • Chengbo Zhai EMAIL logo and Lifang Wei
Published/Copyright: July 10, 2019

Abstract

We study a fractional integro-differential equation subject to multi-point boundary conditions:

D0+αu(t)+f(t,u(t),Tu(t),Su(t))=b, t(0,1),u(0)=u(0)==u(n2)(0)=0,D0+pu(t)|t=1=i=1maiD0+qu(t)|t=ξi,

where α(n1,n], nN, n3, ai0, 0<ξ1<<ξm1, p[1,n2], q[0,p],b>0. By utilizing a new fixed point theorem of increasing ψ(h,r) concave operators defined on special sets in ordered spaces, we demonstrate existence and uniqueness of solutions for this problem. Besides, it is shown that an iterative sequence can be constructed to approximate the unique solution. Finally, the main result is illustrated with the aid of an example.

MSC 2010: 34A08; 34B15; 34B18; 34B40; 45J05

Acknowledgements

This paper was supported financially by the Youth Science Foundation of China (11201272), Shanxi Province Science Foundation (2015011005) and Shanxi Scholarship Council of China (2016–009).

References

[1] B. Ahmad and R. Luca, Existence of solutions for a sequential fractional integro-differential system with coupled integral boundary conditions, Chaos Solitons Fractals 104 (2017), 378–388.10.1016/j.chaos.2017.08.035Search in Google Scholar

[2] K. Balachandran, S. Kiruthika and J. J. Trujillo, Existence results for fractional impulsive integro-differential equation in Banach space, Commun. Nonlinear Sci. Number. Simul. 16 (2011), 1970–1977.10.1016/j.cnsns.2010.08.005Search in Google Scholar

[3] K. Balachandran and J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integro differential equations in Banach spaces, Nonlinear Anal. TMA 72 (2010), 4587–4593.10.1016/j.na.2010.02.035Search in Google Scholar

[4] J. Henderson and R. Luca, Existence of nonnegative solutions for a fractional integro-differential equation, Results Math. 72 (2017), 747–763.10.1007/s00025-017-0655-ySearch in Google Scholar

[5] K. Pei, G. Wang and Y. Sun, Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain, Appl. Math. Comput. 312 (2017), 158–168.10.1016/j.amc.2017.05.056Search in Google Scholar

[6] Y. Wang and L. Liu, Uniqueness and existence of positive solution for the fractional integro-differential equation, Bound. Value Probl. 2017 (2017), 12.10.1186/s13661-016-0741-1Search in Google Scholar

[7] D. Wang and G. Wang, Integro-differential fractional boundary value problem on an unbounded domain, Adv. Differ. Equ. 2016 (2016), 325.10.1186/s13662-016-1051-8Search in Google Scholar

[8] N. Xu and W. Liu, Iterative solutions for a coupled system of fractional differential-integral equations with two-point boundary conditions, Appl. Math. Comput. 244 (2014), 903–911.10.1016/j.amc.2014.07.043Search in Google Scholar

[9] C. Zhai and L. Wei, The unique positive solution for fractional integro-differential equations on infinite intervals, ScienceAsia 44 (2018), 118–124.10.2306/scienceasia1513-1874.2018.44.118Search in Google Scholar

[10] L. Zhang, B. Ahmad, G. Wang and Ravi P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. App. Math. 249 (2013), 51–56.10.1016/j.cam.2013.02.010Search in Google Scholar

[11] Y. Cui and J. Sun, Fixed point theorems for a class of nonlinear operators in Hilbert spaces with Lattice structure and application, Fixed Point Theory Appl. 2013 (2013), 345.10.1186/1687-1812-2013-345Search in Google Scholar

[12] X. Li and Z. Zhao, On a fixed point theorem of mixed monotone operators and applications, Electron. J. Qual. Theory Differ. Equa. 94 (2012), 1–7.10.14232/ejqtde.2011.1.94Search in Google Scholar

[13] J. Sun and Y. Cui, Fixed point theorems for a class of nonlinear operators in Riesz spaces, Fixed Point Theory 14(1) (2013), 185–192.10.1186/1687-1812-2013-119Search in Google Scholar

