# American Institute of Mathematical Sciences

## Existence results of Hilfer integro-differential equations with fractional order

 1 Department of Mathematics, GTN Arts College, Dindigul - 624 004, Tamil Nadu, India 2 PG and Research Department of Mathematics, Kongunadu Arts and Science College(Autonomous), Coimbatore - 641 029, Tamil Nadu, India 3 Department of Mathematics, Sri Eshwar College of Engineering, Coimbatore - 641 202, Tamil Nadu, India 4 Department of Mathematics, Faculty of Education, Harran University, Sanliurfa, Turkey

* Corresponding author: H. M. Baskonus

Received  July 2018 Revised  September 2018 Published  March 2019

The paper is relevance with Hilfer derivative with fractional order which is generalized case of R-L and Caputo's sense. We ensured the solution using noncompact measure and M$\ddot{\text{o}}$nch's fixed point technique. Illustrative examples are included for the applicability of presented technique.

Citation: Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020053
##### References:
 [1] R. P. Agarwal, B. Ahmad, A. Alsaedi and N. Shahzad, Existence and dimension of the set of mild solutions to semilinear fractional differential inclusions, Advance in Difference Equations, 74 (2012), 1-10. doi: 10.1186/1687-1847-2012-74. [2] R. Almeida, What is the best fractional derivative to fit data?, Applicable Analysis and Discrete Mathematics, 11 (2017), 358-368. doi: 10.2298/AADM170428002A. [3] J. Banas and K. Goebel, Measure of Noncompactness in Banach Space, Lecture Notes in Pure and Applied Mathematics, Marcell Dekker, New York, 1980. [4] H. M. Baskonus, T. Mekkaoui, Z. Hammouch and H. Bulut, Active Control of a Chaotic Fractional Order Economic System, Entropy, 17 (2015), 5771-5783. doi: 10.3390/e17064255. [5] A. H. Bhrawy and M. A. Zaky, Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations, Applied Mathematical Modelling, 40 (2016), 832-845. doi: 10.1016/j.apm.2015.06.012. [6] C. Cattani and A. Ciancio, On the fractal distribution of primes and prime-indexed primes by the binary image analysis, Physica A, 460 (2016), 222-229. doi: 10.1016/j.physa.2016.05.013. [7] M. Dokuyucu, E. Celik, H. Bulut and H. M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, The European Physical Journal Plus, 133 (2018), 92. doi: 10.1140/epjp/i2018-11950-y. [8] K. M. Furati, M. D. Kassim and N. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Computers & Mathematics with Applications, 64 (2012), 1612-1626. doi: 10.1016/j.camwa.2012.01.009. [9] R. Garra, R. Gorenflo, F. Polito and Z. Tomovski, Hilfer-Prabhakar derivatives and some applications, Applied Mathematics and Computation, 242 (2014), 576-589. doi: 10.1016/j.amc.2014.05.129. [10] H. Gu and J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Applied Mathematics and Computation, 257 (2015), 344-354. doi: 10.1016/j.amc.2014.10.083. [11] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747. [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. [13] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008), 2677-2682. doi: 10.1016/j.na.2007.08.042. [14] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York, 1993. [15] H. M$\ddot{\text{o}}$nch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 985-999. doi: 10.1016/0362-546X(80)90010-3. [16] I. Podlubny, Fractional Differential Equations, vol., 198, Academic Press, an Diego, 1999. [17] C. Ravichandran and J. J. Trujillo, Controllability of impulsive