Compactness Type Conditions Research Articles

Fuzzy fractional integral equations under compactness type condition

Abstract In this paper we study a fuzzy fractional integral equation. The fractional derivative is considered in the sense of Riemann-Liouville and we establish existence of the solutions of fuzzy fractional integral equations using the Hausdorff measure of noncompactness.

Initial value problems for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, through solving equations step by step, without any assumption of compactness-type conditions, we obtain unique solution of initial value problems of nonlinear second order impulsive integral-differential in Banach spaces. The results obtained generalize and improve the corresponding results of Guo and Chai in papers [D.J. Guo, Initial value problems for nonlinear second-order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl. 200 (1996) 1–13; G.Q. Chai, Initial value problems for nonlinear second order impulsive integro-differential equations in Banach space, Acta Math. Sinica 20 (3) (2000) 351–359 (in Chinese)].

Averaging of Perturbed One Sided Lipschitz Differential Inclusions

We consider one sided Lipschitz differential inclusions perturbed with multimap, satisfying compactness type conditions. The state space is Banach with uniformly convex dual. Averaging result on a finite interval is proved. The averaging of functional differential inclusions is also studied.

Existence and comparison theorems to Volterra fuzzy integral equation in ( En, D) 1

Kaleva (1987) studied the measurability and integrability for the fuzzy set-valued mappings of a real variable whose values are in the fuzzy number space ( E n , D). In this paper, by using the embedding theorem of Kaleva (1990) on ( E n , D), the Darbo fixed-points theorem in the convex cone C[ I, E n ] (where I denotes a bounded closed interval) is given. Further, the criteria for the existence and comparison of solutions to the Volterra fuzzy integral equation in ( E n , D) are established under the compactness-type conditions.

Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions

Kaleva, Fuzzy sets and Systems 35 (1990) 389–396, has concluded that the Peano theorem does not hold for fuzzy differential equations on ( E n , D). This paper deals with the Cauchy problem of fuzzy differential equations, the existence theorems under compactness-type conditions are obtained.

On the existence of weak solutions of differential equations in nonreflexive banach spaces

THE STUDY of the Cauchy problem for differential equations in a Banach space relative to the strong topology has attracted much attention in recent years [l&4]. This study has taken two different directions. One direction is to impose compactness type conditions that guarantee only existence and the corresponding results are extensions of the classical Peano’s Theorem. The other approach is to utilize dissipative type conditions that assure existence and uniqueness of solutions, and the corresponding results are extensions of the classical Picard’s Theorem. However a similar study ofthe Cauchy problem in a Banach space relative to the weak topology has lagged behind. Recently Szep [4] proved Peano’s Theorem in the weak topology for differential equations in a reflexive Banach space. His main tools are the Eberlein-Smulian Theorem and the fact that a subset of a reflexive Banach space is weakly compact if and only if it is weakly closed and norm bounded. In this paper we wish to prove this theorem in arbitrary Banach spaces. Our first approach is to impose weak compactness type conditions in terms of the measure of weak noncompactness developed by De Blasi. We also impose weak dissipative type conditions and prove an existence and uniqueness theorem. Using these existence results and the partial orderings induced by cones, existence of extremal solutions and comparison results relative to the weak topology are also proved.