Remarks on existence and multiplicity of solutions for a class of semilinear elliptic equations

Wu, Xing-Ping; Tang, Chun-Lei

doi:10.1016/j.jmaa.2005.08.079

Cited by 4 publications

(3 citation statements)

References 24 publications

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“…Indeed, since ∥ • ∥ and ∥ • ∥ H are a pair of equivalent norms, Sobolev space H ( [43][44][45]) has the orthogonal decom-…”

Section: Statementmentioning

confidence: 99%

Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks with Time Delays under Dirichlet Zero Boundary Value

Rao¹

2020

Preprint

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In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence results of (globally) exponentially stable positive stationary solution of delayed reaction-diffusion cell neural networks under Dirichlet zero boundary value, including the global stability criteria \textbf{in the classical meaning}. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods.

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“…Indeed, since ∥ • ∥ and ∥ • ∥ H are a pair of equivalent norms, Sobolev space H ( [43][44][45]) has the orthogonal decom-…”

Section: Statementmentioning

confidence: 99%

Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks with Time Delays under Dirichlet Zero Boundary Value

Rao¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Indeed, Due to the orthogonal decomposition of Sobolev space W 1,2 0 (Ω) ( [43][44][45]), we may let H = E(µ 1 )⊕E(µ 1 ) ⊥ , where E(µ k ) represents the eigenfunction space of µ k , and E(µ 1 )…”

Section: )mentioning

confidence: 99%

Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks With Time Delays

Rao¹

2020

Preprint

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Firstly, the existence of asymptotically stable nontrivial stationary solution is derived by the comprehensive applications of Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem, variational methods, and construction of compact operators on a convex set. This new theorem shows that the diffusion is a double-edged sword to the stability, refuting the views in previous literature that the greater the diffusion effect, the more stable the system will be. Next, a series of new theorems are presented one by one, which illustrates that the globally asymptotical stability of ordinary differential equations model for delayed neural networks may be locally stable in actual operation due to the inevitable diffusion. Besides, the non-zero constant equilibrium point is pointed out to be not the solution of delayed reaction diffusion system so that the stability of the non-zero constant equilibrium point of reaction diffusion system must lead to a contradiction. That is, non-zero constant equilibrium points are not in the phase plane of dynamic system. In addition, new theorems are further presented to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks, and thereby one equilibrium solution may become several stationary solutions, even infinitely many positive stationary solutions. Finally, a numerical example illustrates the feasibility of the proposed methods.

show abstract

“…Indeed, since ∥ • ∥ and ∥ • ∥ H are a pair of equivalent norms, Sobolev space H ( [43][44][45]) has the orthogonal decomposition H = E(µ 1 ) ⊕ E(µ 1 ) ⊥ , where E(µ k ) represents the eigenfunction space of µ k , and…”

Section: Corollary 34 (Global Stability Invariancementioning

confidence: 99%

Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks with Time Delays under Dirichlet Zero Boundary Value

Rao¹

2020

Preprint

View full text Add to dashboard Cite

In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence of positive stationary solution of delayed reaction-diffusion cell neural networks under Dirichlet zero boundary value. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods.

show abstract

Remarks on existence and multiplicity of solutions for a class of semilinear elliptic equations

Abstract: The existence and multiplicity results are obtained for solutions of a class of the Dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods, respectively.

Cited by 4 publications

References 24 publications

Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks with Time Delays under Dirichlet Zero Boundary Value

Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks with Time Delays under Dirichlet Zero Boundary Value

Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks With Time Delays

Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks with Time Delays under Dirichlet Zero Boundary Value

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