We study a fractional integro-differential equation subject to multi-point boundary conditions:
where . By utilizing a new fixed point theorem of increasing 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.
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.
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.
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.
J. Henderson and R. Luca, Existence of nonnegative solutions for a fractional integro-differential equation, Results Math. 72 (2017), 747–763.
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.
Y. Wang and L. Liu, Uniqueness and existence of positive solution for the fractional integro-differential equation, Bound. Value Probl. 2017 (2017), 12.
D. Wang and G. Wang, Integro-differential fractional boundary value problem on an unbounded domain, Adv. Differ. Equ. 2016 (2016), 325.
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.
C. Zhai and L. Wei, The unique positive solution for fractional integro-differential equations on infinite intervals, ScienceAsia 44 (2018), 118–124.
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.
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.
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.
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.
C. Zhai, Fixed point theorems for a class of mixed monotone operator with convexity, Fixed Point Theory Appl. 2013 (2013), 119.
C. Zhai and X. Cao, Fixed point theorems for ##InlineEquation:IEq229##$$τ-φ-$$concave operators and applications, Comput. Math. Appl. 59 (2010), 532–538.
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.
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.
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.
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.
C. Zhai and L. Wang, ##InlineEquation:IEq230##$$φ-(h,e)-$$concave operators and applications. J. Math. Anal. Appl. 454 (2017), 571–584.
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.
C. Zhai, C. Yang and X. Zhang, Positive solutions for nonlinear operator equations and several classes of applications, Math. Z. 266 (2010), 43–63.
Z. Zhao, Existence and uniqueness of fixed points for some mixed monotone operators, Nonlinear Anal. 73(6) (2010), 1481–1490.
Z. Zhao, Fixed points of ##InlineEquation:IEq231##$$τ-φ-$$convex operators and applications, Appl. Math. Lett. 23(5) (2010), 561–566.
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.
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.
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.
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.
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.
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.
I. Podlubny, Fractional differential equation, in: Mathematics in Sciences and Engineering, vol. 198, Academic Press, San Diego, 1999.
The International Journal of Nonlinear Sciences and Numerical Simulation publishes original papers on all subjects relevant to nonlinear sciences and numerical simulation. The journal is directed at researchers in nonlinear sciences, engineers, and computational scientists, economists, and others, who either study the nature of nonlinear problems or conduct numerical simulations of nonlinear problems.