Minimization of Bratu-type obstacle boundary value problem and exponential variational inequalities: Existence, uniqueness, and applications

Saudia Jabeen; Bushra Kanwal; Ibtisam Mohammed Aldawish; Sheza El-Deeb; Saudia Jabeen; Bushra Kanwal; Ibtisam Mohammed Aldawish; Sheza El-Deeb

doi:10.3934/math.20261026

AIMS Mathematics

2026, Volume 11, Issue 8: 25601-25626. doi: 10.3934/math.20261026

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Research article

Minimization of Bratu-type obstacle boundary value problem and exponential variational inequalities: Existence, uniqueness, and applications

1.
Department of Mathematics, Air University, E-9, Islamabad 44000, Pakistan
2.
Department of Mathematical Sciences, Fatima Jinnah Women University, Rawalpindi 46000, Pakistan
3.
Mathematics and Statistics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
4.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
5.
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt

Received: 16 April 2026 Revised: 19 July 2026 Accepted: 04 August 2026 Published: 19 August 2026
MSC : 49J40, 90C33

This paper introduces and systematically analyzes a novel class of variational inequalities, called exponential variational inequalities (EVIs), which incorporate exponential nonlinearities through exponentially convex functions. We establish existence results using boundary conditions, coercivity assumptions, and monotone operator techniques. Uniqueness is proved under the condition $ \alpha > \mu\, \ell $, relating the strong monotonicity constant $ \alpha $ of the operator, the exponential bound $ \mu $, and the Lipschitz constant $ \ell $ of the derivative. A contraction mapping argument yields both existence and uniqueness, together with a linearly convergent iterative method under explicit step-size conditions. As an application, we study a Bratu-type obstacle boundary value problem, establish its formulation as an exponential variational inequality, and derive explicit small-domain conditions which ensure existence and uniqueness. These results provide a unified framework for analyzing variational inequalities with exponential nonlinearities arising in combustion, population dynamics, and optimal control.
- exponential variational inequalities,
- Bratu problem,
- obstacle problems,
- existence and uniqueness,
- strong convergence,
- projection methods,
- iterative methods
Citation: Saudia Jabeen, Bushra Kanwal, Ibtisam Mohammed Aldawish, Sheza El-Deeb. Minimization of Bratu-type obstacle boundary value problem and exponential variational inequalities: Existence, uniqueness, and applications[J]. AIMS Mathematics, 2026, 11(8): 25601-25626. doi: 10.3934/math.20261026

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Abstract

This paper introduces and systematically analyzes a novel class of variational inequalities, called exponential variational inequalities (EVIs), which incorporate exponential nonlinearities through exponentially convex functions. We establish existence results using boundary conditions, coercivity assumptions, and monotone operator techniques. Uniqueness is proved under the condition $ \alpha > \mu\, \ell $, relating the strong monotonicity constant $ \alpha $ of the operator, the exponential bound $ \mu $, and the Lipschitz constant $ \ell $ of the derivative. A contraction mapping argument yields both existence and uniqueness, together with a linearly convergent iterative method under explicit step-size conditions. As an application, we study a Bratu-type obstacle boundary value problem, establish its formulation as an exponential variational inequality, and derive explicit small-domain conditions which ensure existence and uniqueness. These results provide a unified framework for analyzing variational inequalities with exponential nonlinearities arising in combustion, population dynamics, and optimal control.

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