A Characterization of Some Class Nonlinear Eigenvalue Problem in VELS

In last the quarter century, many researchers have been interested by the theory of the variable exponent function space and its applications. We well-know that a normal mode analysis of a vibrating mechanical or electrical system gives rise to an eigenvalue problem. We will investigate a characterization of some class nonlinear eigenvalue problem in variable exponent Lebesgue spaces.

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A. Ambrosetti and P. Rabinowitz,(1973),” Dual variational methods in critical point theory and applications”, J. Funct. Anal. 14 , 349–381

D.Cruz-Uribe, SFO and F.I. Mamedov ,(2012), “On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces”, Rev. Mat. Compl., 25(2), 335-367

D.E.Edmunds , P.Gurka and L.Pick ,(1994), ”Compactness of Hardy type integral operators in weighted Banach function spaces”, Studia Math. , 109 (1), 73-90

D.E.Edmunds, V.Kokilashvili and A.Meskhy,(2005), “On the boundedness and compactness of weighted Hardy operator in space L?(?)”, Georgian Math. J., 12(1), 27–44

F.I.Mamedov, Y. Zeren and L. Akın,(2017), “Compactification of weighted Hardy operator in variable exponent Lebesgue spaces”, Asian Journal of Mathematics and Computer Research.17(1), 38-47.

L. Akın, (2018),”On two weight criterions for the Hardy-Littlewood maximal operator in BFS”, Asian Journal of Science and Technology, Vol. 09, Issue: 5, pp.8085-8089.

F.I.Mamedov and Y. Zeren ,(2014), A necessary and sufficient condition for Hardy's operator in the variable Lebesgue space, Abst. Appl. Anal., 5/6, 7 pages.

F.I.Mamedov and Y. Zeren ,(2012), “On equivalent conditions for the general weighted Hardy type inequality in space L?(.)(0, l) “, Zeitsch. fur Anal. und ihre Anwend., 34(1), 55-74

L. Akın, (2018), “ A Characterization of Approximation of Hardy Operators in VLS”, Celal Bayar University Journal of Science, Volume 14, Issue 3, p 333-336

F.I.Mamedov and A.Harman ,(2010), “On a Hardy type general weighted inequality in spaces L?(.)(0, l)”, Integr. Equ. Oper. Th., 66(1), 565-592

M. Willem,(1996),” Minimax Theorems”, Birkhauser, Boston.

V.D.Radulescu,(2015), ”Nonlinear elliptic equations with variable exponent: Old and new”, Nonlinear Analysis, 121, 336–369

E. Piskin, (2018),”Finite time blow up of solutions for a strongly damped nonlinear Klein-Gordon equation with variable exponents”, Honam Mathematical Journal, 40(4), pp. 771-783.

E. Piskin, (2017),”Sobolev Uzayları”, Seçkin Yayıncılık.