fractional functional integro-diffrential equations in Banach spaces, Journal of Function Spaces and Applications, 2013 (2013), Art. ID 812501, 8 pp. doi: 10.1155/2013/812501. [18] C. Ravichandran and D. Baleanu, Existence results for fractional integro-differential evolution equations with infinite delay in Banach spaces, Advances in Difference Equations, 2013 (2013), 1-12. doi: 10.1186/1687-1847-2013-215. [19] C. Ravichandran and D. Baleanu, On the controllability of fractional functional integro-differential systems with an infinite delay in Banach spaces, Advances in Difference Equations, 291 (2013), 1-13. doi: 10.1186/1687-1847-2013-291. [20] C. Ravichandran, K. Jothimani, H. M. Baskonus and N. Valliammal, New results on nondensely characterized integro-differential equations with fractional order, The European Physical Journal Plus, 133 (2018), 1-10. [21] A. Saadatmandi, Bernstein operational matrix of fractional derivatives and its applications, Applied Mathematical Modelling, 38 (2014), 1365-1372. doi: 10.1016/j.apm.2013.08.007. [22] T. Sandev, R. Metzler and Z. Tomoveski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 255203, 21 pp. doi: 10.1088/1751-8113/44/25/255203. [23] A. R. Seadawy, Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: Part Ⅰ, Computers & Mathematics with Applications, 70 (2015), 345-352. doi: 10.1016/j.camwa.2015.04.015. [24] X. B Shu and Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order $1 < \alpha < 2,$, Computers & Mathematics with Applications, 64 (2012), 2100-2110. doi: 10.1016/j.camwa.2012.04.006. [25] R. Subashini, K. Jothimani, S. Saranya and C. Ravichandran, On the results of Hilfer fractional derivative with nonlocal conditions, International Journal of Pure and Applied Mathematics, 118 (2018), 277-289. [26] J. A. Tenreiro Machado and M. Mata, Pseudo Phase Plane and Fractional Calculus modeling of western global economic downturn, Communications in Nonlinear Science and Numerical Simulation, 22 (2015), 396-406. doi: 10.1016/j.cnsns.2014.08.032. [27] J. A. Tenreiro Machado, Fractional dynamics in the Rayleigh's piston, Communications in Nonlinear Science and Numerical Simulation, 31 (2016), 76-82. [28] N. Valliammal, C. Ravichandran and J. H. Park, On the controllability of fractional neutral integrodifferential delay equations with nonlocal conditions, Mathematical Methods in the Applied Sciences, 40 (2017), 5044-5055. doi: 10.1002/mma.4369. [29] V. Vijayakumar, C. Ravichandran, R. Murugesu and J. J. Trujillo, Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators, Applied Mathematics and Computation, 247 (2014), 152-161. doi: 10.1016/j.amc.2014.08.080. [30] J. R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Applied Mathematics and Computation, 266 (2015), 850-859. doi: 10.1016/j.amc.2015.05.144. [31] M. Yang and Q. Wang, Existence of mild solutions for a class of Hilfer fractional evolution equations With nonlocal conditions, Fractional Calculus and Applied Analysis, 20 (2017), 679-705. doi: 10.1515/fca-2017-0036. [32] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069. [33] Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Analysis: Real World Applications, 11 (2010), 4465-4475. doi: 10.1016/j.nonrwa.2010.05.029. [34] Y. Zhou, L. Zhang and X. H. Shen, Existence of mild solutions for fractional evolution equations, Journal of Integral Equations and Applications, 25 (2013), 557-585. doi: 10.1216/JIE-2013-25-4-557.

