Existence results for a class of semilinear differential variational inequalities with nonlocal boundary conditions

Zhenhai Liu, Liang Lu, Xiufeng Guo


In the paper we study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing evolution equations and generalized variational inequalities. Firstly, we show the properties of the solution set for generalized variational inequalities. Then, the existence results are established and proved mainly by the topological degree theory and the Yosida approximations of the generator of $C_0$-semigroup.


Semilinear differential variational inequalities; existence of mild solutions; topological degree theory; Yosida approximations; nonlocal boundary conditions

Full Text:



V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976.

I. Benedetti, N.V. Loi, L. Malaguti and V. Taddei, Nonlocal diffusion second order partial differential equations, J. Differential Equations, 262 (2017), 1499–1523.

I. Benedetti, N.V. Loi and V. Taddei, An approximation solvablility method for nonlocal semiliner differential problems in Banach spaces, Discrete Contin. Dyn. Syst. 37 (2017), no. 6, 2977–2998.

S. Bochner and A.E. Taylor, Linear functionals on certain spaces of abstractly-valued functions, Ann. Math. 39 (1938), 913–944.

L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494–505.

N. Costea and C. Lupu, On a class of variational-hemivariational inequalities involving set valued mappings, Adv. Pure Appl. Math. 1 (2010), no. 2, 233–246.

Y.J. Deng, X.P. Fang and J. Li, Numerical methods for reconstruction of the source term of heat equations from the final overdetermination, Bull. Korean Math. Soc. 52 (2015), no. 5, 1495–1515.

J. Diestel, W.M. Ruess and W. Schachermayer, Weak compactness in L1 (µ, X), Proc. Amer. Math. Soc. 118 (1993), 447–453.

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519–537.

Y.P. Fang and N.J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl. 118 (2003), 327–338.

X.P. Fang, Y.J. Deng and J. Li, Plasmon resonance and heat generation in nanostructures, Math. Methods Appl. Sci. 38 (2015), no. 18, 4663–4672.

C.X. Huang, Z.C. Yang, T.S. Yi and X.F. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations 256 (2014), no. 7, 2101–2114.

M.I. Kamenskiı̆, V.V. Obukhovskiı̆ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space, Walter de Gruyter, Berlin, 2001.

T.D. Ke, N.V. Loi and V. Obukhovskiı̆, Decay solutions for a class of fractional differential variational inequalities, Fract. Calc. Appl. Anal. 18 (2015), 531–553.

J. Li, F. Liu, L. Feng and I. Turner, A novel finite volume method for the Riesz space distributed-order diffusion equation, Comput. Math. Appl. 74(4) (2017) 772-783.

J. Li and F. Wang, Simplified Tikhonov regularization for two kinds of parabolic equations, J. Korean Math. Soc. 48 (2011), no. 2, 311–327.

X.S. Li, N.J. Huang and D. O’Regan, Differential mixed variational inequalities in finite dimensional spaces, Nonlinear Anal. 72 (2010), 3875–3886.

Z.H. Liu and X.W. Li, Approximate controllability for a class of hemivariational inequalities, Nonlinear Anal. Real World Appl. 22 (2015), 581–591.

Z.H. Liu, N.V. Loi and V. Obukhovskiı̆, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23 (2013), Article ID 1350125.

Z.H. Liu, S. Migórski and S.D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations 263 (2017), no. 7, 3989–4006.

Z.H. Liu and S.D. Zeng, Differential variational inequalities in infinite Banach spaces, Acta Math. Sci. 37 (2017), no. 1, 26–32.

Z.H. Liu, S.D. Zeng and Y. Bai, Maximum principles for multi-term space-time variableorder fractional diffusion equations and their applications Fract. Calc. Appl. Anal. 19 (2016), no. 1, 188–211.

Z.H. Liu, S.D. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations 260 (2016), 6787–6799.

N.V. Loi, On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities, Nonlinear Anal. 122 (2015), 83–99.

L. Lu, Z.H. Liu and D. Motreanu, Existence results of semilinear differential variational inequalities without compactness, Optimization 68 (2019), no. 5, 1017–1035.

L. Lu, Z.H. Liu and V. Obukhovskiı̆, Second order differential variational inequalities involving anti-periodic boundary value conditions, J. Math. Anal. Appl. 473 (2019), no. 2, 846–865.

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Adv. Mech. Math., vol. 26, Springer, New York, 2013.

J.S. Pang and D.E. Stewart, Differential variational inequalities, Math. Program. 113 (2008), 345–424.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York, 1983.

G.J. Tang and N.J. Huang, Existence theorems of the variational-hemivariational inequalities, J. Glob. Optim. 56 (2013), no. 2, 605–622.

I.I. Vrabie, C0 -Semigroups and Applications, North-Holland Mathematics Studies 191, North-Holland Publishing Co., Amsterdam, 2003.

J.F. Wang, C.X. Huang and L.H. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle focus type, Nonlinear Anal. Hybrid Syst. 33 (2019), 162–178.

X. Wang, Y.W. Qi, C.Q. Tao and Y.B. Xiao, A class of delay differential variational inequalities, J. Optim. Theory Appl. 172 (2017), no. 1, 56–69

R. Wangkeeree and P. Preechasilp, Existence theorems of the hemivariational inequality governed by a multivalued map perturbed with a nonlinear term in Banach spaces, J. Global Optim. 57 (2013), no. 4, 1447–1464.

C.J. Xu, P.L. Li, Q.M. Xiao and S. Yuan, New results on competition and cooperation model of two enterprises with multiple delays and feedback controls, Bound. Value Probl. 2019 (2019).


  • There are currently no refbacks.

Partnerzy platformy czasopism