STUDY OF FRACTIONAL ORDER DELAY CAUCHY NON-AUTONOMOUS EVOLUTION PROBLEMS VIA DEGREE THEORY

This work is devoted to derive some existence and uniqueness (EU) conditions for the solution to a class of nonlinear delay non-autonomous integro-differential Cauchy evolution problems (CEPs) under Caputo derivative of fractional order. The required results are derived via topological degree method (TDM). TDM is a powerful tool which relaxes strong compact conditions by some weaker ones. Hence, we establish the EU under the situation that the nonlinear function satisfies some appropriate local growth condition as well as of non-compactness measure condition. Furthermore, some results are established for Hyers–Ulam (HU) and generalized HU (GHU) stability. Our results generalize some previous results. At the end, by a pertinent example, the results are verified.

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

The aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form\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

This article is devoted to establish the existence of solution of $\left(\alpha ,\beta \right)$-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.

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 $C^{1}[0,1]$ C 1 [ 0 , 1 ] positive solutions to a second-order singular nonlocal problem $u''(t)+\omega (t)f(t,u(t))=0$ u ″ ( t ) + ω ( t ) f ( t , u ( t ) ) = 0 , $u(0)=u(1)=\int _{0} ^{1}g(t)u(t)\,dt$ u ( 0 ) = u ( 1 ) = ∫ 0 1 g ( t ) u ( t ) d t ; 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.

Generalized Solutions for Nonlocal Elliptic Equations and Systems with Nonlinear Singularities

We use the topological degree method to study the existence of solutions for nonlocal elliptic equations (systems) with a strong singular nonlinearity.