The Extremal Universe Exact Solution from Einstein’s Field Equation Gives the Cosmological Constant Directly

Einstein’s field equation is a highly general equation consisting of sixteen equations. However, the equation itself provides limited information about the universe unless it is solved with different boundary conditions. Multiple solutions have been utilized to predict cosmic scales, and among them, the Friedmann-Lemaître-Robertson-Walker solution that is the back-bone of the development into today standard model of modern cosmology: The Λ-CDM model. However, this is naturally not the only solution to Einstein’s field equation. We will investigate the extremal solutions of the Reissner-Nordström, Kerr, and Kerr-Newman metrics. Interestingly, in their extremal cases, these solutions yield identical predictions for horizons and escape velocity. These solutions can be employed to formulate a new cosmological model that resembles the Friedmann equation. However, a significant distinction arises in the extremal universe solution, which does not necessitate the ad hoc insertion of the cosmological constant;instead, it emerges naturally from the derivation itself. To the best of our knowledge, all other solutions relying on the cosmological constant do so by initially ad hoc inserting it into Einstein’s field equation. This clarification unveils the true nature of the cosmological constant, suggesting that it serves as a correction factor for strong gravitational fields, accurately predicting real-world cosmological phenomena only within the extremal solutions of the discussed metrics, all derived strictly from Einstein’s field equation.

EXISTENCE THEOREMS OF EXTREMAL SOLUTIONS FOR A CLASS OF NONLINEAR INTEGRO－DIFFERENTIAL EQUATIONS

In this paper, the following initial value problem for nonlinear integro-differential equationis considered , whereUsing the method of upper and lower solutions and the monotone iterative technique .We obtain existence results of minimal and maximal solutions .

PERIODIC SYSTEMS WITH TIME DEPENDENT MAXIMAL MONOTONE OPERATORS

We consider a first order periodic system in R^(N),involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation.We prove the existence theorems for both the convex and nonconvex problems.We also show the existence of extremal periodic solutions and provide a strong relaxation theorem.Finally,we provide an application to nonlinear periodic control systems.

GLOBAL SOLUTIONS OF SYSTEMS OF NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

The existence of solutions for systems of nonlinear impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces is studied. Some existence theorems of extremal solutions are obtained, which extend the related results for this class of equations on a finite interval with a finite. number of moments of impulse effect. The results are demonstrated by means of an example of an infinite systems for impulsive integral equations.

Existence of Solutions to Nonlinear Impulsive Volterra Integral Equations in Banach Spaces

In this paper, the existence of solutions is studied for nonlinear impulsive Volterra integral equations with infinite moments of impulse effect on the half line R^+ in Banach spaces.By the use of a new comparison result and recurrence method, the new existence theorems are achieved under a weaker compactness-type condition, which generalize and improve the related results for this class of equations with finite moments of impulse effect on finite interval and infinite moments of impulse effect on infinite interval.

SOLUTIONS FOR A SYSTEM OF NONLINEAR RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS UNDER WEAK TOPOLOGY

In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.

ARBITRARILY SMALL NODAL SOLUTIONS FOR PARAMETRIC ROBIN(p,q)-EQUATIONS PLUS AN INDEFINITE POTENTIAL

We consider a nonlinear Robin problem driven by the(p,q)-Laplacian plus an indefinite potential term and with a parametric reaction term.Under minimal conditions on the reaction function,which concern only its behavior near zero,we show that,for all λ>0 small,the problem has a nodal solution y_(λ)∈C^(1)(Ω)and we have y_(λ)→0 in C^(1)(Ω)asλ→0^(+).

ON NONCOERCIVE (p,q)-EQUATIONS

We consider a nonlinear Dirichlet problem driven by a(p,q)-Laplace differential operator(1<q<p).The reaction is(p-1)-linear near±∞and the problem is noncoercive.Using variational tools and truncation and comparison techniques together with critical groups,we produce five nontrivial smooth solutions all with sign information and ordered.In the particular case when q=2,we produce a second nodal solution for a total of six nontrivial smooth solutions all with sign information.

