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DEMONSTRATIO MATHEMATICAVol. XLIINo 32009Piotr Majcher, Sushil SharmaAPPLICATION OF MEASURES OF WEAKNONCOMPACTNESS TO A NONLOCAL D A R B O U XPROBLEMAbstract. In this paper we study the existence of pseudosolutions of a nonlocalhyperbolic Darboux problem for the equationd2uTj^(x,y)= f {{x,y),u(x,y))with nonlocal boundary conditions u(x, 0) + h\(x,u) = gi(x), u(0, y) + h^iy, u) = g2(y),on the bounded region. The functions considered have values in a Banach space and areweakly-weakly sequentially continuous, and the relevant integrals are Pettis integrals.IntroductionIn this paper we study the existence of pseudosolutions of a nonlocalhyperbolic Darboux problem for functional-differential equations. Methodsof functional analysis together with measures of weak noncompactness andSadovskii's fixed point theorem are applied.We consider the problem' aa&fo(1)kv) = f ((x> v) > u(x> v)) >u(x,0)+ hi(x,u)= gi(x),u(0,y)+ h2(y,u)=g2(y),v)eA>x G [0,ai],yG[0,a2],2where A = [0,ai] x [0,a 2 ] C R , a i , a 2 > 0, / : A x EE, gt G11C7 ([0, ai],E), hi : C (A, E ) E {i = 1,2) (E is a Banach space and E* itstopological dual) are continuous functions.When the functions hi, gi are equal to zero, this problem can be found in[9], [10]. In [9] the properties of the set of solutions
Demonstratio Mathematica – de Gruyter
Published: Jul 1, 2009
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