topological degree method
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Fractals ◽  
2021 ◽  
pp. 2240013
Author(s):  
ZAREEN A. KHAN ◽  
KAMAL SHAH ◽  
IBRAHIM MAHARIQ ◽  
HUSSAM ALRABAIAH

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.



2021 ◽  
Vol 7 (1) ◽  
pp. 50-65
Author(s):  
Mustapha Ait Hammou ◽  
Elhoussine Azroul

AbstractThe 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.



Author(s):  
Usman Riaz ◽  
Akbar Zada

AbstractThis 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.



Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 421-432
Author(s):  
Rahman ur ◽  
Saeed Ahmad ◽  
Fazal Haq

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.



2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shiqi Ma ◽  
Xuemei Zhang

Abstract 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.



2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Youtao Wang ◽  
Guangcun Lu

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



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