Argawal, Ravi  P.

Grace, Said R.

Pinelas, Sandra

Communications in Applied Analysis

English

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Communications in Applied Analysis 13 (2009), no. 1, 93–104ON THE OSCILLATIONS OF FOURTH ORDER FUNCTIONALDIFFERENTIAL EQUATIONSSAID R. GRACE1, RAVI P. AGARWAL2, AND SANDRA PINELAS31Department of Engineering Mathematics, Faculty of EngineeringCairo University, Orman, Giza 12221, EgyptE-mail: srgrace@eng.cu.eg2Department of Mathematical Sciences, Florida Institute of Technology,Melbourne, FL 32901, U. S. A.E-mail: agarwal@fit.edu3Department of Mathematics, Azores University, R. Ma˜e de Deus9500-321 Ponta Delgada, PortugalE-mail: sandra.pinelas@clix.ptABSTRACT. We establish some sufficient conditions for the oscillations of all solutions of fourthorder functional differential equationsddt(a (t)(d3dt3x (t))α)+ q (t) f (x [g (t)]) = 0andddt(a (t)(d3dt3x (t))α)= q (t) f (x [g (t)]) + p (t)h (x [σ (t)])when∫∞a−1/α (s) ds <∞. The case when∫∞a−1/α (s) ds =∞ is also included.1. IntroductionThis paper deals with the oscillatory behavior of solutions of fourth order func-tional differential equationsddt(a (t)(d3dt3x (t))α)+ q (t) f (x [g (t)]) = 0 (1)andddt(a (t)(d3dt3x (t))α)= q (t) f (x [g (t)]) + p (t)h (x [σ (t)]) (2)where the following conditions are assumed to hold:(i) α is the ratio of two positive odd integers;(ii) a (t), p (t) and q (t) ∈ C ([t0,∞) , (0,∞)) ;(iii) g (t) and σ (t) ∈ C1 ([t0,∞) ,R), g (t) < t, σ (t) > t, g′ (t) > 0 and σ′ (t) > 0 fort > t0 and limt→∞g (t) =∞;Received December 25, 2008 1083-2564 $15.00 c©Dynamic Publishers, Inc.

On the oscillations of fourth order functional differential equations

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On the oscillations of fourth order functional differential equations

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