This paper investigates the nonlinear eigenvalue problems (NEPs) for a class of singular Katugampola fractional differential systems with four-point coupled boundary conditions. Firstly, we establish the Green's functions of the aforementioned NEPs and their fundamental analytic properties. Secondly, utilizing some classical fixed-point theorems, some explicit eigenvalue interval-dependent criteria are derived for the existence of at least one positive solution to the addressed NEPs. As applications, some examples are presented to illustrate the feasibility and effectiveness of our main results.
In this paper, we will show the general solution of the Bessel differential equation given by x(2)y"+xy' + (x(2)-v(2))y = 0, where v,x is an element of R and x>0, but only when v= 2m-1/ 2 with m is an element of N. Moreover, contrary to what we found in the literature, our general solution does not depend on a series of functions, our algorithm provides the exact general solution.
We establish the existence of solution for the Navier-Stokes equations with a general damping term, we give examples. We also derive estimates on the asymptotic behavior of the solution, for instance, energy estimate decay in time and extinction in time. We construct a sequence of solutions of auxiliary equations in finite dimension that converges to a genuine solution of the original equations.
A study of the existence and uniqueness of the weak solution to a class of nonlinear elliptic equations governed by the logarithmic perturbation is offered. We exploit interesting properties of the new modular function involving L(p)log(alpha)L-growth and the optimal embedding theorem for Orlicz-Sobolev spaces. We are concerned with these properties in analyzing the existence and uniqueness of the solution by the theory of pseudo-monotone operators proposed in [2,3] combined with variational methods. Our approach deals not only with problems of the p-Laplacian type but also yields a slight extension of the results for more general differential operators with a similar structure.
This article presents sufficient conditions for the complete controllability of generalized semilinear impulsive systems in a finite-dimensional space. The analysis focuses on cases where the nonlinear perturbation functions satisfy the Lipschitz continuity condition. We establish these conditions by leveraging functional analysis techniques and various fixed-point theorems. Furthermore, a numerical example is included to demonstrate the effectiveness of the proposed results.
In this work, we study an initial boundary-value problem for a stochastic Benjamin-Bona-Mahony equation with Riesz-fractional spatial derivative and white noise on the half-line. For the associated linear problem, we construct the Green's function adapting the main ideas of the Fokas method. Then, the main problem will be understood in the Walsh sense and the Picard scheme is used to prove existence and uniqueness of solutions. Moreover, an example is presented to show the results obtained.
In this paper, we establish the existence and the global asymptotic behavior of positive solutions in an exterior domain Omega subset of R-d , d >= 3, {(-Delta) (alpha/ 2) x = f(t)x(p), in Omega, x > 0, in Omega, lim( t ->partial derivative Omega )delta(t)(1- alpha /2) x(t) = 0, lim (|t|->infinity) x(t) = 0, where (-Delta)( alpha /2) is the infinitesimal generator of a killed symmetric alpha -stable process X-Omega on Omega, 0 < alpha < 2, p < 1 and the function f is positive and satisfies the suitable conditions related to the Karamata classes K(0 )and K-infinity. Our approach relies on potential theory, Karamata regular variation theory, and the Schauder fixed point theorem.
We analyze the existence of multiple solutions for a sixth-order boundary value problem. Firstly, we introduce an operator that transforms the problem into a fixed-point problem and delineate its key properties. Subsequently, we investigate the existence of solutions in the functional space C1[0,1], employing the fixed-point theorem of Avery-Peterson. We then provide non-trivial examples and establish a theorem based on the Banach-Piccard theorem, motivating the definition of a numerical method based on the compression principle for the problem. Additionally, we discuss the utilization of nonlinear optimization methods for the problem and compare them with the classical method based on the contraction principle.
For the (p,q)-Laplace equation:-Delta pu-Delta qu=W(x)(alpha|u|p-2u +beta|u|q-2u) in Omega under the Dirichlet boundary condition, we provide Lyapunov-type inequalities using the Sobolev constants or the radius of the maximum inscribed ball. Moreover, we give an existence result for non-trivial and non-negative solutions, and show the optimality of the inequalities.
