# American Institute of Mathematical Sciences

August  2020, 40(8): 4961-4983. doi: 10.3934/dcds.2020207

## Positivity, monotonicity, and convexity for convolution operators

 1 School of Mathematics and Statistics, UNSW Australia, Sydney, NSW 2052, Australia 2 Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Santiago, Chile

* Corresponding author: Carlos Lizama

Received  October 2019 Revised  December 2019 Published  May 2020

Fund Project: The second author is supported by CONICYT under Fondecyt Grant number 1180041

We consider the convolution inequality
 $$$a*u {\ge} v\notag$$$
for given functions
 $a$
and
 $v$
, and we then investigate conditions on
 $a$
and
 $v$
that force the unknown function
 $u$
to be positive or monotone or convex. We demonstrate that these results for abstract convolution equations can be specialized to yield new insights into the qualitative properties of fractional difference and differential operators. Finally, we apply our results to finite difference methods for fractional differential equations, and we show that our results yield insights into the qualitative behavior of these types of numerical approximations.
Citation: Christopher Goodrich, Carlos Lizama. Positivity, monotonicity, and convexity for convolution operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4961-4983. doi: 10.3934/dcds.2020207
##### References:
 [1] L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco, Cesàro sums and algebra homomorphisms of bounded operators, Israel J. Math., 216 (2016), 471-505.  doi: 10.1007/s11856-016-1417-3.  Google Scholar [2] L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco, On well-posedness of vector-valued fractional differential-difference equations, Discr. Cont. Dyn. Systems, Series A, 39 (2019), 2679-2708.  doi: 10.3934/dcds.2019112.  Google Scholar [3] L. Abadias and P. J. Miana, Generalized Cesàro operators, fractional finite differences and Gamma functions, J. Funct. Anal., 274 (2018), 1424-1465.  doi: 10.1016/j.jfa.2017.10.010.  Google Scholar [4] L. Erbe, C. S. Goodrich, J. Baoguo and A. Peterson, Monotonicity results for delta fractional difference revisited, Math. Slovaca, 67 (2017), 895-906.  doi: 10.1515/ms-2017-0018.  Google Scholar [5] J. Bonet, C. Fernández and R. Meise, Characterization of the $\omega$-hypoelliptic convolution operators on ultradistributions, Ann. Acad. Sci. Fenn. Math., 25 (2000), 261-284.   Google Scholar [6] I. Bright, Moving averages of ordinary differential equations via convolution, J. Differential Equations, 250 (2011), 1267-1284.  doi: 10.1016/j.jde.2010.10.011.  Google Scholar [7] Y. Choi, Injective convolution operators on $\ell^{\infty}(\Gamma)$ are surjective, Canad. Math. Bull., 53 (2010), 447-452.  doi: 10.4153/CMB-2010-053-5.  Google Scholar [8] O. Constatin and J. Hargraves, Monotone solutions to a nonlinear integral equation of convolution type, Nonlinear Anal. Real World Appl., 15 (2014), 38-41.  doi: 10.1016/j.nonrwa.2013.05.003.  Google Scholar [9] R. Dahal and C. S. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math. (Basel), 102 (2014), 293-299.  doi: 10.1007/s00013-014-0620-x.  Google Scholar [10] M. Darwish, Monotonic solutions of a convolution functional-integral equation, Appl. Math. Comput., 219 (2013), 10777-10782.  doi: 10.1016/j.amc.2013.05.001.  Google Scholar [11] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.  doi: 10.1016/0362-546X(78)90015-9.  Google Scholar [12] K. Diethelm, Monotonicity of functions and sign changes of their Caputo derivatives, Fract. Calc. Appl. Anal., 19 (2016), 561-566.  doi: 10.1515/fca-2016-0029.  Google Scholar [13] J. Dieudonné, Sur le produit de composition. Ⅱ, J. Math. Pures Appl. (9), 39 (1960), 275-292.   Google Scholar [14] L. Ehrenpreis, Solution of some problems of division. Ⅳ. Invertible and elliptic operators, Amer. J. Math., 82 (1960), 522-588.  