Positive Solutions for Fourth-Order Nonlinear Differential Equation with Integral Boundary Conditions by Qi Wang, Yanping Guo, Yude Ji

“…The associated Green's function for the fourth-order boundary value problems is first given, and the arguments are based on Krasnoselskii's fixed point theorem for operators on a cone.…”
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Positive solutions of higher order quasilinear elliptic equations by Marcelo Montenegro

“…The basic ingredients are the maximum principle, Moser iterative scheme, an eigenvalue problem, a priori estimates by rescalings, sub/supersolutions, and Krasnosel'skiĭ fixed point theorem.…”
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Positive Solutions of a Fractional Boundary Value Problem with Changing Sign Nonlinearity by Yongqing Wang, Lishan Liu, Yonghong Wu

“…We first derive some properties of the associated Green function and then obtain some results on the existence of positive solutions by means of the Krasnoselskii's fixed point theorem in a cone.…”
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Multiple Solutions to Fractional Difference Boundary Value Problems by Huiqin Chen, Yaqiong Cui, Xianglan Zhao

“…The following fractional difference boundary value problems ▵νyt=-ft+ν-1,yt+ν-1, y(ν-2)=y(ν+b+1)=0 are considered, where 1<ν≤2 is a real number and ▵νy(t) is the standard Riemann-Liouville fractional difference. Based on the Krasnosel’skiǐ theorem and the Schauder fixed point theorem, we establish some conditions on f which are able to guarantee that this FBVP has at least two positive solutions and one solution, respectively. …”
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Positive Periodic Solutions for Impulsive Functional Differential Equations with Infinite Delay and Two Parameters by Zhenguo Luo, Liping Luo, Yunhui Zeng

“…We apply the Krasnoselskii’s fixed point theorem to study the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with infinite delay and two parameters. …”
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Existence and Positivity of Solutions for a Second-Order Boundary Value Problem with Integral Condition by Assia Guezane-Lakoud, Hamidane Nacira, Khaldi Rabah

“…The arguments are based on Banach contraction principle, Leray Schauder nonlinear alternative, and Guo-Krasnosel’skii fixed point theorem in cone. Two examples are also given to illustrate the main results.…”
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On the Fractional Variable Order Thermostat Model: Existence Theory on Cones via Piece-Wise Constant Functions by Shahram Rezapour, Mohammed Said Souid, Zoubida Bouazza, Azhar Hussain, Sina Etemad

“…In fact, we will get help from the constant piece-wise functions for transforming our variable order model into an auxiliary standard model of thermostat. By Guo-Krasnoselskii’s fixed point theorem on cones, we derive the required conditions ensuring the existence property for positive solutions. …”
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Solvability of a Second Order Nonlinear Neutral Delay Difference Equation by Zeqing Liu, Liangshi Zhao, Jeong Sheok Ume, Shin Min Kang

“…This paper studies the second-order nonlinear neutral delay difference equation Δ[anΔ(xn+bnxn−τ)+f(n,xf1n,…,xfkn)]+g(n,xg1n,…,xgkn)=cn, n≥n0. By means of the Krasnoselskii and Schauder fixed point theorem and some new techniques, we get the existence results of uncountably many bounded nonoscillatory, positive, and negative solutions for the equation, respectively. …”
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Existence of Solutions for Fractional q-Integrodifference Equations with Nonlocal Fractional q-Integral Conditions by Suphawat Asawasamrit, Jessada Tariboon, Sotiris K. Ntouyas

“…By applying the Banach contraction principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative, the existence and uniqueness of solutions are obtained. …”
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Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation by Jun-Rui Yue, Jian-Ping Sun, Shuqin Zhang

“…By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.…”
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Periodic Solutions for Nonlinear Integro-Differential Systems with Piecewise Constant Argument by Kuo-Shou Chiu

“…Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. …”
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The Eigenvalue Problem for Caputo Type Fractional Differential Equation with Riemann-Stieltjes Integral Boundary Conditions by Wenjie Ma, Yujun Cui

“…By using the Guo-Krasnoselskii’s fixed point theorem on cone and the properties of the Green’s function, some new results on the existence and nonexistence of positive solutions for the fractional differential equation are obtained.…”
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Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations by Yige Zhao, Shurong Sun, Zhenlai Han, Qiuping Li

“…By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. …”
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Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions by Bashir Ahmad, Juan J. Nieto

“…The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.…”
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Global Positive Periodic Solutions of Generalized n-Species Gilpin-Ayala Delayed Competition Systems with Impulses by Zhenguo Luo, Liping Luo, Jianhua Huang, Binxiang Dai

“…We consider the following generalized n-species Lotka-Volterra type and Gilpin-Ayala type competition systems with multiple delays and impulses: xi′(t)=xi(t)[ai(t)-bi(t)xi(t)-∑j=1n‍cij(t)xjαij(t-ρij(t))-∑j=1n‍dij(t)xjβij(t-τij(t))-∑j=1n‍eij(t)∫-ηij0‍kij(s)xjγij(t+s)ds-∑j=1n‍fij(t)∫-θij0‍Kij(ξ)xiδij(t+ξ)xjσij(t+ξ)dξ],a.e, t>0, t≠tk; xi(tk+)-xi(tk-)=hikxi(tk), i=1,2,…,n, k∈Z+. By applying the Krasnoselskii fixed-point theorem in a cone of Banach space, we derive some verifiable necessary and sufficient conditions for the existence of positive periodic solutions of the previously mentioned. …”
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Multipoint BVP for the Langevin Equation under φ-Hilfer Fractional Operator by Mohammed A. Almalahi, Satish K. Panchal, Fahd Jarad

“…Next, we investigate and develop sufficient conditions for the existence and uniqueness of solutions by means of semigroups of operator approach and the Krasnoselskii fixed point theorems as well as Banach contraction principle. …”
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Hilfer-Hadamard Nonlocal Integro-Multipoint Fractional Boundary Value Problems by Chanon Promsakon, Sotiris K. Ntouyas, Jessada Tariboon

“…The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying Schaefer and Krasnoselskii fixed point theorems as well as Leray-Schauder nonlinear alternative. …”
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Boundary Value Problem for the Langevin Equation and Inclusion with the Hilfer Fractional Derivative by Khalid Hilal, Ahmed Kajouni, Hamid Lmou

“…Second of all, we give our first existence result, based on Krasnoselskii’s fixed point, and to deal with the uniqueness result, we use Banach’s contraction principle. …”
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Boundary Value Problem for Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Impulses by Abdelkrim Salim, Mouffak Benchohra, Jamal Eddine Lazreg, Gaston N’Guérékata

“…The results are obtained using the Banach contraction principle and Krasnoselskii’s and Schaefer’s fixed-point theorems.…”
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Qualitative Analyses of Fractional Integrodifferential Equations with a Variable Order under the Mittag-Leffler Power Law by Mdi Begum Jeelani, Abeer S. Alnahdi, Mohammed A. Almalahi, Mohammed S. Abdo, Hanan A. Wahash, Nadiyah Hussain Alharthi

“…Next, we prove the existence and uniqueness of analytic results by application of Krasnoselskii’s and Banach’s fixed point theorems. Besides, the guarantee of the existence of solutions is shown by different types of Ulam-Hyer’s stability. …”
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