The main purpose of this investigation is to revisit solvability process of the Cauchy problems of Caputo fractional two-term initial value problems. To this aim, the Green function technique has chosen to make a bridge between the operator and the fixed point theories. The appeared Green functions in this paper are constructed by the Fox-Wright functions. Our solvability tools include the existence and uniqueness criteria as novel refinements of the Banach contraction principal and Schauder fixed point theorem. This investigation will be finalized by presenting some numerical applications that illustrate proposed solvability criteria.
In this paper, we are interested to discuss the existence of multiple solutions for a class of p ( x )-Kirchhoff type equations with nonhomogeneous Neumann boundary conditions arising in modelling of various phenomena in the study of nonlinear elasticity theory, electro-rheological fluids, and so on. By using a consequence of the local minimum theorem due to Bonanno we look into the existence of one solution under algebraic conditions on the nonlinear term, and two solutions for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz condition on the nonlinear term. Furthermore, by employing a three-critical-point theorem due to Bonanno and Marano, we guarantee the existence of three solutions for the problem in a special case.
The present work deals with a coupled system of fractional differential equations involving four sequential Caputo derivatives in each of its components. The fractional differential system gives rise to a standard coupled system of two ordinary differential equations of order four, which has practical applications in some real-world phenomena such as robotics, aerospace, and electrical engineering. The existence of a unique vector solution for our sequential system is studied. The existence of at least one vector solution for the considered system is also investigated. Some illustrative examples are discussed in detail to show the main results' applicability. The stabilities in the sense of Ulam Hyers for the system is discussed. A conclusion follows at the end.
This article employs the two-step Adomian decomposition method (TSADM) to obtain a solution of a fractional differential equation with complex order using the singular kernel operator.Moreover, we have provided conditions for the existence and uniqueness of a solution with the fixed point theorems.Also, we have mentioned examples and solved them with the help of the proposed method and found the analytical solution in one iteration.
The gradient estimates in the weighted Lorentz spaces for solutions to a class of quasilinear elliptic equations with a measure on the right-hand side are established using the idea of level-set inequality. Regularity results obtained in this paper are concerned with the quasilinear elliptic equations driven by p-Laplacian, under certain smoothness assumptions on the boundary of domain and the data of the problem. Especially, this paper studies the "very singular" case for the growth exponent p , i.e. when 1 < p <= 3n-2/2n-1. As far as we know, the presence of measure source term mu (being a bounded Radon measure) makes the study of regularity theory more challenging due to the notion of solutions and their reasonable existence. The contribution of this paper is the extension of previous results in weighted Lebesgue spaces in [9, 15].
This research paper explores the existence of solution to a higher-order fractional differential equation with a general boundary condition, shedding light on novel extensions beyond existing literature. The equation, characterized by a Caputo fractional derivative exhibits non- linearity and resonance, making it a compelling subject of study. The investigation employs coincidence degree theory, a robust tool for the examination of differential equations and the identification of solution. Notably, this paper delves into nonlinear growth patterns of function. The main results of the research are accompanied by an illustrative example to clarify the concepts discussed.
In this paper, we investigate a class of p-Hamiltonian systems. By means of the Mountain Pass Lemma, we obtain the existence of one nontrivial periodic solution under some new conditions.
This paper aims to establish a global estimate for solutions to non-uniformly elliptic double obstacle problems in Lorentz and Orlicz-Sobolev spaces. In this study, we build upon the technique introduced M-alpha by Tran and Nguyen in their paper [28]. This technique relies on the concept of the good - lambda inequality proposed by Mingione and the definition of the distribution function by Grafakos. We make use of certain familiar assumptions about non-smooth domains. Additionally, we employ function spaces, inequalities, and several lemmas to support our proof.
