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March  2021, 26(3): 1723-1747. doi: 10.3934/dcdsb.2020180

## Existence results for fractional differential equations in presence of upper and lower solutions

 1 Université de Monastir, Faculté des Sciences de Monastir, , LR18ES17, 5000 Monastir, Tunisie 2 Universidade de Santiago de Compostela, Instituto de Matemáticas, Santiago de Compostela, 15782, Spain

* Corresponding author: Universidade de Santiago de Compostela, Instituto de Matemáticas, Santiago de Compostela, 15782, Spain

Received  August 2019 Revised  April 2020 Published  March 2021 Early access  June 2020

Fund Project: The second author is supported by grant numbers MTM2016-75140-P (AEI/FEDER, UE) and ED431C 2019/02 (GRC Xunta de Galicia)

In this paper, we study some existence results for fractional differential equations subject to some kind of initial conditions. First, we focus on the linear problem and we give an explicit form of solutions by reduction to an integral problem. We analyze some properties of the solutions to the linear problem in terms of its coefficients. Then we provide examples of application of the mentioned properties. Secondly, with the help of this theory, we study the nonlinear problem subject to initial value conditions. By using the upper and lower solutions method and the monotone iterative algorithm, we show the existence and localization of solutions to the nonlinear fractional differential equation.

Citation: Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180
##### References:
 [1] T. M. Atanacković, S. Pilipović and B. Stanković, A theory of linear differential equations with fractional derivatives, Bull. Cl. Sci. Math. Nat. Sci. Math., 37 (2012), 71-95. [2] T. M. Atanacković and B. Stanković, Linear fractional differential equation with variable coefficients Ⅰ, Bull. Cl. Sci. Math. Nat. Sci. Math., 38 (2013), 27-42. [3] T. M. Atanacković and and B. Stanković, Linear fractional differential equation with variable coefficients Ⅱ, Bull. Cl. Sci. Math. Nat. Sci. Math., 39 (2014), 53-78. [4] C. Bai, Infinitely many solutions for a perturbed nonlinear fractional boundary-value problem, Electron. J. Differential Equations, 136 (2013), 12 pp. [5] R. Bourguiba and F. Toumi, Existence results for semipositone fractional differential equation, Rocky Mountain J. Math., 49 (2019), 2495-2512.  doi: 10.1216/RMJ-2019-49-8-2495. [6] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/3779. [7] F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62 (2011), 1181-1199.  doi: 10.1016/j.camwa.2011.03.086. [8] A. A. Kilbas and M. Saǐgo, Solution in closed form of a class of linear differential equations of fractional order, Differ. Uravn., 33 (1997), 195-204. [9] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics studies, 204, Elsevier Science B.V., Amsterdam, 2006. [10] Q. Kong and M. Wang, Positive solutions of nonlinear fractional boundary value problems with Dirichlet boundary conditions, Electron. J. Qual. Theory Differ. Equ., 17 (2012), 13 pp. doi: 10.14232/ejqtde.2012.1.17. [11] M. A. Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$, Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964. [12] V. Lakshmikantham and J. V. Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1 (2008), 38-45. [13] F. Li, M. Jia, X. Liu, C. Li and G. Li, Existence and uniqueness of solutions of second-order three-point boundary-value problems with upper and lower solutions in the reverse order, Nonlinear Anal., 68 (2008), 2381-2388.  doi: 10.1016/j.na.2007.01.065. [14] J. J. Nieto, Differential inequalities for functional perturbations of first-order ordinary differential equations, Appl. Math. Lett., 15 (2002), 173-179.  doi: 10.1016/S0893-9659(01)00114-8. [15] J. J. Nieto and R. Rodríguez-López, Monotone method for first-order functional differential equations, Comput. Math. Appl., 52 (2006), 471-484.  doi: 10.1016/j.camwa.2006.01.012. [16] I. Podlubny, The Laplace Transform Method for Linear Differential Equations of the Fractional Order, Inst. Expe. Phys., Slov. Acad. Sci., UEF-02-94, Kosice, 1994. [17] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. [18] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367-386. [19] W. Pogorzelski, Integral Equation and their Applications Vol. I, International Series of Monographs in Pure and Applied Mathematics, 88, Pergamon Press, Oxford-New York-Frankfurt; PWN-Polish Scientific Publishers, Warsaw, 1966. [20] H. Qin, X. Zuo, J.Liu and L. Liu, Approximate controllability and optimal controls of fractional dynamical systems of order $1 < q < 2$ in Banach spaces, Adv. Difference Equ., 73 (2015), 17 pp. doi: 10.1186/s13662-015-0399-5. [21] T. Ren, S. Li, X. Zhang and L. Liu, Maximum and minimum solutions for a nonlocal $p$-Laplacian fractional differential system from economical processes, Bound. Value Probl., 118 (2017), 15 pp. doi: 10.1186/s13661-017-0849-y. [22] M. Rivero, L. Rodríguez-Germá and J. J. Trujillo, Linear fractional differential equations with variable coefficients, Appl. Math. Lett., 21 (2008), 892-897.  doi: 10.1016/j.aml.2007.09.010. [23] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. doi: 10.1007/978-1-4612-0873-0. [24] V. E. Tarasov, Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Nonlinear Physical Science, Springer, Heidelberg; Higher Education Press, Beijing, 2010. doi: 10.1007/978-3-642-14003-7. [25] J. Wu, X. Zhang, L. Liu and Y. Wu, Positive solution of singular fractional differential system with nonlocal boundary conditions, Adv. Difference Equ., 323 (2014), 15 pp. doi: 10.1186/1687-1847-2014-323. [26] 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.  doi: 10.1016/j.amc.2014.07.043. [27] X. Zhang, L. Liu and Y. Wu, Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives, Appl. Math. Comput., 19 (2012), 1420-1433.  doi: 10.1016/j.amc.2012.07.046. [28] X. Zhang, Y. Wu and Y. Cui, Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator, Appl. Math. Lett., 82 (2018), 85-91.  doi: 10.1016/j.aml.2018.02.019.

