Analysis of the solvability and stability of the operator-valued Fredholm integral equation in Hölder space

In this paper, the solvability of an operator-valued integral equation in Hölder spaces, i.e., \begin{equation*} \label{fredholm} w(\zeta_1) = y(\zeta_1)+w(\zeta_1)\int_{\bf J}\kappa(\zeta_1, \varphi)(T_1w)(\varphi)d\varphi+z(\zeta_1)\int_{\bf J}h(\varphi, (T_2w)(\varphi))d\varphi, \end{equation*} for $\zeta_1\in{\bf J} = [0, 1],$ is studied by using Darbo's fixed point theorem (FPT). The process of the measure of noncompactness of the operators which constitute an intermediary of contraction and compact mappings can be explained with the help of Darbo's FPT. The greater effectiveness of Darbo's FPT due to its non-involvement of the compactness property gives a better scope when dealing with the Schauder FPT, where compactness is an essential property. To obtain a unique solution, we apply the Banach fixed point theorem and discuss the Hyers-Ulam stability of the integral equation. We also give some important examples to illustrate the existence and uniqueness of the results.</p

An analysis on the approximate controllability results for Caputo fractional hemivariational inequalities of order 1 < r < 2 using sectorial operators

In this paper, we investigate the effect of hemivariational inequalities on the approximate controllability of Caputo fractional differential systems. The main results of this study are tested by using multivalued maps, sectorial operators of type (P, η, r, γ ), fractional calculus, and the fixed point theorem. Initially, we introduce the idea of mild solution for fractional hemivariational inequalities. Next, the approximate controllability results of semilinear control problems were then established. Moreover, we will move on to the system involving nonlocal conditions. Finally, an example is provided in support of the main results we acquired

A New Approach to the Solution of the Fredholm Integral Equation via a Fixed Point on Extended b-Metric Spaces

It is very well known that real-life applications of fixed point theory are restricted with the transformation of the problem in the form of f ( x ) = x . (1) The Knaster&ndash;Tarski fixed point theorem underlies various approaches of checking the correctness of programs. (2) The Brouwer fixed point theorem is used to prove the existence of Nash equilibria in games. (3) Dlala et al. proposed a solution for magnetic field problems via the fixed point approach

Some Fixed-Point Theorems in <i>b</i>-Dislocated Metric Space and Applications

In this article, we prove some fixed-point theorems in b-dislocated metric space. Thereafter, we propose a simple and efficient solution for a non-linear integral equation and non-linear fractional differential equations of Caputo type by using the technique of fixed point

Applying periodic and anti-periodic boundary conditions in existence results of fractional differential equations via nonlinear contractive mappings

Abstract We introduce a notion of nonlinear cyclic orbital ( ξ − F ) $(\xi -\mathscr{F})$ -contraction and prove related results. With these results, we address the existence and uniqueness results with periodic/anti-periodic boundary conditions for: 1. The nonlinear multi-order fractional differential equation L ( D ) θ ( ς ) = σ ( ς , θ ( ς ) ) , ς ∈ J = [ 0 , A ] , A > 0 , $$\mathcal{L}(\mathcal{D})\theta (\varsigma )=\sigma \bigl(\varsigma , \theta ( \varsigma ) \bigr), \quad \varsigma \in \mathscr{J}=[0,\mathscr{A}], \mathscr{A}>0,$$ where L ( D ) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R ( ♭ = 0 , 1 , 2 , 3 , … , w ) , γ w ≠ 0 , 0 ≤ δ 0 0 ; θ ( ς ) = σ ¯ ( ς ) , ς ∈ [ − τ , 0 ] , $$\begin{aligned} &\mathcal{L}(\mathcal{D})\theta (\varsigma ) =\sigma \bigl(\varsigma , \theta ( \varsigma ),\theta (\varsigma -\tau ) \bigr), \quad \varsigma \in \mathscr{J}=[0, \mathscr{A}], \mathscr{A}>0; \\ &\theta (\varsigma ) =\bar{\sigma}(\varsigma ),\quad \varsigma \in [-\tau ,0], \end{aligned}$$ where L ( D ) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R ( ♭ = 0 , 1 , 2 , 3 , … , w ) , γ w ≠ 0 , 0 ≤ δ 0 < δ 1 < δ 2 < ⋯ < δ w − 1 < δ w < 1 ; $$\begin{aligned} &\mathcal{L}(\mathcal{D})=\gamma _{w} \,{}^{c} \mathcal{D}^{\delta _{w}}+ \gamma _{w-1} \,{}^{c} \mathcal{D}^{\delta _{w-1}}+\cdots+\gamma _{1} \,{}^{c} \mathcal{D}^{\delta _{1}}+\gamma _{0} \,{}^{c} \mathcal{D}^{\delta _{0}},\\ &\gamma _{\flat}\in \mathbb{R}\quad (\flat =0,1,2,3,\ldots,w), \qquad \gamma _{w} \neq 0, \\ &0\leq \delta _{0}< \delta _{1}< \delta _{2}< \cdots< \delta _{w-1}< \delta _{w}< 1; \end{aligned}$$ moreover, here D δ c ${}^{c}\mathcal{D}^{\delta}$ is predominantly called Caputo fractional derivative of order δ

On uniform stability and numerical simulations of complex valued neural networks involving generalized Caputo fractional order

Abstract The dynamics and existence results of generalized Caputo fractional derivatives have been studied by several authors. Uniform stability and equilibrium in fractional-order neural networks with generalized Caputo derivatives in real-valued settings, however, have not been extensively studied. In contrast to earlier studies, we first investigate the uniform stability and equilibrium results for complex-valued neural networks within the framework of a generalized Caputo fractional derivative. We investigate the intermittent behavior of complex-valued neural networks in generalized Caputo fractional-order contexts. Numerical results are supplied to demonstrate the viability and accuracy of the presented results. At the end of the article, a few open questions are posed

A New Approach to the Solution of Non-Linear Integral Equations via Various <i>F_Be</i>-Contractions

In this article, we introduce and establish various approaches related to the F-contraction using new sorts of contractions, namely the extended F B e -contraction, the extended F B e -expanding contraction, and the extended generalized F B e -contraction. Thereafter, we propose a simple and efficient solution for non-linear integral equations using the fixed point technique in the setting of a B e -metric space. Moreover, to address conceptual depth within this approach, we supply illustrative examples where necessary

Sehgal Type Contractions on Dislocated Spaces

In this paper, we investigate the contractive type inequalities for the iteration of the mapping at a given point in the setting of dislocated metric space. We consider an example to illustrate the validity of the given result. Further, as an application, we propose a solution for a boundary value problem of the second order differential equation

Some Valid Generalizations of Boyd and Wong Inequality and ψ,ϕ-Weak Contraction in Partially Ordered b−Metric Spaces

In this manuscript, we use ψ,ϕ-weak contraction to generalize coincidence point results which are established in the context of partially ordered b-metric spaces. The presented work explicitly generalized some recent results from the existing literature. Examples are also provided to show the authenticity of the established work