doi: 10.2307/2372971.  Google Scholar [15] C. Fernández, A. Galbis and D. Jornet, Perturbations of surjective convolution operators, Proc. Amer. Math. Soc., 130 (2002), 2377-2381.  doi: 10.1090/S0002-9939-02-06359-1.  Google Scholar [16] M. Gómez-Callado and E. Jordá, Regularity of solutions of convolution equations, J. Math. Anal. Appl., 338 (2008), 873-884.  doi: 10.1016/j.jmaa.2007.05.053.  Google Scholar [17] C. S. Goodrich, A convexity result for fractional differences, Appl. Math. Lett., 35 (2014), 58-62.  doi: 10.1016/j.aml.2014.04.013.  Google Scholar [18] C. S. Goodrich, A note on convexity, concavity, and growth conditions in discrete fractional calculus with delta difference, Math. Inequal. Appl., 19 (2016), 769-779.  doi: 10.7153/mia-19-57.  Google Scholar [19] C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: Fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589. doi: 10.1007/s11856-020-1991-2.  Google Scholar [20] C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer International Publishing, 2015. doi: 10.1007/978-3-319-25562-0.  Google Scholar [21] A. Hanyga, On solutions of matrix-valued convolution equations, CM-derivatives and their applications in linear and nonlinar anisotropic viscoelasticity, Z. Angew. Math. Phys., 70 (2019), art. 103, 13 pp. Google Scholar [22] B. Jia, L. Erbe and A. Peterson, Two monotonicity results for nabla and delta fractional differences, Arch. Math. (Basel), 104 (2015), 589-597.  doi: 10.1007/s00013-015-0765-2.  Google Scholar [23] B. Jin, B. Li and Z. Zhou, Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math., 138 (2018), 101-131.  doi: 10.1007/s00211-017-0904-8.  Google Scholar [24] R. Kamocki, A new representation formula for the Hilfer fractional derivative and its application, J. Comp. Appl. Math., 308 (2016), 39-45.  doi: 10.1016/j.cam.2016.05.014.  Google Scholar [25] J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.  Google Scholar [26] A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024.  Google Scholar [27] A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equations Oper. Theory, 71 (2011), 583-600.   Google Scholar [28] O. Lipovan, Asymptotic properties of solutions to some nonlinear integral equations of convolution type, Nonlinear Anal., 69 (2008), 2179-2183.  doi: 10.1016/j.na.2007.07.056.  Google Scholar [29] C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.  Google Scholar [30] C. Lizama, $\ell_p$-maximal regularity for fractional difference equations on UMD spaces, Math. Nachr., 288 (2015), 2079-2092.  doi: 10.1002/mana.201400326.  Google Scholar [31] C. Lizama and M. Murillo-Arcila, Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations, J. Differential Equations., 263 (2017), 3175-3196.  doi: 10.1016/j.jde.2017.04.035.  Google Scholar [32] G. Lv, H. Gao, J. Wei and J.-L. Wu, BMO and Morrey-Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.  doi: 10.1016/j.jde.2018.08.042.  Google Scholar [33] S. G. Samko and R. P. Cardoso, Integral equations of the first kind of Sonine type, Int. J. Math. and Math. Sci., 57 (2003), 3609-3632.   Google Scholar [34] K. Schumacher, Traveling-front solutions for integro-differential equations. Ⅰ., J. Reine Angew. Math., 316 (1980), 54-70.  doi: 10.1515/crll.1980.316.54.  Google Scholar [35] K. Strom, On convolutions of B-splines, J. Comp. Appl. Math., 55 (1994), 1-29.  doi: 10.1016/0377-0427(94)90182-1.  Google Scholar [36] V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.  Google Scholar [37] Y. Wang, L. Liu and Y. Wu, Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity, Nonlinear Analysis, 74 (2011), 6434-6441.  doi: 10.1016/j.na.2011.06.026.  Google Scholar [38] Z. Xu and C. Wu, Monostable waves in a class of non-local convolution differential equation, J. Math. Anal. Appl., 462 (2018), 1205-1224.  doi: 10.1016/j.jmaa.2018.02.036.  Google Scholar

