# American Institute of Mathematical Sciences

July  2009, 23(3): 733-753. doi: 10.3934/dcds.2009.23.733

## Differential equations in metric spaces with applications

 1 Dipartimento di Matematica, Via Branze, 38 – 25123 Brescia, Italy 2 Dipartimento di Matematica e Applicazioni, Via Bicocca degli Arcimboldi, 8 – 20126 Milano

Received  December 2007 Revised  February 2008 Published  November 2008

This paper proves the local well posedness of differential equations in metric spaces under assumptions that allow to comprise several different applications. We consider below a system of balance laws with a dissipative non local source, the Hille-Yosida Theorem, a generalization of a recent result on nonlinear operator splitting, an extension of Trotter formula for linear semigroups and the heat equation.
Citation: Rinaldo M. Colombo, Graziano Guerra. Differential equations in metric spaces with applications. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 733-753. doi: 10.3934/dcds.2009.23.733
 [1] Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027 [2] Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055 [3] Chaudry Masood Khalique, Muhammad Usman, Maria Luz Gandarais. Nonlinear differential equations: Lie symmetries, conservation laws and other approaches of solving. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : i-ii. doi: 10.3934/dcdss.2020415 [4] Daniel Grieser. A natural differential operator on conic spaces. Conference Publications, 2011, 2011 (Special) : 568-577. doi: 10.3934/proc.2011.2011.568 [5] Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247 [6] Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629 [7] Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271 [8] Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595 [9] A. Alamo, J. M. Sanz-Serna. Word combinatorics for stochastic differential equations: Splitting integrators. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2163-2195. doi: 10.3934/cpaa.2019097 [10] Alexander Pankov. Nonlinear Schrödinger Equations on Periodic Metric Graphs. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030 [11] Yucheng Bu, Yujun Dong. Existence of solutions for nonlinear operator equations. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180 [12] Graziano Crasta, Benedetto Piccoli. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete & Continuous Dynamical Systems, 1997, 3 (4) : 477-502. doi: 10.3934/dcds.1997.3.477 [13] Rinaldo M. Colombo, Graziano Guerra. Hyperbolic balance laws with a dissipative non local source. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1077-1090. doi: 10.3934/cpaa.2008.7.1077 [14] Laura Caravenna. Regularity estimates for continuous solutions of α-convex balance laws. Communications on Pure & Applied Analysis, 2017, 16 (2) : 629-644. doi: 10.3934/cpaa.2017031 [15] Yunmei Chen, Xianqi Li, Yuyuan Ouyang, Eduardo Pasiliao. Accelerated bregman operator splitting with backtracking. Inverse Problems & Imaging, 2017, 11 (6) : 1047-1070. doi: 10.3934/ipi.2017048 [16] Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773 [17] X. Xiang, Y. Peng, W. Wei. A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces. Conference Publications, 2005, 2005 (Special) : 911-919. doi: 10.3934/proc.2005.2005.911 [18] Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018 [19] Stephan Gerster, Michael Herty. Discretized feedback control for systems of linearized hyperbolic balance laws. Mathematical Control & Related Fields, 2019, 9 (3) : 517-539. doi: 10.3934/mcrf.2019024 [20] Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035

2020 Impact Factor: 1.392