Sufficient Conditions For Existence Research Articles

Results on Implicit Fractional Pantograph Equations with Mittag‐Leffler Kernel and Nonlocal Condition

In this study, the main focus is on an investigation of the sufficient conditions of existence and uniqueness of solution for two‐classess of nonlinear implicit fractional pantograph equations with nonlocal conditions via Atangana–Baleanu–Riemann–Liouville (ABR) and Atangana–Baleanu–Caputo (ABC) fractional derivative with order σ ∈ (1,2]. We introduce the properties of solutions as well as stability results for the proposed problem without using the semigroup property. In the beginning, we convert the given problems into equivalent fractional integral equations. Then, by employing some fixed‐point theorems such as Krasnoselskii and Banach’s techniques, we examine the existence and uniqueness of solutions to proposed problems. Moreover, by using techniques of nonlinear functional analysis, we analyze Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability results. As an application, we provide some examples to illustrate the validity of our results.

On degree theory for non-monotone type fractional order delay differential equations

<abstract><p>In this paper, we establish a qualitative theory for implicit fractional order differential equations (IFODEs) with nonlocal initial condition (NIC) with delay term. Because area related to investigate existence and uniqueness of solution is important field in recent times. Also researchers are using existence theory to derive some prior results about a dynamical problem weather it exists or not in reality. In literature, we have different tools to study qualitative nature of a problem. On the same line the exact solution of every problem is difficult to determined. Therefore, we use technique of numerical analysis to approximate the solutions, where stability analysis is an important aspect. Therefore, we use a tool from non-linear analysis known as topological degree theory to develop sufficient conditions for existence and uniqueness of solution to the considered problem. Further, we also develop sufficient conditions for Hyers- Ulam type stability for the considered problem. To justify our results, we also give an illustrative example.</p></abstract>

A nonlinear population dynamics model of patient diagnosis and treatment involving in two level medical institutions and its qualitative analysis of positive singularity.

In this article, we firstly establish a nonlinear population dynamical model to describe the changes and interaction of the density of patient population of China's primary medical institutions (PHCIs) and hospitals in China's medical system. Next we get some sufficient conditions of existence of positive singularity by utilising homotopy invariance theorem of topological degree. Meanwhile, we study the qualitative properties of positive singularity based on Perron's first theorem. Furthermore, we briefly analyze the significance and function of the mathematical results obtained in this paper in practical application. As verifications, some numerical examples are ultimately exploited the correctness of our main results. Combined with the numerical simulation results and practical application, we give some corresponding suggestions. Our research can provide a certain theoretical basis for government departments to formulate relevant policies.

On multi-step methods for singular fractional q-integro-differential equations

Abstract The objective of this paper is to investigate, by applying the standard Caputo fractional q-derivative of order α \alpha , the existence of solutions for the singular fractional q-integro-differential equation D q α [ k ] ( t ) = Ω ( t , k 1 , k 2 , k 3 , k 4 ) {{\mathcal{D}}}_{q}^{\alpha }\left[k]\left(t)=\Omega \left(t,{k}_{1},{k}_{2},{k}_{3},{k}_{4}) , under some boundary conditions where Ω \Omega is singular at some point 0 ≤ t ≤ 1 0\le t\le 1 , on a time scale T t 0 = { t : t = t 0 q n } ∪ { 0 } {{\mathbb{T}}}_{{t}_{0}}=\left\{t:t={t}_{0}{q}^{n}\right\}\cup \left\{0\right\} , for n ∈ N n\in {\mathbb{N}} where t 0 ∈ R {t}_{0}\in {\mathbb{R}} and q ∈ ( 0 , 1 ) q\in \left(0,1) . We consider the compact map and avail the Lebesgue dominated theorem for finding solutions of the addressed problem. Besides, we prove the main results in context of completely continuous functions. Our attention is concentrated on fractional multi-step methods of both implicit and explicit type, for which sufficient existence conditions are investigated. Finally, we present some examples involving graphs, tables and algorithms to illustrate the validity of our theoretical findings.

Fixed points of hesitant fuzzy set-valued maps with applications

In this paper, the notion of hesitant fuzzy fixed points is introduced. To this end, we define Suzuki-type $(\alpha,\beta)$-weak contractions in the framework of hesitant fuzzy set-valued maps, thereby establishing some corresponding fixed point theorems. The presented concept herein is an extension of fuzzy set-valued and multi-valued mappings in the corresponding literature. Examples are provided to support the assertions and generality of our obtained ideas. Moreover, one of our results is applied to investigate sufficient conditions for existence of a class of functional equation arising in dynamic programming.

Kobayashi–Warren–Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation

In this paper we study the Dirichlet problem for the Kobayashi–Warren–Carter system. This system of parabolic PDE’s models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the ω-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the ω-limit set belong to the space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constant solutions.

Consumer Optimization and a First-Order PDE with a Non-Smooth System

We study a first-order nonlinear partial differential equation and present a necessary and sufficient condition for the global existence of its solution in a non-smooth environment. Using this result, we prove a local existence theorem for a solution to this differential equation. Moreover, we present two applications of this result. The first concerns an inverse problem called the integrability problem in microeconomic theory and the second concerns an extension of Frobenius’ theorem.

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.

To investigate a class of the singular fractional integro-differential quantum equations with multi-step methods

The objective of this paper is to investigate, by applying the standard Caputo fractional $q$--derivative of order $\alpha$, the existence of solutions for the singular fractional $q$--integro-differential equation $\mathcal{D}_q^\alpha [k](t) = \Omega (t , k(t), k'(t), \mathcal{D}_q^\beta [k](t), \int_0^t f(r) k(r) \, {\mathrm d}r )$, under some boundary conditions where $\Omega(t, k_1, k_2, k_3, k_4)$ is singular at some point $0 \leq t\leq 1$, on a time scale $\mathbb{T}_{ t_0} = \{ t : t =t_0q^n\}\cup \{0\}$, for $n\in \mathbb{N}$ where $t_0 \in \mathbb{R}$ and $q \in (0,1)$. We consider the compact map and avail the Lebesgue dominated theorem for finding solutions of the addressed problem. Besides, we prove the main results in context of completely continuous functions. Our attention is concentrated on fractional multi-step methods of both implicit and explicit type, for which sufficient existence conditions are investigated. Lastly, we present some examples involving graphs, tables and algorithms to illustrate the validity of our theoretical findings.

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.

Multi-leader-follower potential games

In this paper, we discuss a particular class of Nash games, where the participants of the game (the players) are divided into two groups (leaders and followers) according to their position or influence on the other players. Moreover, we consider the case, when the leaders’ and/or the followers’ game can be described as a potential game. This is a subclass of Nash games that has been introduced by Monderer and Shapley in 1996 and has beneficial properties to reformulate the bilevel Nash game. We develope necessary and sufficient conditions for Nash equilibria and present existence and uniqueness results. Furthermore, we discuss some Examples to illustrate our results. In this paper, we discussed analytical properties for multi-leader follower potential games, that form a subclass of hierarchical Nash games. The application of these theoretical results to various fields of applications are a future research topic. Moreover, they are meant to serve as a starting point for the developement of efficient numerical solution methods for multi-leader-follower games.

Fixed Points of $p$-Hybrid $L$-Fuzzy Contractions

In this paper, the notion of $p$-hybrid $L$-fuzzy contractions in the framework of $b$-metric space is introduced. Sufficient conditions for existence of common $L$-fuzzy fixed points under such contractions are also investigated. The established ideas are generalizations of many concepts in fuzzy mathematics. In the case where our postulates are reduced to their classical variants, the concept presented herein merges and extends several significant and well-known fixed point theorems in the setting of both single-valued and multi-valued mappings in the corresponding literature of discrete and computational mathematics. A few of these special cases are pointed out and discussed. In support of our main hypotheses, a nontrivial example is provided.

On the inhomogeneous NLS with inverse-square potential

We consider the inhomogeneous nonlinear Schr\"odinger equation with inverse-square potential in $\mathbb{R}^N$ $$ i u_t + \mathcal{L}_a u+\lambda |x|^{-b}|u|^\alpha u = 0,\;\;\mathcal{L}_a=\Delta -\frac{a}{|x|^2}, $$ where $\lambda=\pm1$, $\alpha,b>0$ and $a>-\frac{(N-2)^2}{4}$. We first establish sufficient conditions for global existence and blow-up in $H^1_a(\mathbb{R}^N)$ for $\lambda=1$, using a Gagliardo-Nirenberg-type estimate. In the sequel, we study local and global well-posedness in $H^1_a(\mathbb{R}^N)$ in the $H^1$-subcritical case, applying the standard Strichartz estimates combined with the fixed point argument. The key to do that is to establish good estimates on the nonlinearity. Making use of these estimates, we also show a scattering criterion and construct a wave operator in $H^1_a(\mathbb{R}^N)$, for the mass-supercritical and energy-subcritical case.

