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A nonlinear fractional partial integro‐differential equation with nonlocal initial value conditions
Mathematical Methods in the Applied Sciences, 2023In this work, we study a new nonlinear partial integro‐differential equation with nonlocal initial value conditions and investigate the solutions of this equation. By considering an equivalent implicit integral equation via series, we prove the uniqueness of solutions of the equation by Babenko's approach, Banach's contraction principle, and the ...
Chenkuan Li +4 more
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Existence of Semilinear Differential Equations with Nonlocal Initial Conditions
Acta Mathematica Sinica, English Series, 2006The author considers the existence of mild solutions for semilinear Cauchy problems \[ u'(t) =Au(t) +f(t,u(t)), \quad t\in [0,b]\text{ a.e., }u(0) =g(u) +u_0, \] where \(A\) is an infinitesimal generator of a strongly continuous semigroup \(T(t)\) of bounded linear operators in a Banach space \(X\), \(f: [0,b] \times X\to X\), \(g\in C([0,b] ;X)\) are ...
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Existence results for systems with coupled nonlocal initial conditions
Nonlinear Analysis: Theory, Methods & Applications, 2014In this interesting paper the authors discuss the system of the first order differential equations \[ u_j'=f_j(t,u_1,\dots,u_n), \, j=1, 2,\dots,n, \] a.e. on the interval \([0,1]\) associated with the conditions of the form \[ u_j(0)=\sum_{i=1}^n\alpha_{ji}[u_i], \] where the symbol \(\alpha_{ji}[u_i]\) stands for the known Riemann-Stieltjes integral \
Bolojan Nica O +2 more
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Constrained differential inclusions with nonlocal initial conditions
2018We show existence for the perturbed sweeping process with nonlocal initial conditions under very general hypotheses. Periodic, anti-periodic, mean value and multipoints conditions are included in this study. We give abstract results for differential inclusions with nonlocal initial conditions through bounding functions and tangential conditions.
Jourani, Abderrahim, Vilches, Emilio
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Global Solutions for Nonlinear Delay Evolution Inclusions with Nonlocal Initial Conditions
Set-Valued and Variational Analysis, 2012Under suitable assumptions, the author obtains a sufficient condition for the existence of global \(C^{0}\)-solutions for the following nonlinear functional evolution equation \[ \left\{\begin{aligned} u^{\prime }(t)&\in Au(t)+f(t), \;t\in \mathbb{R}_{+}, \\ f(t)&\in F(t,u(t),u_{t}), \;t\in \mathbb{R}_{+}, \\ u(t)&=g(u)(t), \;t\in [ -\tau ,0], \end ...
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On the Cauchy Problems of Fractional Evolution Equations with Nonlocal Initial Conditions
Results in Mathematics, 2011The authors study Cauchy problems with nonlocal conditions involving fractional derivatives. Criteria for compactness and Lipschitz continuity of the nonlocal term are relaxed in order to prove existence and uniqueness of a solution. The compactness of the semigroup is used to relax the conditions on the nonlocal term.
Wang, Rong-Nian, Yang, Yan-Hong
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Delay evolution equations with mixed nonlocal plus local initial conditions
Communications in Contemporary Mathematics, 2015We consider the delay differential equation u′(t) ∈ Au(t) + f(t, ut), t ∈ ℝ+, where A is the infinitesimal generator of a nonlinear semigroup of contractions in a Banach space X and f is continuous, subjected to a general mixed nonlocal + local initial condition of the form u(t) = g(u)(t) + ψ(t), t ∈ [-τ, 0].
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Semilinear Delay Evolution Equations with Nonlocal Initial Conditions
2014An existence and asymptotic behaviour result for a class of semilinear delay evolution equations subjected to nonlocal initial conditions is established. An application to a semilinear wave equation is also discussed.
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Existence for nonlinear evolution inclusions with nonlocal retarded initial conditions
Nonlinear Analysis: Theory, Methods & Applications, 2011The author proves a sufficient condition for the global existence of bounded \(C^0\)-solutions for a class of nonlinear functional differential evolution equations of the form \[ u'(t)\in Au(t)+f(t), \,\, t\in [0,+\infty ), \] \[ f(t)\in F(t,u(t),u(t-\tau_1),\dots,u(t-\tau_n)), \,\, t\in [0,+\infty ) \] \[ u(t)=g(u)(t), \,\, t\in [-\tau ,0], \] where \(
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