# American Institute of Mathematical Sciences

December  2020, 9(4): 1057-1071. doi: 10.3934/eect.2020056

## Differential inclusion problems with convolution and discontinuous nonlinearities

 1 Department of Mathematics, Yulin Normal University, Yulin, Guangxi, 537000, China 2 College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, China 3 Department of Mathematics, Université de Perpignan, 52 Avenue Paul Alduy, 66860, Perpignan, France

* Corresponding author

Received  October 2019 Revised  January 2020 Published  December 2020 Early access  May 2020

The paper investigates a new type of differential inclusion problem driven by a weighted (p, q)-Laplacian and subject to Dirichlet boundary condition. The problem fully depends on the solution and its gradient. The main novelty is that the problem exhibits simultaneously a nonlocal term involving convolution with the solution and a multivalued term describing discontinuous nonlinearities for the solution. Results stating existence, uniqueness and dependence on parameters are established.

Citation: Yongjian Liu, Zhenhai Liu, Dumitru Motreanu. Differential inclusion problems with convolution and discontinuous nonlinearities. Evolution Equations and Control Theory, 2020, 9 (4) : 1057-1071. doi: 10.3934/eect.2020056
##### References:
 [1] D. Averna, D. Motreanu and E. Tornatore, Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence, Appl. Math. Lett., 61 (2016), 102-107.  doi: 10.1016/j.aml.2016.05.009. [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [3] S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3. [4] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0. [5] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983. [6] Z. Liu and D. Motreanu, Inclusion problems via subsolution-supersolution method with applications to hemivariational inequalities, Appl. Anal., 97 (2018), 1454-1465.  doi: 10.1080/00036811.2017.1408076. [7] S. A. Marano and P. Winkert, On a quasilinear elliptic problem with convection term and nonlinear boundary condition, Nonlinear Anal., 187 (2019), 159-169.  doi: 10.1016/j.na.2019.04.008. [8] D. Motreanu and V. V. Motreanu, Non-variational elliptic equations involving $(p, q)$-Laplacian, convection and convolution, Pure Appl. Funct. Anal., preprint. [9] D. Motreanu and P. D. Panagiotopoulos, Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, in Nonconvex Optimization and its Applications, 29, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4615-4064-9. [10] D. Motreanu and Z. Peng, Doubly coupled systems of parabolic hemivariational inequalities: Existence and extremal solutions, Nonlinear Anal., 181 (2019), 101-118.  doi: 10.1016/j.na.2018.11.005. [11] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, in Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997. [12] M. Sofonea, W. Han and M. Shillor, Analysis and approximation of contact problems with adhesion or damage, in Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006. [13] M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. [14] E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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##### References:
 [1] D. Averna, D. Motreanu and E. Tornatore, Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence, Appl. Math. Lett., 61 (2016), 102-107.  doi: 10.1016/j.aml.2016.05.009. [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [3] S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3. [4] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0. [5] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983. [6] Z. Liu and D. Motreanu, Inclusion problems via subsolution-supersolution method with applications to hemivariational inequalities, Appl. Anal., 97 (2018), 1454-1465.  doi: 10.1080/00036811.2017.1408076. [7] S. A. Marano and P. Winkert, On a quasilinear elliptic problem with convection term and nonlinear boundary condition, Nonlinear Anal., 187 (2019), 159-169.  doi: 10.1016/j.na.2019.04.008. [8] D. Motreanu and V. V. Motreanu, Non-variational elliptic equations involving $(p, q)$-Laplacian, convection and convolution, Pure Appl. Funct. Anal., preprint. [9] D. Motreanu and P. D. Panagiotopoulos, Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, in Nonconvex Optimization and its Applications, 29, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4615-4064-9. [10] D. Motreanu and Z. Peng, Doubly coupled systems of parabolic hemivariational inequalities: Existence and extremal solutions, Nonlinear Anal., 181 (2019), 101-118.  doi: 10.1016/j.na.2018.11.005. [11] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, in Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997. [12] M. Sofonea, W. Han and M. Shillor, Analysis and approximation of contact problems with adhesion or damage, in Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006. [13] M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. [14] E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
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