[14] C. Zhai, Fixed point theorems for a class of mixed monotone operator with convexity, Fixed Point Theory Appl. 2013 (2013), 119.10.1186/1687-1812-2013-119Search in Google Scholar

[15] C. Zhai and X. Cao, Fixed point theorems for ##InlineEquation:IEq229##$$&#x03C4;-&#x03C6;-$$concave operators and applications, Comput. Math. Appl. 59 (2010), 532–538.10.1016/j.camwa.2009.06.016Search in Google Scholar

[16] C. Zhai and D. R. Anderson, A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations, J. Math. Anal. Appl. 375 (2011), 388–400.10.1016/j.jmaa.2010.09.017Search in Google Scholar

[17] C. Zhai and M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems, Nonlinear Anal. 75 (2012), 2542–2551.10.1016/j.na.2011.10.048Search in Google Scholar

[18] C. 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(2) (2017), 625–645.10.12775/TMNA.2016.095Search in Google Scholar

[19] C. Zhai and F. Wang, Properties of positive solutions for the operator equation Ax = λx and applications to fractional differential equations with integral boundary conditions, Adv. Differ. Equ. 2015 (2015), 366.10.1186/s13662-015-0704-3Search in Google Scholar

[20] C. Zhai and L. Wang, ##InlineEquation:IEq230##$$&#x03C6;-(h,e)-$$concave operators and applications. J. Math. Anal. Appl. 454 (2017), 571–584.10.1016/j.jmaa.2017.05.010Search in Google Scholar

[21] C. Zhai and 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), 594–614.10.1016/j.jmaa.2011.04.066Search in Google Scholar

[22] C. Zhai, C. Yang and X. Zhang, Positive solutions for nonlinear operator equations and several classes of applications, Math. Z. 266 (2010), 43–63.10.1007/s00209-009-0553-4Search in Google Scholar

[23] Z. Zhao, Existence and uniqueness of fixed points for some mixed monotone operators, Nonlinear Anal. 73(6) (2010), 1481–1490.10.1016/j.na.2010.04.008Search in Google Scholar

[24] Z. Zhao, Fixed points of ##InlineEquation:IEq231##$$&#x03C4;-&#x03C6;-$$convex operators and applications, Appl. Math. Lett. 23(5) (2010), 561–566.10.1016/j.aml.2010.01.011Search in Google Scholar

[25] Z. Zhao, Existence of fixed points for some convex operators and applications to multi-point boundary value problems, Appl. Math. Comput. 215(8) (2009), 2971–2977.10.1016/j.amc.2009.09.044Search in Google Scholar

[26] X. Zhang and Q. Zhong, Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions, Fract. Calc. Appl. Anal. 20(6) (2017), 1471–1484.10.1515/fca-2017-0077Search in Google Scholar

[27] X. Zhang and Q. Zhong, Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables, Appl. Math. Lett. 80 (2018), 12–19.10.1016/j.aml.2017.12.022Search in Google Scholar

[28] X. Zhang, Z. Shao, Q. Zhong and Z. Zhao, Triple positive solutions for semipositone fractional differential equations m-point boundary value problems with singularities and ##InlineEquation:IEq232##$$p-q-$$order derivatives, Nonlinear Anal. Model. Control 23(6) (2018), 889–903.10.15388/NA.2018.6.5Search in Google Scholar

[29] R. Pu, X. Zhang, Y. Cui, P. Li and W. Wang, Positive solutions for singular semipositone fractional differential equation subject to multipoint boundary conditions, J. Funct. Spaces 2017 (2017), Article ID 5892616.10.1155/2017/5892616Search in Google Scholar

[30] J. Henderson and R. Luca, Existence of positive solutions for a singular fractional boundary value problem, Nonlinear Anal. Model. Control 22(1) (2017), 99–114.10.15388/NA.2017.1.7Search in Google Scholar

[31] I. Podlubny, Fractional differential equation, in: Mathematics in Sciences and Engineering, vol. 198, Academic Press, San Diego, 1999.Search in Google Scholar

Received: 2019-01-28
Accepted: 2019-06-25
Published Online: 2019-07-10
Published in Print: 2020-04-26

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