show all references

##### References:
 [1] T. M. Atanacković, S. Pilipović and B. Stanković, A theory of linear differential equations with fractional derivatives, Bull. Cl. Sci. Math. Nat. Sci. Math., 37 (2012), 71-95. [2] T. M. Atanacković and B. Stanković, Linear fractional differential equation with variable coefficients Ⅰ, Bull. Cl. Sci. Math. Nat. Sci. Math., 38 (2013), 27-42. [3] T. M. Atanacković and and B. Stanković, Linear fractional differential equation with variable coefficients Ⅱ, Bull. Cl. Sci. Math. Nat. Sci. Math., 39 (2014), 53-78. [4] C. Bai, Infinitely many solutions for a perturbed nonlinear fractional boundary-value problem, Electron. J. Differential Equations, 136 (2013), 12 pp. [5] R. Bourguiba and F. Toumi, Existence results for semipositone fractional differential equation, Rocky Mountain J. Math., 49 (2019), 2495-2512.  doi: 10.1216/RMJ-2019-49-8-2495. [6] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/3779. [7] F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62 (2011), 1181-1199.  doi: 10.1016/j.camwa.2011.03.086. [8] A. A. Kilbas and M. Saǐgo, Solution in closed form of a class of linear differential equations of fractional order, Differ. Uravn., 33 (1997), 195-204. [9] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics studies, 204, Elsevier Science B.V., Amsterdam, 2006. [10] Q. Kong and M. Wang, Positive solutions of nonlinear fractional boundary value problems with Dirichlet boundary conditions, Electron. J. Qual. Theory Differ. Equ., 17 (2012), 13 pp. doi: 10.14232/ejqtde.2012.1.17. [11] M. A. Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$, Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964. [12] V. Lakshmikantham and J. V. Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1 (2008), 38-45. [13] F. Li, M. Jia, X. Liu, C. Li and G. Li, Existence and uniqueness of solutions of second-order three-point boundary-value problems with upper and lower solutions in the reverse order, Nonlinear Anal., 68 (2008), 2381-2388.  doi: 10.1016/j.na.2007.01.065. [14] J. J. Nieto, Differential inequalities for functional perturbations of first-order ordinary differential equations, Appl. Math. Lett., 15 (2002), 173-179.  doi: 10.1016/S0893-9659(01)00114-8. [15] J. J. Nieto and R. Rodríguez-López, Monotone method for first-order functional differential equations, Comput. Math. Appl., 52 (2006), 471-484.  