arXiv subjects
Publications and source records attributed to Shalmali Bandyopadhyay.
We establish the existence of weak solutions of coupled systems of elliptic partial differential equations with quasimonotone nonlinearities in the differential equation and on the boundary.
We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings. Under a Lipschitz condition on the nonlinearity bounded by the first eigenvalue, we prove existence and uniqueness using the contraction mapping theorem. Under a weaker one-sided growth condition, we establish existence using the Leray--Schauder fixed point theorem. To apply these functional analytic methods, we reformulate the problem as an operator equation, which requires developing the spectral theory for the Dirichlet Laplacian with mixed nabla-delta derivatives. We establish self-adjointness, positivity, and completeness of eigenfunctions, and the product eigenfunctions form a complete orthonormal basis in the n-dimensional setting.
We study a second-order differential equation involving a quasi-derivative, leading to a non-self-adjoint Sturm--Liouville-type problem with four coefficient functions. To analyze this equation, we develop a generalized Prüfer transformation that expresses solutions in terms of amplitude and phase variables. We further prove the monotonicity of eigenfunction zeros with respect to the spectral parameter and derive upper and lower bounds for the eigenvalues.
We consider elliptic systems with superlinear and subcritical boundary conditions and a bifurcation parameter as a multiplicative factor. By combining the rescaling method with degree theory and elliptic regularity theory, we prove the existence of a connected branch of positive weak solutions that bifurcates from infinity as the parameter approaches zero. Furthermore, under additional conditions on the nonlinearities near zero, we obtain a global connected branch of positive solutions bifurcating from zero, which possesses a unique bifurcation point from infinity when the parameter is zero. Finally, we analyze the behavior of this branch and discuss the number of positive weak solutions with respect to the parameter using bifurcation theory, degree theory, and sub- and super-solution methods.
We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-Δu + u = 0$ in $Ω$ with $\frac{\partial u}{\partial η} = λf(u)$ on $\partialΩ$ as well as an extension to a corresponding system of equations. While existence, uniqueness, nonexistence, and multiplicity results for such problems are well-established, their numerical treatment presents computational challenges due to the absence of comparison principles and complex bifurcation phenomena. We present finite difference formulations for both single equations and coupled systems with cross-coupling boundary conditions, establishing admissibility results for the finite difference method. We derive principal eigenvalue analysis for the linearized problems to determine unique bifurcation points from trivial solutions. The eigenvalue analysis provides additional insight into the theoretical properties of the problem while also providing intuition for computing approximate solutions based on the proposed finite difference formulation. We combine our finite difference methods with continuation methods to trace complete bifurcation curves, validating established existence and uniqueness results and consistent with the results of the principle eigenvalue analysis.
This paper establishes an existence theory for discrete second-order boundary value problems on non-uniform time grids using the upper and lower solution method. We consider difference equations of the form $u^{ΔΔ}(t_{i-1}) + f(t_i, u(t_i), u^Δ(t_{i-1})) = 0$ on a non-uniform time grid ${t_0, t_1, \ldots, t_{n+2}}$ with mixed boundary conditions $u^Δ(t_0) = 0$ and $u(t_{n+2}) = g(t_{n+2})$. This extends previous work on homogeneous boundary conditions to the non-homogeneous case, requiring a sophisticated functional analytic framework to handle the resulting affine function spaces. Our approach employs a decomposition strategy that separates boundary effects from the differential structure, enabling the application of Brouwer's Fixed Point Theorem to establish existence with solutions bounded between upper and lower functions.
We explore singular second-order boundary value problems with mixed boundary conditions on a general time scale. Using the lower and upper solutions method combined with the Brouwer fixed point theorem we demonstrate the existence of a positive solution and obtain the desired solution by using a sequence of solutions to a sequence of nonsingular second-order equations and passing to the limit.
This paper investigates the existence of positive solutions for regular discrete second-order single-variable boundary value problems with mixed boundary conditions, including a nonhomogeneous Dirichlet boundary condition, of the form: \begin{equation*} u^{ΔΔ}(t-1)+h(t,\ u(t),\ u^Δ(t-1))=0 \mbox{ for }t\in[1,\ T+1];~u^Δ(0)=0;~u(T+2)=g(T+2) \end{equation*} where h is continuous on $[1, T + 1] \times \mathbb{R}^2$ and $g: [0, T + 2] \to \mathbb{R}^+$ is continuous. Using the concept of upper and lower solutions, we establish conditions under which the boundary value problem admits at least one positive solution. Our approach involves constructing an auxiliary problem with a modified nonlinearity and applying Brouwer Fixed Point Theorem to a carefully defined solution operator. We prove that any solution to this auxiliary problem that remains within the bounds of the upper and lower solutions is equivalent to a solution of the original problem.
This paper investigates the existence of solutions for a class of nonlinear higher-order dynamic equations subject to mixed boundary conditions. We consider boundary value problems in which the nonlinear reaction functions satisfy polynomial growth conditions both in the interior of the domain and on the boundary. Our analysis employs a systematic approach based on fixed-point theorems for expansive mappings combined with completely continuous operators to establish stronger existence results. Under appropriate growth conditions on the nonlinear terms, we first prove the existence of at least one classical solution, which is not guaranteed to be nonnegative. We then strengthen our hypotheses to establish the existence of at least three nonnegative solutions. The theoretical framework relies on cone theory and carefully constructed open bounded subsets within function spaces equipped with appropriate norms. Our methodology provides a unified approach to multiple solution problems for higher-order
We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory.