Conditions For Existence Of Solutions Research Articles

Generalized Boussinesq System with Energy Dissipation: Existence of Stationary Solutions

In this paper, we investigate the solvability of a boundary value problem for a heat and mass transfer model with the spatially averaged Rayleigh function. The considered model describes the 3D steady-state non-isothermal flow of a generalized Newtonian fluid (with shear-dependent viscosity) in a bounded domain with Lipschitz boundary. The main novelty of our work is that we do not neglect the viscous dissipation effect in contrast to the classical Boussinesq approximation, and hence, deal with a system of strongly nonlinear partial differential equations. Using the properties of the averaging operation and d-monotone operators as well as the Leray–Schauder alternative for completely continuous mappings, we prove the existence of weak solutions without any smallness assumptions for model data. Moreover, it is shown that the set of all weak solutions is compact, and each solution from this set satisfies some energy equalities.

Necessary and sufficient condition for existence for a case of eigenvalues of multiplicity two

We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/09/abstr.html

Stability Results for Nonlinear Implicit ϑ-Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions

This article examines the necessary conditions for the unique existence of solutions to nonlinear implicit ϑ-Caputo fractional differential equations accompanied by fractional order integral boundary conditions. The analysis draws upon Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Furthermore, the circumstances leading to the attainment of Ulam–Hyers–Rassias forms of stability are established. An illustrative example is provided to demonstrate the derived findings.

Solvable limits of a class of generalized vector nonlocal nonlinear schrödinger equation with balanced loss-gain

We consider a class of one dimensional Vector Nonlocal Non-linear Schrödinger equation (VNNLSE) in an external complex potential with time-modulated Balanced Loss-Gain(BLG) and Linear Coupling(LC) among the components of Schrödinger fields, and space-time dependent nonlinear strength. The system admits Lagrangian and Hamiltonian formulations under certain conditions. It is shown that various dynamical variables like total power, -symmetric Hamiltonian, width of the wave-packet and its speed of growth, etc are real-valued despite the Hamiltonian density being complex-valued. We study the exact solvability of the generic VNNLSE with or without a Hamiltonian formulation. In the first part, we study time-evolution of moments which are analogous to space-integrals of Stokes variables and find condition for existence of solutions which are bounded in time. In the second part, we use a non-unitary transformation followed by a coordinate transformation to map the VNNLSE to various solvable equations. The coordinate transformation is not required at all for the limiting case when non-unitary transformation reduces to pseudo-unitary transformation. The exact solutions are bounded in time for the same condition which is obtained through the study of time-evolution of moments. Various exact solutions of the VNNLSE are presented.

On the Fisher‐KPP model with nonlocal nonlinear sources

AbstractThe Cauchy problem considered in this paper is the following: where . When the coefficient remains positive, (1) is analogous to It is well known that when , the local solution of (2) blows up in finite time as long as the initial value is nontrivial. The main aim of this paper is to obtain sufficient conditions for global existence of solutions to (1) for and thus it forms a contrast to (2). The exponent produces an exact balance in the mass‐invariant scaling of diffusion and growth in (1). In particular, we prove that if , and where is the sharp constant of the inequality , then (1) admits a global classical solution. Moreover, the long time behavior of the solution is also discussed. In addition, we prove that when , the problem is solvable globally in time without any restriction on the initial data.

Solution Existence Theory for Artificial Neural Networks

Artificial neural networks (NNs) have been widely used in creation of various machine learning (ML) models. Training an NN for a given dataset is essentially finding a set of weights and biases, known as learning (or training) parameters. An NN uses the affine transformations to encode the features in the dataset in the form of learning parameters. One may naturally ask fundamental questions: (1) can the NN be trained, meaning that can a solution for a set of learning parameters can be found? and (2) is the solution unique? This paper addresses these questions. First, detailed formulation is presented for a simple NN with just one basic affine transformation unit (ATU) for linear regression problems. Conditions for solution existence and uniqueness will be examined, leading to the Solution Existence Theory. A discussion will be presented when this Solution Existence Theory is extended to general NNs for complex problems, in addition to a discussion on predictability of NNs.

Malmquist-type theorems on some complex differential-difference equations

Abstract This article is devoted to study the existence conditions of solutions to several complex differential-difference equations. We obtain some Malmquist theorems related to complex differential-difference equations with a more general form than the previous equations given by Zhang, Huang, and others. Moreover, some examples are provided to demonstrate why some restrictive conditions in some of our theorems cannot be removed.

