Abstract:In [4] and [5], the author studied strong maximum principles for nonlinear parabolic problems with initial and nonlocal inequalities, respectively. Our purpose here is to extend results in [4] and [5] to strong maximum principles for nonlinear parabolic problems with nonlocal inequalities together with integrals. The results obtained in this paper can be applied in the theories of diffusion and heat conduction, since considered here integrals in nonlocal inequalities can be interpreted as mean amounts of the… Show more
“…For example, the system (1.1) with the nonlocal initial conditions (1.4) can be used to model a reaction-diffusion process of a little amount of gases in a transparent tube, in a more appropriate way than with the usual initial-value condition; for more insight on the physical interpretation of this system see the paper by Byszewski [3]. We mention that a physical motivation for the integral form of the initial condition is presented in [4,5,23]. Moreover, the system (1.1) can be used to describe the periodic solutions in the case α(u, v) = u(T ), β(u, v) = v(T ), where T > 0 and f and g are T -periodic.…”
Abstract. In this paper we study the existence, localization and multiplicity of positive solutions for parabolic systems with nonlocal initial conditions. In order to do this, we extend an abstract theory that was recently developed by the authors jointly with Radu Precup, related to the existence of fixed points of nonlinear operators satisfying some upper and lower bounds. Our main tool is the Granas fixed point index theory. We also provide a non-existence result and some examples to illustrate our theory.
“…For example, the system (1.1) with the nonlocal initial conditions (1.4) can be used to model a reaction-diffusion process of a little amount of gases in a transparent tube, in a more appropriate way than with the usual initial-value condition; for more insight on the physical interpretation of this system see the paper by Byszewski [3]. We mention that a physical motivation for the integral form of the initial condition is presented in [4,5,23]. Moreover, the system (1.1) can be used to describe the periodic solutions in the case α(u, v) = u(T ), β(u, v) = v(T ), where T > 0 and f and g are T -periodic.…”
Abstract. In this paper we study the existence, localization and multiplicity of positive solutions for parabolic systems with nonlocal initial conditions. In order to do this, we extend an abstract theory that was recently developed by the authors jointly with Radu Precup, related to the existence of fixed points of nonlinear operators satisfying some upper and lower bounds. Our main tool is the Granas fixed point index theory. We also provide a non-existence result and some examples to illustrate our theory.
“…where −A is the infinitesimal generator of a strongly continuous semigroup T (t), t > 0, on a Banach space X, 0 t 0 < t 1 < • • • < t p t 0 +a, a > 0, u 0 ∈ X, f : [t 0 , t 0 +a]×X → X and g : [t 0 , t 0 +a] p ×X → X are given functions. Subsequently, Byszewski investigated the same type of problem for a different class of evolution equations in a Banach space [4][5][6][7][8][9][10][11]. Moreover, Corduneanu [12] and Gripenberg et al .…”
The aim of this paper is to prove the existence and uniqueness of mild and classical solutions of the non-local Cauchy problem for a semilinear integrodifferential equation with deviating argument. The results are established by using the method of semigroups and the contraction mapping principle. The paper generalizes certain results of Lin and Liu.
“…(1) u(to) + .q(tl, t2,..., tp, u( )) Uo, (2) where -A is the infinitesimal generator of a C0-semigrou p T(t), t >_ 0, on a Banach space X, 0<_t 0<tl<t2<...<tp<_t o+a,,a>O, and u 0EX and f'[t0, t 0+a]x XX, g(tl,...,tp,. )'XX are given functions.…”
Section: Introductionmentioning
confidence: 99%
“…)'XX are given functions. Subsequently, Byszewski [2][3][4][6][7][8] investigated the same type of problem stated to a different class of evolution equations in Banach space.…”
In this paper we prove the existence and uniqueness of local and global solutions of a nonlocal Cauchy problem for a class of integrodifferential equation. The method of semigroups and the contraction mapping principle are used to establish the results
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