Existence of common best proximity point for single and multivalued non-self mappings

In this article, we introduce a new concept called proximal E. A property for single and multivalued mappings. We prove existence of proximally coincidence point for such class of mappings. We provide interesting example which illustrates our main results. Finally, we provide sufficient condition for existence of common best proximity point for this class of mappings.

Dynamics of Fractional Holling Type-II Predator-Prey Model with Prey Refuge and Additional Food to Predator

Prey refuge and additional food provided to predator help in balancing the population in ecology. In this paper, we have analysed a fractional Holling Type-II predator-prey model with prey refuge and additional food to predator. Existence and uniqueness of solution is established with the help of existing theory of fractional calculus. Sufficient conditions for existence and stability of equilibrium points are derived. Effect of prey refuge and quality of additional food to predator on balancing the population is crucially analyzed. Theoretical results are supported by numerical simulations.

Impulsive control and global stabilization of reaction‐diffusion epidemic model

In this paper, by using the variational method, a sufficient condition for the unique existence of the stationary solution of the reaction‐diffusion ecosystem is obtained, which directly leads to the global asymptotic stability of the unique equilibrium point. Moreover, delayed feedback ecosystem with reaction‐diffusion item is considered, and the above sufficient condition may not warrant the global stability for it. Utilizing impulse control results in its globally exponential stability. Besides, one of numerical examples shows that applying Laplacian semigroup in a newly obtained criterion is better than applying impulsive differential inequations, for the impulse condition becomes less harsh. It is worth mentioning that the Neumann zero‐boundary value that the infected and the susceptible people or animals should be controlled in the epidemic prevention area and not allowed to cross the border, which is a good simulation of the actual situation of epidemic prevention. And numerical simulations indicate the effectiveness of impulse control, which has a certain enlightening effect on the actual epidemic prevention work. That is, in the face of the epidemic situation, taking a certain frequency of positive and effective epidemic prevention measures is conducive to the stability and control of the epidemic situation. Particularly, the newly obtained theorems quantifies this feasible step.

Metrizability of Pseudo Topological Vector Spaces

In the present work, we introduce the notion of pseudo-seminorm, then we established, a criterion for the metrizability of a pseudo vector space \U0001d6ef with a pseudo topology \U0001d70f. It is specified, that for the metrizability of \U0001d6ef is necessary and sufficient that \U0001d6ef is first countable. Moreover, a sufficient condition for existence a base for the filter of neighborhoods of zero is proved and demonstrated that this condition introduces a compatible pseudo topology with the algebraic vector space structures.

ON SOLUTIONS TO ITERATIVE DIFFERENTIAL EQUATIONS

We present existence and uniqueness of solution for iterative differential equation y’=f(x,y,y(g(y))) using Picard’s iteration method by imposing some conditions on f and g. These conditions are sufficient conditions (not necessary conditions) for existence of solution. Furthermore, examples are provided to clarify existence and uniqueness of solution.

Life cycle of startup financing

I characterize an optimal, incentive compatible, and renegotiation proof contract of venture capital (VC ) financing of a startup (that may be successful or not) whose rate of arrival of success is a function of the accumulated investment stock. The contract depends on the startup valuation, prior probability of success, and initial capital. Sufficient conditions for existence of such a contract are specified. The paper explains why the startup has to rely on different ways of financing in different stages of its life, and why VC financing is not feasible in early stages of development of the startup.

Stability results of positive solutions for a system ofψ-Hilfer fractional differential equations

The major objective of this work is to investigate sufficient conditions of existence and uniqueness of positive solutions for a finite system of ψ-Hilfer fractional differential equations. The gained results are obtained by building the upper and lower control functions of the nonlinear expression with the help of fixed point theorems such as Banach and Schauder. Furthermore, we establish various kinds of Ulam stability results by applying the techniques of nonlinear functional analysis. A pertinent example is provided to corroboration of the results obtained.