Topological Degree Method Research Articles

Some Qualitative Analyses of Neutral Functional Delay Differential Equation with Generalized Caputo Operator

In this paper, a new class of a neutral functional delay differential equation involving the generalized ψ -Caputo derivative is investigated on a partially ordered Banach space. The existence and uniqueness results to the given boundary value problem are established with the help of the Dhage’s technique and Banach contraction principle. Also, we prove other existence criteria by means of the topological degree method. Finally, Ulam-Hyers type stability and its generalized version are studied. Two illustrative examples are presented to demonstrate the validity of our obtained results.

Existence results for some nonlinear elliptic equations via topological degree methods

Existence results for some nonlinear elliptic equations via topological degree methods

On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space

We obtained results on the existence and uniqueness of a mild solution for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a Carathéodory-type multivalued nonlinearity satisfying some monotonicity condition in the phase variables. We used the Yosida approximations of the linear part of the inclusion, the method of a priori estimates of solutions, and the topological degree method for condensing vector fields. As an example, we considered the existence and uniqueness of a solution to the Cauchy problem for a system governed by a perturbed subdifferential inclusion of a fractional diffusion type.

Topological degree methods for partial differential operators in generalized Sobolev spaces

The main aim of this paper is to prove, by using the topological degree methods, the existence of solutions for nonlinear elliptic equation Au = f where Au is partial dierential operators of general divergence form.

Existence result for a nonlinear elliptic problem by topological degree in Sobolev spaces with variable exponent

Abstract The aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form { A ( u ) = f i n Ω u = 0 o n ∂ Ω \left\{ {\matrix{{A\left( u \right) = f} \hfill & {in} \hfill & \Omega \hfill \cr {u = 0} \hfill & {on} \hfill & {\partial \Omega } \hfill \cr } } \right. where A(u) = −diva(x, u, ∇u) is a Leray-Lions operator and f ∈ W−1,p′ (.)(Ω) with p(x) ∈ (1, ∞). Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.

Analysis of (α,β)-order coupled implicit Caputo fractional differential equations using topological degree method

AbstractThis article is devoted to establish the existence of solution of(α,β)$\left(\alpha ,\beta \right)$-order coupled implicit fractional differential equation with initial conditions, using Laplace transform method. The topological degree theory is used to obtain sufficient conditions for uniqueness and at least one solution of the considered system. Beside this, Ulam’s type stabilities are discussed for the proposed system. To support our main results, we present an example.

Topological degree methods for a Neumann problem governed by nonlinear elliptic equation

Abstract In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation - d i v a ( x , u , ∇ u ) = b ( x ) | u | p - 2 u + λ H ( x , u , ∇ u ) , - div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right), where Ω is a bounded smooth domain of 𝕉 N .

Study of impulsive fractional differential equation under Robin boundary conditions by topological degree method

This research work is dedicated to investigating a class of impulsive fractional order differential equations under the Robin boundary conditions via the application of topological degree theory (TDT). We establish some adequate results for the existence of at most one solution for the consider problem. Further, the whole analysis is illustrated by providing a pertinent example. We keep in mind that the conditions we develop by using TDT are much weaker than using ordinary fixed point theory. Hence TDT can be used as powerful tool for the theoretical analysis of many linear and nonlinear problems.

On qualitative theory of fractional order delay evolution equation via the prior estimate method

In this manuscript we use prior estimate method also known as the topological degree method to investigate the existence and uniqueness of solution for a class of evolution fractional order differential equations (FODEs) with proportional delay using Caputo derivative. In the same line, we will also study different kinds of Ulam stability such as Ulam‐Hyers (UH) stability, generalized Ulam‐Hyers (GUH) stability, Ulam‐Hyers‐Rassias (UHR) stability, and generalized Ulam‐Hyers‐Rassias (GUHR) stability for the considered problem. To justify our results, we provide a pertinent example.

Application of topological degree method in quantitative behavior of fractional differential equations

In the present manuscript we incorporate fractional order Caputo derivative to study a class of non-integer order differential equation. For existence and uniqueness of solution some results from fixed point theory is on our disposal. The method used for exploring these existence results is topological degree method and some auxiliary conditions are developed for stability analysis. For further elaboration an illustrative example is provided in the last part of the research article.

Positive solutions to second-order singular nonlocal problems: existence and sharp conditions

In this paper we consider sharp conditions on ω and f for the existence of C^{1}[0,1] positive solutions to a second-order singular nonlocal problem u''(t)+omega (t)f(t,u(t))=0, u(0)=u(1)=int _{0} ^{1}g(t)u(t),dt; it turns out that this case is more difficult to handle than two point boundary value problems and needs some new ingredients in the arguments. On the technical level, we adopt the topological degree method.

