• Title/Summary/Keyword: Leray-Schauder Alternative

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GLOBAL EXISTENCE FOR VOLTERRA-FREDHOLM TYPE FUNCTIONAL IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS

  • Vijayakumar, V.;Prakash, K. Alagiri;Murugesu, R.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.17 no.1
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    • pp.17-28
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    • 2013
  • In this paper, we study the global existence of solutions for the initial value problems for Volterra-Fredholm type functional impulsive integrodifferential equations. Using the Leray-Schauder Alternative, we derive conditions under which a solution exists globally.

Existence and Stability Results on Nonlinear Delay Integro-Differential Equations with Random Impulses

  • Vinodkumar, Arumugam;Gowrisankar, Muthusamy;Mohankumar, Prathiban
    • Kyungpook Mathematical Journal
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    • v.56 no.2
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    • pp.431-450
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    • 2016
  • In this paper, the existence, uniqueness, stability via continuous dependence and Ulam stabilities of nonlinear integro-differential equations with random impulses are studied under sufficient condition. The results are obtained by using Leray-Schauder alternative fixed point theorem and Banach contraction principle.

NONTRIVIAL SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

  • Guo, Yingxin
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.1
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    • pp.81-87
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    • 2010
  • In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem(BVP) $-D_0^{\alpha}+u(t)=\lambda[f(t, u(t))+q(t)]$, 0 < t < 1 u(0) = u(1) = 0, where $\lambda$ > 0 is a parameter, 1 < $\alpha$ $\leq$ 2, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville differentiation, f : [0, 1] ${\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is continuous, and q(t) : (0, 1) $\rightarrow$ [0, $+\infty$] is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when $\lambda$ in some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on f.

CONTROLLABILITY OF IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH INFINITE DELAY IN BANACH SPACES

  • Chang, Yong-Kui
    • Journal of applied mathematics & informatics
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    • v.25 no.1_2
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    • pp.137-154
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    • 2007
  • In this paper, we establish a sufficient condition for the controllability of the first-order impulsive functional differential inclusions with infinite delay in Banach spaces. The approach used is the nonlinear alternative of Leray-Schauder type for multivalued maps. An example is also given to illustrate our result.

Some Nonlinear Alternatives in Banach Algebras with Applications II

  • Dhage, B.C.
    • Kyungpook Mathematical Journal
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    • v.45 no.2
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    • pp.281-292
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    • 2005
  • In this paper a nonlinear alternative of Leray-Schauder type is proved in a Banach algebra involving three operators and it is further applied to a functional nonlinear integral equation of mixed type $$x(t)=k(t,x({\mu}(t)))+[f(t,x({\theta}(t)))]\(q(t)+{\int}_0{^{\sigma}^{(t)}}v(t,s)g(s,x({\eta}\(s)))ds\)$$ for proving the existence results in Banach algebras under generalized Lipschitz and $Carath{\acute{e}}odory$ conditions.

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EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SINGULAR SYSTEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

  • Wang, Lin;Lu, Xinyi
    • Journal of applied mathematics & informatics
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    • v.31 no.5_6
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    • pp.877-894
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    • 2013
  • In this paper, we study the existence and uniqueness of solutions for a singular system of nonlinear fractional differential equations with integral boundary conditions. We obtain existence and uniqueness results of solutions by using the properties of the Green's function, a nonlinear alternative of Leray-Schauder type, Guo-Krasnoselskii's fixed point theorem in a cone. Some examples are included to show the applicability of our results.

APPROXIMATE CONTROLLABILITY OF SECOND-ORDER NONLOCAL IMPULSIVE FUNCTIONAL INTEGRO-DIFFERENTIAL SYSTEMS IN BANACH SPACES

  • Baleanu, Dumitru;Arjunan, Mani Mallika;Nagaraj, Mahalingam;Suganya, Selvaraj
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.4
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    • pp.1065-1092
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    • 2018
  • This manuscript is involved with a category of second-order impulsive functional integro-differential equations with nonlocal conditions in Banach spaces. Sufficient conditions for existence and approximate controllability of mild solutions are acquired by making use of the theory of cosine family, Banach contraction principle and Leray-Schauder nonlinear alternative fixed point theorem. An illustration is additionally furnished to prove the attained principles.