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##### References:
 [1] R. P. Agarwal, B. Ahmad, A. Alsaedi and N. Shahzad, Existence and dimension of the set of mild solutions to semilinear fractional differential inclusions, Advance in Difference Equations, 74 (2012), 1-10. doi: 10.1186/1687-1847-2012-74. [2] R. Almeida, What is the best fractional derivative to fit data?, Applicable Analysis and Discrete Mathematics, 11 (2017), 358-368. doi: 10.2298/AADM170428002A. [3] J. Banas and K. Goebel, Measure of Noncompactness in Banach Space, Lecture Notes in Pure and Applied Mathematics, Marcell Dekker, New York, 1980. [4] H. M. Baskonus, T. Mekkaoui, Z. Hammouch and H. Bulut, Active Control of a Chaotic Fractional Order Economic System, Entropy, 17 (2015), 5771-5783. doi: 10.3390/e17064255. [5] A. H. Bhrawy and M. A. Zaky, Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations, Applied Mathematical Modelling, 40 (2016), 832-845. doi: 10.1016/j.apm.2015.06.012. [6] C. Cattani and A. Ciancio, On the fractal distribution of primes and prime-indexed primes by the binary image analysis, Physica A, 460 (2016), 222-229. doi: 10.1016/j.physa.2016.05.013. [7] M. Dokuyucu, E. Celik, H. Bulut and H. M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, The European Physical Journal Plus, 133 (2018), 92. doi: 10.1140/epjp/i2018-11950-y. [8] K. M. Furati, M. D. Kassim and N. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Computers & Mathematics with Applications, 64 (2012), 1612-1626. doi: 10.1016/j.camwa.2012.01.009. [9] R. Garra, R. Gorenflo, F. Polito and Z. Tomovski, Hilfer-Prabhakar derivatives and some applications, Applied Mathematics and Computation, 242 (2014), 576-589. doi: 10.1016/j.amc.2014.05.129. [10] H. Gu and J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Applied Mathematics and Computation, 257 (2015), 344-354. doi: 10.1016/j.amc.2014.10.083. [11] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747. [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. [13] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008), 2677-2682. doi: 10.1016/j.na.2007.08.042. [14] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York, 1993. [15] H. M$\ddot{\text{o}}$nch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 985-999. doi: 10.1016/0362-546X(80)90010-3. [16] I. Podlubny, Fractional Differential Equations, vol., 198, Academic Press, an Diego, 1999. [17] C. Ravichandran and J. J. Trujillo, Controllability of impulsive fractional functional integro-diffrential equations in Banach spaces, Journal of Function Spaces and Applications, 2013 (2013), Art. ID 812501, 8 pp. doi: 10.1155/2013/812501. [18] C. Ravichandran and D. Baleanu, Existence results for fractional integro-differential evolution equations with infinite delay in Banach spaces, Advances in Difference Equations, 2013 (2013), 1-12. doi: 10.1186/1687-1847-2013-215. [19] C. Ravichandran and D. Baleanu, On the controllability of fractional functional integro-differential systems with an infinite delay in Banach spaces, Advances in Difference Equations, 291 (2013), 1-13. doi: 10.1186/1687-1847-2013-291. [20] C. Ravichandran, K. Jothimani, H. M. Baskonus and N. Valliammal, New results on nondensely characterized integro-differential equations with fractional order, The European Physical Journal Plus, 133 (2018), 1-10. [21] A. Saadatmandi, Bernstein operational matrix of fractional derivatives and its applications, Applied Mathematical Modelling, 38 (2014), 1365-1372. doi: 10.1016/j.apm.2013.08.007. [22] T. Sandev, R. Metzler and Z. Tomoveski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 255203, 21 pp. doi: 10.1088/1751-8113/44/25/255203. [23] A. R. Seadawy, Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: Part Ⅰ, Computers & Mathematics with Applications, 70 (2015), 345-352. doi: 10.1016/j.camwa.2015.04.015. [24] X. B Shu and Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order $1 < \alpha < 2,$, Computers & Mathematics with Applications, 64 (2012), 2100-2110. doi: 10.1016/j.camwa.2012.04.006. [25] R. Subashini, K. Jothimani, S. Saranya and C. Ravichandran, On the results of Hilfer fractional derivative with nonlocal conditions, International Journal of Pure and Applied Mathematics, 118 (2018), 277-289. [26] J. A. Tenreiro Machado and M. Mata, Pseudo Phase Plane and Fractional Calculus modeling of western global economic downturn, Communications in Nonlinear Science and Numerical Simulation, 22 (2015), 396-406. doi: 10.1016/j.cnsns.2014.08.032. [27] J. A. Tenreiro Machado, Fractional dynamics in the Rayleigh's piston, Communications in Nonlinear Science and Numerical Simulation, 31 (2016), 76-82. [28] N. Valliammal, C. Ravichandran and J. H. Park, On the controllability of fractional neutral integrodifferential delay equations with nonlocal conditions, Mathematical Methods in the Applied Sciences, 40 (2017), 5044-5055. doi: 10.1002/mma.4369. [29] V. Vijayakumar, C. Ravichandran, R. Murugesu and J. J. Trujillo, Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators, Applied Mathematics and Computation, 247 (2014), 152-161. doi: 10.1016/j.amc.2014.08.080. [30] J. R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Applied Mathematics and Computation, 266 (2015), 850-859. doi: 10.1016/j.amc.2015.05.144. [31] M. Yang and Q. Wang, Existence of mild solutions for a class of Hilfer fractional evolution equations With nonlocal conditions, Fractional Calculus and Applied Analysis, 20 (2017), 679-705. doi: 10.1515/fca-2017-0036. [32] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069. [33] Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Analysis: Real World Applications, 11 (2010), 4465-4475. doi: 10.1016/j.nonrwa.2010.05.029. [34] Y. Zhou, L. Zhang and X. H. Shen, Existence of mild solutions for fractional evolution equations, Journal of Integral Equations and Applications, 25 (2013), 557-585. doi: 10.1216/JIE-2013-25-4-557.
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