doi: 10.1016/j.camwa.2006.01.012. [16] I. Podlubny, The Laplace Transform Method for Linear Differential Equations of the Fractional Order, Inst. Expe. Phys., Slov. Acad. Sci., UEF-02-94, Kosice, 1994. [17] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. [18] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367-386. [19] W. Pogorzelski, Integral Equation and their Applications Vol. I, International Series of Monographs in Pure and Applied Mathematics, 88, Pergamon Press, Oxford-New York-Frankfurt; PWN-Polish Scientific Publishers, Warsaw, 1966. [20] H. Qin, X. Zuo, J.Liu and L. Liu, Approximate controllability and optimal controls of fractional dynamical systems of order $1 < q < 2$ in Banach spaces, Adv. Difference Equ., 73 (2015), 17 pp. doi: 10.1186/s13662-015-0399-5. [21] T. Ren, S. Li, X. Zhang and L. Liu, Maximum and minimum solutions for a nonlocal $p$-Laplacian fractional differential system from economical processes, Bound. Value Probl., 118 (2017), 15 pp. doi: 10.1186/s13661-017-0849-y. [22] M. Rivero, L. Rodríguez-Germá and J. J. Trujillo, Linear fractional differential equations with variable coefficients, Appl. Math. Lett., 21 (2008), 892-897.  doi: 10.1016/j.aml.2007.09.010. [23] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. doi: 10.1007/978-1-4612-0873-0. [24] V. E. Tarasov, Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Nonlinear Physical Science, Springer, Heidelberg; Higher Education Press, Beijing, 2010. doi: 10.1007/978-3-642-14003-7. [25] J. Wu, X. Zhang, L. Liu and Y. Wu, Positive solution of singular fractional differential system with nonlocal boundary conditions, Adv. Difference Equ., 323 (2014), 15 pp. doi: 10.1186/1687-1847-2014-323. [26] 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.  doi: 10.1016/j.amc.2014.07.043. [27] X. Zhang, L. Liu and Y. Wu, Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives, Appl. Math. Comput., 19 (2012), 1420-1433.  doi: 10.1016/j.amc.2012.07.046. [28] X. Zhang, Y. Wu and Y. Cui, Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator, Appl. Math. Lett., 82 (2018), 85-91.  doi: 10.1016/j.aml.2018.02.019.
Graph of the solution to (17)
Graph of the solution for a negative forcing term
Sufficient conditions for the nonnegativity of solutions
 $\begin{array}{c} f \geq 0 \mbox{ on } [0,b] \\ c\geq 0 \\ \lambda \geq 0 \\ R\geq 0 \end{array}$ $\begin{array}{c} f \geq 0 \mbox{ on } [0,b] \\ c\geq 0 \\ \lambda < 0 \\ R\leq 0\end{array}$
 $\begin{array}{c} f \geq 0 \mbox{ on } [0,b] \\ c\geq 0 \\ \lambda \geq 0 \\ R\geq 0 \end{array}$ $\begin{array}{c} f \geq 0 \mbox{ on } [0,b] \\ c\geq 0 \\ \lambda < 0 \\ R\leq 0\end{array}$
Sufficient conditions for the nonpositivity of solutions
 $\begin{array}{c} f \leq 0 \mbox{ on } [0,b] \\ c\leq 0 \\ \lambda \geq 0 \\ R\geq 0 \end{array}$ $\begin{array}{c} f \leq 0 \mbox{ on } [0,b] \\ c\leq 0 \\ \lambda\geq 0 \\ R\leq 0\end{array}$
 $\begin{array}{c} f \leq 0 \mbox{ on } [0,b] \\ c\leq 0 \\ \lambda \geq 0 \\ R\geq 0 \end{array}$ $\begin{array}{c} f \leq 0 \mbox{ on } [0,b] \\ c\leq 0 \\ \lambda\geq 0 \\ R\leq 0\end{array}$
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