A class of nonlinear boundary value problems without Landesman-Lazer condition

Kannan, R.; Nieto, Juan J.; Ray, M.B.

doi:10.1016/0022-247x(85)90093-9

Cited by 22 publications

(8 citation statements)

References 9 publications

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“…In this paper we shall use coincidence degree [8,18] to present an extension of Nieto's result when N grows linearly and C is a wedge. Our result implies the Granas fixed point theorem and some results of Cesari and Kannan [3,6] which have been extensively used in differential equations [3,4,5,7,15,16]. We shall also apply our abstract results to discuss the existence of nonnegative solutions to some boundary value problems when the nonlinearity is a Carathéodory function and has at most linear growth.…”

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confidence: 90%

Existence of Nonnegative Solutions of a Semilinear Equation at Resonance with Linear Growth

Santanilla¹

1989

Proceedings of the American Mathematical Society

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Abstract.A coincidence degree result is established to study sufficient conditions for the existence of nonnegative solutions of a semilinear equation at resonance in which the nonlinearity has at most linear growth. Nonnegative solutions to some boundary value problems are obtained to illustrate the theory.The problem of existence of solutions in a convex set, or nonnegative solutions, for abstract semilinear equations at resonance has been recently considered by Nieto [20], Gaines and Santanilla [9], Mawhin and Rybakowski [19], and Santanilla [22]. They have considered the problem of existence of solutions to(1) Lu = Nu in a convex set, where L: dorn LcI->Z is a Fredholm operator of index zero, N:X -» Z is not necessarily linear and satisfies a compactness property relative to L, and X , Z are real Banach spaces. Using the alternative method, Nieto [20] introduced sufficient conditions for the existence of solutions to Equation ( 1 ) in a cone, when the nonlinearity N is bounded. In this paper we shall use coincidence degree [8,18] to present an extension of Nieto's result when N grows linearly and C is a wedge. Our result implies the Granas fixed point theorem and some results of Cesari and Kannan [3,6] which have been extensively used in differential equations [3,4,5,7,15,16]. We shall also apply our abstract results to discuss the existence of nonnegative solutions to some boundary value problems when the nonlinearity is a Carathéodory function and has at most linear growth.

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confidence: 90%

Existence of Nonnegative Solutions of a Semilinear Equation at Resonance with Linear Growth

Santanilla¹

1989

Proceedings of the American Mathematical Society

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“…Let us mention here, for bounded nonlinearities, the papers by Cesari and Pucci [9], de Figueiredo and Ni [12], Gonçalves [16], Kannan, Nieto and Ray [22], Schaaf and Schmitt [31]; and, for unbounded nonlinearities in the case of ordinary differential equations, those ones by Gupta [18] and Iannacci and Nkashama [21].…”

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confidence: 99%

Nonlinear Second Order Elliptic Partial Differential Equations at Resonance

Iannacci¹,

Nkashama²,

Ward³

1989

Transactions of the American Mathematical Society

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Abstract.In this paper we study the solvability of boundary value problems for semilinear second order elliptic partial differential equations of resonance type in which the nonlinear perturbation is not (necessarily) required to satisfy the Landesman-Lazer condition or the monotonicity assumption. The nonlinearity may be unbounded and some crossing of eigenvalues is allowed. Selfadjoint and nonselfadjoint resonance problems are considered.

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“…For instance, if f = f (t, x) = h(t) − g(t, x), where h ∈ L 2 (I), g ∈ C(I × R; R), then in accordance with the well-known Landesman-Lazer theorem [6] there exists at least one solution of the problem (1.1), (1.2) if the following inequality holds: Some authors (see, for example, [5,7]) consider that a key condition, for the existence of at least one solution of the problem (1.1), (1.2), is that the function f satisfies a monotonicity assumption with respect to the variable x. Other authors provide another solvability conditions for this problem [1,2,4].…”

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confidence: 87%