Existence of Solutions for Implicit Obstacle Problems of Fractional Laplacian Type Involving Set-Valued Operators
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Existence of Solutions for Implicit Obstacle Problems of Fractional Laplacian Type Involving Set-Valued Operators Dumitru Motreanu1,2 · Van Thien Nguyen3 · Shengda Zeng1,4 Received: 19 August 2019 / Accepted: 9 September 2020 © The Author(s) 2020
Abstract The paper is devoted to a new kind of implicit obstacle problem given by a fractional Laplacian-type operator and a set-valued term, which is described by a generalized gradient. An existence theorem for the considered implicit obstacle problem is established, using a surjectivity theorem for set-valued mappings, Kluge’s fixed point principle and nonsmooth analysis. Keywords Implicit obstacle problem · Surjectivity theorem · Generalized fractional Laplacian · Generalized gradient · Fixed point theorem Mathematics Subject Classification 35R11 · 35J50 · 35J60 · 26E25 · 47J22
1 Introduction Partial differential equations, involving nonlocal operators, have recently received much attention since the nonlocal operators, which are infinitesimal generators of Lévy-type stochastic processes, describe precisely various phenomena in such fields
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Shengda Zeng [email protected] Dumitru Motreanu [email protected] Van Thien Nguyen [email protected]
1
Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China
2
Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France
3
Departement of Mathematics, FPT University, Education zone, Hoa Lac High Tech Park, Km29 Thang Long highway, Thach That ward, Hanoi, Vietnam
4
Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Lojasiewicza 6, 30348 Kraków, Poland
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Journal of Optimization Theory and Applications
as population dynamics, game theory, finance, image processing (see [1–5] and the references therein). On the other hand, in many physical processes and engineering applications, the mathematical models are formulated as inequalities instead of equations, extensively appearing in the form of variational inequalities and hemivariational inequalities. Roughly speaking, the variational inequalities arise in a convex framework, whereas the hemivariational inequalities address systems with nonconvex and nonsmooth structure (see [6–17]). Recent works focus on systems governed by nonlocal operators and exhibiting setvalued terms in the form of generalized gradient of a locally Lipschitz function. Frassu et al. [18] proved the existence of three nontrivial solutions for a pseudo-differential inclusion driven by a nonlocal anisotropic operator and with generalized gradient of a locally Lipschitz potential. Teng [19] and Xi et al. [20] applied the nonsmooth critical point theory to obtain multiplicity results for nonlocal elliptic hemivariational inequalities. Liu and Tan [21] employed a surjectivity theorem for pseudomonotone and coercive operators to explore a nonlocal hemivariational inequality. For related works, we refer to [13,14]. In relevant situations encountered in eng
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