    ## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020180
##### References:
  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. Google Scholar  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. Google Scholar  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. Google Scholar  C. Bai, Infinitely many solutions for a perturbed nonlinear fractional boundary-value problem, Electron. J. Differential Equations, 136 (2013), 12 pp. Google Scholar  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.  Google Scholar  R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/3779.  Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  M. A. Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$, Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964. Google Scholar  V. Lakshmikantham and J. V. Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1 (2008), 38-45. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367-386. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar

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##### References:
  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. Google Scholar  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. Google Scholar  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. Google Scholar  C. Bai, Infinitely many solutions for a perturbed nonlinear fractional boundary-value problem, Electron. J. Differential Equations, 136 (2013), 12 pp. Google Scholar  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.  Google Scholar  R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/3779.  Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  M. A. Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$, Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964. Google Scholar  V. Lakshmikantham and J. V. Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1 (2008), 38-45. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367-386. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar
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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