Document Type : Research Paper


Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.


In this paper, we investigate the existence of a solution for the fractional q-integro-differential inclusion with new double sum and product boundary conditions. One of the most recent techniques of fixed point theory, i.e. endpoints property, and inequalities, plays a central role in proving the main results. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables and some figures. The paper ends with an example.


Main Subjects

[1] C.R. Adams, The General Theory of a Class of Linear Partial $q$-Difference Equations, Trans. Amer. Math. Soc., 26(3) (1924) , pp. 283-312.
[2] R.P. Agarwal, B. Ahmad, A. Alsaedi and N. Shahzad, Existence and dimension of the set of mild solutions to semilinear fractional differential inclusions, Adv. Difference Equ., 74 (2012) pp. 1-10.
[3] B. Ahmad, A. Alsaedi and S.K. Ntouyas, A study of second-order q-difference equations with boundary conditions, Nonlinear Anal., 35 (2012), pp. 1-10.
[4] H. Akca, J. Benbourenane and H. Eleuch, The q-derivative and differential equation, J. Phys. Conf. Ser., 1411(1) (2019), 012002.
[5] A. Alalyani and S. Saber, Stability analysis and numerical simulations of the fractional COVID-19 pandemic model, Int. J. Nonlinear Sci. Numer. Simul., 24(3) (2022), pp. 989-1002. 
[6] A. Amini-Harandi, Endpoints of set-valued contractions in metric spaces, Nonlinear Anal., 72(1) (2010), pp. 132-134.
[7] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20(2) (2016), pp. 763-769. 
[8] M. Bohner, O. Tunc and C. Tunc, Qualitative analysis of caputo fractional integro-differential equations with constant delays, Comp. Appl. Math., 6(40) (2021), pp. 214.
[9] A. Boutiara, J. Alzabut, M. Ghaderi and Sh. Rezapour, On a coupled system of fractional $(p, q)$-differential equation with Lipschitzian matrix in generalized metric space, AIMS Math., 8(1) (2022), pp. 1566-1591.
[10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differentiation App., 1(2) (2015), pp. 73-85.
[11] A. Carpinteri and F. Mainardi, Fractals and fractional calculus in continuum mechanics, Springer-Verlag Wien, New York, 1997.
[12] H. Covitz and S.B. Nadler, Multivalued contraction mappings in generalized metric spaces, Israel J. Math., 8 (1970), pp. 5-11. 
[13] A. Din, Y. Li and M.A. Shah, The complex dynamics of hepatitis B infected individuals with optimal control, J. Syst. Sci. Complex., 34(4) (2021), pp. 1301-1323. 
[14] F.Z. El-Emam, Convolution conditions for two subclasses of analytic functions defined by Jackson q-difference operator, J. Egypt. Math. Soc., 7 (2022), pp. 1-10.
[15] M. El-Shahed and F.M. Al-Askar, Positive Solutions for Boundary Value Problem of Nonlinear Fractional $q$-Difference Equation, ISRN Math. Anal., (2011), 385459.
[16] M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Rational Mech. Anal., 198(1) (2010), pp. 189-232.
[17] R. George, F. Al-shammari, M. Ghaderi and Sh. Rezapour, On the boundedness of the solution set for the $\psi$-Caputo fractional pantograph equation with a measure of non-compactness via simulation analysis, AIMS Math., 8(9) (2023), pp. 20125-20142.
[18] R. George, M. Aydogan, F.M. Sakar, M. Ghaderi and Sh. Rezapour, A study on the existence of numerical and analytical solutions for fractional integrodifferential equations in Hilfer type with simulation, AIMS Math., 8(5) (2023), pp. 10665-10684.
[19] R. George, M. Houas, M. Ghaderi, Sh. Rezapour and S.K. Elagan, On a coupled system of pantograph problem with three sequential fractional derivatives by using positive contraction-type inequalities, Results Phys., 39 (2022), pp. 105687.
