Existence results for semi-linear differential equations with nonlocal boundary value conditions
DOI:
https://doi.org/10.12775/TMNA.2025.042Keywords
Semi-linear elliptic equation, nonlocal boundary conditions, convergent matricesAbstract
The paper deals with a semi-linear elliptic equation with non-local boundary conditions. In order to tackle problem under consideration we employ the classical Schauder theorem and the theory of convergent matrices. Firstly, we examine the solvability of the auxiliary problem \begin{equation*} \begin{cases} -\Delta u(\boldsymbol{x}) = f(\boldsymbol{x},u(\boldsymbol{x})) & \text{for }\boldsymbol{x}\in \Omega, \\ \displaystyle u(\boldsymbol{x}) = \int_\Omega K(\boldsymbol{x},\boldsymbol{\xi}) h(\boldsymbol{\xi},u(\boldsymbol{\xi})) d\boldsymbol{\xi} & \text{for } \boldsymbol{x}\in \partial \Omega, \end{cases} \end{equation*} where the growth of $f$, $h$ and the kernel $K$ are limited and where $\Omega$ is open and bounded set with a Lipschitz bondary. Next, under the assumption of a sublinear growth for $f$ and $g$, and using the approximation methods, the solvability of the problem of the form \begin{equation*} \begin{cases} -\Delta u(\boldsymbol{x}) = f(\boldsymbol{x}, u(\boldsymbol{x})) & \text{for }\boldsymbol{x}\in \Omega, \\ u(\boldsymbol{x}) = \alpha(\boldsymbol{x}) g (u(\boldsymbol{p}_1),\ldots, u(\boldsymbol{p}_m)) & \text{for }\boldsymbol{x}\in \partial \Omega, \end{cases} \end{equation*} is examined in the $H^2$-sense, where $\Omega$ is an open and bounded subset of $\mathbb{R}^2$ or $\mathbb{R}^3$ with a $C^2$ boundary and where $\boldsymbol{p}_1,\ldots, \boldsymbol{p}_m \in \Omega$. We focus mainly on having the nonlocal condition fixed at one point and then proceed to the multipoint case.References
C. Avramescu and C. Vladimirescu, Sur un theoreme de point fixe, Stud. Cerc. Mat. 22 (1970), 215–220.
M. Beldziński, M. Galewski and I. Kossowski, On a version of hybrid existence result for a system of nonlinear equations, Adv. Nonlinear Stud. 23 (2023), 20230112.
I. Benedetti, T. Cardinali and R. Precup, Fixed point-critical point hybrid theorems and application to systems with partial variational structure, J. Fixed Point Theory Appl. 23 (2021), no. 4, 63.
S. Biagi, A. Calamai, G. Infante, Nonzero positive solutions of elliptic systems with gradient dependence and functional BCs, Adv. Nonlinear Stud. 20 (2020), no. 4, 911–931.
H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, NY, 2011, xiii+599 pp.
A. Calamai and G. Infante, On the solvability of parameter-dependent elliptic functional BVPs on annular-like domains, Discrete Contin. Dyn. Syst. Ser. B (2025), DOI: 10.3934/dcdsb.2024193.
W.A. Day, Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982), no. 3, 319–330.
W.A. Day, Heat Conduction within Linear Thermoelastacity, Springer Tracts in Natural Philosophy, vol. 30, Springer–Verlag, New York, 1982, viii+84 pp.
P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, 2nd ed., Birkhäuser Adv. Texts, Springer, Basel, 2013, x+649 pp.
L.C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Providence, RI, American Mathematical Society, 2010, xxi+749 pp.
D.G. de Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours, Lectures on Mathematics and Physics, Mathematics, Tata Institute of Fundamental Research, vol. 81, Springer–Verlag, Berlin etc., 1989, vi+96 pp.
A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44 (1986), no. 3, 401–407.
M. Galewski, Basic Monotonicity Methods with Some Applications, Compact Textb. Math., Birkhäuser Cham, 2021, x+180 pp.
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser, Basel, 2006, x+202 pp.
G. Infante, Nonzero positive solutions of a multi-parameter elliptic system with functional BCs, Topol. Methods Nonlinear Anal 52 (2018), 665–675.
G. Infante, Nonzero positive solutions of nonlocal elliptic systems with functional BCs, J. Elliptic Parabol. Equ. 5 (2019), 493–505.
G. Infante, G. Mascali, J. Rodrı́guez-López, A hybrid Krasnosel’skiı̆--Schauder fixed point theorem for systems, Nonlinear Anal. 80 (2024), 104165.
M.D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fundamenta 22 (1934), 77–108.
I. Kossowski, Radial solutions for nonlinear elliptic equation with nonlinear nonlocal boundary conditions, Opuscula Math. 43 (2023), no. 5, 675–687.
P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Stud. Adv. Math., vol. 44, Cambridge University Press, Cambridge, 1999, xii+343 pp.
J. Marschall, The trace of Sobolev–Slobodeckiuı spaces on Lipschitz domains, Manuscr. Math. 58 (1987), 47–65.
D. Medková, The Laplace Equation. Boundary Value Problems on Bounded and Unbounded Lipschitz Domains, Springer, Cham, 2018, xiv+660 pp.
C.V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 88 (1998), no. 1, 225–238.
N.S. Papageorgiou, V. Rǎdulescu and D. Repovš, Nonlinear Analysis – Theory and Methods, Springer Monogr. Math., Springer, Cham, 2019, xi+577 pp.
R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modelling 49 (2009), no. 3–4, 703–708.
D.R. Smart, Fixed Point Theorems, Cambridge Tracts in Math., vol. VIII, Cambridge University Press, 1980, 93 pp.
M.E. Taylor, Measure Theory and Integration, Grad. Stud. Math., vol. 76, 2006, xiii+319 pp.
E. Zeidler, Nonlinear Functional Analysis and its Applications, II/A: Linear Monotone Operators, Springer–Verlag, 1990, xviii+467 pp.
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