MONOTONE ITERATIVE TECHNIQUE FOR NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

The objective of this paper is to develop monotone techniques for obtaining extremal solutions of initial value problem for nonlinear neutral delay differential equations.

A BIFURCATION PROBLEM ASSOCIATED TO AN ASYMPTOTICALLY LINEAR FUNCTION

We study the existence of positive solutions to a two-order semilineav elliptic problem with Dirichlet boundary condition（Pλ）{-div（c（x）△u=λf（u）inΩ,u=0 on ЭΩ where ΩCR n 〉 2 is a smooth bounded domain; f is a positive, increasing and convex source term and c（x） is a smooth bounded positive function on Ω, We also prove the existence of critical value and claim the uniqueness of extremal solutions.

VISCOSITY BEHAVIOR OF PS/TOLUENE SOLUTION IN DILUTE AND EXTREMELY DILUTE CONCENTRATION REGION

The influences of both the volume of PS/toluene solution in the Ubbelohde viscometer and the precision of the time measuring on the viscosity behavior in dilute and extremely dilute concentration region are investigated. It was found that the influence of the former can neglect, but that of the latter is so prominent that the data fluctuate bitterly and linearity of the curve of the reduced viscosity vs. concentration (hsp/c^c) becomes too bad to obey the Huggins equation down to the extremely dilute region, despite the error of the flow times Dt 0.2s, which is permitted by the conventional method of viscosity measurement. Through strict mathematical analyses, it was found that the error (E) of the reduced viscosity is in proportion and inverse proportion to Dt and concentration c, respectively. So the less the concentration, the more the error is. Consequently, a lowest concentration limit cL corresponding to given experimental error may exist and it will be meaningless for further operation below cL because of the great fluctuation of the data. Therefore, it needs to seriously reconsider the application of the conventional method of Ubbelohde viscosity measurement in the extremely dilute polymer solution under traditional conditions because of the great influence of the experimental error.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SOME NONLINEAR DIFFERENCE EQUATIONS

In this paper, we investigate the asymptotic behavior of the extremal solutions of a difference equation and their character and prove the existence of the non-extremal solutions.

Electrolyte-gated SrCoO_(x) FET sensor for highly sensitive detecting pH in extreme alkalinity solution

The pH monitoring is significantly important in chemical industry,biological process,and pollution treatment.However,it remains a great challenge to measure pH in extreme alkalinity conditions.Herein,we employ an electrolyte-gated field-effect-transistor(FET)strategy using non-stoichiometric SrCoO_(x) with rich oxygen-vacancy defects as channel materials for detecting extreme alkalinity.The corresponding channel can provide effective oxygen-ion-migration sites for reversible transformation of OH-↔O_(2)-+H^(+)driven by electric field.The resultant electrolyte-gated FET sensor exhibits a sensitive linear response to high concentrations of alkaline solution,1–20 M.Significantly,the sensor has the ability to directly indicate the pH values ranging from 14.0 to 17.0 in consideration of ion-activity coefficient data.This work offers a great possibility for directly detecting base concentration as well as pH values in extreme alkaline solutions.

Uniqueness of the Weak Extremal Solution to Biharmonic Equation with Logarithmically Convex Nonlinearities

In this note, we investigate the existence of the minimal solution and the uniqueness of the weak extremal （probably singular） solution to the biharmonic equation △2ω=λg（ω）with Dirichlet boundary condition in the unit ball in Rn, where the source term is logarithmically convex. An example is also given to illustrate that the logarithmical convexity is not a necessary condition to ensure the uniqueness of the extremal solution.

Second Order Nonlinear Evolution Inclusions Existence and Relaxation Results

This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.

Existence Theorems for Periodic Differential Inclusions in IR^N

We study the periodic problem for differential inclusions in R^N.First we look for extremal periodicsolutions.Using techniques from multivalued analysis and a fixed point argument we establish an existencetheorem under some general hypotheses.We also consider the“nonconvex periodic problem”under lowersemicontinuity hypotheses,and the“convex periodic problem”under general upper semicontinuity hypotheseson the multivalued vector field.For both problems,we prove existence theorems under very general hypotheses.Our approach extends existing results in the literature and appear to be the most general results on the nonconvexperiodic problem.