[20] F. Guo, Sh. Kang and F. Chen, Existence and uniqueness results to positive solutions of integral boundary value problem for fractional $q$-derivatives, Adv. Difference Equ., 379 (2018), pp. 1-15.
[21] J. R. Graef and L. Kong, Positive solutions for a class of higher order boundary value problems with fractional $q$-derivatives, Trans. Amer. Math. Soc., 218(19) (1924), pp. 9682-9689.
[22] J. Hadamard, Essai sur l'etude des fonctions, donnees par leur developpement de Taylor, Gauthier-Villars, 1892.
[23] Z. Heydarpour, M.N. Parizi, R. Ghorbanian, M. Ghaderi, Sh. Rezapour and A. mosavi, A study on a special case of the Sturm-Liouville equation using the Mittag-Leffler function and a new type of contraction, AIMS Math., 7(10) (2022), pp. 10665-10684.
[24] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. phys., 284(1-2) (2002), pp. 399-408.
[25] R. Hilfer, Applications of fractional calculus in physics. World Scientific, Singapore, 2000.
[26] F.H. Jackson, $q$-Difference equation, Amer. J. Math., 32(4) (1910), pp. 305-314. 
[27] F.H. Jackson, On $Q$-Definite Integrals, Quart. J. Pure Appl. Math., 41 (1910), pp. 193-203. 
[28] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic, Dordrecht, 1991.
[29] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[30] Z.G. Lio, On the $q$-derivative and $q$-series expansions, Inter. J. Number Theory, 9(8) (2013), pp. 2069-2089.
[31] R.P. Meilanov and R.A. Magomedov, Thermodynamics in fractional calculus, J. Eng. phys. thermophy., 87(6) (2014), pp. 1521-1531.
[32] Sh. Mahmood, M. Jabeen, S.N. Malik, H.M. Srivastava, R. Manzoor and S. M. J. Riaz, Some Coefficient Inequalities of $q$-Starlike Functions Associated with Conic Domain Defined by $q$-Derivative, J. Funct. Spaces, (2018), 8492072.
[33] K.M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 133(1) (2018), pp. 1-13.
[34] L. Podlubny, Fractional differential equations, AcademicPress, San Diego, 1999.
[35] H. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), pp. 213-231.
[36] J.A. Tenreiro Machado, F.M. Silva, S.R. Barbosa, I.S. Jesus, M.C. Reis, M.G. Marcos and A.F. Galhano, Some applications of fractional calculus in engineering, Math. probl. Eng., (2010), 639801.
[37] C, Tunc and O. Tunc, On the stability, integrability and boundedness analyses of systems of integro-differential equations with time-delay retardation, RACSAM., 115(3) (2021), pp. 115.
[38] O. Tunc, O. Atan, C, Tunc and J.C. Yao, Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov–Razumikhin Method, Axioms , 10(2) (2021), pp. 58.
[39] O. Tunc and C. Tunc, Solution estimates to Caputo proportional fractional derivative delay integro-differential equations, RACSAM., 117(1) (2023), pp. 12.
[40] L. Vazquez, J.J. Trujillo and M. Pilar Velasco, Fractional heat equation and the second law of thermodynamics, Fract. Calc. Appl. Anal., 14 (2011), pp. 334-342.
[41] B.J. West and P. Grigolini, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 1998.
[42] K. Wlodarczyk, D. Klim and R. Plebaniak, Existence and uniqueness of endpoints of closed set-valued asymptotic contractions in metric spaces, J. Math. Anal. Appl., 328(1) (2007), pp. 46-57.
[43] D. Wardowski, Endpoints and fixed points of set-valued contractions in cone metric spaces, Nonlinear Anal., 71(1-2) (2009), pp. 512-516.
[44] Ch. Yo and J. Wang, Existence of solutions for nonlinear second-order q-difference equations with first-order $q$-derivatives, Adv. Diff. Equ., 124 (2013), pp. 1-11.
[45] Y. Zhao, H. Chen and Q. Zhang, Existence and multiplicity of positive solutions for nonhomogeneous boundary value problems with fractional $q$-derivatives, Bound. Value Probl., 103 (2013), pp. 1-16.
[46] Y. Zhao, H. Chen and Q. Zhang, Existence results for fractional $q$-difference equations with nonlocal q-integral boundary conditions, Adv. Diff. Equ., 74 (2013), pp. 1-15.