Positivity, monotonicity, and convexity for convolution operators

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

Positivity, monotonicity, and convexity for convolution operators

Christopher Goodrich ^1, and Carlos Lizama ^2,,

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

$ \begin{equation} a*u {\ge} v\notag \end{equation} $

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.

Keywords: Convolution, positivity, monotonicity, convexity, finite difference.

Mathematics Subject Classification: Primary: 42A85, 44A35; Secondary: 26A33, 26A48, 26A51, 39A70, 65L12.

Citation: Christopher Goodrich, Carlos Lizama. Positivity, monotonicity, and convexity for convolution operators. Discrete and Continuous Dynamical Systems, 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	J. Dieudonné, Sur le produit de composition. Ⅱ, J. Math. Pures Appl. (9), 39 (1960), 275-292.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer International Publishing, 2015. doi: 10.1007/978-3-319-25562-0.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equations Oper. Theory, 71 (2011), 583-600.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	K. Strom, On convolutions of B-splines, J. Comp. Appl. Math., 55 (1994), 1-29. doi: 10.1016/0377-0427(94)90182-1.
[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.
[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.
[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.

show all references

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	J. Dieudonné, Sur le produit de composition. Ⅱ, J. Math. Pures Appl. (9), 39 (1960), 275-292.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer International Publishing, 2015. doi: 10.1007/978-3-319-25562-0.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equations Oper. Theory, 71 (2011), 583-600.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	K. Strom, On convolutions of B-splines, J. Comp. Appl. Math., 55 (1994), 1-29. doi: 10.1016/0377-0427(94)90182-1.
[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.
[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.
[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.

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