Homogeneous Sylvester equations of lower triangular matrices

We study the homogeneous Sylvester equation AX=XB of lower triangular matrices with 1's along the diagonal. A necessary and sufficient condition for existence of solution is presented in terms of computable linear systems.

A parallel type decomposition scheme for quasi-linear abstract hyperbolic equation

Abstract The Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is the sum of operators A 1 , A 2 , … , A m {A_{1},A_{2},\ldots,A_{m}} . Each summand is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval the classical difference problems are solved in parallel (independently from each other) with the operators A 1 , A 2 , … , A m {A_{1},A_{2},\ldots,A_{m}} . The weighted average of the obtained solutions is announced as an approximate solution at the right end of the local interval. The convergence of the proposed scheme is proved and the approximate solution error is estimated, as well as the error of the difference analogue for the first-order derivative for the case when the initial problem data satisfy the natural sufficient conditions for solution existence.

Spinning gravimagnetic particles in Schwarzschild-like black holes

We study the motion of a spinning particle with gravimagnetic moment in Schwarzschild-like spacetimes with a metric $d{s}^{2}=\ensuremath{-}f(r)d{t}^{2}+{f}^{\ensuremath{-}1}(r)d{r}^{2}+{r}^{2}d{\mathrm{\ensuremath{\Omega}}}^{2}$, specifically we deal with Schwarzschild, Reissner-Nordstrom black holes as well as Ayon-Beato-Garcia and Bardeen regular spacetimes. First, we introduce the Hamiltonian system of equations which describes such kind of particles. In the case of null gravimagnetic moment, the equations are equivalent to the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations. Working in the equatorial plane, using the constants of motion generated by the symmetries of the considered spacetimes and the spin supplementary conditions (SSC), we change the problem of solving six differential equations for the momenta and the nonvanishing spin-tensor components to solving six algebraic equations. We show that the equation for the ${P}_{0}(r)$ component totally decouples, ${P}_{0}(r)$ can be found by solving a 6th order polynomial. We analyze the conditions for existence of solutions of this algebraic system for the relevant cases of gravimagnetic moment equal to unit, which corresponds to a gravimagnetic particle, and zero which corresponds to the MPTD system. A numerical algorithm to generate solutions of the momenta ${P}_{\ensuremath{\mu}}$ is provided and some solutions are generated.

New Techniques for Solutions of Fuzzy Volterra Integral Equations via Soft Set-Valued Maps

In this paper, new generalized concepts of set-valued maps under the name soft set-valued maps are established. The presented results herein are soft set extensions of various existing fuzzy set-valued and crisp multivalued fixed point theorems. Examples are provided to authenticate the hypotheses of our obtained results. Moreover, our main result is applied to establish conditions for existence of solutions to fuzzy Volterra integral equations.

A global uniqueness of solutions implies global existence for (l+1)-point boundary value problems

Let n≥2 denote an integer, and let l∈{1,…,n−1}. A family of (l+1)-point boundary value problems for an n-th order ordinary differential equation is studied and sufficient conditions for existence of solutions are obtained. In particular, it is shown that a type of global uniqueness of solutions implies global existence of solutions. Then explicit conditions in terms of monotonicity of the nonlinear term are obtained to imply the global uniqueness of solutions.

Fuzzy fixed point theorems and Ulam-Hyers stability of fuzzy set-valued maps

Abstract In this paper, new common fuzzy fixed point theorems for sequence of fuzzy set-valued maps in the framework of complete b-metric spaces are established. Consequently, corresponding fixed point theorems in the setting of point-to-set-valued and single-valued mappings are deduced. A few nontrivial examples which dwell upon the generality of our results are provided. Moreover, following the fact that most available Ulam-Hyers type stability results deal with crisp mappings, we initiate the study of stability and well-posedness of functional inclusions involving fuzzy set-valued maps. It is well-known that solution of any functional inclusion is a subset of an appropriate ambient space. With this information, fuzzy fixed point problem for which the right-hand-side is a cut set of a fuzzy set-valued map is introduced. Furthermore, sufficient conditions for existence of solutions of Cantilever Beam Problem and integral inclusions are investigated to indicate the usability of our obtained results.

Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS

We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in $${\mathbb {R}}^N$$ , $$\begin{aligned} i \partial _t u +\Delta ^2 u -|x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$ where $$\sigma > 0$$ and $$b > 0$$ . We first study the local well-posedness in $${\dot{H}}^{s_c}\cap \dot{H}^2 $$ , for $$N\ge 5$$ and $$0<s_c<2$$ , where $$s_c=\frac{N}{2}-\frac{4-b}{2\sigma }$$ . Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $$\dot{H}^{s_c}\cap \dot{H}^2$$ with $$0\le s_c<2$$ . Finally, we study the phenomenon of $$L^{\sigma _c}$$ -norm concentration for finite time blow up solutions with bounded $$\dot{H}^{s_c}$$ -norm, where $$\sigma _c=\frac{2N\sigma }{4-b}$$ . Our main tool is the compact embedding of $$\dot{L}^p\cap \dot{H}^2$$ into a weighted $$L^{2\sigma +2}$$ space, which may be seen of independent interest.

Existence of solutions for third order multi point impulsive boundary value problems on time scales

In this paper, we obtain sufficient conditions for existence of solutions of a third order m-point impulsive boundary value problem on time scales. To the best of our knowledge, there is hardly any work dealing with third order multi point dynamic impulsive BVPs. The reason may be the complex arguments caused by both impulsive perturbations and calculations on time scales. As an application, we give an example demonstrating our results.

Necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and solution representation

In this paper, we investigate the necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and the solution representation by using the multivariate Mittag-Leffler function. First we convert fuzzy initial value problem into the cut problem (system of fractional differential equations with inequality constraints) and obtain existence results for the solution of the cut problem under (1,1)- differentiability. Next we study the conditions for the solutions of the cut problem to constitute the solution of a fuzzy initial value problem and suggest a necessary and sufficient condition for the (1,1)-solution. Also, some examples are given to verify the effectiveness of our proposed method. The necessary and sufficient condition, solution representation for (1,2)-solution of initial value problem of fuzzy fractional Bagley-Torvik equation are shown in Appendix.

On Well-Posedness and Concentration of Blow-Up Solutions for the Intercritical Inhomogeneous NLS Equation

We consider the focusing inhomogeneous nonlinear Schrödinger (INLS) equation in \({\mathbb {R}}^N\)$$\begin{aligned} i \partial _t u +\Delta u + |x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$where \(N\ge 2\) and \(\sigma \), \(b>0\). We first obtain a small data global result in \(H^1\), which, in the two spatial dimensional case, improves the third author result in [22] on the range of b. For \(N\ge 3\) and \(\frac{2-b}{N}<\sigma <\frac{2-b}{N-2}\), we also study the local well posedness in \(\dot{H}^{s_c}\cap \dot{H}^1 \), where \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma }\). Sufficient conditions for global existence of solutions in \(\dot{H}^{s_c}\cap \dot{H}^1\) are also established, using a Gagliardo-Nirenberg type estimate. Finally, we study the \(L^{\sigma _c}\)-norm concentration phenomenon, where \(\sigma _c=\frac{2N\sigma }{2-b}\), for finite time blow-up solutions in \(\dot{H}^{s_c}\cap \dot{H}^1\) with bounded \(\dot{H}^{s_c}\)-norm. Our approach is based on the compact embedding of \(\dot{H}^{s_c}\cap \dot{H}^1\) into a weighted \(L^{2\sigma +2}\) space.

Stationary Non-equilibrium Solutions for Coagulation Systems

We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.

Real scalar field solutions of the EKG equations including matter

Static (not stationary) solutions of the Einstein–Klein–Gordon (EKG) equations including matter are obtained for real scalar fields. The scalar field interaction with matter is considered. The introduced coupling allows the existence of static solutions in contraposition with the case of the simpler EKG equations for real scalar fields and gravity. Surprisingly, when the considered matter is a photon-like gas, it turns out that the gravitational field intensity at large radial distances becomes nearly a constant, exerting an approximately fixed force to small bodies at any distance. The effect is clearly related with the massless character of the photon-like field. It is also argued that the gravitational field can generate a bounding attraction, that could avoid the unlimited increase in mass with the radius of the obtained here solution. This phenomenon, if verified, may furnish a possible mechanism for explaining how the increasing gravitational potential associated to dark matter, finally decays at large distances from the galaxies. A method for evaluating these photon bounding effects is just formulated in order to be further investigated.

The existence and stability of stationary solutions of the inflow problem for full compressible Navier-Stokes-Poisson system

In this paper, we consider an inflow problem for the non-isentropic Navier-Stokes-Poisson system in a half line (0, ∞). For the general gas including ideal polytropic gas, we first give some results for the existence of the stationary solution with the aid of center manifold theory on a 4 × 4 system of autonomous ordinary differential equations. We also show the time asymptotic stability of the stationary solutions with small strength under smallness assumptions on the initial perturbations by using an elementary energy method.