Narrow Results

This paper is concerned with the existence and controllability of solutions for infinite delay functional differential systems with multi-valued impulses in Banach space. Sufficient conditions for the existence are obtained by using a fixed point theorem for multi-valued maps due to Dhage. An example is also given to illustrate our results.

This paper is devoted to system of semilinear heat equations with exponential-growth nonlinearity in two-dimensional space which is the analogue of the scalar model problem studied in [S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Local well posedness of a 2D semilinear heat equation, Bull. Belg. Math. Soc.21 (2014) 1–17]. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space (H¹ × H¹)(ℝ²). The uniqueness part is nontrivial although it follows Brezis–Cazenave's proof [H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math.68 (1996) 73–90] in the case of monomial nonlinearity in dimension d ≥ 3. Next, we show that in the defocusing case our solution is bounded, and therefore exists globally in time. Finally, for this system, we treat the question of blow-up in finite time under the negativity condition on the energy functional. The technique to be used is adapted from [Bull. Belg. Math. Soc. 21 (2014) 1–17].

In this paper, we consider two types of set-valued Volterra–Hammerstein integral equations and prove the existence and uniqueness theorem.

In this paper, we study the existence of mild solutions of impulsive evolution fractional functional differential equation of order $0<\alpha <1$ involving a Lipschitz condition on term $I_k$. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators due to Burton and Kirk.