Investigating a Class of Nonlinear Fractional Differential Equations and Its Hyers-Ulam Stability by Means of Topological Degree Theory

The aim of this article is to seek some adequate conditions via a prior estimate method (topological degree method) to derive the existence of solution to a nonlinear boundary value problem of fractional differential equations (FDEs). With the help of topological degree method which has been applied in many articles, we establish the required results for existence and uniqueness of solution to a class of FDEs. Moreover, we also formulate sufficient conditions for Hyers-Ulam stability to the solution of the considered problem. Finally, an appropriate example is provided to justify the relevant results.

Topological degree methods for a Strongly nonlinear p(x)-elliptic problem

This article is devoted to study the existence of weak solutions for the strongly nonlinear p(x)-elliptic problem Our technical approach is based on the recent Berkovits topological degree.

The supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations

In this paper, we provide supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations ut−Δum=a(x,t)up with m > 0, p > 0. By using a new method to deal with the “blow up sequence”, we obtain the L∞ estimate with optimal exponent 1<pm<ps, where ps is the Sobolev critical exponent, then we apply topological degree method to prove the existence of positive periodic solutions. If pm≥ps,N≥3 and Ω is star–shaped, we establish Rellich–Pohozaev type identity to prove the nonexistence of positive periodic solutions. We improve the results of Mizoguchi [10, Therem 1] by filling the gap m>1,pm≥1+2mN. In addition, we also give quite complete results about the existence of periodic solutions for the fast diffusion case 0 < m < 1. As far as we know, there are no results about the fast diffusion case before present work.

Existence of solutions for fractional difference equations via topological degree methods

In this paper, a class of nonlinear fractional difference equations with Caputo-like difference operator is considered. Some existence results for the given equation are obtained by means of topological degree methods. Three examples are constructed for the illustration of the obtained theory.

Existence and uniqueness of solutions for coupled system of fractional differential equation by means of topological degree method

Existence and uniqueness of solutions for coupled system of fractional differential equation by means of topological degree method

Study of a class of arbitrary order differential equations by a coincidence degree method

In this manuscript, we investigate some appropriate conditions which ensure the existence of at least one solution to a class of fractional order differential equations (FDEs) provided by \t\t\t{−CDqz(t)=θ(t,z(t));t∈J=[0,1],q∈(1,2],z(t)|t=0=ϕ(z),z(1)=δCDpz(η),p,η∈(0,1).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} -{}^{C}\\mathbf{D}^{q} z(t)=\\theta (t,z(t)); \\quad t\\in \\mathfrak{J}=[0,1], q\\in (1, 2], \\\\ z(t)\\vert _{t=0}=\\phi (z),\\qquad z(1)=\\delta{}^{C}\\mathbf{D}^{p} z(\\eta ),\\quad p, \\eta \\in (0,1). \\end{cases} $$\\end{document} The nonlinear function theta :mathfrak{J}times mathbf{R} rightarrow mathbf{R} is continuous. Further, delta in (0, 1) and phi in C(mathfrak{J},mathbf{R}) is a non-local function. We establish some adequate conditions for the existence of at least one solution to the considered problem by using Grönwall’s inequality and a priori estimate tools called the topological degree method. We provide two examples to verify the obtained results.

A Lazer-Leach-type condition for singular differential equations with a deviating argument at resonance

In this paper, we establish a Lazer-Leach-type condition depending on the delay for the existence of positive periodic solutions for singular differential equations with a deviating argument at resonance. The proof of the main result is based on the phase-plane analysis and topological degree methods.

Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations

We established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows:D cαxt-ft,xt=gt,yt,Iαyt,t∈J=[0,1],D cαyt-ft,yt=gt,xt,Iαxt,t∈J=0,1,D cpx0=ψxη1,x′0=0,…,xn-20=0,D cpx1=ψxη2,D cpy0=ψyη1,y′0=0,…,yn-20=0,D cpy1=ψyη2, wherep,η1,η2∈0,1,ψis linear,D cαis Caputo fractional derivative of orderα, withn-1&lt;α≤n,n∈N, andIαis fractional integral of orderα. The nonlinear functionsf,gare continuous. For obtaining sufficient conditions on existence and uniqueness of positive solutions to the above system, we used the technique of topological degree theory. Finally, we illustrated the main results by a concrete example.

Existence and multiplicity of periodic solutions for a generalized hematopoiesis model

A generalization of the nonautonomous Mackey–Glass equation for the regulation of the hematopoiesis with several non-constant delays is studied. Using topological degree methods we prove, under appropriate conditions, the existence of multiple positive periodic solutions. Moreover, we show that the conditions are necessary, in the sense that if some sort of complementary conditions are assumed then the trivial equilibrium is a global attractor for the positive solutions and hence periodic solutions do not exist.