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##### References:
 [1] L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco, Cesàro sums and algebra homomorphisms of bounded operators, Israel J. Math., 216 (2016), 471-505.  doi: 10.1007/s11856-016-1417-3.  Google Scholar [2] L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco, On well-posedness of vector-valued fractional differential-difference equations, Discr. Cont. Dyn. Systems, Series A, 39 (2019), 2679-2708.  doi: 10.3934/dcds.2019112.  Google Scholar [3] L. Abadias and P. J. Miana, Generalized Cesàro operators, fractional finite differences and Gamma functions, J. Funct. Anal., 274 (2018), 1424-1465.  doi: 10.1016/j.jfa.2017.10.010.  Google Scholar [4] L. Erbe, C. S. Goodrich, J. Baoguo and A. Peterson, Monotonicity results for delta fractional difference revisited, Math. Slovaca, 67 (2017), 895-906.  doi: 10.1515/ms-2017-0018.  Google Scholar [5] J. Bonet, C. Fernández and R. Meise, Characterization of the $\omega$-hypoelliptic convolution operators on ultradistributions, Ann. Acad. Sci. Fenn. Math., 25 (2000), 261-284.   Google Scholar [6] I. Bright, Moving averages of ordinary differential equations via convolution, J. Differential Equations, 250 (2011), 1267-1284.  doi: 10.1016/j.jde.2010.10.011.  Google Scholar [7] Y. Choi, Injective convolution operators on $\ell^{\infty}(\Gamma)$ are surjective, Canad. Math. Bull., 53 (2010), 447-452.  doi: 10.4153/CMB-2010-053-5.  Google Scholar [8] O. Constatin and J. Hargraves, Monotone solutions to a nonlinear integral equation of convolution type, Nonlinear Anal. Real World Appl., 15 (2014), 38-41.  doi: 10.1016/j.nonrwa.2013.05.003.  Google Scholar [9] R. Dahal and C. S. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math. (Basel), 102 (2014), 293-299.  doi: 10.1007/s00013-014-0620-x.  Google Scholar [10] M. Darwish, Monotonic solutions of a convolution functional-integral equation, Appl. Math. Comput., 219 (2013), 10777-10782.  doi: 10.1016/j.amc.2013.05.001.  Google Scholar [11] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.  doi: 10.1016/0362-546X(78)90015-9.  Google Scholar [12] K. Diethelm, Monotonicity of functions and sign changes of their Caputo derivatives, Fract. Calc. Appl. Anal., 19 (2016), 561-566.  doi: 10.1515/fca-2016-0029.  Google Scholar [13] J. Dieudonné, Sur le produit de composition. Ⅱ, J. Math. Pures Appl. (9), 39 (1960), 275-292.   Google Scholar [14] L. Ehrenpreis, Solution of some problems of division. Ⅳ. Invertible and elliptic operators, Amer. J. Math., 82 (1960), 522-588.  doi: 10.2307/2372971.  Google Scholar [15] C. Fernández, A. Galbis and D. Jornet, Perturbations of surjective convolution operators, Proc. Amer. Math. Soc., 130 (2002), 2377-2381.  doi: 10.1090/S0002-9939-02-06359-1.  Google Scholar [16] M. Gómez-Callado and E. Jordá, Regularity of solutions of convolution equations, J. Math. Anal. Appl., 338 (2008), 873-884.  doi: 10.1016/j.jmaa.2007.05.053.  Google Scholar [17] C. S. Goodrich, A convexity result for fractional differences, Appl. Math. Lett., 35 (2014), 58-62.  doi: 10.1016/j.aml.2014.04.013.  Google Scholar [18] C. S. Goodrich, A note on convexity, concavity, and growth conditions in discrete fractional calculus with delta difference, Math. Inequal. Appl., 19 (2016), 769-779.  doi: 10.7153/mia-19-57.  Google Scholar [19] C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: Fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589. doi: 10.1007/s11856-020-1991-2.  Google Scholar [20] C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer International Publishing, 2015. doi: 10.1007/978-3-319-25562-0.  Google Scholar [21] A. Hanyga, On solutions of matrix-valued convolution equations, CM-derivatives and their applications in linear and nonlinar anisotropic viscoelasticity, Z. Angew. Math. Phys., 70 (2019), art. 103, 13 pp. Google Scholar [22] B. Jia, L. Erbe and A. Peterson, Two monotonicity results for nabla and delta fractional differences, Arch. Math. (Basel), 104 (2015), 589-597.  doi: 10.1007/s00013-015-0765-2.  Google Scholar [23] B. Jin, B. Li and Z. Zhou, Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math., 138 (2018), 101-131.  doi: 10.1007/s00211-017-0904-8.  Google Scholar [24] R. Kamocki, A new representation formula for the Hilfer fractional derivative and its application, J. Comp. Appl. Math., 308 (2016), 39-45.  doi: 10.1016/j.cam.2016.05.014.  Google Scholar [25] J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.  Google Scholar [26] A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024.  Google Scholar [27] A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equations Oper. Theory, 71 (2011), 583-600.   Google Scholar [28] O. Lipovan, Asymptotic properties of solutions to some nonlinear integral equations of convolution type, Nonlinear Anal., 69 (2008), 2179-2183.  doi: 10.1016/j.na.2007.07.056.  Google Scholar [29] C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.  Google Scholar [30] C. Lizama, $\ell_p$-maximal regularity for fractional difference equations on UMD spaces, Math. Nachr., 288 (2015), 2079-2092.  doi: 10.1002/mana.201400326.  Google Scholar [31] C. Lizama and M. Murillo-Arcila, Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations, J. Differential Equations., 263 (2017), 3175-3196.  doi: 10.1016/j.jde.2017.04.035.  Google Scholar [32] G. Lv, H. Gao, J. Wei and J.-L. Wu, BMO and Morrey-Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.  doi: 10.1016/j.jde.2018.08.042.  Google Scholar [33] S. G. Samko and R. P. Cardoso, Integral equations of the first kind of Sonine type, Int. J. Math. and Math. Sci., 57 (2003), 3609-3632.   Google Scholar [34] K. Schumacher, Traveling-front solutions for integro-differential equations. Ⅰ., J. Reine Angew. Math., 316 (1980), 54-70.  doi: 10.1515/crll.1980.316.54.  Google Scholar [35] K. Strom, On convolutions of B-splines, J. Comp. Appl. Math., 55 (1994), 1-29.  doi: 10.1016/0377-0427(94)90182-1.  Google Scholar [36] V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.  Google Scholar [37] Y. Wang, L. Liu and Y. Wu, Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity, Nonlinear Analysis, 74 (2011), 6434-6441.  doi: 10.1016/j.na.2011.06.026.  Google Scholar [38] Z. Xu and C. Wu, Monostable waves in a class of non-local convolution differential equation, J. Math. Anal. Appl., 462 (2018), 1205-1224.  doi: 10.1016/j.jmaa.2018.02.036.  Google Scholar
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